UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Dynamics and control of orbiting flexible systems : a formulation with applications Ng, Chun-Ki Alfred 1992

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


831-ubc_1992_spring_ng_chun_ki_alfred.pdf [ 7.05MB ]
JSON: 831-1.0080824.json
JSON-LD: 831-1.0080824-ld.json
RDF/XML (Pretty): 831-1.0080824-rdf.xml
RDF/JSON: 831-1.0080824-rdf.json
Turtle: 831-1.0080824-turtle.txt
N-Triples: 831-1.0080824-rdf-ntriples.txt
Original Record: 831-1.0080824-source.json
Full Text

Full Text

DYNAMICS AND CONTROL OF ORBITING FLEXIBLESYSTEMS: A FORMULATION WITH APPLICATIONSCHUN—KI ALFRED NGB.A.Sc., University of British Columbia, 1984M.A.Sc., University of British Columbia, 1987A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate StudiesDepartment of Mechanical EngineeringWe accept this thesis as conformingtothe required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© Alfred Chun—Ki Ng, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of my depart-ment or by his or her representatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without my written permission.Department of Mechanical EngineeringThe University of British ColumbiaVancouver, B.C. CANADADate: April 1992ABSTRACTA relatively general formulation for studying the nonlinear dynamics and controlof spacecraft with interconnected flexible members in a tree-type topology is devel-oped. The distinctive features of the formulation include the following:(i) It is applicable to a large class of present and future spacecraft with flexiblebeam and plate type appendages, arbitrary in number and orientation.(ii) The members are free to undergo predefined slewing maneuvers to facilitatemodelling of sun tracking solar panels and large angle maneuvers of spacebased robots.(iii) Solar radiation induced thermal deformations of flexible members are incor-porated in the study.(iv) The governing equations of motion are highly nonlinear, nonautonomous andcoupled. They are programmed in a modular fashion to help isolate the effectsof flexibility, librational motion, thermal deformations, slewing maneuvers,shifting center of mass, higher modes, initial conditions, etc.The first chapter of the thesis presents a general background to the subject anda brief review of the relevant literature on multibody dynamics. This is followed bythe kinematics and kinetics of the problem leading to the Lagrangian equations ofmotion. The third chapter focuses on methodology and development of the computercode suitable for parametric dynamical study and control.Next, versatility of the general formulation is illustrated through the analysisof five spacecraft configurations of contemporary interest: the next generation ofmulti-purpose communications spacecraft represented by the INdian SATellite II (IN-iiSAT II); the First Element Launch (FEL) and the Permanently Manned Configuration(PMC) of the proposed Space Station Freedom; the Mobile Servicing System (MSS)to be developed by Canada for operation on the Space Station; and the Space FlyerUnit (SFU) to be launched by Japan in mid-nineties. In the FEL study, the attentionis directed towards interactions between the librational and vibrational dynamics.During the PMC investigation, effects of the thermal deformation and orbital eccen-tricity are introduced and the microgravity environment around the station centerof mass explored. The MSS study assesses pointing errors arising from inplane andout-of-plane maneuvers of the robotic arms. The SFU represents a challenging con-figuration to assess deployment and retrieval dynamics associated with a solar array.Parameters considered here include symmetry, orientation and duration of the de-ployment/retrieval maneuvers.Results of the dynamical study clearly shows that, under critical combinationsof parameters, the systems can become unstable. Obviously, the next logical step isto explore control strategies to restore equilibrium. To that end, feasibility of thenonlinear control based on the Feedback Linearization Technique (FLT) is exploredwith reference to the INSAT II and the MSS. Results show the procedure to be quitepromising in controlling the INSAT II over a range of disturbances, including thethermal effects. Application of the control to the MSS reduced the pointing errorinduced by robotic arm maneuvers significantly.The amount of information obtained through a planned parametric analysis ofthe system dynamics and control is indeed enormous. More significant results aresummarized in the concluding chapter together with a few recommendations for futurestudy.iiiTABLE OF CONTENTSABSTRACT ^  iiLIST OF SYMBOLS ^  viiiLIST OF FIGURES  xvLIST OF TABLES ^ xxivACKNOWLEDGEMENT  xxv1. INTRODUCTION ^ 11.1 Preliminary Remarks 11.2 A Brief Review of the Relevant Literature ^ 71.3 Scope of the Present Investigation 142. FORMULATION OF THE PROBLEM ^ 182.1 Preliminary Remarks ^ 182.2 Kinematics of the Problem 202.2.1^Configuration selection ^ 202.2.2^Coordinate system 202.2.3^Position of spacecraft in space ^ 242.2.4^Solar radiation incidence angles 272.2.5^Shift in the center of mass ^ 292.3 Elastic and Thermal Deformation 312.3.1^Background ^ 31iv2.3.2 Substructure equations of motion ^  352.3.3 Thermal deformation ^  372.3.4 Transverse vibration  39^2.4^Kinetics ^  472.4.1 Rotation matrices ^  472.4.2 Kinetic energy  492.4.3 Potential energy ^  522.5^Lagrangian Formulation  543. NUMERICAL IMPLEMENTATION ^  573.1^Preliminary Remarks ^  573.2^Numerical Integration Subroutine ^  583.3^Computational Flowchart  593.4^Subroutine FCN ^  623.4.1 Background  623.4.2 Definitions of new operators ^  643.4.3 Subprograms in FCN  683.5^Program Functions ^  773.6^Program Verification  794. PARAMETRIC STUDIES ^ 954.1 Preliminary Remarks 954.2 First Element Launch ^ 964.3 Permanently Manned Configuration ^ 1124.3.1^Dynamic response ^ 1124.3.2^Thermal deformation and eccentricity ^ 1294.3.3^Velocities and accelerations ^ 1314.4 Mobile Servicing System ^ 1474.5 Space Flyer Unit 1634.6 Closing Comments ^ 1875. NONLINEAR CONTROL 1885.1 Preliminary Remarks ^ 1885.2 Feedback Linearization Technique ^ 1905.3 Quasi-Open and Quasi-Closed Loop Control ^ 2005.3.1^Quasi-open loop control ^ 2025.3.2^Quasi-closed loop control    2025.4 Application of the Quasi-closed Loop Control to INSAT II .^. 2055.5 Application of the Quasi-closed Loop Control to MSS ^ 2165.6 Closing Comments ^ 230vi6. CONCLUDING REMARKS ^  2326.1^Conclusions ^  2326.2^Recommendations for Future Work ^  235BIBLIOGRAPHY^  238APPENDICESI^DETAILS OF Tsys , Isys , AND Hsys ^  249II^REPRESENTATION OF Tsys , Isys , AND Hsys IN TERMS OFTHE NEW OPERATORS ^  255III A SAMPLE OF THE COMPUTER PROGRAM FOR THEEVALUATION OF 071sys IN,Isys AND Hsys ^ 263viiLIST OF SYMBOLSab^ beam radiusa', 4 power boom accelerations in 17, and Z, directions, respec-tively; Eq. (4.3)a l , a 2 , a3^constants written in terms of p, w, and i; Eq. (2.4)bb beam wall thicknessb 1 , b2 , b3^components of it along X0 , Y0 , and Z, axes, respectively;Eq. (2.3)di, dij^position vectors from 0, to 0, and Oi to^respectively;Fig. 2-2dmi, dmij^elemental mass in body .8,, Bi, and Bi,j, respectivelyfc,^ fundamental frequency of bodies Bc , B,, and Bi,j, respec-tivelyh^angular momentum per unit mass of spacecrafti orbit inclination with respect to the ecliptic plane; Fig. 2-34^unit vectors in the directions of Xk, Yk, and Zk axes, respec-tively; k = p, c, i or i, jkb^thermal conductivity of an appendagedirection cosines of Rim with respect to Xp ,Yp ,Zp axes;Eq. (2.39)ib, 1p^beam and plate lengths, respectively/* /*p^thermal reference lengths of the beam and plate, respectively;Eqs. (2.16) and (2.19)viiiMPniqf , qr(qr)dqsttcukb^bV ,WWPXk, Yk, ZkCcmClm61 7nlength of bodies Be , Bi, and Bio , respectivelybeam mass per unit lengthmass of the bodies Be , Bi, and Bio , respectivelymass per unit area of the platenumber of^bodies attached to body Bivector representing flexible and rigid generalized coordinatesvector representing the desired rigid generalized coordinatessolar radiation intensity; W/m, 2timeplate thicknessunit vector representing direction of the solar radiation; Eq. (2.3)unit vectors^kk}T; k = c,i or i, jtransverse vibration of a beam in its Y and Z directions,respectivelytransverse vibration of the plate in its Z directioncoordinates of a point along Xk,Yk,Zk directions, respec-tively; k = c, i, or i, jtransformation matrices defining orientation of Fi andrelative to Fe , respectivelyposition vector from C i to the instantaneous centre of massof spacecraft; Eq. (2.12b)position vector from 0, to the centre of mass of undeformedixC i , CfFkEib,zzspacecraft; Eq. (2.12a)centres of mass of the undeformed and deformed configura-tions of spacecraft, respectivelyflexural rigidity of the plateflexural rigidity of a beam about its Y and Z axes, respec-tivelyreference frame for coordinate axes Xk,Yk,Zk; k = o, c, i orH8 ' t^generalized coordinate associated with the s th and t th modesin its X and Y directions, respectively, for a plate undergoingtransverse vibrations; k = c, i, or i, jdimensionless generalized coordinate; irk ' t = Ht t //kHsys^ angular momentum of spacecraft with respect to X,,Y,,Z,axes; Eq. (2.36)Lys^ inertia matrix of spacecraft with respect to the X,,K,Zc axes;Eq. (2.37)(-rxx)k, (iYY^(Izz)k principal inertia of Body Bk about Xk, Yk, and Zk axes,respectively; k = c, i, or i , jKp Kv^ displacement and velocity gain matrices, Eq. (5.5)MN, Nioc, Oa, 0 27jpr Q k rtotal mass of spacecrafttotal number of Bi and Bi j bodies, respectivelyorigins of the coordinate axes for bodies 33,, Ba , and Bi ,j ,respectivelygeneralized coordinates associated with the r th transverse vi-bration mode of a beam in its Y and Z directions, respec-tively; k = c, i or i, jPrk,^ dimensionless generalized coordinates; Prk = Pj Ilk , •-•••",1, •-Q f, QrQ0, Q0, QiRcrn,Rc , Ri, RijRcm, Rc, Ri, Ri,iSx , Sy, SZQrklikcontrol effort vectors for flexible and rigid coordinates, re-spectively; Eq. (5.6)control effort for pitch, roll and yaw degrees of freedom, re-spectivelyposition vector from the centre of force to the instantaneouscentre of mass of spacecraft; Figure 2-2position vectors of the mass elements dmc , dmi, and dm i ,j,respectively as measured from the centre of force; Eq. (2.9)magnitudes of Rcm , .13,, Ri, and Rij, respectivelypointing errors of the manipulator in the orbit normal, localvertical, and local horizontal directions, respectively; Eq. (4.5).\/Stotal kinetic energy of spacecraftsystem kinetic energy due to various coupling effects;Eq. (2.35)potential energy of spacecraft; Ue Ugunit matrixstrain energy of spacecraft; Eq. (2.40)gravitational potential energy of the spacecraft; Eq. (2.38)body coordinate axes associated with B c , Bi, and Bi j , re-stotTTsysUUUeUgXk,Yk,Zkxi,x"k , 'kspectively; k^c, i, or i, jX0 ,Y0 ,Z0^inertial coordinate system located at the earth's centerXp ,Yp , Zp^coordinate axes with origin at Cf and parallel to x,,yc ,z,,respectivelyZS^orbital frame with X, in the direction of the orbit-normal, Y,along the local vertical, and Z3 towards the local horizontal;Figure 2-4a s , at^absorptivity and coefficient of thermal expansion of the ap-pendage material, respectivelyas i, am^slewed angle and maximum slew angle; Eq. (4.4)6c, Si, Oij^vectors representing transverse vibration of dm c , dmi, anddmi,j, respectively; Figure 2-2ax, ay , az^angular accelerations of the system about X,, Y, and Z, axes,respectively61, ö^tip deflection of a beam element in the Yk and Zk directions,respectively; k = c, i, or i, j; Eq. (4.1)eccentricityend deflection of a plate centerline in Xk and Yk directions,respectively; k = c, i, or i, j; Eq. (4.2)rotation about the local horizontal axis, Z 1 , of the interme-diate frame X 1 ,Y1 ,Z1 ; Figure 2-4solar radiation incidence angles measured with respect to Xk,Yk, and Zk axes, respectively; k = c, i or i, jcolumn matrix representation of the direction cosines of solarxiiradiation incidence angles; {cos 0',^kcos Ov ,^kcos Oz}T • k =— cor i, j;A^ rotation about the local vertical axis, Y2 , of the intermediateframe X2 ,Y2 ,Z2 ; Figure 2-4gravitational constantmatrix representing slewing motion of body Bi0^true anomalyp longitude of the ascending node; Figure 2-3Pc, Pi, Pio^vectors denoting positions of dm c , dmi, and dmi,j, respec-tively, in the undeformed configuration of the spacecraft; Fig-ure 2-2Stefan-Boltzman constantTst^ slewing period; Eq. (4.4)vectors denoting thermal deformations of dmc , dmi, anddmi,j, respectively; Figure 2-2argument of the perigee point; Figure 2-3co^librational velocity vector with respect to Xp ,Yp ,Zp axes;Eq. (2.2)rotation about the orbit normal, Xs ; Figure 2-4ABBREVIATIONS c.m.^center of massp-p peak-to-peakDOF^Degrees Of FreedomFEL^First Element LaunchFEM Finite Element MethodFLT^Feedback Linearization TechniqueIP InPlaneINSAT II^INdian SATellite IIMSS Mobile Servicing SystemOP^Out-of-PlanePMC Permanently Manned ConfigurationPV^PhotoVoltaicQCLC Quasi-Closed-Loop ControlQOLC^Quasi-Open-Loop ControlSAP Solar Array PedalSFU^Space Flyer UnitSSM Substructure Synthesis MethodDot ( • ) and prime ( ) represent differentiations with respect to time tand true anomaly 9, respectively. Subscripts o and e indicate initial and equilibriumconditions, respectively. Unless stated otherwise, overbar ( ) represents a vector;boldfaced symbol, a matrix; and underbar ( _ ) refers to a dimensionless quantity.xivLIST OF FIGURES1-1^The European Space Agency's L—SAT (Olympus) launched in 1989.^. . 21-2^SCOLE configuration showing the main components: shuttle, mast, andplate-type reflector antenna. ^  31-3^The proposed Space Station Freedom configuration as of 1988. ^ 51-4^The layout of the thesis showing the four major parts of the present study ^ 172-1^A schematic diagram of the spacecraft model. ^  212-2^The system of coordinates used in the formulation.  232-3^Orbital elements defining position of the center of mass of spacecraft.^252-4^Modified Eulerian rotations V), 0, and A defining an arbitrary spatialorientation of spacecraft.   262-5^Solar radiation incidence angles op, 4, and Op. ^ 282-6^An illustration of the Eulerian rotation^  503-1^Flowchart showing numerical approach to multibody dynamics simulation. 603-2^Flowchart of a numerical integration subroutine. ^ 633-3^Flowchart of the subroutine FCN. ^  693-4^Librational response and energy variation of a rigid satellite subjected toan initial pitch disturbance: (a) c, = 1°; (b) 0 /0 = 0.03. ^ 813-5^The effect of roll disturbance on a rigid satellite's librational responseand energy variation: (a) 00 = 1°; (b) 0/0 = 0.2.   823-6^Dynamical response and energy variation of a rigid satellite with aninitial condition in the yaw degree of freedom: (a) A, = 1°; (b) Ao = 0.02. 83xv^3-7^The spacecraft model used to assess accuracy of the present formulation. 853-8^A typical response obtained by the present formulation simulating thespacecraft model studied by Ng [90]. The results showed perfectagreement.   863-9^Schematic diagram of the space station based MSS studied byChan [15]: (a) coordinate systems; (b) design configuration. It is usedhere to assess validity of the present general formulation. ^ 873-10 A comparison between the MSS response obtained by Chan [15] andthe present formulation^  893-11 Dynamics of the MSS using one, two, three, and four assumed modes inthe simulations:(a) librational response; ^  90(b) the first mode vibrational response; ^  91(c) the second mode vibrational response;  92(d) the third mode vibrational response. ^  924-1^Configuration of the FEL used in the numerical simulation:(a) coordinate systems; (b) design configuration. ^  984-2^Librational response of the rigid FEL showing the inherent unstablecharacter of its orientation.   1004-3^Librational behavior of the rigid FEL showing the unstable response inthe presence of an external disturbance in pitch, roll or yaw of 0.1°.^. 1014-4^Dynamical response of the FEL in absence of an external initialdisturbance. ^  1034-5^Effect of an initial 1 cm tip deflection of the power boom on theFEL dynamics:xvi(a) initial deflection in the local vertical direction, (P3) 0 = 0.826x10 -4; ^  104(b) initial deflection in the local horizontal direction, (0 0 = 0.826x10-4 ^  1064-6^Librational and vibrational dynamics of the FEL subjected to an initial1 cm tip deflection of the stinger:(a) initial deflection in the local vertical direction, (P1) o = 0.1872x10-3; ^  108(b) initial deflection in the local vertical direction, (0 0 = 0.1872x10-3 ^  1094-7^The response of the FEL with an initial condition on the PV array orthe radiator:(a) PV array radiator with a tip deflection of 0.5 cm, (H 21^= 0.2485 x10 -3; ^  110(b) PV array with a tip deflection of 0.5 cm, (H3' 1 ) 0 = 0.866x10-3 ^  1114-8^Schematic diagram of the PMC used in the numerical simulations:(a) coordinate systems; (b) design configuration. ^ 1144-9^Libration response of the PMC due to deviation from the equlibrium^configuration.   1164-10 Rigid body response of the PMC to rotational disturbances. ^ 1174-11 Response of the PMC due to deviation from the equilibrium orientation. 1184-12 The influence of the power boom initial tip displacement of 1 cm on thePMC response:(a) displacement in the local vertical direction, (PD 0 = 0.4348 x10 -4 ; . 120xvii(b) displacement in the local horizontal direction, (Q,1 )0 =0.4348 x 10-4 ^  1214-13 The response of the PMC subjected to an initial 1 cm tip deflection ofthe stinger:(a) deflection in the local vertical direction, (P1) 0 = 0.1872x 10 -3 ; .^123(b) deflection in the orbit normal direction, (Q1), = 0.1872x 10 -3 .^1244-14 Effect of station radiator, PV array, and radiator disturbance on thePMC dynamics:(a) station radiator subjected to a tip deflection of 0.5 cm, (H 214 ) 0 =0.2485 x 10 -3; ^  125(b) PV array radiator subjected to a tip deflection of 0.5 cm,(H 14 ' 1 ) 0 = 0.2485x 10 -3 ; ^  127(c) PV array subjected to a tip deflection of 0.5 cm, (4 4 ) 0 = 0.866x10-4 ^  1284-15 Librational and vibrational responses of the PMC in absence of externalexcitation showing the effect of thermally deformed PV arrays^ 1304-16 Response of the PMC in eccentric orbit (€ = 0.02). Initial disturbance^is due to deviation from the equilibrium orientation.   1324-17 Variation of angular velocities as well as linear and angular accelerationfields for the PMC in absence of active disturbances. ^ 1344-18 The effect of power boom and stinger disturbance on the PMC velocitiesand accelerations:(a) initial power boom tip displacement of 1 cm in the local verticaldirection; ^  136(b) initial power boom tip displacement of 1 cm in the local horizontalxviiidirection; ^  137(c) initial stinger tip displacement of 1 cm in the local vertical direction; 138(d) initial stinger tip displacement of 1 cm in the orbit normal direction. 1394-19 Variations of PMC velocities and accelerations showing the effect ofstation radiator, PV array and radiator disturbances:(a) station radiator with a tip deflection of 0.5 cm^  141(b) PV array with a tip deflection of 0.5 cm.  142^(c) PV radiator with a tip deflection of 0.5 cm.   1434-20 Influence of the thermal deformations and orbital eccentricity on thePMC velocities and accelerations:(a) thermally deformed PV arrays; ^  144(b) orbital eccentricity (€ = 0.02).  1464-21 Coordinate systems and design configuration used in the simulations ofMSS slewing maneuvers.   1494-22 System response with the lower link of the MSS undergoing a 180° slewin 5 minutes -   1514-23 Forced oscillations of the space station showing the effect of a 180 ° slewin 7.5 minutes of the lower link of the MSS: (a) out-of-plane maneuver;(b) inplane maneuver.   1524-24 Response of the space station with the MSS subjected to a 180°maneuver of the lower link in 10 minutes: (a) out-of-plane maneuver;(b) inplane maneuver^  1534-25 Effect of the maneuver time on the pointing error with the lower linkslewing through 180°:(a) 5-minute maneuver; ^  155xix(b) 7.5-minute maneuver; ^  157(c) 10-minute maneuver.  1584-26 Dynamical response of the space station with the MSS located at 50 mfrom the central body c.m.: (a) out-of-plane maneuver; (b) inplanemaneuver^  1604-27 Pointing error of the MSS, located 50 m from the central body c.m., withthe lower link undergoing a 180 ° maneuver. ^  1614-28 Effect of the increased MSS stiffness on the response of the space stationwith the lower link undergoing a 180° maneuver in 5 minutes:(a) out-of-plane maneuver; (b) inplane maneuver.   1624-29 Pointing error of the MSS, of increased stiffness, with the lower linkundergoing a 180° maneuver.   1644-30 Coordinate systems and the two possible design configurations for theSFU central body and arrays. ^  1684-31 Dynamical response of the SFU during deployment of the solar arraypedals:(a) inplane deployment;   171(b) out-of-plane deployment.   1724-32 Response characteristics of the SFU during deployment of the solar arraypedal B2:(a) inplane deployment; ^  173(b) out-of-plane deployment.   1754-33 Librational and vibrational responses of the SFU during the out-of-planeretrieval maneuver of the solar array pedals:(a) 5-minute retrieval period; ^  176xx(b) 10-minute retrieval period; ^  177(c) 20-minute retrieval period  1784-34 System response of the SFU showing unstable motion induced by theinplane retrieval of the solar array pedals:(a) 5-minute retrieval period; ^  179(b) 10-minute retrieval period;  181(c) 20-minute retrieval period^  1824-35 Out-of-plane retrieval of one of the solar array pedals (B2) showinginstability of the system:(a) 5-minute retrieval period; ^  183(b) 10-minute retrieval period;  184(c) 20-minute retrieval period^  185^5-1^Coordinate systems and design configuration used in simulations of theINSAT II. ^  1925-2^Uncontrolled response of the rigid INSAT II showing instability of thesystem: (a) 77G0 = 00 = A o = 1 ° ; (b) 010^010^1. ^ 1945-3^Controlled librational response of the INSAT II for three different setsof gains:(a) 0o=^= Ao = °; ^  196(b)O io = (Y0 = Afo = 1.   1975-4^Comparison of control efforts for three different sets of gains used in theINSAT II attitude control:(a) 77b0 = 00 = A o = 1 ° ; ^  198(b) O fo = Olo = AO = 1•   1995-5^Block diagram for the control of flexible INSAT II:xxi(a) quasi-open loop control; ^  203(b) quasi-closed loop control.   204^5-6^Librational and vibrational responses of the uncontrolled INSAT II withinitial condition of qp o =^= Ao = 1 °• ^ 2065-7^Variation of tip deflections of the INSAT II flexible appendages showingthe effect of control gains. ^  2075-8^Control effort variations for the INSAT II subjected to three differentcombinations of control gains. ^  2085-9^Response of the uncontrolled INSAT II, with reduced stiffness of theappendages, to the initial conditions of /Po = cbo = Ao = 1°   2105-10 Vibrational response of the INSAT II with flexible appendages ofreduced stiffness. ^  2115-11 Comparison of the control effort variations for the INSAT II withappendages of reduced stiffness.   2135-12 Librational and vibrational responses of the INSAT II showing theeffect of thermally deformed appendages. ^  2145-13 Dynamics of the INSAT II with thermally deformed appendages andinitial conditions of '0 0 =^= Ao = 1°^  2155-14 Tip deflections of thermally deformed INSAT II appendages showingthe effect of control gains. ^  2175-15 Plots of control efforts required for the INSAT II with thermallydeformed appendages.   2185-16 Controlled system performance of the MSS undergoing a 5-minute OPmaneuver: (a) libration response; (b) vibration response; (c) controleffort time histories; (d) pointing error variation. ^  2205-17 Controlled system performance of the MSS undergoing a 10-minute OPmaneuver: (a) libration response; (b) vibration response; (c) controleffort time histories; (d) pointing error variation. ^  2225-18 The performance of the MSS controller in the presence of offset and a5-minute OP maneuver: (a) libration response; (b) vibration response;(c) control effort time histories; (d) pointing error variation.   2245-19 The performance of the MSS controller in the presence of offset and a10-minute OP maneuver: (a) libration response; (b) vibration response;(c) control effort time histories; (d) pointing error variation. ^ 2265-20 The performance of the MSS undergoing a 5-minute OP maneuver withthe controller implemented at the beginning of the slew: (a) librationresponse; (b) vibration response; (c) control effort time histories;(d) pointing error variation. ^  2285-21 MSS performance in the presence of offset and undergoing a 5-minuteOP maneuver with the controller implemented at the beginning of theslew: (a) libration response; (b) vibration response; (c) control efforttime histories; (d) pointing error variation.   229LIST OF TABLES2-1 A comparison between Warburton [94] approximate and Gorman [96]exact eigenvalues of cantilever and free-free plates. ^ 463-1 List of T, A, and F operations executed in subroutine RMAT^.^. 723-2 List of all the 0 operations executed in subroutine VECTOR^.^. 733-3 Spacecraft data used to assess accuracy of the present formulation^. 853-4 Data for the space station based MSS studied by Chan [15] 883-5 Comparison of the CPU time required using one, two, three and fourassumed modes in the simulation of the MSS ^ 944-1 Physical parameters of the major components of the FEL ^ 994-2 Physical parameters of the major components of the PMC ^ 1154-3 Data for the space station based MSS used in the simulation^.^. 1484-4 System performance vs. the type of maneuver and its period^.^.^. 1654-5 Data for the SFU used in the simulation  ^1694-6 Summary of the SFU stability with respect to deployment/retrievalperiod and orientation  ^1865-1 The INSAT II data ^ 1915-2 Summary of maximum Q.0, Q0, and QA required for the cases studied 2165-3 Summary of the system performance for the controlled MSS without-of-plane maneuver ^ 227xxivACKNOWLEDGEMENTSI would like to thank Prof. V.J. Modi for his guidance throughout the preparationof the thesis.The thesis has benefited from discussions and consultation among colleagues. Inparticular, I would like to express my gratitude to Mr. J. Chan, Mr. H. Mah, andMr. A. Suleman for exchange of ideas. Also, Dr. Karray has provided useful tutorialinto the nonlinear control study. His help is deeply appreciated.During the course of my Ph.D. study, program execution went through a transitionfrom the mainframe to decentralized workstation. This was facilitated through thetechnical advice offered by Mr. A. Steeves. Special thanks to Prof. D. Cherchas forthe use of VaxStation 32_00.The research project is supported by the Natural Sciences and Engineering Re-search Council of Canada, Grant No. A-2181 and the Center of Excellence Program,Grant No. IRIS/C-8, 5-55380, both held by Prof. Modi.1. INTRODUCTION1.1 Preliminary RemarksEver since the launching of the first satellite Sputnik in 1957, there is a generaltrend towards larger and more complex spacecraft designs. This is as a consequence ofthe ever increasing demand on spacecraft capabilities in telecommunications, weatherforecasting, surveillance, remote sensing and others. Most of the designs can beidentified as consisting of a main body with appendages attached to it. The mainbody can be rigid or flexible whereas the appendages in the form of solar arrays,antennas, robotic arms, etc. are usually flexible. The following few examples ofcontemporary spacecraft illustrate this point:(i) The European Space Agency's L—SAT (Large SATellite system, Olympus,Figure 1-1 ), launched in 1989, represents a new generation of communicationssatellite. It has two solar panels, each 25 m in length, connected to a centralbody.(ii) To better understand the behaviour of flexible structures in space and theircontrol, the NASA's Langley Research Centre once designed an Orbiter basedexperiment called COFS (Control Of Flexible Structures). Its first phase,called SCOLE (Structural COntrol Laboratory Experiment, [1]), involved lab-oratory simulation of a flexible configuration as shown in Figure 1-2 followedby flight verification on board the Shuttle. The experimental setup consistsof a mast around 40 m in length, with its one end attached to the shuttle.At the other end, there is an asymmetrically mounted reflector plate-typeantenna, 22.8 m in diameter.(iii) The U.S. led Space Station Freedom, scheduled to be completed around 1998,SolarPanelCentralBodyFigure 1 - 1^The European Space Agency's L-SAT (Olympus) launched in 1989.2ReflectorPlateMastFigure 1-2^SCOLE configuration showing the main components: shuttle, mast,and plate-type reflector antenna.3will have a main truss of 155 m in length. Equipment attached to the trussincludes habitat, command and other modules, power generation equipmentand system control assembly, stinger and resistojet, photovoltaic (PV) arrays,PV array and station radiators, etc. (Figure 1-3 ).The space dynamicists have traditionally relied mostly on mathematical tools todesign spacecraft. Ground based experiments have been of limited value due to prac-tical difficulty in simulating environmental forces (gravity gradient, magnetic, freemolecular, solar radiation, etc.) and structural flexibility. This has led to increasingdependence on numerical methods, particularly with larger and more complex space-craft configurations. A general formulation applicable to a large class of systems isalways attractive although it usually demands more time and effort. On the otherhand, once the governing equations are established and the associated integration pro-gram is operational, it becomes a powerful versatile tool. Several, relatively general,approaches for multibody dynamics simulation have been developed. In general, theytreat systems of interconnected rigid/flexible bodies of tree-type topology. As can beexpected, each approach has its own special features. Some of them are touched uponbelow:(i) Treetops, developed by Singh et al. [2], is based on Kane's method in formu-lating the equations of motion. Deformations are described by a modal setwhich satisfies the kinematic boundary conditions. It has the capability tocarry out numerical linearization and control simulation.(ii) ALLFLEX, designed by Ho et al. [3,4], formulates the equations of motionusing Ho's direct path method [5] in conjunction with the Lagrangian orNewtonian approach. Substructure modal data are obtained through a fi-nite element program. Besides control simulation capability, the program has4\......^A^\\ \`\\ \\\\\U MAIM filAR U Ufk^vv 4^v \\\^AvAvA\40. wilinixt ".\\m\ -Ar■ aim ,,,„\\^\\\\\\\\NASIMS \ ==V ►►ta provision to accept subroutines accounting for the environmental distur-bances.(iii) DISCOS (Dynamic Interaction Simulation of COntrols and Structures) is anindustry standard software for multibody dynamics simulations. The equa-tions of motion are derived using Lagrange's approach. The program hasseveral features of both the Treetops and ALLFLEX.A measure of the program efficiency is the order of the algorithm, nx, where nis the number of degrees of freedom and x is the index of the order. Physically,the order relates to the computational time. As the number of degrees of freedomdoubles, the execution time is expected to increase by 2' times. Obviously, a smallindex is desirable. The order of the Treetops is not revealed in literature whereasALLFLEX and DISCOS have orders of n 2 and n3 . Recent developments in DISCOShave reduced the order to n for some special cases [6].These three programs are described as easy to use and portable among comput-ers. A number of users would not agree with this assertion. Experience suggeststhat making a program operational and applicable to a problem in hand often in-volves enormous time and effort. Perhaps the major limitation is their 'black-box'character which makes any enlighten variations virtually impossible. Thus one is leftwith a feeling of a bystander pulling levers and collecting massive output of numbersrather than actively participating in the investigation. There are two other aspectsto consider:(i) Besides portability of the program, its size is a critical factor in determiningwhether the program can be executed on a new machine. The limitation onsize is particularly important in time-sharing computer systems.(ii) To take advantage of the feature that allows user-supplied subroutines, one6has to have a thorough understanding of the formulation procedures and pro-gram architecture. Unfortunately, this information is not provided because ofthe proprietary considerations. Even when available, their thorough under-standing would entail substantial effort and time.This being the case, considerable interest exists in the community of space dy-namicists and control engineers to develop efficient, relatively general formulations,applicable to a class of systems of contemporary interest, and associated algorithmsthat are user-friendly. The present thesis takes a small step in that direction.1.2 A Brief Review of the Relevant LiteratureOver the past thirty years, the amount of literature accumulated on the subjectof spacecraft dynamics is literally enormous. It can be classified into four broad cat-egories: formulation; dynamics and control; environmental effects; and experimentalvalidation. The thesis is primarily concerned with the first three aspects: formula-tion, dynamics and environmental effects (thermal); hence, the focus of the review isin these areas. Likins [7], Modi et al. [8-10], Roberson [11], and Markland [12] haveprovided excellent overview of the subject. Theses by Lips [13], Ibrahim [14], Chan[15] and others have also reviewed relevant aspects at great lengths to acquaint aninterested researcher with the status of the field.Perhaps year 1965 marked the beginning of interest in the multibody formula-tion methodology. Early literature was limited to rigid spacecraft with hinged joints.Hooker and Margulies [16] derived the attitude equations for n rigid bodies intercon-nected to form a topological tree configuration. The formulation was based on theNewton-Euler approach and accounted for dissipative and elastic joints. This prob-lem was also studied by Roberson and Wittenburg [17] who introduced the idea ofsystem-graph, which made the implementation of the equations on a digital computer7easier.Roberson [18] later extended the study to interconnected nonrigid bodies wheretranslational motion between the members was allowed. Ho [5] considered a tree-type topological configuration with the end bodies flexible. He selected the direct pathmethod and derived the equations using both the Newton-Euler as well as Lagrangianprocedures. The latter formulation was found to be faster from the mathematicalpoint of view as the lengthy procedure to eliminate the constraint forces and torqueswas not necessary. To overcome this drawback of Newton-Euler method, Hooker[19] has shown that it is possible to use Ho's method together with Newton-Eulerapproach to derive equations of motion in which constraint forces and torques do notappear explicitly.By introducing path and reference matrices that describe the topology of n-bodyconfigurations, Jerkovsky [20] presented the equations of motion for both tree-typeand closed-loop configurations. A comparison with the multibody formulations byRoberson and Witterburg [17], Hooker [21], Ho [5], and several others was also given.Hughes [22] derived the equations of motion for a chain of flexible bodies withterminal members rigid. The Newton-Euler approach was used and the resultingequations were linear in the angular rates as well as elastic deformations. The equa-tions were tailored for control system design.Kane and Levinson [23] compared the pros and cons of different formulation tech-niques such as Newton-Euler method, D'Alemberts' principle, Hamilton's canonicalequations and Lagrangian procedure. They then introduced what is referenced as"Kane's method" using two classes of quantities known as "partial angular veloci-ties" and "partial velocities." The authors have concluded that "Kane's method"leads to the simplest equations.8Using the Lagrangian procedure, Modi and Ibrahim [24] presented the generalequations of motion for studying librational and vibrational dynamics of a large classof spacecraft during deployment of flexible members. The equations accounted for thegravitational effects, shifting center of mass, changing rigid body inertia, appendageoffset and transverse oscillation. Appendages with variable mass density, flexuralrigidity, and cross-sectional area along its length can also be accommodated.For a maneuvring spacecraft, the general equations of motion were obtained byMeirovitch and Quinn [25] using the Lagrangian approach in conjunction with thecomponent mode synthesis. In order to improve computational efficiency, a pertur-bation approach was then applied. This resulted in a set of equations governing therigid-body motion (unperturbed system), and a system of time-varying linear equa-tions for small elastic motions as well as deviations from the prescribed rigid-bodymaneuver (perturbed system).Vu-Quoc and Simo [26] studied the dynamics of satellites with flexible componentsby referring the motion directly to the inertial frame. In order to avoid numericalproblems associated with ill-conditioned matrices, the dynamics of far field (attitudemotion) and near field (elastic deformations) were treated separately through intro-duction of a rotationally-fixed floating frame.Recent contributions in multibody formulation include studies by Keat [27] usingthe velocity transform method, Huston [28] employing Kane's method, and Kurdila[29] relying on Maggi's approach. Unlike the earlier studies, the emphasis here is onthe numerical efficiency rather than methodology.The literature on spacecraft control is primarily concerned with two aspects: at-titude control and vibration control. Contributions in the attitude control field havebeen reviewed by Roberson [11] whereas Balas [30], and Meirovitch and Oz [31] have9provided overviews of the vibration control problems. A few studies aimed at controlalgorithms for flexible spacecraft are briefly touched upon here.Flexible spacecraft, being a distributed parameter system, needs to be discretizedto study the associated dynamics and its control. The discrete system can be trans-formed further into the modal-space using modal matrices. Meirovitch and Oz [32]have found that the control as applied to the transformed system (modal-space con-trol) is more efficient than that applied to the discrete system ("actual space" con-trol). Subsequent contributions by the same authors [33-35] study application of themodal-space control to different spacecraft configurations.Balas [36] examined the undesirable effects of applying control to a flexible struc-ture discretized by a limited number of modes. This effect, known as spillover, resultsin the excitation of unmodeled modes. Using modal-space control, the author [37,38] has derived the conditions under which spillover effect can be eliminated. Theidea was later extended to attitude stabilization of flexible spacecraft [39]. Balas hasalso discussed advantages of the Direct Velocity FeedBack (DVFB) procedure [40];however, implementation of the DVFB was found to be quite involved.Wie and Bryson [41] modeled flexible space structures using single-input single-output transcendental transfer functions. The models were simple enough for polesand zeroes to be determined analytically. The results were then used in the pre-liminary controller design. Wie [42] applied this approach to Control of FlexibleStructures (COFS—I) mast flight system . Chu et al. [43] employed the same ideain modeling and designing the Space Station attitude controller. Using multi-inputmulti-output transfer functions and numerical algorithms, Kida et al. [44] designedthe controller of flexible spacecraft with constrained and unconstrained modes.Goh and Caughey [45] explored the idea of stiffness modification in vibration1 0suppression of flexible structures. The control scheme guaranteed global stability byvirtue of the positive definite rate of energy decay. The implementation required notthe conventional actuators but rather transducers which converted strain into controlsignals and then into electronic damping. This was considered to be a favourablefeature.In understanding the problem at a fundamental level and progressively makingit more complicated to approach a real-life situation, the study by Reddy et al. [46]concerning attitude and vibration control of plate-like platform is particularly useful.Equally informative are the contributions by Yedavalli [47] and Sundarnarayan et al.[48] on the robustness of controllers for Large space Structures (LSS) applications.In general, the studies pertaining to environmental influence on spacecraft dy-namics and control are relatively few; however, contributions on solar radiation ef-fects are not lacking. Among them, research by Modi et al. [49-541, Beletsky andStarostin [55], Goldman [56], Yu [57], Frisch [58,59], Tsuchiya [60,61], and Bainumet al. [62-66] are worth-mentioning.Modi and Brereton [49] studied the planar librational stability of a slender flex-ible satellite under the influence of solar heating. Using the quasi-static solutionsdescribing a thermally flexed configuration and the concept of integral manifold, theauthors developed charts as functions of initial conditions and eccentricity showinglimiting stability of the satellite. In general, thermal heating was found to decreasethe size of the stability region.A similar approach was adopted by Modi and Flanagan [50-52] to study the in-fluence of solar radiation pressure on gravity oriented satellites at different altitudes.It was observed that the solar radiation was a significant disturbance except at lowaltitudes (less than 1000 km) where aerodynamic drag became dominant. Introduc-11ing the solar parameter proportional to the satellite length, the difference betweenmaterial reflectivity and transmissivity, offset of the center of pressure, etc., stabilitycharts were obtained which clearly indicated the effect of solar radiation. The sig-nificance of the solar parameter was two-fold. On the one hand, the stability regiondecreased as the parameter value increased. On the other hand, for a given initialcondition, there existed a value of the solar parameter for which librational motionwas minimum. By a judicious choice of the solar parameter, the authors [53] showedthat solar radiation can be utilized to damp the librational motion effectively.Modi and Kumar [54] studied the solar heating effect on gravity oriented satel-lites with flexible plate-type appendages. Employing the integral manifold concept,stability charts were obtained. Appendage flexibility was found to cause a substantialreduction in the size of the stability region. The destabilizing influence was particu-larly severe in presence of the solar radiation pressure and orbit eccentricity. Similarproblem was also studied by Beletsky and Starostin [55]. The emphasis was on theexistence and stability of symmetrical and nonsymmetrical periodic solutions. Theproblem was solved by both the analytical (Volosov-Morgunov averaging method)and numerical means.Using a quasi-static approach, Goldman [56] studied the influence of solar heatingon the dynamic stability of a satellite. Equilibrium position of the booms in thepresence of differential heating was determined without consideration of the transversedeformations. The simulation results helped to confirm that solar heating contributedto the anomalous behavior of the Naval Research Laboratory's Gravity GradientSatellite 164.Yu [57] investigated thermally induced vibration of spacecraft booms with a tipmass. The problem was formulated using Hamilton's principle. In absence of damp-12ing, the motion of the boom was found to be stable if pointed away from the sun.Viscous-fluid damper incorporated into the tip mass also proved to be effective insuppressing thermal flutter. However, the results are controversial: using other ap-proaches, Jordan [67] and Augusti [68] concluded that the boom motion was stableonly when pointed towards the sun.Frisch [58] studied coupled nonplanar transverse and torsional vibration of booms.Numerical simulation results helped to explain the anomalous behaviour of severalthree-axis-stabilized satellites with long extendable booms. The results showed thatthermally induced vibration can be eliminated by increasing torsional rigidity of theboom. The author also presented approaches based on finite element and finite dif-ference methods to include thermal effect in multibody formulations [59].Tsuchiya analyzed the effect of thermally induced appendage vibration on a spin-ning satellite [60], and a satellite with a rotor [61]. The criteria for appendage res-onance were derived. Amplitudes of the vibrational and nutational motions weredetermined by the method of averaging. Numerical simulation of the system near theresonance was also presented.Krishna and Bainum examined the effects of solar radiation pressure on the rigidand flexible modes of a beam [62] and a square plate [63]. The dynamical responseindicated that induced rigid body response was more significant than the flexiblemotion. Control laws based on linear quadratic Gaussian technique were found to beeffective in providing both shape and orientation control. The authors [64-66] laterextended the study to include thermal deformation of the structure. In general, theeffect of solar radiation pressure acting on structures undergoing thermal deformationswas found to be relatively more important.Sikka et al. [69] studied the static problem of thermal deformation. The tem-13perature distribution and curvature produced in long, solid circular and rectangularcross-section cylinders were obtained using approximate analytical methods such asthe least-square fitting. Modeling a lattice-type space structures using finite elementmethod, Lutz et al. [70] carried out a thermoelastic analysis of the structural mem-bers. Although thermal moments were found to be very small, thermal axial forcesinduced significant mechanical bending in the structure.1.3 Scope of the Present InvestigationWith this as background, this thesis presents a relatively general formulationparticularly suitable for studying dynamics of evolving structures such as the SpaceStation. In essence, it is a Lagrangian formulation based on the direct path method.The emphasis is on the applicability of the formulation to study complex dynamicsof large space structures using relatively simple mathematical models to gain betterphysical understanding of interactions between librational dynamics and flexibility.The formulation has the following distinctive features:(i) It is applicable to an arbitrary number of lumped masses, beam and plate typestructural members, in any desired orbit, interconnected to form a tree-typetopology.(ii) It takes into account slewing of solar panels and transverse vibrations of thebodies. Thus, it is possible to study the complex system dynamics due to in-teractions between librational motion, transverse vibration, and slewing ma-neuvers.(iii) The shift in the centre of mass due to transverse vibrations and slewing ma-neuvers is incorporated in the formulation.(iv) The formulation accounts for the thermal deformation of both beam and plate14type appendages explicitly. Effect of the free molecular environment can beintroduced quite readily through generalized forces.(v) The governing nonlinear, nonautonomous and coupled equations are pro-grammed in such a way that the effects of flexibility, librational motion,thermal deformation, slewing maneuvers, shifting c.m., higher modes, initialconditions, etc. can be isolated easily.The thesis can be divided into four parts: relatively general formulation of theproblem; program implementation; numerical simulation; and control study.The problem formulation begins with the study of kinematics of a spacecraft withinterconnected flexible bodies. The discretization of elastic deflection and evaluationof the thermal deformation are discussed next, leading to the expressions for kineticand potential energies. Using the Lagrangian procedure, general equations of motion,applicable to a large class of systems, are obtained.The second part discusses implementation of the equations of motion into com-puter codes. It briefly describes transformation of the equations into a form suitablefor numerical analysis and develops associated algorithm. The emphasis is on themethodology that results in algorithms which are both easy to program and debug.Validity of the program is first established through comparison of dynamical simu-lation results, for two particular configurations, obtained by Ng [71] and Chan [15].The convergence of the numerical solution is also demonstrated.In the third part, response simulation of four distinctively different spacecraftconfigurations of contemporary interests is carried out. The objective here is tostudy complex interactions between librational and vibrational dynamics, flexibilityand initial disturbances. Effects of thermal deformations are also discussed. Theamount of information obtained is literally enormous; however, for brevity, only some15typical results useful in establishing trends are presented here.The next logical step is to explore effectiveness of a control strategy applicableto such a formidable class of problems. The last part studies the control using thefeedback linearization approach to selected spacecraft models.The concluding chapter summarizes more important results and presents recom-mendations for future studies. An overview of the thesis layout is shown in Figure 1-416Parametric Studies* First Element Launch (FEL)* Permanently Manned Configuration (PMC)* Mobile Servicing System (MSS)* Space Flyer Unit (SFU)Dynamics and Control of Orbiting Flexible Systems:A Formulation with ApplicationsProblem Formulation• Kinematics• Elastic & Thermal Deformations• Kinetics• Lagrangian FormulationNumerical Implementation• Numerical Integration SubroutineComputation FlowchartSubroutine FCN• Program Implementation• Program VerificationNonlinear Control• Feedback Linearization Technique* Quasi Open-loop ControlQuasi Closed-loop Control• Application to Indian Satellite IIand MSSFigure 1 - 4^The layout of the thesis showing the four major parts of the present study.2. FORMULATION OF THE PROBLEM2.1 Preliminary RemarksFrom the literature review, it is apparent that the the Newton-Euler method andthe Lagrangian approach are more commonly used procedures in the dynamical for-mulation of multibody systems. The Newton-Euler method is based on the principleof angular momentum whereas the Lagrangian approach relies on the system energy.Attractive features of the Newton-Euler method include relatively less time and ef-fort as well as more compact form of the governing equations. This also makes theNewton-Euler method computationally more attractive. However, the method hastwo major drawbacks:(i) It requires the introduction and subsequent elimination of the constraintforces.(ii) The principle of angular momentum has to be applied at the centre of massof the system; hence, for a system with shifting center of mass, its applicationcan lead to inaccuracies.In the present study, due to the complex character of the system, the eliminationof constraint forces is indeed quite involved. Also, due to the thermal deformationsand transverse vibrations of the appendages, the centre of mass is shifting. Appli-cation of the Newton-Euler method is, therefore, not particularly attractive. On theother hand, Lagrangian approach does not suffer from these limitations. Further-more, Silver [72] has shown that, at least for ground based robot manipulators, theLagrangian formulation with a proper choice of generalized coordinates can be numer-ically as efficient as the Newton-Euler approach. However, the Lagrangian procedurehas its own undesirable features —it involves enormous amount of algebra leading to18lengthy equations of motion. This is especially true in the case of multibody systemswhere position vectors are represented as a product of a matrix (e.g. A) and a vector(say, U). Differentiation of either A or U is not difficult to handle; however, differen-tiation of the product of AU would involve enormous amount of algebra, especiallywhen second derivative is required. Fortunately, with the advent of computers andrefined softwares, the problem is manageable. The user supplies the derivatives of Aand U and let the computer evaluate the derivatives of products by numerical means.This approach is appealing in cases where closed-form solution to the problem doesnot exist and a numerical simulation is the only alternative. In the present case, thegoverning equations of motion are expected to be highly nonlinear, nonautonomous,and coupled; hence, closed-form solution is not expected to exist. The Lagrangianprocedure is therefore selected to assure accuracy of the governing equations.This chapter can be divided into four sections: kinematics, substructure defor-mations, kinetics, and the Lagrangian formulation. The kinematics begin with adiscussion of the system geometry and reference coordinate systems used to identifythe deformed configuration. The spatial orientation of the system as described bya set of orbital elements and modified Eulerian rotations is presented next, togetherwith the solar radiation incidence angles. Finally, the shift in the center of massdue to deformations, and associated rotation matrices, are discussed. In the follow-ing section, determination of thermal and elastic deformations of substructures areexplained and application of the assumed mode method to represent elastic defor-mations examined. The kinetics of the problem deals with evaluation of the kineticand potential energies. Using the Lagrangian procedure, the governing equations ofmotion are finally derived.192.2 Kinematics of the Problem2.2.1 Configuration selectionThe system model selected for study consists of flexible bodies connected to forma branched geometry: central body B, is connected to bodies Bi (B1,... , Bar). Inturn, each Bi is connected to bodies (Bo, . . . , Bi,„,) as shown in Figure 2-1. Altogether, there are Ni (= I 1 ni) Bi bodies. The number and locations ofbodies are kept arbitrary so that the configuration can be used to study a large class ofpresent and future spacecraft. For instance, to simulate the European Space Agency'sOlympus (L—SAT, Figure 1-1), the satellite's central rigid body and two solar panelsare represented by bodies B, and Bi, respectively. As for the SCOLE configurationmentioned earlier, the dynamic simulation may be performed treating the Orbiter,mast and reflector antenna as bodies .13,, Bi , and Bi,j, respectively. When appliedto the proposed Space Station Freedom, the central body B, may simulate the maintruss of the Space Station with the modules, power generation equipment and systemcontrol assembly treated as lumped masses. The stinger, station radiators, PV arraysand radiators are represented by bodies Bi.2.2.2 Coordinate systemConsider the spacecraft model in Figure 2-2 . The centres of mass of the unde-formed and deformed configurations of the system are located at C i and Cf. , respec-tively. Let XQ ,YQ ,ZO be the inertial coordinate system located at the earth's centre.Attached to each member of the model is a body coordinate system helpful in definingrelative motion between the members. Thus reference frame F e is attached to bodyB, at an arbitrary point O. Frame Fi , with origin at 0i, is attached to body Bi atthe joint between body Bi and Bc . In addition, for defining attitude and solar radi-20•• PerigeeEartht.>FlexibleBodyFlexibleBody B1,2I - - —  ' • ...I^•^•$ •♦ •• ♦i ♦ ♦s %^•• • •♦ •%^ •^-• — ••••s s%•• .^• ,^• so$ ♦^•^I •• • • ••i ^ ♦i^♦ s• ♦^ •s ♦ ••1 •^• s ..^, •,,♦ •••• Flexible• • Body B iL•FlexibleBody th,,,,i \ OrbitFlexibleBody B2FlexibleBody BNiRem• • — ••• ••♦Figure 2-1^A schematic diagram of the spacecraft model.ation incidence angles, a reference frame is located at Cf such that the axes Xp , Yp ,and zp are parallel to X,, K, and Z,, respectively. Note, an arbitrary mass elementdmi on body Bi can be reached through a direct path from 0, via Oi. 0,, in turn, islocated with respect to the instantaneous center of mass Cf and the inertial referenceframe, Fo . Thus motion of dmi caused by librational and vibrational motions of B,and Bi can be expressed in terms of the inertial coordinate system. Similarly, frameFij is attached to body Bi j and has its origin (Oij) at the joint between Bi and Bij .The relative position of Oi with respect to 0, is denoted by the vector di wherebydi defines the position of 0,,j relative to O i .The location of the elemental mass of the central body, dm,, relative to 0, isdefined by a series of vectors. p L indicates the undeformed position of the element.Thermal deformation of the element is represented by T c . Finally, the transversevibration of the element, 6,, shifts the element to the end position. Similarly, pi, ,and Si define the location of the elemental mass dmi, in body Bi, relative to Oi. Forthe elemental mass dmi j of body Bi,j , its position relative to is defined by p-i ,j ,ri ,j , and Si,j.Orientation of the coordinate axes Xi,Y„Zi and X,^j relative to X,,Y,,Z,is defined by the matrices Ci and CU./Li, respectively such that^Up = tic =ui =^)14^ (2.1)where iti j is the matrix denoting the motion of body Bi ,j relative to body Bi.(k = p, c, i, or i, j) is the column vector representing the unit vectors in the corre-sponding coordinate axes; for instance,^=^k,}T. It should be noted that thethermal deformation and transverse vibration of B, and Bi result in the time-varyingcharacteristics of Ci and Ci j , respectively.22FlexibleBody BiRiZoRip;^%, Perigee,^...Orbit.,‘.. Flexible \.^... , Body Bi^ss ,^Central. . Body Be,Figure 2-2^The system of coordinates used in the formulation.2.2.3 Position of spacecraft in spaceConsider a spacecraft with its instantaneous centre of mass at Cf negotiatingan arbitrary trajectory about the centre of force coinciding with the homogeneous,spherical earth's centre. At any instant, the position of Cf is determined by the orbitalelements p, i , co, 6, R,,, and 0. Here, p is the longitude of the ascending node; i,the inclination of the orbit with respect to the ecliptic plane; w, the argument of theperigee point; 6, the eccentricity of the orbit; RC1Z , the distance from the center ofthe earth to Cf ; and 0, the true anomaly of the orbit. In general, p, i , w, and 6, arefixed while Rem and 0 are functions of time (Figure 2-3 ).As the spacecraft has finite dimensions, i.e. it has mass as well as inertia, inaddition to negotiating the trajectory, it is free to undergo librational motion about itscenter of mass. Let X„ Y„ Z, represent moving coordinates along the orbit normal,local vertical, and local horizontal, respectively. Any spatial orientation of Xp ,Yp ,Zpwith respect to X,,Y8 ,Z, can be described by three modified Eulerian rotations inthe following sequence: a pitch motion, '0, about the Xs-axis giving rise to the firstset of intermediate axes Xi ; a roll motion 0 about the Z 1 -axis generating theintermediate axes X2,Y2 ,Z2 ; and finally, a yaw motion, A, about the Y2-axis yieldingXp ,Yp ,Zp (Figure 2-4 ). From the figure, it can be seen that the librational velocityvector, cD, is given by,=^sin A + (e 1 .p) cos cos A]i p +^— (a + 1p) sin cbjcos A + (e + '0 cos 0 sin A] kp , (2.2)where e represents the orbital rate of the spacecraft.24Figure 2-3^Orbital elements defining position of the center of mass of spacecraft.25 Z2NJA ZsZpLocal HorizontalY2 Yp.411111111111114111111111111111111111J11111111111111111MM111•11•111111111111111IMIM•111111111111111111111•MII■11111111111111111M■111M11111111111■11111111111111•NOME■11111111111•11'WINENM&111111111.NOM■1111111111■(1) YiXs ,Orbit NormalLocal VerticalFigure 2 - 4^Modified Eulerian rotations , t, and A defining an arbitrary spatialorientation of spacecraft.262.2.4 Solar radiation incidence anglesPosition of the spacecraft with respect to the sun is defined by the solar radia-tion incidence angles, Op', or,, and Op'. They are defined as angles between the unitvector, ft, representing the direction of solar radiation, and the Xp , Yp , and Zp axes,respectively (Figure 2-5 ).With reference to the moving coordinate system X s , Ys , Zs , the unit vector itcan be written as=^{—al cos^a2 sin 0] j, + [a l sin^a2 cos 9] k,= Nis + b23 + b3 k8 , (2.3)where:a l = cos p cos co + sin q cos i sin co ;a2 = cos p sin co — sin p cos i cos co ;a3 = sin p sin i .^ (2.4)Now, in terms of the coordinates Xp ,Yp ,Zp :is = — sin Op + cos q sin Ajp + cos cos Aizp ;js = (cos 0 cos O)ip + (cos 0 sin O. sin A — sin 0 cos A)jp+ (cos 0 sin cos A + sin 1p sin A) kp ;ks = (sin 0 cos 0.)ip + (sin 0 sin sin A + cos cos A) jl,+ (sin 0 sin q cos A — cos sin A)^(2.5)Hence, substituting from Eq. (2.5) into Eq. (2.3), it can be rewritten as= cos Ox + cos 0Y5 + cos qP^P P^P P= ;T1 tt-P 13 (2.6)27OrbitYpEarth Figure 2-5^Solar radiation incidence angles O p', q) , and (6.28Perigee Iwhere:pT = {cos Op , cos Ov cos q5z} •p ,^Pcos 0' = b i (cos sin 0 cos A + sin '0 sin A) b2 (sin ,tp sin q cos) — cos sin A)• b3 cos 0 cos A ;cos 0v = bi cos cos 0 b2 sin '0 cos 0 — b3 sin 0 ;cos 0z = b i (cos '0 sin 0 sin A — sin cos A) b2 (sin sin 0 sin A + cos cos A)▪ b3 cos 0 sin A .^ (2.7)It can be seen that the angles are functions of both the orbital elements (p, i, co) andthe libration angles (0, 0, A). Using Eq. (2.1), the solar radiation incidence angleswith respect to coordinate frames attached to 13,, Bi and Bi ,j bodies can be obtained:Oc — Op (2.8a)Ot t^(C)2' OP (2.8b)= (C j)T Op (2.8c)where^0i and ki are defined similar to q  as above.2.2.5 Shift in the center of massThe centre of mass of the spacecraft is the reference point to describe the space-craft libration and orbital motion. For a rigid system, where the centre of massremains stationary, it can be determined easily; however, this is no longer true fora flexible system. The general expression describing a shift in the centre of mass isderived below.Consider the spacecraft in Figure 2-2. Here, Ci and Cf represent the centres ofmass of the undeformed and deformed configurations of the system, respectively. The29Rcm = —U R, dm, + E[f Ri dmimc^rni =1 mi,jI _rii ,j drn i,d} ,^(2.10)nivector^which denotes the position of Cf relative to C i , represents the shift inthe instantaneous centre of mass of the spacecraft due to its deformation. This vectorwill be necessary in evaluation of the kinetic and potential energies of the system.From the figure, with reference to Xp ,Y-p ,Zp axes, the vectors from the center ofthe earth to elemental mass dmc , dmi, and dmi,i represented by Pc , Ri , andrespectively, can be written as:Rc = Rcm CL.„ —^pc + Tc be= Rcni — Clm —^+ di + q(pi^Si) ;Ri = Rcm —^+ di + fi,j Si,j) . (2.9)Taking moment about the centre of force giveswhere M is the mass of the spacecraft. Substituting Eq. (2.9) into Eq. (2.10) yields1Ccm = m^{ Pc + 7-c + Sc Chnc E[f {di C, [pi + + Si] dm iTlIc^ 2=1^Mini+ E f {di+ Ci ^fi,i Siii1} dm i^, (2.11)where:Clem =0":7-2, + Ccm ;CIL, =position vector of C i , the centre of mass of the undeformedspacecraft, relative to 0,,CIL =position vector of Cf relative to C i ;ni =number of Bi bodies attached to body Bi;N =number of Bi bodies;30niM =total mass of the spacecraft; rn, E [m i +SinceLC"c77, = m^c{ dmc Ni= 1 krnific(0i) f CFfiirnii=}{mi,j [fic(Or) + CFiji (0i j)1 + f CUP"i j dini 5j }] ,mid(2.12a)an can be simplified as164 dm,+ E [{ mi[1- ,(00 + 6,(0i)]+ f CFV-i + dmii=1^ mini,j [J--,(oi)+ sc ow + cf(fi (oi ,j )+ si(oij))]f qi[( 111 ,i^Iti,jTi,j RAJ] i,j}]^(2.12b)miwhere 7k(0/) and 4(00 (k c,i; 1 i or i, j) represent 1--k and 6k, respectively,evaluated at the coordinates of Oi; and U is the unit matrix. Equation (2.11) is thegeneral expression valid for both rigid as well as flexible systems; however, for a rigidsystem, Eq. (2.12a) is adequate.2.3 Elastic and Thermal Deformations2.3.1 BackgroundEvaluation of the kinetic and potential energies requires expressions for the systemdeformation. In the past, considerable effort has been directed to this end; hence, itwould be appropriate to briefly review the relevant literature.31For a multibody system, two distinct approaches have been popular to estimateelastic deformations: the Finite Element Method (FEM) and Substructure SynthesisMethod (SSM). In the FEM, the system is first subdivided into finite elements withdegrees of freedom at the nodes. Using the local degrees of freedom as generalizedcoordinates, the mass and stiffness matrices of the element can be derived readily.Applying the boundary conditions for the system and compatibility requirementsbetween adjacent elements, the system mass and stiffness matrices can be assembledfrom the corresponding matrices of the elements. The system modes can then beevaluated numerically using finite element subroutines such as NASTRAN.In the SSM, the system's flexural motion is represented in terms of the compo-nents' dynamics. The first step is to obtain the series of admissible functions, bysolving the eigenvalue problem for each component, representing its elastic defor-mation. These functions are referred to as "component modes" by Hurty [73], whopioneered the SSM approach. It should be pointed out that the FEM can be used inderiving the component modes. Compared to the FEM, the subsubstructure eigen-value problem can represent a tremendous saving in the computational effort. This isespecially true when the components are geometrically similar or uniform elements.In the latter category, analytical solutions for beam and plate type elements are avail-able. The second step is to assemble the component admissible functions in such away that each component does not vibrate as an independent body but rather as apart of the system. Hurty [73] implemented this idea by enforcing geometric compat-ibility between adjacent elements at pre-selected points. Subsequent research in theSSM is aimed at selection of the substructure modes and improvements in geometriccompatibility at internal boundaries.Craig and Bampton [74] used the FEM to obtain the component shape functions.32Geometric compatibility between elements can be satisfied easily by selecting appro-priate boundary conditions for components. Benfield and Hruda [75] derived thecomponent admissible functions based on free boundaries. Geometric compatibilityis then enforced by applying inertial and stiffness loadings at internal boundaries.MacNeal [76] selects hybrid modes to represent component elastic motions, i.e. themodes are evaluated by assuming the boundaries to be free, fixed, or free in onepart and fixed in the rest. The advantage of this approach is the generality of theboundary conditions under which the component modes are calculated. Using thesame approach, Rubin [77] includes the contribution of residual modes to improveaccuracy of the system elastic deformation.Hughes [78] suggested another approach in obtaining the component modes. Hedistinguished two kinds of component modes: "unconstrained" and "constrained."In the former category, the component mode of a substructure is calculated withoutimposing any restriction on the adjacent body. The component modes obtained byHurty [73], Craig and Bampton [74], Benfield and Hruda [75], MacNeal [76], andRubin [77] belong to "unconstrained" category. In contrast, "constrained" modesof a component are derived by holding the adjacent body stationary. Hughes [79]pointed out that "unconstrained" modes are commonly used in the study of flexibleaircraft whereas "constrained" modes are often used for flexible spacecraft. Defining"modal identities " which are integrals of mode shapes relating to linear and angularmomentum, Hughes [79] has shown the existence of relations between these integralsof "constrained" and "unconstrained" modes. These identities can be used to assessthe influence of each substructure on the system. Using a spacecraft with a rigidcentral body and flexible appendages, Hablani [80] studied the convergence of both"constrained" and "unconstrained" modes. He concluded that as the rigid portionincreases, the convergence of "constrained" and "unconstrained" modes improves.33For "constrained" modes, the contribution to system dynamics comes from the firstfew modes whereas the importance of "unconstrained" modes is usually not orderedby frequency.The next question confronting space dynamicists is the number of modes required.Traditionally, high frequency modes are dropped; however, Hablani [80] and Hughes[81] have pointed out that frequency alone is not a sufficient condition in modaltruncation unless the modes are "constrained" . Hughes [81] recommended using"modal identities" of linear and angular momentum as criteria. Similarly, Hughesand Skelton [82] studied, besides frequency consideration, completeness of inertialindices, controllability and observability of component modes in modal truncation.Another approach, adopted by Gregory [83], is the application of Moore's [84] internalbalancing theory. The objective here is to transform the state-space form of thesystem's equations of motion into the balanced form such that the controllabilityand observability grammians are equal and diagonal. Generalized coordinates whichhave small diagonal elements are least controllable or observable; hence, they canbe discarded. Using the same idea, Spanos and Tsuha [85] reduced the model orderobtained by Hurty et al. [73-77]. The effectiveness of the approach was illustratedthrough the dynamical study of the Galileo spacecraft.The SSM discussed so far, though computationally less intensive than the FEM,still involves considerable effort. It stems from the three steps required: componentmodes derivation, geometric compatibility satisfaction, and model order reduction.Visualizing the SSM from another perspective, Meirovitch and Hale [86,87] simpli-fied the procedure further. The SSM can be regarded as a Rayleigh-Ritz procedureapplied to the "intermediate" substructure. The difference between an actual and"intermediate" substructure lies in geometric compatibility: the former is geomet-34rically compatible along the boundaries with the adjacent elements; whereas thelatter satisfies geometric compatibility at a limited number of points on the internalboundaries. The convergence of the intermediate substructure to the actual one isguaranteed by the Rayleigh-Ritz method provided the admissible functions form acomplete set. The convergence can be achieved by: (i) increasing the number of ad-missible functions; (ii) confirming to the requirement of geometric compatibility; and(iii) implementing both (1) and (ii) simultaneously. This point of view suggests thatit is theoretically correct to use any admissible functions, such as polynomials, pro-vided they satisfy the kinematic boundary conditions and constitute a complete set.Considering a plate-type problem, Meirovitch and Hale [86,87] illustrated the ideaby showing the convergence of eigenvalues using low-order polynomials as admissiblefunctions. The results agree with those obtained using 36 terms of the transcendentalfunctions as admissible functions. The same idea was applied to a truss-like structureby Meirovitch and Kwak [88]. Here, the authors pointed out an inconsistency in theRayleigh-Ritz theory: individual admissible function has to satisfy all the boundaryconditions although the solution is based on the linear combination of the admissiblefunctions. Using this argument, the authors presented a method to select the admis-sible functions. These are called "quasi-comparison functions" which differ from theadmissible functions in that the former satisfy only some of the natural boundaryconditions. However, when synthesized they result in a faster convergence.2.3.2 Substructure equations of motionYu [57] has shown that the equation for transverse vibration of a thermally flexedappendage is given bya4wb 02 Mt^02 w bMb (^7x4X ^02 + mb 8t 2= 0,^(2.13)35with the appropriate boundary conditions. For instance, for a beam cantilevered atx = 0 and free at x = /b, the boundary conditions are:Wb = ^^OX =-- 0^at x = 0 ;02 wbaxe^53wb ambEIb^ + m: . Erb  ax3^OX+  ^= uA at X = it, .Here EIb is the bending stiffness of the beam; mb, the mass per unit length of thebeam; and MP, the thermal bending moment given byMt . Area Ea tT(x, y, z)z dA ,where T(x,y, z) is the difference between the ambient temperature and the temper-ature at a point on the appendage with coordinates (x, y, z); and a t is the thermalexpansion coefficient of the beam material. The integral is over the cross sectionalarea of the appendage. For a thermally deformed plate, Johns [89] has shown thatthe equation of motion is given bya2 wpDV4wP 1 v 172 71lvin-P^P ate^ = 0 ,^(2.14)t—where V is the Laplacian; mp , the plate mass per unit length; D and v are the flexuralrigidity and Poisson's ratio, respectively; and Mr, the thermal bending moment ofthe plate defined similar to M. For a cantilevered plate with the built-in edge atx = 0 and free edges at x = 1p , y = +wp/2, the boundary conditions are given by:Mx = 0,My = 0,wP = OwP = 0OxOMx 2OMxY = 0,Ox^ayamy am^ 2 ^xY = 0,ay^ax(mxy)i — (mxy)2(m.y) i — (mxy)2at x^0 ;= 0^at x = /p ;= 0 at y = ±wp/2 ;where subscripts 1 and 2 refer to values of the twisting moment, Mxy , on the sidesawb36forming the corner.The analytical solution for Eqs. (2.13) and (2.14) is difficult to obtain even whenpossible. The problem is overcome by first assuming that the thermal and elastic de-formation are independent, i.e. the solution for thermal deformation can be obtainedindependent of the elastic displacement, and vice versa. The thermal deformationfunction is obtained as a solution of the heat balance equation, for a beam or a plate,with the assumption that the element is not undergoing any transverse vibration.Similarly, elastic deformation is represented by admissible functions obtained fromthe beam or plate equation without thermal deformation.It should be pointed out that the longitudinal vibrations, torsional oscillationsand foreshortening effects are purposely not considered in the formulation. This doesnot imply that they are difficult to incorporate or negligible. The formulation evenwithout these effects is indeed quite challenging. The main objective is to assessthe influence of thermal deformations which is anticipated to be dominant comparedto the parameters mentioned above. Further complication of the problem, it is felt,will only mask appreciation of interactions between librational dynamics, flexibility,thermal effects and initial conditions.2.3.3 Thermal deformationsModi and Brereton [49] obtained the heat balance equation of an appendage underthe influence of the solar radiation. The time constant of the equation was found tobe very small for most appendage materials; hence at each orientation, the appendagewas assumed to attain the steady state instantaneously. The steady state solutionshowed that the shape of the centreline of a thermally flexed appendage is given by:1* = —ln [cos (—/*)] cos e ;37—6z = —ln [cos (-11 )1 cos Oz ; (2.15)/*^1*where/*btab kbbb^qsas ) 3 /4 (8 — €b) (2.16)as atqs 4iEbatab 7rebat) — eb ) •Here 6y , 6z represent deflections in the Y and Z directions, respectively; Ov and Ozare solar radiation incidence angles in the appendage reference Y and Z directions,respectively; 97 is the distance along the appendage axis from the centre point; andlb, the thermal reference length of the appendage, is a function of the solar radiationintensity (TO, Stefan-Boltzman constant (U t ), appendage dimension (ab , bb), andappendage physical properties (as , at , Eb, kb). For common appendage materialssuch as steel or beryllium copper, lb is found to be over 100 m.Evaluation of the kinetic and potential energies involves integration of Eq. (2.15).However, the transcendental character of the equation makes the integration difficult.Ng and Modi [90] showed that the solution to Eq. (2.15) can be approximated to aparabolic form as follows:6v^2( ^cos OY •6z2= () 2 COS Oz (2.17)The difference between Eqs. (2.15) and (2.17) is found to be negligible^3%) for< 0.6, which corresponds to an appendage of up to 60 m in length. Thus, in anumber of situations, Eq. (2.17) can be used without incurring significant error.Similar to Eq. (2.17), Krishna and Bainum [66] used an approximate parabolicsolution to describe the deflection of a thermally flexed plate,wherebz1* = 2 —/* cos Oz , (2.18)38t,1* =atAT •(2.19)Here 6, is the deflection of the plate; /p*, the thermal reference length of the plate; t e ,the thickness of the plate; and AT (= T7 — T2), the temperature difference betweenthe top (T1) and bottom (T2) of the plate, is obtained by solving numerically the heatbalance equations:Ebo t(Tt + 21) = a s q, ;EbOtT24 k b= (Ti — T2) •t cEquations (2.17) and (2.18) are used in the formulation to represent thermaldeformations of beams and plates, respectively. For instance, the thermal deformationof body Bi (say, a beam), Ti, and of body Bid (say, a plate), tid , are given by,=^cos,2 \ /17^(xi )^q^— cos o, " (xj ) 2 f}T2 V:22X* ^COS Of j^.2^1i,7=- {0 , 0 , — (2.20)2.3.4 Transverse vibrationsThe transverse vibration of the substructures can be obtained by the SSM. How-ever, as pointed out by Meirovitch et al. [86-88], convergence of any set of admissiblefunctions to the actual solution is guaranteed by Rayleigh-Ritz procedure providedthe admissible functions satisfy the kinematic boundary conditions and form a com-plete set. With this in mind, the vibrational displacements of beam-type elementsare represented in terms of modes of the Euler-Bernoulli beams For body Bc , theadmissible functions used are similar to those of a free-free beam [91],(x)^cosh(Or5--, ) cos(Or) — -yr isinh(Or ) + sin(Pr x )] ,lb^lb^lb^lb39r = 1,2,...^(2.21)where f3r/b is the solution of the equationcosh(Pr) cos(13r) — 1 = 0 ,and -yr is given byyr — sin(i3r) + sinh(Or)—cos(pr) + cosh(pr) •For Bi and Bi,j, cantilevered modes are selected,Or (x) = cosh(f3r —x ) — cos(Or —, ) — -yr {si11h(f3r —x ) — sin(f3r x )]lb^tb^lb^lbr = 1, 2, .. . (2.22)where Or is the solution of the equationcosh(/3r) cos(O r) + 1 = 0 ,and -yr is given by= sin(Or) — sinh(/3r)cos(13r) — cosh(13r) •Using Eqs. (2.21) and (2.22), the transverse vibrations of a beam-type substructurein its reference Y and Z directions, vb and w b , respectively, can be written as:nV =^Pr (t)Or (x)r=1nwb =^Qr(t )Or (x);^ (2.23)r=1where Pr(t) and Cr(t) are the generalized coordinates associated with vibrations inthe Y and Z directions, respectively; and Or (X) is the admissible function.The equation of motion for a plate of length /p and width wp (aspect ratio ar =40/p/wp ) undergoing transverse vibration is given by [92]04w^94w^04w^02wOx4 + 2 ox2ay2 ay4 D at2 = 0 ,where D = Eh3 112(1 — v 2 ). The boundary conditions are given by:(1) Simply supported along edge x = 1pw(x, y) = 02 w(x, y) = nOx 2(2.24)at^x = /p ;(ii) Clamped along edge x = 1pw(x,Y) = Ow(x,y)= 0^at^x = /p ;(iii) Free along edge x = 1p02W(X , y^02W(X, y)^ + V \ ^=0 at^x = /pOx2^ay2Except for plates with at least two opposite edges simply supported, the solutionto Eq. (2.24) is difficult to obtain, even when possible. Approximate solutions havebeen reported by numerous researchers and reviewed by Leissa [93]. In general, theapproximate solution does not satisfy completely either the boundary conditions orthe governing differential equation.Warburton [94] put forward an approximate solution by assuming the displace-ment to be,mwP(x, y ) = E^Hs ' t (t)cbs (x)1P t (y) •^(2.25)s=1 t=1The shape functions for opposite edges that are fixed-free and free-free are given by:(i) Fixed at x = 0 and free at x = 1pOs(x) = cos(P 8 —x ) — cosh(Ps—x ) 73 [sin ps (-1 ) — sinh(P 8 —x )]/p^1 ^ 1P41s = 1, 2, 3, ...^(2.26)wheres= sin /3 3 — sinh /3 8cos /3 8 — cosh /3 8and^cos /3 3 cosh 3 8 = — 1 .(ii) Free at x = 0 and free at x = /p(x) = sin [3 8 ( -1;x — -2-1 )]^-ys sinh [/3 8 ( — )]s = 1, 3, 5, .. .^(2.27a)sin 1 /3 8^1^1where^-ys = sinh 8^and^tan —2/3 8 — tank —2/3 8 = 0 ;1 /3and^Os (x) = cos [Os ( — )] + ys cosh [Os ( —/^2ps = 2, 4, 6, . . .^(2.27b)— sin 1 )3 8^1^1with^78 = ^2^and^tan —,3 8 + tanh — /3 8 = 0 .sinh1 /3 8 2 2Note that rigid body rotation of the plate is not considered in (ii). It should bepointed out that the shape functions of (i) satisfy the boundary conditions completelywhereas those of (ii) are only approximate.In order to overcome the shortcoming of the approximate solution. Gorman[95] introduced the method of superposition to solve the plate equation. The idea issimilar to the SSM. The original plate problem is treated as a combination of buildingblocks. Each building block is a plate problem where the solution is readily available.The solutions of these building blocks are then synthesized. Finally, the constants inthe individual building block solutions are constrained so as to satisfy the boundaryconditions of the original plate problem. This results in a set of algebraic equationswhere the nontrivial solution gives the natural frequencies and mode shapes desired.Gorman [96] tabulated the results for different boundary conditions. For a cantilever42plate, for instance, the shape functions are found by three building blocks such that:w(x, y) =^(x, y) + w2(x, Y) + w3(x, Y) ;^(2.28)wherewl (x, Y) =^Fim^ y^ (cosh P„ x + 9b ,m, cos 7 x„ ) sin M71Oa^m=1,3,5 - 'm 2 wPm=km+20  9E m^x^y(Cosh m —(P ci ,m cosh 7,—/) sin — —M712 wPwith On , m, = [var2 (m7/2) 2 —^cosh/3m + eb,m [var2 (7n7/2) 2 + -y2 ] cos'}',[(1 — v)a2r (mir/2) 2 Pm2 sinh0 d,m^[—(1 — v)agm7/2) 2 +^sinh7m= ar0! (m7/2) 2 ;7m, = a, VA! — ( m7r/2) 2^or^ar V(m7/2) 2 — A! ;k *w2 (x, y) =^En(sinh On—/^Ob,n sin 7'n x )cos mr—wn=0,1^'nwith ect,n = [var2 (n,702 — ,Th2]Obn = ^on[_ (1 — 0 D apr2 ( (sinh in) h2h  : n + 07n[(1 — v)agn70 2 + -y,i] cos 'ynn=14+1 ovc,n^On 7 Th00 En^x^X^Yo-2 ] +:00, nsd h, [nvosani :2 (hn 7rn) —2 +) cos ] n: ri n7 n—w ; ;^IP 1P^Pec,n = [var2(m1 )2 — ,Th2]p sinh Pn — 9d , n [— va2, (n7) 2 + 72 ] sinh 'Yn ;On[ — ( 1 — ii)agn7 ) 2 + 1%2 ] cosh OnOb,m, =7m [(l — v)a,!,(m7/2) 2 + -y72,2] sin7,Ocon = [va2, (m7r/2) 2 —^cosh Om + 9d ,m [vcer2 (m7/2)2 +72] cosh 7,, ;0m [( 1 _ o cer2(m7/ 2 )2 _ 0m} sinh P,^-ed,n ='Yn [— ( 1 — v)agn7r)2 oi] cosh7n43Nn = (1/ar )\/(arA,) 2 (n7r) 2 ;Yn = (liar ) /(arAe)2— (n7r) 2^or^War )1/( 777 ) 2 (arAe) 2 ;k *Ep [cos ofp ( 1 — x—1)p=0 (7 + )3P ) cos 'ypP^cosh Pp (1 — —x)] COS p71- —Ycos -ypcosh /4^1Poo — Ep^[cosh -yp (1 —p= 4+1(7P2 — pp2 ) cos h eypcosh -ypcosh Pp cosh Pp (1 — 1—P )] cosp7r—Y ;Pp = (liar) \/(arAe) 2 + (P7 ) 2 ;'Yp = ( 1 Iar)V(arAe) 2 — (P702^Or^(1/Ctr) V (P7 ) 2 (arAe) 2Here, A, represents the eigenvalues of the plate. The values of k: (i = in , n, p)are the maximum integral values of i such that [A 2e — (m7r) 2 ], [(arAe)2 (nir)21,and Rcer A e ) 2 — (p7r) 21 are positive. The general solution of Eq. (2.28) satisfies someof the boundary conditions already. However, there are three additional boundaryconditions to be satisfied: (i) zero bending moment at y = wp; (ii) zero slope atx = 0; and (iii) zero bending moment at x = /p . The Fourier coefficients Eni , En ,and Ep are determined by constraining the overall solution to satisfy these threeboundary conditions simultaneously. It is achieved in the following manner. First,the number of terms (say k) in w i , w2 and w3 are selected. Using the boundaryconditions, three equations with 3k unknowns are obtained. Since En,,, En , and Epare Fourier coefficients, one can make use of the orthogonality conditions to obtain3k — 3 equations. Let Eq. (2.28) be rewritten in a formw(x, y)^Emf m(13m, 7,i ) sin 2 W-Y-n=0,1 En f n(Pn, 7n) cos n7rm=1,3,5^-Y—w3 (x,y) =with44+ E Epfp (Op ,-yp ) cos xr .p=0Multiplying the equation by sin iry/2w and integrating over 0 to 1 would eliminateEni for m 0 1. This procedure can be repeated 3k times to obtain 3k equations.Finally, the coefficient matrix of dimension 3k x 3k in terms of Ern , En , and Epis obtained. The matrix equation can then be solved to determine the eigenvaluesand eigenvectors. Note, since the method gives a solution that satisfies the platedifferential equation and all the boundary conditions, it can be regarded as exact.The next question is the accuracy of the approximate solution. Leissa [93] studiedthe convergence of eigenvalues obtained by different researchers using Rayleigh-Ritzmethod and concluded that in general the accuracy deteriorates as the number offree edges increases. Also, accuracy is further affected by the existence of a diagonalsymmetry such as in the case of square free-free plate. These observations are verifiedin Table 2-1 . Here, a comparison is made between the first five eigenvalues obtainedby Warburton (approximate) and Gorman (exact) for cantilever and free-free plates.It can be seen that the approximate solution predicts the eigenvalues with goodaccuracy for the cantilever plate; however, this is no longer true for free-free plates.Despite the accuracy of Gorman's solution, it is not readily available. Consider-able amount of numerical work, as outlined above, has to be performed first. Also,complexity of the solution makes programming task much more difficult. In viewof this, Warburton's approximate shape functions are adopted in the formulation.It should be pointed out that, if higher accuracy in admissible functions is desired,Gorman's solution can always be accommodated in the general formulation at theexpense of programming effort and computing cost.Using Eqs. (2.23) and (2.25), Si and Si ,j (say, for a beam and a plate, respectively)45Table 2-1^A comparison between Warburton [94] approximate and Gorman [96]exact eigenvalues of cantilever and free-free plates.CANTILEVER PLATEMode l/w = 1 l/w = 1.5 l/w = 21st W 3.52^(5) 3.52^(5) 3.52^(5)G 3.46^(S) 3.44^(S) 3.42^(S)2nd W 9.32^(A) 13.41^(A) 17.61^(A)G 8.36^(A) 11.43^(A) 14.50^(A)3rd W 22.03^(5) 22.03^(5) 22.03^(S)G 21.09^(5) 21.32^(5) 21.28^(5)4th W 28.52^(5) 40.65^(A) 50.60^(A)G 27.06^(S) 38.70^(A) 47.32^(A)5th W 31.69^(A) 56.78^(S) 61.69^(S)G 30.55^(A) 53.06^(5) 59.76^(5)FREE-FREE PLATEMode l/w = 1 l/w = 1.5 l/w = 21st W 13.86^(A) 20.79^(A) 22.38^(S)G 3.29^(A) 4.91^(A) 5.31^(5)2nd W 19.36^(5) 22.38^(5) 27.72^(A)G 4.81^(S) 5.29^(S) 6.49^(A)3rd W 24.73^(S) 47.80^(S/A) 60.59^(S/A)G 6.11^(5) 11.38^(S/A) 14.34^(5/A)4th W 35.97^(S/A) 50.37^(5) 61.69^(S/A)G 8.56^(S/A) 12.45^(S) 21.92^(S)5th W 61.69^(5/A) 61.69^(S/A) 89.54^(5)G 15.23^(S/A) 21.17^(A) 24.98^(A)W: Warburton's results G: Gorman's results(S) Symmetric mode (A) Antisymmetric mode(S/A) Symmetric-Antisymmetric modecan be written as,n= {o,^(t)01(xi) ,^Q;(t)0 7z(Xi)I T ;r=1^ r=146m nSij = 0 , 0, EE^(00:si,j(Xi,j)01,j(Y i,j)} T •^(2.29)s=1 t=12.4 Kinetics2.4.1 Rotation matricesMatrix^as defined in Eq. (2.1) denotes orientation of the frame F, relative tothe frame F. Two rotation sequences are needed to determine q: the first one,C ci 'r , defines the rigid body orientation of F, with respect to F, whereas the secondone, Cci 'f , defines the rotation of frame F, relative to Fc due to elastic and thermaldeformations of the body Bc . A modified Eulerian rotation of the following sequenceis selected: a rotation Oxr, about Xe-axis, followed by 0 yr c about Ye-axis, and finallyOzrc about Zc-axis, i.e.,Cr =cos 9;,sin Orz'c[— sin OLcos OL0 10 0 1cos a[^0^0 sin Orcy1^01. — sin Oyr, 0 cos Ourc1 0 0x 0 cos O'x', — sin Oxr,0 sin 611x', cos O'x'c[cos Oyrc cos Ozrc sin Oxrc sin Oyrc cos Ozr, — cos Oxrc sin OL= cos Brc sin OLsin Br sinsi  Br sin Ozr, + cos Oxr, cos Br— sin OrYc^ sin Oxrc cos Oyrccos Oxrc sin Elyrc cos Ozrc + sin Oxr, sin Ozrc^cos Oxrc sin^sin Ozrc — sin Oxr, cos Ozr,cos Or cos Orx,^YcSimilarly,[cos Of cos Of sin OL si n tgc^ccos Ol — cos Olc sinO lyc^zcCic, = cos Of sin Of sin Of sin Of sin Of + cos Of cos Ofyc^zc^x,^yc^zc^xc^zc— sin 9- c^sin Of cos OfYc(2.30a)472 '7r— •O 'x'c ^cos Olc sin Ofc^zc^xcos Of + sin Ofc sin OlcY^cos 9 sin By ^Of — sin Of cos Ofxc^Yc^zc^xc^zccos OSc cos Bye(2.300andCi = cr xThe choice of the rotation sequence is somewhat arbitrary because the multiplicationof rotation matrices is commutative for small rotations; however, as pointed out byHughes [97], the concern is the location of the singularity. For the sequence selected,the singularity is located at O yrc = 7r/2 or Otcc = 7r/2. If this leads to a problem, it iseasy to select another rotation sequence with minor changes to the computer program.Note that BSc , Bye , and Olc are functions of the thermal and elastic deformations ofBe. Consider the spacecraft model in Figure 2-6 with a beam type central bodyattached to another body at coordinates (dxe , dvc , dzc ). Initially, in absence of anydeformations:Or = 0YcO'z'c = 0 .^ (2.31a)With the inclusion of thermal and elastic deformations, 0.1 c , Bye , and Olc can beevaluated using Eqs. (2.20) and (2.29):ofc =7F2 'Of = — s Qc (0 dYc'Yc^le dx Ixc =dxcr=1dc cos cb cz ;=^(t) dOr 1^dx c--d xcr=1^c c(1X c cos cbg . (2.31b)48m ns=1 t=1n (t)^dx1^dx —d Otc(dyc)^cos og ;, ,H 1,^d,(t)^de^dxc'Cfr:(dxc) --- 1yc=c1 —^coss=1 t=1Ofe = —YIf B, is a free plate, then the appropriate expressions would be:(2.31c)As for qj , which is the orientation matrix of frame Fij relative to Fc , it is expressedas the product of^and qj , i.e.^cF = cF^.lj^1 ljwhere CI  is defined similar to Ci in Eq. (2.12) with subscripts xe, ye,by xi, yi , and zi , respectively.2.4.2 Kinetic energyThe kinetic energy, T, of the spacecraft is given by(2.32)z, replacedT= 1 { f Rc•Re dm,c +1-i• dmi + ni•.Rij drni ,j]^, (2.33)2 me^ j=1 m j=i rni,jwhere R,, P i , and Ri j are obtained by differentiating Eq. (2.9) with respect to time:-hc = itcm — afc m, — Ccm + T c + Sc + co x [—CL, —^+ Pc + Tc 6c1•Ri = Rem, — ucm, — Ccm + di + q(4 i + 6i ) +^+ 6i)+Co x [—Clm —^+ di + q(fii + + 6i)] ;= Rcm — acj — a icm + di +ri,i^8i,j)^+ i,j)+ cv x [—Olm — Ccm + di +j^Ti,j^6i,j)] •^ (2.34)49(a)(b)Figure 2 - 6^An illustration of the Eulerian rotation.50Substituting Eq. (2.34) into (2.33), the kinetic energy expression can be writtenin the form^T = Torb Tcm Th^Tt^Th,jr Th,t Thm Tjr,t Tjr,v1 —TTt4.) -Lsys W + CDT Hays1= Torb Tsys + ,TTh fisys^2^Y8^— (2. 35)where C4.) is the libration velocity vector; L ys , the inertia matrix; _L ys , the angularmomentum with respect to the Fc frame; 1C4FIsys c-4.), the kinetic energy due to purerotation; and C4.) 27-/sys , the kinetic energy due to coupling between rotational motion,transverse vibration, and thermal deformation. Lys represents the kinetic energycontributions due to various effects with the subscripts involved defined below:orb^orbital motion;cm^centre of mass motion;h^hinge position between body B, and Bi or between body Bi and Bij;jr^joint rotation due to elastic and thermal deformation;t^thermal deformation;v^transverse vibration.For instance, Tt ,,„ refers to the contribution of kinetic energy due to the rate of thermaldeformation and transverse vibration velocity. Details of the kinetic energy expressionare given in Appendix I. The evaluation of the integrals require a priori knowledgeof pk  Tk , 4, and dk (k = c, i, or i, j); hence, the configuration and location of eachbody must be specified before the evaluation can proceed. The matrix L ys , whichrepresents inertia of the system, is time-dependent and consists of several components,Lys = Icm + Ih + Ir + I t ++ Ih , r + Th,t + Ih , v + Tr,t + Ir ,v + It , v ,^(2. 36)51where the subscripts cm, h, t, and v have the same meaning as before. Subscriptr denotes contribution from the rigid body component. For example, I r , t representscontribution to the inertia by the rigid component and thermal deformation.The angular momentum vector, Lys , can be written as1-1,y, = licm + Hh^Ht^+ Hh,rHkt Hh,v+ Hr,t + Hr,v Ht,v^(2. 37)with the subscripts defined as before. Similar to L ys , FI,y, is a time-dependentquantity. The details of Isy, and Tins are given in Appendix I.2.4.3 Potential energyThe potential energy, U, of the spacecraft has contribution from two sources:gravitational potential energy, Uy , and stain energy due to transverse vibration andthermal deformation, Ue ,U = Ue + Uy .The potential energy due to gravity gradient is given byUy =^dm,^dm,^dmiite^f c R,f^U^R• R• •ji=1 j=1 mi,j^i'3Substituting the expressions for R,, Ri , and Rij from Eq. (2.9), and ignoring theterms of order 1//4m and higher, Uy can be written asU = PeM ^tr [Is s ] +^/^,^(2.38)—g^2/cm^Y^3//e2RL^Ywhere^is the gravitational constant and [ represents the direction cosine vector ofRem with reference to Xp ,Yp ,Zp axes. From Figure 2-4, / is given by/ = (cos sin cb cos A + sin sin A)i, + cos cos 0,52+ (cos /P sin q sin A — sin cos A)k, .^(2.39)The strain energy expression for a beam and a plate are [24]:DfAp{(aa2:2P) 2 + 2 v (952:: 302:12P )2+2(1 — V) (aa2WP) dAp ;xay1 f {EI„ ( a 2Wb ) 2 -4- EI„( a2Vb ) 22^} dib ;i b^ax2^ax2Ue,plate =Ue,beam(a2wp) 2ay2(2.40)where D and v are the flexural rigidity and Poisson's ratio of the plate, respectively;and Elyp and EIzz are the bending stiffness of the beam about Y and Z axes,respectively. By specifying the configuration of each body constituting the system,the strain energy can be evaluated. For instance, a system with bodies 13,, Bi beamsand Bi j plates, the strain energy expression can be written asr a 2^12 r a2Ue = —2 fic {EI,,yy[ ax2 ((T,) z + 0,M] + EI,,„1:,u;i ((7-,) y + (61 2)y) ] } dl,N 1 i^a2xi+ E [-2^{EIi,yy [,., 2 ((ri), + (6ai=1^i ) z ) ]2 52Eii,zz[v + (8i) )]Y^2+^a ((Ti)2x i2 A.^ {[;7. 2 + (oi,i)z)] 2j (("Ti,i)zj=1r^a2((Ti,j)z02[ax 2 (Si,i)z)i [a 2^((r,j)zYi,j ( 6i,i)z)]+ L a2991 2^ ((Ti i)z^( 6i,j)z)1 202 2+ 2(1 — Vi,j) [^u^((Ti,j)z^(Si,j) z )]^dAi,j],3Yi,iwhere (m)y (Sk) y and (7-k) z (6k) z (k = c, i, or i, j) are the Y and Z componentsof Tk + 8k, respectively.532.5 Lagrangian. FormulationUsing the Lagrangian procedure, the governing equations of motion can be ob-tained fromd t OT \ _ OT OUdt‘ 04 ) Oq^aq Fqwhere q and Fq are the generalized coordinate and generalized force, respectively.The number of generalized coordinates depends on the system configuration, i.e., thedegrees of freedom involved. For instance, consider a central beam-type body Be ,undergoing general librational motion, with N beams (body Bi) attached to it. Inturn, let each Bi body be attached to ni plates (Bi,3 bodies). Then q will beq = Rem , 9, IP, 0, A,^Qic'e, Pi e , Q iri ,with:^rc = 1 , • • • I nre ;ri = 1, .^, nri ;s = 1,...,n, ;t = 1,^, nt ;where nre and nri denote the number of modes used to represent the transversevibration of Be and Bi , respectively; and n s and n t represent the number of modesof transverse vibration of Bi j in the plate reference X and Y directions, respectively.The total number of generalized coordinates is now given asNg = 5 + 2 x n rc + 2 x (N x nri ) +^x ns x nt)Nwherei=1In general, the effect of librational and vibrational motion on the orbital motionis small unless the system dimension is comparable to Rcm [98,99]. Hence, the orbitcan be represented by the classical Keplerian relations:h2Rcm =^E COS 9)Rc2 771. 9 = h ; (2.41)where h is the angular momentum per unit mass of the system, A, is the gravitationalconstant, and E is the eccentricity of the orbit. Therefore, q and Ng would reduce to:q =^0, A, Pcc ,^Pii, Q iri,^;^Ng = 3 + 2 x n„ + 2 x (N x nri )^x ns x nt)Here Prcc, Qrt P i ^t, and H::; are the nondimensional values Pcrc11,, Q crc11,,Q iri//i, and Hisjiki , respectively. In parametric studies of spacecraft dy-namics, dimensionless parameters and independent variable are desirable. For in-stance, simulation results using time as the independent variable from t = Tinitiai tot T final are applicable for a particular orbit only. In contrast, using true anomalyas the independent variable, simulations for 9 = Oinitia/ to 9 = 9 final are valid forsimilar orbits at different altitudes. In the present study, all masses and lengths arenondimensionalized by the characteristic mass and length, respectively. As the sim-ulation results are independent of the characteristic mass and length, their choice isarbitrary. Variables with a dimension of time, such as frequencies, are nondimension-alized by the orbital rate at the perigee point, G. As for the time derivatives, theirtrue anomaly counterparts are obtained using the following relations:^d^d^dt^dBd2^d2^2€ sin 9  ddt 2^d92^1 + E cos 0 dewhere Eq. (2.42b) is derived from Eqs. (2.41) and (2.42a).(2.42a)(2.42b)oil K.KMlib,vibA" CA Ka[Mlib1\4 1Tib,vib^Mvib^1 q1 Cq1 Kq1• • •nv qnv qnvQ,pQ4Qa• • •QqiQqnvThe general equations of motion can now be written as:or M(q)q" C (q, q', 0) K (q, 0) = Q(0) ; (2.43)where primes denote differentiation with respect to the true anomaly; n, is the totalnumber of vibrational degrees of freedom such that Ng = 3 + nv . Here M is a non-singular symmetric matrix of dimension Ng x Ng . The entries in M come from secondorder terms of d I clO(OT 0q 1 ). C is a Ng x 1 vector representing the gyroscopic termsof the system. They come from two sources: from the Coriolis terms of d/c/O(8T/0q 1 )and from 017 0q. K, also a Ng x 1 vector, denotes the stiffness of the system. Its solecontribution is from (OU 00. 0, the generalized force vector of dimension Ng x 1, isevaluated using the virtual work principle. Note, nonlinear entries in M together withnonlinear and time varying components of 0, K, and Q result in a set of coupled,nonlinear, and nonautonomous equations of motion.563. NUMERICAL IMPLEMENTATION3.1 Preliminary RemarksThe computer code required to simulate a general system of interconnected flex-ible bodies is expected to be complicated and lengthy. In implementing such a com-puter program, the following points must be taken into consideration:(i) Each computer has a constraint on the acceptable program size beyond whichits execution is at a suboptimal speed. This limit on size is especially stringentin time-sharing systems.(ii) The numerical integration subroutine determines, to a certain extent, thespeed of execution and accuracy of the numerical solution. The ideal fast,higher-order subroutine is based on the multi-value method.(iii) The architecture of the code is critical in program management and debug-ging. A computer code with simple and easy to follow algorithm is alwaysdesirable.With the advances in computer technology, the importance of the first pointhas diminished; however, the other two aspects remain critical for the success of amultibody simulation program.The chapter begins with a discussion of the numerical integration subroutineand program flowchart. This is followed by details of the program and subroutinestructures. The emphasis is on the effort required by the user, the architecture andalgorithm of the computer code that facilitates simulations of a large class of space-craft.573.2 Numerical Integration SubroutineIn order to solve the equations of motion numerically, they have to be first rear-ranged into a set of first order ordinary differential equations. Consider Eq. (2.43), aset of N second order differential equations, which can be rewritten asqll = m --1 (00(0)— C (q, q1 , 0) — K (q, 0)}= F (q, q', 0) .^ (3.1)Now letthen(3.2)which represents a set of 2N first order differential equations to be integrated numer-ically.The subroutine IMSL:DGEAR is chosen to perform the numerical integration ofthe equations of motion. The advantages of this subroutine include: (a) automaticadjustments of the iteration step-size and order of the iteration formulae; (b) user-supplied or numerical evaluation of the system Jacobian matrix; and (c) user selectedintegration methods (Adams or Gear method). The latter feature is particularlyappealing as the two methods complement each other. Adams method is particularlysuitable for non-stiff equations whereas Gear method handles stiff systems efficiently[100,101].In order to understand the program architecture, a little background on themethodology of numerical integration is appropriate. Consider Eq. (3.2) with gn,as the solution at the nth integration interval (T), such that 0 = nT . Two methods,one-value or multi-value, can be used to obtain g m . The former uses onl y gn_ i while58the latter uses k previous values^, -g7i — k . The advantage of the multi-valuemethod over the one-value approach is in the accuracy and reliability of the solution;however, the multi-value method has a disadvantage in the execution speed.The multi-value method consists of three stages: prediction, error test and correc-tion. Given iion_ 1 ,^, gn — k , the predicted value of gn , denoted by gn,,o , is obtained bya linear interpolation method. An error test is then performed on gn , 0 to determinewhether the error is within the user-specified tolerance. A negative result of the testprompts the corrector formula to determine a refined value gn , i which is subjected tothe error test again. The error test and correction procedures are repeated rn, timesuntil satisfies the tolerance giving gn3.3 Computational FlowchartThe flowchart showing computational steps involved for multibody simulation ispresented in Figure 3-1 . The input phase requires the user to supply the followinginformation:(i) orbital elements p, i, w (used for the determination of solar radiation incidenceangles), and eccentricity, c;(ii) simulation period and IMSL: DGEAR parameters such as tolerance, method,initial step size, etc.;(iii) inclusion/exclusion of thermal deformation effects;(iv) initial conditions representing disturbances.This is followed by the calling of the subroutines MODEL and MODE which arebriefly explained below.Subroutine MODELThe subroutine reads in the data relating to the system configuration. The data59Call IMSL : DGEARNumericalIntegration117 / Output /VIncrement TOrbitalElementsSystemConfigurationsSubstructuresPhysical PropertiesEnvironmentalDisturbances (Y/N)SimulationPeriod^0 -> T(final)IMSL:DGEARParameters STOPFigure 3- 1^Flowchart showing numerical approach to multibody dynamics sim-ulation.60include:(i) number of Bi and Bi j bodies;(ii) number of modes for beam and plate shape functions;(iii) geometry of each body: rigid body, flexible beam or plate-type appendage;(iv) for body Bk: mass (mk), length (/k), length/width ratio (r k) , thermal lengthratio (L), and stiffness (wk);(v) principal and cross-product of inertias of body Bk:^(IYY)k, (izz)k)(Ixy )k, (Ixz )k, and (/yz)k;(vi) orientation of Tic with respect to the orbital axes, of Fi w.r.t. Fc , and Fijw.r.t. Fi ;(vii) location of Oi relative to 0,, and of Oi relative to Oi.Note that Bs can be either a rigid body, free-free beam or completely free platewhereas Bi and B, j can be rigid bodies, cantilevered beams, cantilevered plates orcombinations of them. For a parametric study, all quantities in (iv) and (v) areintroduced in dimensionless forms.Subroutine MODE Once the geometry of each body is known, the modal integrals for each bodycan be determined. These integrals, which are used in the evaluation of kinetic andpotential energies, include:rCu(C) dCf Cu( (Or (0) 2 dCCl_ f" d2 Or (C)^;5^j_4-1 Ck2s^f"^d2 (C) 4°76 ' =^OW^dC—C161Cu4. 3 = f COr (C)dC ;= f C 2 Ch(C)dC ;— Ctf61 ClOr (() d^ s (()(14b7 's = ^d(^d(—The limits of the integrals are either (-0.5 < < 0.5) for free-free shape functionsor (0 < C < 1) for the cantilevered case. The first four integrals ((VC, , 403', and4.4) are used in the evaluation of the kinetic energy expressions. For instance, usingEq. (2.20) and (2.29) to evaluate Tt,, in Eq. (2.35), 44 would be required. Thelast three integrals are used in the evaluation of strain energies as can be seen fromEq. (2.40). For plate type appendages, the integrals for shape functions in both Xand Y directions are evaluated.After executing the subroutines MODEL and MODE, the program begins thenumerical integration of the equations of motion which are coded in the subroutineFCN. At the completion of each integration step, the program can evaluate the ener-gies of the system if such information is desired. Let the solution p-n_ i at 0 (n —1)Tbe known. The determination of gn can proceed as indicated in Figure 3-2 . Theimportance of the subroutine FCN is now apparent; it is required at every predictionor correction step. Hence, it would be useful to understand in greater details thearchitecture of FCN.3.4 Subroutine FCN3.4.1 BackgroundIn general, evaluation of the kinetic and potential energies is straightforward;however, determination of their derivatives is not necessarily so. For instance,1 ^1Tv = 2— fc (5 lc • dm, + 2- ^[f (cps'i) • (cps'i ) dmii=1m62-1/Prediction : Call FCN, Y r1 , 0FailCorrection : Call FCN, Y_ n,mPassFigure 3-2^Flowchart of a numerical integration subroutine.ddO aq'(CF,Pi,i 8ii7) • (q,jiti,Ai)dmi ji;3=1 rni,3f^ae,^d^dm,^d  aq'^dO aq'+E[f { (crei +c i7s:/). a(c(*+(c18:)i=i^aq'^d9 a 6:)qid a(q ni(3.3a)dmi63ni{(CfijttijO lii^Cic^-7/^°(Cittijaiii) Oqi7=1 m.L 73ddmide NI (3.3b) At first glance, numerical evaluation of Eq. (3.3b) does not seem to be much moredifficult than that of Eq. (3.3a). Only the number of terms has increased from 3 to9. The problem lies in a dramatic increase in vector dot products, multiplicationsbetween matrices and between matrices and vectors. In evaluating Tv according toEq. (3.3a), 3 dot products, 1 product of matrices and 2 multiplications between amatrix and a vector are required. The respective numbers of operations in Eq. (3.3b)for evaluating dId0(0T,1 OV) are 6, 7, and 13. Obviously, the increase in the effortrequired for the operations is phenomenal. Although the operations can be executedmanually, the resulting computer codes for Eq. (3.3b) are lengthy and hence demandconsiderable effort to manage and debug. An effective way to evaluate derivatives of113y3, and Hsys is therefore necessary. One such approach is presented next. Theidea is to assemble terms in Eq. (3.3b) in stages, each according to a simple algorithm.The significant advantage of this approach is that the user is only required to performsimple differentiation and supply this information to the program.3A.2 Definitions of new operatorsConsider matrices A, B and vectors C, D such thatan^a1 2 a13 bil b12 b13A ==a21a 31[CiC2 }C3a22a32a23a33= [am] ; B =D =_b21[b31did2d3b22b32;b23b33[bk i] ;64Four operations on these matrices and vectors are defined:(an bil + a21 b21 a31b31)(anbil a22b21 a32b31)(a13b11 a23 b21 a33b31)(an biz + a21 b22 a31b32)T(A,B) = (an biz a22 b22 a32 b32 )(a13 b12 a23 b22 a33 b32 )(an b13 a21 b23 + a31 b33 )(a12b13 a22 b23 a32 b33 )(a i3 b13 a23 b23 a33 b33 )_= [vm ] , m = 1,^,9; (3.4a)all [bk l ]r(A, B) = an [bkl]a31 [bkl]a 1 2 [bkl ]a22 [bkl]a32 [bkl]a13 [bkl]a23 [bkl ]a33 [bkl][-Yrnn]^m = 1,^,9; n = 1,. ••,9;^ (3.4b)(b21a31 — b31a21) (b3ia11 — b11a31) (bnan — bmaii)(b21a32 — b31a22) (b31a12 — 191032) (bnan — b21a12)(b21a33 — b31 a23 ) (b31a13 — b 11 a33 ) (b11 a23 — b21a13)(b22 a31 — b32 a21 ) (b32 a 11 — b12 a31 ) (b12 a21 — b22a11)A(A,B) = (b22a32 — b32a22) (b32a12 — b12a32) (b12a22 — b22a12)(b22a33 — b32a23) (b32a13 — b12a33) (b12a23 — b22a13)(b23 a31 — b33 a21 ) (b33 a11 — b13 a31 ) (b13 a21 — b23an)(b23a32 — b33a22) (b33a12 — b13a32) (b13a22 — b23a12)(b23a33 — b33a23) (b33a13 — b13a33) (bi3a23 — b23a13)_= [Amn] m = 1,^,9; n = 1,^, 3; (3.4c)C142ci d3c2 d1C2 d2C2 d3c3 d iC3 d2C3 d3 -e(C , D) =65E9k=1 Akl Okv.9La k= 1 Ak2UDk^•E 91c= 1 Ak3 ekUsing these operators, Tt ,,, It ,,, and -It ,, can be written as:A(A, B) o 0(C, D)= [Gm] ,^m = 1,^, 9; (3.4d)where T, r, A, and 0 have the dimensions of 9 x 1, 9 x 9, 9 x 3 and 9 x 1, respectively.The next step is to operate r, A, and T with O giving:vkek ;k=1[4,171kekkE9k= 07kOk(3.5a)(3.5b)(3.5c)T(A, B) o Co(C, D) =r(A, B) o e(C, D)EEE9k=i72k 0 k9k175kOk9k=f78kOkE 9k=173 k OkE9k=176kOkE9k=179kOkTt ,,„ = fT(U, U) 0 e(7,',6c1 ) dmcmc+ E [f T(q, c?) o 0(rft,SD dmii=1 mini+ E^.1) o^dmi,j]j=1 rni,j= f {2T (13 U) o 0(7fc , 6,)Umc- r(u, u) 0 [0(7c, 6c) + 0(6c, 'Tc)] dmcN+ [f {2T(q, q) 0 0(-fi, soui=i^rni- r(q, q) o [clef, 6i) + (Si, ;7=i)i} dmini+ E [f {2T(cu.ti, j , C?j ,u,,, i ) 0^ojiyu- r(cui i cuL i j )^+^dmi,j]66Htm = fmc {A(U, U) o [0(rfc, Sc) + 0(8c , i-D1 dm,+^ [f {A(q,^o^+ 0(6i,j)i} dmii=1^977'ini+ E^{ A (q, ipi, j,^0 [e^S'ij )^dmi,j]j.iThe expressions appearing in Eqs. (I-1), (I-2), and (I-3) are rewritten in terms ofthe new operators in Appendix II. For example, consider Eq. (3.3b), which can beexpressed as:d 0T,dO aq' = Lc T(U, U) o 6',)} dm,de Oqi+^f -d-T(q,cp 0 -P-0(4 si )i=1 Limi ( de^Dvd 0T(q,^o -5179 -570(S li , ei )} dmin.{^ j) 0 -670(8ii ,8ii)^a^_ _mz,3d a^e, c ,0 --(-1,3 ,-; t9 dmi, (3.6)Merit of the new operators is not quite apparent through a direct comparison ofEqs. (3.3b) and (3.6); however, there are two subtle advantages. First, once rotationalmatrices q and are determined, all the required T, F, and A operationscan be easily programmed using simple algorithms. Similarly, all the required 0operations can be coded once the position vectors are determined. The evaluations ofTsys , Lys , fins and their derivatives is just a matter of assembling the appropriateT, F or A with the suitable O. The second advantage is best illustrated using theexpressions of I r., It , and I,:Ir = f {T(u,u) 0 e(pc, Pc)u — r(u,u) 0 o(pc, fic)} &Ticmc67+E^{T(q, cD 0 e(pi, -fix - r(q, o^mii=1ni+E f^CicApi,j) oj.1— o^dmi,j] ;It ={T(U,U) o^— r(u, u) e(T--,,,-01 dm,n7,c+E [fm {T(cF, cD 0^;ix - r(q, o^dmii=1ni+E f^60-ii,Tii)uj=1—r(q itt i^0 e(Tii , ,T-ii )} dmi ,j1 ;= f {T(u,u) 0(6c , 6 c )U — r(u, o 0(6c , 6,)} dm,+E [f {T(q, cD 0 e(Si, Si)U — r(q, c,..) 0 8(6i, Si)i=1ni+E f {T(cu.tij , CLi fti,j) oi=1- r(czp i , j , c f,. ;ii i , j ) 0 e(sii , 6,j )} dm i ,] •^(3.7)Note, the same 1' and r operations are used in each expression. This results in asignificant saving of the computational effort.3.4.3 Subprograms in FCNFigure 3-3 shows the flowchart for subroutine FCN. Contributions of the varioussubprograms are briefly explained below.Subprogram DIRANG The subroutine evaluates the angular velocities (.7.) of Eq. (2.2), direction cosines68Parallel ProcessingFeasibleSubprogram SLEWL^SubprogramKINENESubprogramINEMATSubprogramANGMOMSubprogramPOTENEEvaluate Y'YES SubprogramCNTROLSubroutine FCNSubprogram DIRANGSubprogram SOLROTSubprogram VECTORSubprogram RMATSubprogramGBNIFORAssembly ofEquations of MotionFigure 3-3^Flowchart of the subroutine FCN.69/ of Eq. (2.39), and their derivatives. These include OcD10q, OcD10q 1 , dIcle(OColOqi),and &Dia (q = 0,0, and A). These derivatives are evaluated manually and thenprogrammed into the subroutine. In evaluating dcD/c/O, it is further separated intotwo parts: one contains v", 0", and A" while the other contains the coupling among'0 1 , 0', and A'. This procedure is necessary in the assembly of the system mass matrix.Subprogram SOLROTThe objective of the subroutine is to evaluate the solar radiation incidence angles(0-e , and ki), the rotation matrices (q and q j ) and their derivatives. Toillustrate, the procedure for evaluation of ik and Cf is explained below:(i) Using the orbital elements and libration angles as input, :;15, as given byEq. (2.7) is calculated. The derivatives (00,10q, 01c , dlc10(00,10q'), andare evaluated based on the explicit coded expressions.(ii) Oxe , Oye , and 0,c are evaluated using Eq. (2.31). Note that depending onthe configuration of 13,, the subroutine selects the correct expression fromEqs. (2.31a), (2.31b), and (2.31c).(iii) The rotation matrix as given by Eq. (2.30) and its derivatives are deter-mined next. Again, explicit expressions for the derivatives are coded for thispurpose.The same procedure is repeated for q2 and Ow using Eq. (2.8a) and (2.30), re-spectively. Finally CF  is determined from Eq. (2.32) and substituted into Eq. (2.8b)to give the required 02,j.Subprogram SLEWIn this subroutine, Ajj, j , and qi are coded explicitly. Using and itsderivatives from subroutine SOLROT, the subprogram then evaluates^and70the associated derivatives.Subprogram RMATThe subprogram evaluates all the necessary T, r, and A matrices given byEq. (3.4a), (3.4b) and (3.4c), respectively. These matrices are shown in Table 3-1. The alphanumeric name in the first column identifies the array in the computerprogram; and its location in the array is denoted by the subscript of T, r, or A.For instance, A(U, Cr) is represented by R93i(4) in the program. The requiredderivatives for each operation are:ay OT d ar dTT :Oq^8q'' de Oq'^dear dr:Oq^dOOA OA d OA dAAlthough there are a number of operations involved, they all can be executedusing similar algorithms. Besides, the derivatives of the operators do not require anymanual effort. For instance,OA(CF, CF)^OCF^OCF A( Oqi , Cfl A(CF, Oqi )aqwith OCF/Oq provided by the subroutine SOLROT.Subprogram VECTORThe subroutine evaluates all the required 0 column matrices as given by Eq. (3.4d).Table 3-2 represents their complete list. Each 0 column matrix occupies a uniqueposition in the program which is indicated by its array name in the first column andits array position in the subscript of O. For instance, DDij(3) refers to 0(di,A :aq^Oq'^dO Oq'^d071Table 3-1^List of 1', A, and r operations executed in subroutine RMATR91c T I (U, U)R91i T i (Cr, Cr) T2(Cr, q) T3(U, Cr)T4 (q, q) T5 (U, C?)R91ij T i(Cr , (q, i mi,i)') T2(Cr ,C4u,i, j) T3(q , (q PU)')T4 (C, CF jpi,j)R91 j T i((q,iiii,iY , (^,,ptli,i) i ) T2((q, pi, V , CF,ikti,i) T3(U , (Cil-ii,i) i )T4(C,iiti,i,q,,jili,j) T5 (U, C i pi,j)R99c r i (u, U)R99i r i (q, cn r2(u, q)R99ij ri(q, qj iii,j)R99j ri(cuti J , cutij ) r2 (u, c4, i , j )R93c A i (U, U)R93i A i (Cr, Cr) A2 (Cr, q) A3 (U, Cr)AM, q) A5 (U, q)R93ij Ai (Cr,^,.iiii, V) A2(Cr, qii-ti,i) A3(C?, (CULLi)')A4(C i j^p,i,j)R93j A i ((q jp,i,j)', (CULi,in A2 ((qjiti, j) i , CULL j) A3 (U, (CULL j) 1 )A4 (q, ji-ti,j, CU-1U) A5 (U, CF,i iti, j)fk, gk, and hk are defined as follows:fk^Pk &ink ;•mk mkgk - f Tk amk ;mk mkhk = 1^Sk d'ink ;mk Lkwhere k = c, i, or i, j. The required derivatives include 00140010V , d1c10(00 Oq i ),and de/dB. Only the derivatives of dk , gk , hk, 'fk, and 6k (k c, i, or i, j) are calcu-lated beforehand and coded. Using this information, the subroutine is able to evaluate72Table 3-2^List of all the 0 operations executed in subroutine VECTORCMC 01(0cm, Ccm, ) 02(C/cm, Ccrn) 03(Ccm, Ccra) 04(Ccm, Ccm)DDi 01(cl'i, di ) 02 (d2, di) 03(di, di) 04(di, di)DDij 401(di, dij ) 02(cei, di,j) 03(di,di,j) 04(di, di,j)DD j 01(d2,j,di,j) 02(dij, cii,j) 03(di,j, di,j ) 04(di,j, di,j)DFi ei(di, ii) 02 (di, fi)D Fij 01(d2, fi,j) 02(4 fi,j)DF j ei(cei,j, fi,j) 02(di,J, fi,j)DGi 01(d2, g2) 02(d2, -gi) 03(di, .--gii) 04(di, gi)DGij 01 (d2, yi,j) 02 (d2, ji,j) 03 (di, Yi,j ) 04(di, gi,j)DGj 01(d2,.7, g2,.7) 402(di,j, ji,j) 03(di ,j, pl ,i ) 04(di ,j, .0 i,j)D Hi 01(d2, Wi ) 02(d2, hi) 03(di, h li) 04(4 hi)D Hij 01(d, Wij ) 02(4 hid) 03(di, hi, j) 04(di, hi^)DH j 01(di,j,h/i ,i ) 02(di,j,hi,j) 03(di,j,14,9) 04(di,j,hi,j)RRc 01(Pc, Pc)RRi el (Pi, Pi)RRj el (Pi,j, Pi,j)RTc e1(fic,t-c1) 02(t, Pc) 03 (Pc, Tc)RTi e1(Pi,7f2) 02(T', Pi) 03(Pi, Ti)RT j e1(Pi,j,71,9) e2 (71,j, Pi,i) e3(fii,i, Ti,i)RV c ei(fic,ec) 02(61c, Pc) 03 (loc, 6c)RV i el (Pi, 61 ) 02 ( 62, Pi) 03(-0i,6i)RV j ei(Pi,j, 67,9) 02(61,9, Pi,j) 03(Pi,9 , 62,9)TT c 01(T- , TD 02('T., 'Tc) 03(Tc, '7- ) (1)4(Tc) 'fc)TTi e1 (f',J1) 02(ft,7i) 03 (Ti, Ti) 04 (Ti, Ti)TT j 01(71,i, 'T,i) 02(71,i, ri,j) 03(Ti,9,71,9) 04(T2,9,T2,9)TV c e1(t.,61c) 02('T.,6c) 03(tc, 61c ) 04(Tc, 6c)TV i e1(f',6i) 02(T2, 6i) 03(ti,6D 04(T-i, Si)TV j 01(T2 ,9 , e2111,j, (52,9) e3(Ti,9, (5i,j) 04 ( T2 ,9 , 6i, )VVC 01(81c, (51c ) 02(ec,6c) 03(6c, 6c) 04(6c, 6c)VVi 01(6"A) 02(67, 6i) 03(6i, 6i) 04(6i, 6i)VVj el (62,j , 61i ,j ) 02(61i,j , 6i,J) 03(6i,i, 6i,j) 04(62,i, oi,i)73all the required derivatives. Take 0(di, di,j)/aq as an example, it can be evaluatedas:a^_490(di,di^d, -^ad.i)= 0(^,di .)+0(di , ^z73 ).aq ag aqThe 0 vector for the shift in the centre of mass is evaluated in two stages. First,61,, and their derivatives are determined using Eq. (2.12). Now, the procedureas explained earlier is used to evaluate O.Subprogram KINENEThe evaluation of aTsys laq and dIdO(OTsys laq i ), where Lys is given by Eq. (2.35),is the objective of this subroutine. In order to facilitate the assembly of system massmatrix, did9(5Tsy3 /0V) is separated into components of double derivative and cou-pling among velocities (Coriolis effect). The evaluation is done using 'I' and 8 oper-ations as given in Eq. (3.5a). A portion of the computer program for the assembly ofaTsys lOq is presented in Appendix II.Subprogram INEMATHere, the mass inertia matrix^of the system, as given by Eq. (2.36), is as-sembled using the operation between F and 0 in Eq. (3.5b). In addition, the twoderivatives, 0I sys /0q and dIsys idO, are also evaluated. Appendix II shows the sourcecode for the assembly of Lys .Subprogram ANGMOMThis subroutine evaluates angular momentum fins given by Eq. (2.37) and therelated derivatives including OH,sys iOq, 01-/sys ik, d/d0(01-/sys /OV), and dilsys IdO.Similar to dC,Vd61 , dfisys IdO is separated into two parts with one consisting of thedouble derivative terms and the other the Coriolis terms. The evaluation is justa matter of assembly of A and 0 operations in Eq. (3.5c). The source code for74calculation of Hsys is listed in Appendix II.Subprogram POTENEThe subroutine calculates the DU/aq term in Lagrange's equation. Here U repre-sents sum of the gravitational potential energy and the strain energy. The former isevaluated using Eq. (2.38) while the latter with Eq. (2.40). Depending on the beamor plate type appendage, the subroutine selects the correct expression for calculatingthe strain energy derivative.Subprogram GENFORThis subroutine calculates the generalized force based on user-supplied explicitexpressions. For instance, the generalized force for the Space Station can be theactuator output, solar pressure disturbance, aerodynamic drag, astronaut motion,etc.Subprogram CNTROLIf a control study is undertaken, this subprogram would calculate the desiredcontrolled states and control effort using the Feedback Linearization Technique.Assembly of equations of motionAfter the execution of the aforementioned subprograms, the next step is to as-semble the equations of motion in the form of Eq. (3.2). From Eq. (2.35), the kineticenergy of the system, T, is given byT = T„^Yb + Ts s +21 oTis Ys + Hsys •Taking the derivative with respect to q and q' gives:T aTsys^a H sysaT aTsys ^ risyso + fins] 6-)T ^aq^aq^aq L L aq^aq(2.35 )(3.8)75OT^.911,sys + awl' [is _ .4_, + fisyd +LDTallaqs,y, aq' (3.9)Note since the Keplerian relations (Eq. 2.41) are assumed to be valid, Tor t) does notcontribute to the equations of motion; hence, it is discarded in the differentiation.Differentiating Eq. (3.9) with respect to B givesd OTd aT,sy^d aDT=^s ^ [Lysw + Hsys ]d0 aq'^d0^dB aq '&DT dins _ T dc-i) dHsys i dcDT aH,sys8q' [ d0 w j'sYs d0^d0^d0 aq ' + (DT d aHsYs d0 aq'which can be separated into the first and second order components:d aT ) = Oi( d aT,ys ) d ac-oTQ1 ( d0 aq'^dB aq' ) + d0 aq' [ sYs+ OcDT r dins , i. + c i 'D +: 01.sYsi d0 )] dHsys m^u.) + J.,„„ s (t.Ji dalaq' L d0 d0^.1dC)T )aHsys _T aHsys .+ 01( d0^aq'^d0 aq' '^(3.10a)O2 U(0 aq'OT ) = 02 ( d0 aq'd aT,y3 ) +aaq' I_c-DT FT s ys02(d6.c10 ) + 02 ( dHasys  )](ddcp eT ) 0 ..F1:97 + 02^ (3.10b)Here 0 1 and 02 represent the first and second order components of the functionin parenthesis. Equation (3.1) can now be assembled with ease. M is given byEq. (3.10b) and Q by the subroutine GENFOR. C is the sum of Eqs. (3.8) and(3.10a), and K is given by OU/aq of the subroutine POTENE. Using a suitablematrix inversion subroutine, M -1 can be calculated; hence, -4" can be obtained. Itis then rearranged in the form of Eq. (3.2) and ready for the numerical integration.Finally, a few general comments concerning implementation of the computer pro-gram are appropriate.(i) To help assess validity of the computer code, the program checks for the76symmetric character of M at every integration step. Any indication of non-symmetric entries in M would halt further execution.(ii) The program requires minimal effort in terms of explicit differentiation by theuser. Only subroutines DIRANG, SOLROT, and VECTOR require the userto supply explicit expressions of differentiation which, in general, are simple.In evaluating the potential and kinetic energies, no further differentiationeffort is required from the user.(iii) Each subroutine is programmed with simple and repetitive algorithm. Thisfeature greatly aids in the debugging process of the program.(iv) The modification of the program can be undertaken quite easily due to itsmodular architecture. This is particularly useful if other shape functions aredesired. One just has to replace the subroutine MODE with the new shapefunctions, while the rest of the program remains intact.(v) Although the source code is being programmed for sequential execution pur-pose, the code can always be modified for parallel execution if it is supportedby the hardware. This would result in a potential saving in execution time.The possibility of parallel execution exists at two places. First, it can beimplemented in processing RMAT and VECTOR subroutines. Second, thesubroutines KINENE, INEMAT, and ANGMOM can be executed simultane-ously.3.5 Program FunctionsThe usefulness of the computer program is not limited to simulating runs for anyinitial conditions; the program also serves as a powerful tool to study the effect offlexibility, librational motion, thermal deformation, shift in c.m., slewing maneuvers,77etc. For instance, in order to investigate the thermal effect, one has to supply theappropriate index for the thermal parameter in the input of the main program: '0'for exclusion and '1' for inclusion of the thermal effect. If '0' is assumed, the programwould essentially set all components of Tsys , Lys , and fins associated with thethermal effect to zero, i.e.Tsys = Torb + Tem -I- Th Tjr + Tv + Th,ir Th,t) Tir,v ;Lys =lc, -F1h^-F1h,r^+^;Hsys = Hcm, Hh Hjr + Hv Hh,jr + + Hh,v + Hr,v •A second run is then made with index '1' which can be compared with the previousresult to reveal the effect of thermal deformations. Similar procedure can be used tostudy the effect of shifting c.m. and slewing maneuvers.The effect of flexibility and higher modes is assessed by manipulating input tothe subroutine MODEL. In order to study the effect of flexibility, one first needs toobtain simulation results for the rigid body case. This can be achieved easily byputting flexible degrees of freedom for each body to be zero. The higher mode effectscan be studied by carrying out simulation runs with an increasing number of assumedmodes. Once again, this number is supplied at the subroutine MODEL input phase.The program can also be used to study gyroscopic effect and the influence ofthe choice of shape functions. Unlike the aforementioned cases, these studies requireminor modifications to the program. To isolate the gyroscopic effect, C in Eq. (3.1)is set to zero; whereas to study the choice of shape functions, one has to replacesubroutine MODE with a new set of admissible functions.783.6 Program VerificationIn modelling a system of such complexity, an obvious question concerning itsvalidity arises. In absence of similar equations obtained independently by others, twoavenues are available:(1) check energy and angular momentum functions;(ii) comparison with particular cases if reported in literature.Both methods are pursued here. A check on the energy variation is performedfirst. An arbitrary rigid satellite with the following nondimensional inertia diadic ischosen:Isys =2719.8^0^00^10.53^—13.14 .0^—13.14 2717.2 The satellite is aligned in the gravity gradient stabilized orientation, i.e. the maximuminertia, 1-Ix , is about the orbit normal. Assuming the nondimensional gravitationalpotential energy at 9 = 0 and 0 = Of be denoted by U0 and Uf, respectively, SU =Uf — Uo . Similarly, ST = Tf — To is the difference in the nondimensional kineticenergy of the satellite at 0 = 0 and 0 = Of. The variation of SU and ST for thesatellite subjected to three different librational disturbances is shown in Figures 3-4to 3-6. Two simulation runs were carried out for each initial condition. In the firstcase, denoted by the solid line in each plot, e 0 0 and C, -) as given by Eq. (2.2) is used.In contrast, the second case, represented by the dotted line, is a fictitious physicalsituation with the orbital rate assumed to be zero, i.e., the orbital frame is stationary(e = 0). With this assumption,= [— 0' sin A + 0' cos 0 cos A] ip + [A' — sin 0] 5 p+ [0 1 cos A + Vi cos 0 sin A] kp .^ (3.11)79Figure 3-4 shows the system response and energy variation for an initial pitchdisplacement of 1° (Figure 3-4a) and pitch velocity disturbance of 0.03 (Figure 3-4b).The figure shows that although there is no visible difference in the librational responsefor both the cases, the energy variation is different. Note that, the magnitude of STand SU are immaterial, instead, it is the trend that is useful. In 0 = 0 case, thesystem is conservative as ST + SU is zero; however, this is not true for 0 0 0 case. Itcan be regarded as only quasi-conservative.For an initial condition in the roll degree of freedom (0 0 = 1° in Figure 3-5 aand eo = 0.2 in Figure 3-5b), Figure 3-5 shows that the two cases have differentlibrational responses as well as energy variations. Once again, only the fictitioussystem is conservative. Similar observation can be made for the response associatedwith an initial yaw displacement of 1° (Figure 3-6 a) and yaw rate of 0.02 (Figure 3-6b).In theory, both the cases should be conservative. The lack of conservation ofenergy for 8 0 0 case is attributed to the assumption that the Keplerian relations,Eq. (2.41), are valid. For a rigid satellite, the total energy of the system as given byEqs. (2.34) and (2.35) is:and1 -^- ^heM^/eT U = —M fee • Ric^tr [I, Im, 2RL^YsF —1(DT^+^ T Is2^Ys^2./IL^vsST + SU =I^— I2^f sys f^o sys o3ite  (-T I^-^-T I+^1 1 „To )2/iL f sYs 1 f — o Y(3.12)(3.13)In the 0 0 0 case, the first three terms of Eq. (3.12) are forced to be constants by theassumption of the Keplerian relations; hence, it is not surprising that the change in80o ---  6=0X 05.0 5.0(a) .00 = 1°21p0c500-110^1 ^0.0^5 -^6U o --5 ^120OT 0-1201206E 0- 1200.00X 020 02ro0-11boo^---  e=o5.0-10.0(b) 1b 1. = 0.030, No. of Orbits^0, No. of Orbits0, No. of Orbits^0, No. of OrbitsFigure 3-4^Librational response and energy variation of a rigid satellite sub-jected to an initial pitch disturbance: (a) O. = 1°; (b) zY„ = 0.03.810, No. of Orbits 0, No. of Orbits9 =09 0(b) 0 10 = 0.20, No. of Orbits 6, No. of Orbitsb=o6U—2,10Figure 3 - 5^The effect of roll disturbance on a rigid satellite libration and energyvariation: (a) 0,, = 1°; (b) 0 1,, = 0.2.8240-40^0.00, No. of Orbits0, No. of Orbitseg o ---  boo5.0^0.0 5.00.2— — — boo60 obU 0.0-0.240bE 0(b) ) = 0.021I/Jo0 WV\AAAAA0, No. of Orbits^0, No. of OrbitsFigure 3-6^Dynamical response and energy variation of a rigid satellite withan initial condition in the yaw degree of freedom: (a) A c,^1°;(b)^0.02.83energies as given by Eq. (3.13) is not zero. Even though this implies that the systemis not truly conservative, Eq. (2.41) is not a poor assumption. With the first threeterms in Eq. (3.12) of several orders of magnitude higher than the remaining two,the percentage change in total energy (ST + SU)/(T U) is hardly significant. Forthe present satellite orbiting at 300 km altitude, the change is less than 10 -8 %. Theconclusion from this study is that energy variation can be used to verify the computercode by setting the orbital rate to zero.Next, a comparative study with two cases reported in the literature was under-taken. Ng [90] studied the librational and vibrational dynamics of a gravity-gradientstabilized satellite with two thermally flexed appendages (Figure 3-7 ). The presentformulation can handle this model by treating the rigid central body to be B, andthe two appendages as body Bi (i = 1, 2). Using the data in Table 3-3 for a satellitein a 90-minute orbit, the present formulation gives results identical to those obtainedby Ng [90].A set of typical response plots are presented in Figure 3-8 . Here, the generalizedcoordinates P1 and P2 correspond to the inplane vibration of B 1 and B2 in the firstmode, respectively. The initial condition applied is (P1) e = 0.05 and the satellite istaken to be orbiting in an eccentric orbit with 6 = 0.2.The second model used for verification is based on the two arm flexible manipula-tor as reported by Chan [15]. Here, the space station is treated as a rigid body (B e )whereas the robot arms (B 1 and B1 , 1 ) are taken to be cantilever beams supported ona mobile base (Figure 3-9). As the present formulation does not account for trans-lational motion of any member, the base was considered stationary. The numericalvalues used in the simulation are listed in Table 3-4 . A 100-minute circular orbit isassumed in the simulation. The two formulations gave almost identical results (Figure84Table 3-3^Spacecraft data used to assess accuracy of the present formulationCentral Body (Be) me^=^42,000 kg _ -lc^= 10 m(_Tzx ) c^=^100,000 kg m 2(iyy ) c^= 400,000 kg m 2(Izz ) c^= 400,000 kg m 2Upper and Lower Appendages (B 1 and B2) m1^m2^10 kg^(Ixx)i^=^(1-xx)2^0/2^= 100 m^(Iyy)i^= (402 33,333 kg m 2f2^=^0.003 Hz^(Izz )i^=^(rzz)2^33,333 kg m 2Local VerticalUpper Appendage (B2)Central Body (Be )Lower Appendage (B1)Figure 3-7^The spacecraft model used to assess accuracy of the present formu-lation.Orbit85P 1_4 0.01 1t III MI1 3-20 —-40 — 1^10.1—0.1 —^00. 1P 1 o o-0, No. of OrbitsFigure 3-8^A typical response obtained by the present formulation simulatingthe spacecraft model studied by Ng [90]. The results showed perfectagreement.86/LocalVertical(b) OrbitNormalLocalHorizontalOrbit(a) C)1LocalVerticalAx,AYc , X1^Z,, Z1^VY1LocalHorizontalzOrbitNormalz1,1^VY1,1Figure 3-9 Schematic diagram of the space station based MSS studied by Chan[15]: (a) coordinate systems; (b) design configuration. It is used hereto assess validity of the present general formulation.87Table 3-4^Data of the space station based MSS studied by Chan [15]Space Station (Be) me = 240,120 kg/,^=^115.35 m8 x 10 5 kg m 22.67 x 10 8 kg m 22.67 x 10 8 kg m 2Upper and Lower Appendages (B 1 and B2) mi^mi,i^=^10 kg^(1-xx)1^("xx)2^011^=^1_ Li^=^100 M^(Iyy)1^=- \-yy, 2^33,333 kg m2fi,i^=^0.003 Hz^(1-,,,)1^(izz)2^=^33,333 kg m23-10 ). Minor discrepancies may be attributed to the modified cantilevered modesassumed by Chan [15].The generalized coordinates for this model are:.q =^, A ,^Qri /2'6 Q1,1 'r = 1,^, nri ;where nri represents the number of modes selected. In Figure 3-10, using only onemode and with the initial condition being applied inplane ((n o = 0.05), only 0, Piand P 1 1 are excited.--1,A study of convergence in terms of the number of shape functions required isappropriate here. Four simulations are performed using one, two, three and fourmodes to represent beam vibrations of the MSS configuration. The results are shownin Figure 3-11 . For simplicity, the following discussion uses symbols I, II, III, and IVto indicate simulations and the associated number of modes. In general, irrespectiveof the number of modes chosen, the trend in response remains essentially the same.Discrepancies appear in the phase and amplitude.88 Present Formulation Chan [15] FormulationFigure 3-10 A comparison between the MSS response obtained by Chan [15] andthe present formulation.89x10—0.00^0 05c5011XOx10 —_ - -x10 -3-10.0011^1]0.05 1 mode 2 modes^ 2 modes 3 modesX0X06, Orbit^0, Orbit3 modes 4 modes0, OrbitFigure 3-11 Dynamics of the MSS using one, two, three, and four assumed modesin the simulations: (a) librational response.903 modes1 mode — — — 2 modes2X10 -52 modes- 21(2 1—14P1,141x10-5---Th/f\frA,V ', "27k.;(b)x 10 -1x10-1eit0 02(1cir-•—'10.0010- 12Q14pl- 410.00^ 0.02^0.00^0 0261 , Orbit^0, Orbit 3 modes 4 modes2x10 -56, OrbitFigure 3-11 Dynamics of the MSS using one, two, three, and four assumed modesin the simulations: (b) the first mode vibrational response.911p211Q211—11Q 2—1(d)1P3112Q 21—21P312Q 31 12—11Q2X10 -2—10.010.00 0.010.00— — — 2 modesx10-5d7NY,x10 -5PrA,'x 10- 5p2—.14 modes—11/--)2M1,10, Orbit 6, Orbit3 modes 4 modesx10-6---"'"'vN,Al • i(\. •xio-2‘APV-1M .A,,;vdi:WA \ AP' Pfif-!'• .(c)0.00^0.010, OrbitFigure 3-11 Dynamics of the MSS using one, two, three, and four assumed modesin the simulations: (c) the second mode vibrational response; (d) thethird mode vibrational response.92The convergence of the libration angles is shown in Figure 3-11a. Comparing Iand II, the following observations can be made:(i) convergence of the pitch response, both in phase and amplitude, is signifi-cantly improved with an increase in the number of modes.(ii) roll angle is essentially identical for both simulations;(iii) although the yaw response has the largest amplitude, the difference in yawresponse is confined to the phase only.Simulation III shows a dramatic improvement in the pitch response prediction, both inphase and amplitude. As expected, the roll response continues to display insignificantdiscrepancy, while the phase error in the yaw is markedly reduced. With four modes(simulation IV), the librational response appears to have converged to the true value.The convergence of the first mode response (P1, Q 1 , P1 ,1 and 9 1 1 ) is shown inFigure 3-lib. Similar to the pitch response, a comparison between I and II showsdiscrepancies both in phase and amplitude. However, the deviations become smallerwith the inclusion of the third mode; and finally become negligible in simulation IV.Figure 3-11c shows the convergence of the generalized coordinates (PL Q 21 , /1 1 andQ 2 ) associated with the second mode. A comparison between II and III shows er-rors in phase and amplitude. The situation improves with the addition of the fourthmode; however, the error still persists suggesting that higher modes are necessary toachieve better convergence. Finally, Figure 3-11d compares response of the gener-alized coordinates associated with the third mode (Pt Q 31 , /1 1 and 931 1 ) using 3and 4 modes in the simulation. The discrepancy between the two cases indicates thenecessity to include higher modes for improving convergence to the true dynamicsFrom the simulation results it can be concluded that, in general, both rigid andflexible dynamics converge as the number of modes in the simulation increases. Hence93the cantilever modes, used here as admissible functions, can predict the system dy-namics with accuracy. In the present case, convergence of the librational degreesof freedom and the first generalized coordinates (generalized coordinates associatedwith the first mode) need at least three modes in the simulation. Convergence ofthe second generalized coordinates is only marginally improved with addition of thefourth mode and the convergence of the third generalized coordinates would certainlyrequire higher modes.Table 3-5 compares the CPU time required in the four cases. The number ofequations increases from 14 (for one mode, Nq = 7) to 38 (with four modes, Nq = 19),with the CPU time required showing an increase of 34 fold. The enormous rise inthe computational effort is attributed to the stiff equations associated with highermodes; hence, there is a tradeoff between accuracy and CPU time. In the presentstudy, without sacrificing the physics of the- problem, one mode is always assumedin the simulation. Higher accuracy, if necessary, can always be obtained using moremodes and hence requiring more computational effort.Table 3-5 Comparison of CPU time required using one, two, three and fourassumed modes in the simulation of the MSSNo. of Modes Nq CPU time sec * CPU/CPUref**1 7 5,540.11 1.0002 11 28,347.89 5.1223 15 88,155.92 15.9124 19 189,525.45 34.210* based on VaxStation 320**CPUref represents the time required using 1 mode944. PARAMETRIC STUDIES4.1 Preliminary RemarksThe objective of this chapter is to illustrate versatility of the relatively generalformulation. To this end, dynamic simulations of five spacecraft models, of contem-porary interest, were carried out. The three models are related to the proposed SpaceStation. It should be pointed out that the station is an extremely complex, highlyflexible platform with diverse interconnected structural elements (beam, plate, shell,etc.) It will have a size of the soccer field. Of course, such a gigantic structure can-not be carried to the operational altitude in its entirety. It will be constructed instages through integration of modular subassemblies extending over 30 flights of theSpace Shuttle. Thus the proposed Space Station represents an evolving structure withtime dependent geometry, inertia, flexibility, damping, and other properties duringits constructional phase. Of course, the Space Station will operate in presence of avariety of disturbances induced by the environment, on board operations and interac-tions with the Shuttle flights. Thus each stage of its development would represent achallenging dynamical problem in design, dynamics, stability and control. Here, thedynamics of the FEL (First Element Launch) configuration, the PMC (PermanentlyManned Configuration), and the on board MSS (Mobile Servicing System) which willassist in construction and operation of the station, are investigated at some length.Next generation of communications satellites represent a large class of configura-tions of significant importance. After all the revolution in communication is the giftof the space age that has affected the entire world. A rather novel design of theINdian SATellite II (INSAT II), the multipurpose Indian communications satellite tobe launched this year, is considered for study here. Finally, dynamics of the SFU95(Space Flyer Unit), a Japanese experiment to be launched in 1993, is studied duringdeployment and retrieval of the solar arrays.The amount of information obtained is enormous and only a sample of it is pre-sented here in a condensed form. However, it does suggest trends, and lays foundationfor the design and development of control strategies.4.2 First Element LaunchThe United States led Space Station Freedom program is currently in the designand development phase. The backbone of the station is the power boom. It is essen-tially a truss structure to which modules, equipment and subsystems are attached.There are five major modules (two Habitat, two Laboratory and one Logistics) clus-tered around the geometric centre of the power boom. The subsystems include thephotovoltaic (PV) arrays and the associated power generation equipment, anten-nas, Attitude Control Assembly (ACA), Reaction Control System (RCS), and MobileRemote Manipulator System (MRMS). Solar and stellar sensors, satellite servicingprovisions, Orbital Maneuvring Vehicle (OMV), and shuttle berthing ports are someexamples of the user equipment. In order to maximize solar energy input, the PV ar-rays can undergo predefined rotation via alpha and beta joints. Total power generatedby each pair of PV array is 18.75 kW. Heat rejection of the station is achieved by twostation radiators and four PV array radiators whereas waste disposal is accomplishedby the resistojet located at the end of the stinger.As pointed out before, the Space Station will be constructed utilizing aroundthirty Space Transportation System (STS) flights [102,103]. The first flight wouldresult in construction of the First Element Launch (FEL) configuration. It will havean overall length of 60 m and a mass of 17,680 kg. Major equipment installed in theFEL configuration includes two PV arrays, an alpha joint, fuel storage tanks, stinger96and resistojet, avionics, and RCS.The design configuration of the FEL would be such that the axial directions ofthe power boom and PV arrays are parallel to the orbit normal and local vertical,respectively. It should be pointed out that, in general, the design configuration isnot identical to the equilibrium configuration. The FEL is simulated here by thepower boom, a free-free beam, with lumped masses representing the alpha joint, fuelstorage tanks, avionics, and RCS. The stinger and the resistojet are treated as acantilever beam and point mass, respectively. The PV arrays and PV array radiatorare represented as cantilevered plates. Figure 4-1 shows the coordinate system usedin the numerical simulation. Considering only the first mode of vibration for theflexible elements, the generalized coordinates and the degrees of freedom are:,1^1,,—1 r)1 pl^1^1B:4q=0, 0, A, c^1 Q^1 1 .cNg = 10 .The numerical data used in the simulations are obtained or estimated from the NASAreports [102,103] and are summarized in Table 4-1 . The nondimensional inertia diadicof the system is found to be,^1.1715^0^—0.1088Isys =^0^9.2108^0—0.1088^0^9.5538Consider first the rigid body dynamics of the FEL. Here, the power boom as wellas the attached appendages are assumed to be rigid. The equilibrium configurationof the FEL corresponds to: We = 0°, ce = 0.74°, and A, = 0°, which is different fromthe design configuration (power boom along the orbit normal, solar panels parallelto the local vertical). Hence the system, even in absence of any external or internaldisturbance, would exhibit some motion as shown in Figure 4-2. Introduction of an97(b)LocalHorizontalOrbitOrbitNormalLocalVertical(a)X4 411^Z3X3Body B4 Y3^Body B3z,Orbit NormalLocal Horizontal^111Local VerticalY2Body BcBody B2Body B1BODY DESCRIPTIONB, Power BoomBi StingerB2 PV Array RadiatorB3, B4^_ PV ArraysFigure 4-1^Configuration of the FEL used in the numerical simulation: (a) co-ordinate systems; (b) design configuration.98Table 4-1^Physical parameters of the major components of the FELPower Boom (Body Bc ) Stinger (Body B1)lc 60 m /1 26.7 mme 15,840^kg 270 kg1.936^Hzm1-1 xl x)10.5^Hz(1-xx)c 0.15 x106 kg-m2 f( 0(-Tyy)c(1-zz) c4.37 x 1064.28 x 106kg-m2kg-m2 (1-1 zYYz))1164,16064,160kg-m2kg-m2PV Radiator (Body B2) PV Array (Body Bi, i = 3,4)/2 11.5^m /i 33 mm2 450^kg mi 444^kg0.1^Hz f 0.1^Hz(1-xx)2 50^kg-m 2 (rxx)i 1,332^kg-m2(402 19,837^kg-m2 (40i 161,172^kg-m2(1-zz)2 19,887^kg-m2 (Izz)i 162,504^kg-m2initial disturbance of 0.1° in pitch, roll or yaw only accentuates the unstable responseand the system starts to tumble in less than one orbit (Figure 4-3 ).DefiningIzz IXX= Tlyy= 0.91 ;1"xx — ibtyA 2 = Tizz= —0.84;Kane et al. [104] have shown that stable motion is possible if and only if K1 < 0 andK2 > —K1. This confirms unstable orientation of the FEL and points to the need ofa suitable control.The effect of flexibility on the system response is shown in Figures 4-4 to 4-8.99Sy —kr=1nSize8, OrbitFigure 4-2^Librational response of the rigid FEL showing the inherent unstablecharacter of its orientation.For better appreciation of the response, the vibration degrees of freedom are plottedin terms of the tip deflection of the centerline. Using Eq. (2.23), the tip deflections 61and SL of a beam element (e.g. Bk) in Yk and Zk directions, respectively, are givenby:13T(t)01,(1k);Q rk(t )O lk. ( 1 k) •^ ( 4 . 1)r=1For a plate element Bk, the displacement is a function of both the Xk and Yk coordi-nates. Defining and EYk to be the tip displacements of the line Yk = 0 and Xk = 0,respectively, and using Eq. (2.25), and EZ are then given bynk = E TIV(t)OUlk)Ot(o) (4.2a)s= 1 t= 1100(a) 'tko = 0.1°900-90900-9090- ^0-90 -0.00, Orbit1.0(c)^= 0.1°90090900--9090-- ^0ipo00No(b) O. = 0.1°90- 90 ^^90 -0° 0 ^- 9090A°-90'^0.0^1.0^0.0 1.00, Orbit^0, OrbitLibrational behavior of the rigid FEL showing the unstable responsein the presence of an external disturbance in pitch, roll or yaw of0.1°.- 90Figure 4-3101m n= E^Htt(t)oz(0)0t(wk)t=1(4. 2 b)Note that, for a cantilever plate, since ¢4(0) = 0, EYk, is identically zero. For instance,61 of Figure 4-4 represents the tip deformation of B2 (station radiator) of the lineYk = O. The range of the deformation plotted is —0.5 x 10 -5 cm and +0.5 x 10 -5 cm.Figure 4-4 shows response of the system initially in the design configuration. Thisresult can be used as a reference for other cases with initial disturbances. Here, devi-ation from the equilibrium configuration serves as a disturbance to the system. Evenwith such a small disturbance, the system gradually undergoes unstable librationalmotion. The periodic oscillations of the flexible degrees of freedom in the direction ofthe local horizontal (Scz, E3 , and €T) or the orbit normal (Si) indicate their non-zeroequilibrium position. On the other hand, the other degrees of freedom (sg, Si', and€D, which are parallel to the local vertical, have zero equilibrium position. Due tothe coupling effect with librational and vibrational degrees of freedom, their responseamplitudes gradually increase with time.The effect of the power boom disturbance is shown in Figure 4-5. In Figure 4-5a, the power boom is initially deformed in the first mode with a tip deflection of1 cm in the local vertical direction ((/31,) 0 = 0.826 x 10 -4 ). Even with this smalldisturbance, the pitch response is excited significantly with high frequency harmonicsof 0.02° in amplitude. Similar trend is observed in the roll response. Although theamplitude is lower than that in the pitch response, the roll motion is excited ata higher frequency. Under this initial condition, the power boom is expected to bevibrating symmetrically about the local vertical; hence, it is not surprising to see thatS' and roll have similar response trend with the distinct beat phenomenon present.Modi and Ng [105] have investigated the beat phenomenon of a gravity-gradient1020.0 0.20.2sg 0.0-0.20.1Sf 0.0--0.10.2Si 10 - 2 c0.020.2^0.00^0.011^o0.00 ^-0.01 ^0.5 ^0.0-0 50.5A ° 0.0-0.5X 10 -4 cmx10 -7 c^0.5 ^E2 0.0- 0.5 ^0.5 ^E3€  0.0- 0.50.5efi^0.0- 0.50.00 0.020, Orbit0, Orbit0, Orbit EZFigure 4-4^Dynamical response of the FEL in absence of an external initial dis-turbance.1030.2sg 0- 0.20.2Sf 0.0- 0.20.5Si 0.0- 0.5(a) P i ). = 0.826 x 10 -40.021P o oo-^-0.02^0.01 00400010014100o 0.00_ 44-0.01x10 cm0.01 0.1x1 0-A ° o oo af 0.0--0.1^0.02^0.00-0.01^0.00 0 026, Orbit 6, Orbit0.10.0- 0.20.2efi 0.0- 0.20.00 0.026, OrbitFigure 4-5 Effect of an initial 1 cm tip deflection of the power boom on theFEL dynamics: (a) initial deflection in the local vertical direction,) = 0.826 x 10-4.104oriented satellite with two appendages. It is found that the beat is caused by thedifferential gravitational torques acting on the two appendages. In the present case,the beat response is attributed to asymmetrical loading on the power boom coupledwith the gravitational disparity. As the arrays and the stinger become smaller, thebeat gradually disappers (not shown). Compared to Figure 4-4, the yaw responseremains unchanged. The flexible element experiencing the largest deflection is thestinger. Its maximum amplitude is about 3 cm in the local vertical direction.In Figure 4-5b, the power boom is initially disturbed in the local horizontal direc-tion ((Q 1 ),, = 0.826 x 10 -4 ). Since this initial condition gives rise to predominantlysymmetrical motion about the local horizontal, the yaw motion is excited with highfrequency harmonics. However, as can be expected, both pitch and roll remain essen-tially unexcited. Compare to Figure 4-5a, the beat phenomenon is no longer presentin the power boom or rigid body response. Also, note that in Figure 4-5a, the maxi-mum value of S is about 0.1 x 10 -3 cm when excited by the initial disturbance in IT,direction; whereas in the present case, sg is an order of magnitude higher when thesame disturbance is applied in the Ze direction. These show the directional charac-teristics of the FEL power boom: given the response in direction B when disturbed indirection A, one cannot predict the response in direction A when disturbed in direc-tion B. Once again, the stinger experiences the largest deflection with an amplitudeof 5 cm in the orbit normal direction. Note that the PV array response has the sameamplitude as the previous case; however, now the low frequency component is morepronounced.Response of the system with the stinger subjected to an initial disturbance isshown in Figure 4-6 . The stinger is deformed initially in the first mode of a cantileverbeam with a tip deflection of 1 cm in either the local vertical direction ((,P1),, =1050.2 x io -2 c(b) (Q 1 )0 = 0.826 x 10'0.01 0.00v.' 0. 00- 0. 01 -0.01 -^rho-t- 0.000.020, Orbit 0, Orbit10 cmOWL- 0.50.18f 0.0- 0.10.02^0.00- 0.01 ,^0.01XO0.000.5q 0.0- 0.50.2E3• 0.0- cmx10 cm0.00^ 0.020, OrbitExtifbiFigure 4-5 The effect of an initial 1 cm tip deflection of the power boom on theFEL dynamics: (b) initial deflection in the local horizontal direction,(Qic\ 0) = 0.826 x 10-4.1060.1872 x 10 -3 , Figure 4-6a) or along the orbit normal ((Q 1 ) 0 = 0.1872 x 10 -3 ,Figure 4-6b). Similar to the power boom disturbance case, Figure 4-6a shows thatboth pitch and roll responses have high frequency harmonics although the roll motionis hardly noticeable. Also, the initial condition has no effect on the yaw response atall. As for the power boom, it is slightly excited with a peak amplitude of about0.005 cm in the local vertical direction. The excitation of other flexible members arealso small: 0.25 cm for EZ and 0.5 cm for E3 and ET. Figure 4-6b shows that disturbancealong the orbit normal has very little impact on librational and vibrational motions ofthe system. Compared to Figure 4-4, the librational response remains essentially thesame. The effect on the power boom is again minor while amplitudes of the PV arrayand radiator vibrations are one and three orders of magnitude smaller, respectively.Also, analogous to the power boom response of Figure 4-5, (5T has a peak amplitudeof only 0.1 x 10 -4 cm when excited by the initial condition on Qi whereas 6f is twoorders of magnitude higher when excited by the initial disturbance of P.Figure 4-7 shows the influence of PV array and radiator disturbance on the systemresponse. In Figure 4-7a, the PV array radiator is initially deflected in the first modeof a cantilever plate with a tip deflection of 0.5 cm ((H 21 ' 1 ) 0 = 0.2485 x 10 -3 ). Note,both roll and yaw responses are hardly affected. Pitch response shows low frequencyharmonics of a small amplitude. The excitation of the power boom and the stinger isalso small: the peak amplitudes of and sy being only 0.25 x 10 -4 cm and 0.02 cm,respectively. The only flexible member that shows noticeable deformation is the PVarray with a peak amplitude of about 0.25 cm. With the same initial disturbanceapplied to the PV array ((H 1 ' 1 ) 0 = 0.866 x 10 -4 ), Figure 4-7b shows response resultssimilar to those observed in Figure 4-6a: the rigid body dynamics as well as powerboom and stinger motions are virtually absent. However, it is of interest to note adegree of similarity in the response trends of (5'. and E3 (or €T) in Figure 4-4b, where107x 10 —1 cm(ED. = 0.1872 x 10 -30.16cz 0.00. 10.2x 10 CMSi 0 0--0.20.20020. 00 0 020.1Sy 0.0-0.1-xio-2 cSl o.o-0.2 ^0.00(a)0.01 ^1110.00 1/- 0.010.01r 0fl`1-' 0.00- 0. 010 .0100.00- 0.010.00 0.02—0.10, Orbit0. 10.0-0 10, Orbit^ 0, OrbitFigure 4-6^Librational and vibrational dynamics of the FEL subjected to aninitial 1 cm tip deflection of the stinger: (a) initial deflection in thelocal vertical direction, (no = 0.1872 x 10-3.108-0.5 -0.01^-0.2^0.2 0.00= 0.1872 x 10'(b) (Q I0, Orbit 0, Orbitbitaik•NON0.01IP° 0.00-0.010.5se^0.0-0.5 ^0.50.0of 0.0-0.10.2-0 . o o-0.1o.o0.020.1 x l^cmo o-x l^cm- 0.1 ,^0.00Figure 4-6 Librational and vibrational dynamics of the FEL subjected to aninitial 1 cm tip deflection of the stinger: (b) initial deflection in thelocal vertical direction, (Q 1 ),, = 0.1872 x 10-3.0.026, OrbitEZ1090.01- 0.010. 01rhO0.00- 0.010. 0100. 00- 0.01(a) (IP2 ' 1 ). = 0.2485 x 10 -30.00 0 020.0 \MA'0 .1X10 -4 cm0.5-0 .10.26,Y. 0.0-0.20.5x1 0 -10, Orbit^0.1^cf 0.0-^-0.1 ,^^0.5^-^0.0-0.5 ^0.5Eq 0.061/EZ.111-••Orbit^ 6, OrbitFigure 4-7^The response of the FEL with an initial condition on the PV arrayor the radiator: (a) PV array radiator with a tip deflection of 0.5 cm,(H 12 '1 ),, = 0.2485 x 10-3.110MVp0°0.00-0.010.0 100. 00-0 .01 -^0.10.0-0.10.1-^(5f 0.0xio-3 C0 .00 0 02(b)^-= 0.866 x 10 -30.26f 0.0xio - 2 c(vv0.2^0.00 0 02x 1 0 — ' c0.2E 0 . 0--0.20. 1E 0. 03—0.10.1x10 cm64 0.0—0.1^0.00 0.020, Orbitx10 cm0, Orbit^0, OrbitFigure 4-7 The response of the FEL with an initial condition on the PV array orthe radiator: (b) PV array with a tip deflection of 0.5 cm, (H 1 ' 1 ),, =0.866 x 10-3.111the initial condition corresponds to Ocz = 1 cm. This points to the coupling of powerboom vibration in the local horizontal direction with the PV array deformation.In summary, Figures 4-1 to 4-7 demonstrate that flexibility effects on the FELresponse cannot be overlooked. A small disturbance applied to any flexible membercan affect the rigid body motion significantly. The power boom disturbance in thelocal vertical direction is the most critical one as the resulting high frequency modu-lated pitch and roll responses would require high bandwidth controllers. Furthermore,with the power boom or the stinger subjected to a given magnitude of disturbance,its direction is critical in predicting the rigid body as well as vibratory responses ofother components.4.3 Permanently Manned Configuration4.3.1 Dynamic responseThe PMC will be established after fifteen STS flights. It will be 115 m in lengthand 160,972 kg in mass. The major difference from the FEL configuration is an ad-ditional pair of PV arrays and their radiator, two station radiators, and the modules.The two pairs of PV arrays are expected to provide a total power of 37.5 kW for thestation. The orientation of the PMC will be similar to that of the FEL: the powerboom is aligned with its axis parallel to the orbit normal; the axial directions of radi-ators (station and PV arrays) and the stinger are parallel to the local horizontal; andthe PV arrays in stationary mode have their axes parallel to the local vertical. Thecoordinate systems for the numerical simulation are shown in Figure 4-8 . Here thePMC is idealized as a free-free beam (power boom) supporting the stinger modelledas a cantilever beam and eight cantilever plates representing arrays and radiators (4PV arrays, 2 PV array radiators and 2 station radiators). Considering only the first112mode of vibration, the generalized coordinates and the number of degrees of freedomare:q =^ Pl, ,^p 1 ni al"^u1,1--L 2H"1 , -14_/1^"-1,1 H1 ' 1^Hu1, and " ,•,^, 9 Ng = 15 .The numerical data for the components are the same as before except for the powerboom and station radiators and are summarized in Table 4-2 .To begin with, the rigid body dynamics of the PMC is considered. The equilib-rium configuration of the PMC is found to be at oe = 0°, q5e = —0.002°, and A, = 0°.Even in the absence of any disturbance, the PMC exhibits some librational motion asshown in Figure 4-9. An initial displacement of 0.1° in pitch, roll or yaw is applied.Like the FEL, Figure 4-10 shows that the PMC is inherently in an unstable orienta-tion. However, unlike the FEL, librational disturbances result in different responsetrends. Figure 4-10a shows that an initial pitch disturbance results in a relativelygradual librational instability with the pitch, roll, and yaw amplitudes reaching 10°,7°, and 5°, respectively, at the end of one orbit 90 min). With an initial roll dis-placement, Figure 4-10b shows that the divergence sets in much faster. Actually, thePMC starts to tumble after one orbit when the yaw angle reaches —110° although thepitch and roll angles (34° and 29°, respectively) are still less than 90°. Finally, withan initial yaw disturbance, Figure 4-10c shows that the response is similar to that inthe pitch disturbance case; however, the magnitudes in roll (39°) and yaw (22°) aresignificantly higher. The nonclimensional inertia diadic of the PMC is found to be0.6706^0^0.0046Lys =^0^13.4346^00.0046^0^13.2783113Orbit NormalXe,Zi^A^Local HorizontalI • Ye' Yi ^_^Local VerticalV OrbitNormalFigure 4-8^Schematic diagram of the PMC used in the numerical simulations:(a) coordinate systems; (b) design configuration.114Power Boom (Body Be )lcfi( IX X Cc(Izz)e(Ixz)c115 m- 154,583 kg- 0.193 Hz- 1.44 x 106 kg-m2- 43.26 x 10 6 kg-m243.26 x 10 6 kg-m 20.18 x 10 5 kg-m 2Stinger (Body B 1 ) Station Radiator (Body Bi, i = 2, 3) 26.7 m270 kg0.5 Hz3,844 kg-m 264,160 kg-m264,160 kg,m 211.5 m1,395 kg0.1 Hzti^061,496 kg-m 2- 65,340 kg-m 2PV Radiator (Body Bi, i = 4,5) PV Array (Body Bi, i = 6, ... 9)- 11.5 m- 450 kg0.1 Hz50 kg-m 219,837 kg-m 2- 19,887 kg-m 2• 33 m444 kg0.1 Hz• 1,332 kg-m 2161,172 kg-m 2• 162,504 kg-m 2Table 4-2^Physical parameters of the major components of the PMCgiving K1 = 0.94 and K2 = —0.961. Although these values are similar to those of theFEL (KJ. = 0.91 and K2 = —0.84), the trend of the libration responses for the twocases are different. This points to the difficulty in predicting response trends basedon K1 and K2 values only.The system response of the PMC with the inclusion of component flexibility isshown in Figures 4-11 to 4-14. For conciseness, only the response of one station115A45, o00—7500—750.0^ 1.00, OrbitFigure 4-9^Libration response of the PMC due to deviation from the equlibriumconfiguration.radiator (B2), PV array radiator (B 4 ) and PV array (B 6 ) are plotted. The systembehaviour indicates some similarities as well as dissimilarities with the correspondingFEL responses.Figure 4-11 shows response of the system initially in the design configuration.The motion ensues due to deviation of the system orientation from its equilibriumconfiguration. Note, the pitch motion is essentially unexcited. Although the roll andyaw degrees of freedom are excited, their amplitudes increase very slowly. Unlikethe FEL, only the power boom vibration in the local horizontal direction (6) andthe station radiator (q) have non-zero equilibrium positions. In the power boomresponse, the component in the local vertical direction (6g) is hardly excited. Thestinger experiences a peak vibrational amplitude of about 0.1 x 10 -5 cm. Note that,unlike the FEL, the stinger is stiffer than the power boom and hence has the highest116(a) Oc. = 0.1°10—1010—101 00—1 00.0 1 .00, Orbit(b) = 0.1° (c)45 100—45 —1045 450 0—45 4590 4590 450°A= 0.1°0.0^ 1.0^0.00, Orbit^0, OrbitFigure 4-10 Rigid body response of the PINK to rotational disturbances.1171180.2Of^0 . 0X 10 -3 cm0.2o.o0.0-0.20.2sy^0.0-0.2x1 0 '° c0.02^0.00^ 0 020, Orbit^ 0, OrbitFigure 4-11^Response of the PMC due to deviation from the equilibrium orien-tation.-0.2 1^0.000.01x10-00.00^-0.01^^0.01 ^rhO0.00-^-0.010.01xo0.00-0.010.0 0.20, Orbit6420.50.0- 0.10.2Eg 0.0-0.2X 10 -4 cmX 10 -2 cmnumber of oscillations per unit time.Next, the system is subjected to an initial disturbance corresponding to the powerboom deformed in the first mode with a tip deflection of 1 cm. With the initial de-flection in the local vertical direction, (P 1,),, = 0.4348 x 10 -4 , Figure 4-12 showsthe system response over 0.02 orbit. Note that similar to the FEL, the power boomdisturbance in the local vertical direction excites the libration motion with high fre-quency modulations superposed on it. Yet, unlike the FEL, only the pitch motion isexcited in the present case, with a considerably lower frequency due to a more flexiblepower boom. Also, the beat phenomenom no longer exists since the powerboom isnow loaded symmetrically. The stinger, though excited, vibrates predominantly inthe local vertical direction with a magnitude of 0.6 cm peak-to-peak. Other flexiblecomponents oscillate with frequencies of the power boom vibration. Even with thissmall initial disturbance, the PV array radiators experience a large deflection with apeak amplitude of 2.5 cm. In Figure 4-12b, response of the PMC is to an initial dis-turbance in the local horizontal direction ((Q 1 ),, = 0.4348 x 10 -4 ). In the FEL case,yaw is the only librational motion excited. Here, only the pitch response is presentwith a negligible amplitude. The stinger and the PV array radiator responses arealso small with peak amplitudes of 0.01 cm and 0.1 cm, respectively. It is interestingto note that both the station radiators and PV arrays have similar response trendsbut with different amplitudes: 1 cm for the station radiators and 2.5 cm for the PVarrays, although the latter are more massive.The focus is now turned to the stinger disturbance. The initial condition of thestinger is identical to that of Figure 4-6. The system behaviour may be expected tobe similar to that observed for the FEL case. Yet, since the PMC is about ten timesas heavy as the FEL, the forced response to the stinger disturbance is expected to be119x10 cmx10 cm0.011/00.00(a) (121),, = 0.4348 x 10 -4-0. 010.01rhO0.00E4x10 -20.00^ 0.020, Orbit0.26Yc^0.0--0.2x10^cm0.2 xl^cmOf^0.0- 0.20.5 xi^cmbl^0.0 - vS €4-0.50.5x 10 - 6 cbi^0.0--0.50.00Figure 4-12-0.010.01xo0.00-0. 010.50.0- 0.1^002^0.00 0020, Orbit^ 8, OrbitThe influence of the power boom initial tip displacement of 1 cm onthe PMC response: (a) displacement in the local vertical direction,(pci \ 0) = 0.4348 x 10-4.120121Figure 4-1 2 The influence of the power boom initial tip displacement of 1 cm onthe PMC response: (b) displacement in the local horizontal direction,(Qci \ 0) = 0.4348 x 10-4.0.01X00.00—0.010 020.00(b) (Q 1 ). = 0.4348 x 10 -40.010.00f 0f,`/-' 0.00—0.01x10-0, Orbit0.2 x cm6' 0.0—0.20.00 0.020, Orbit 0, Orbit0.2di oo^—0.2^^0.5 xi0 -6 cmdl o .o WNW0.2EZ 0.0—0.20.24 0.00.2x10- cm—0.5xl^cmEysmaller in magnitude. Figure 443 a verifies these predictions. With the initial stingerdisturbance in the local vertical direction, the figure shows that the PMC is pitchingabout the orbit normal at a higher frequency. However, the amplitude is considerablysmaller than that for the FEL case. The power boom, PV array and radiator oscillatewith maximum amplitudes of 0.003 cm, 0.1 cm and 0.05 cm, respectively. Thesevalues are about an order of magnitude smaller than the corresponding ones in theFEL case. Note that the stinger response in the orbit normal direction attained amaximum amplitude of 0.001 cm in the FEL case, whereas now the same degree offreedom is hardly excited. The response of the PMC with initial stinger disturbancein the orbit normal direction is shown in Figure 4-13b. Similar to the FEL, thelibrational motion is unaffected by the stinger disturbance. The PMC librates as if noinitial condition was applied to the system (other than deviation from the equilibriumorientation, Figure 441). Likewise, the power boom and the stinger deformations inthe local vertical direction are excited slightly. The response trends of the PV arrayand radiator are also similar to those in the FEL case; however, the amplitudes ofthe response in the PMC case are considerably smaller. In the FEL configuration,the maximum array displacement was 0.05 cm whereas it is only 0.01 cm in the PMCcase.The flexibility effects of the station radiator, as well as PV array and radiatorare shown in Figure 4-14. In Figure 4-14a, an initial tip displacement of 0.5 cm isapplied to one of the station radiators ((H 12-1 ),, = 0.2485 x 10 -3 ). As can be seen, thisdisturbance has no influence on the librational response of the PMC. For the powerboom, the response 8: is identical to EZ in Figure 4-12b where the initial conditionwas applied to This illustrates the coupling of Q 1 and 11 21-1 degrees of freedom.--cThe peak amplitude of the power boom motion is found to be 0.002 cm. As expected,the forced oscillation of the stinger is predominantly in the direction parallel to the122X00.00—^–0.01 -0. 00 0 02(a) (El). = 0.1872 x 10 -30.020.000, Orbitx10 -,0.5 ^0 0.0 --0.50.5of 0. 0-0.50.26,Y.^0.0-0.2xlionommi0, OrbitEz0.5-^EZ 0.0^-0.5^,^0.1E4 0.0 -1^- 0.1 ^xi0.2 ^Cm0.020, OrbitFigure 4-13 Response of the PMC subjected to an initial 1 cm tip deflection ofthe stinger: (a) deflection in the local vertical direction, (P1)0 =0.1872 x 10-3.123(b) (Q 1 ), = 0.1872 x 10 -30.01 x^10-311/30.00-0.2 ,^0.000.026, Orbit6, Orbit^0.01^^0°0000 ^- 0.0 10.56T 0.0-0.50.26f 0.0 -)fxliNgle\ Malt0. 10 - 1 cmX 10 4 cm0.0--0.2^0.00 0 020, OrbitFigure 4-13 The response of the PMC subjected to an initial 1 cm tip deflectionof the stinger: (b) deflection in the orbit normal direction, (Q 1 ) ='^°0.1872 x 10-3.124(a) (H22 '1 ). = 0,2485 x 10 -30.011/30.00-0.010.01,ot,`f-' 0.00-0.01 -0.01 ^X00.000.5x1 0 -\ V^V(5.f o.o--0.50.00 0 02bl o o- -3 cmEqa-EZ- 0.10.1 x 10 -2 cm€4 0.0 N./^- 0.1 ^0.2 xl^cm^es 0.0^0.2^0.00 0.020.1x100.0 -0, Orbit0, Orbit^ 0, OrbitFigure 4-14 Effect of the station radiator, PV array, and radiator disturbance onthe PIVIC dynamics: (a) station radiator subjected to a tip deflectionof 0.5 cm, (H21,1,0) = 0.2485 x 10-3.125radiator disturbance, i.e. along the orbit normal. Since the stinger is stiffer thanthe power boom, the amplitude of the stinger motion is only 0.0005 cm. It shouldbe pointed out that since E2 is perpendicular to eft or E6 , the PV array and radiatorshould not be excited. However, since is excited gradually, the coupling effectinduces the response €,T and 66. At the end of 0.02 orbit, the magnitudes of vibrationare 0.001 cm and 0.2 cm for the PV array and radiator, respectively.The disturbance is now applied to the PV array radiator with a tip deflection of0.5 cm ((H1,1)0 = 0.2485 x 10 -3 , Figure 4-14b). Similar to the FEL, the PV arrayradiator disturbance has minor effect on the librational motion. It is interesting tonote that sg exhibits the same response trend as that of €,T in Figure 4-12a. Onceagain, this shows the coupling between PV array radiator and the power boom de-formation in the local vertical direction. The forced oscillation of the stinger is nowpredominantly in the local vertical direction, which is the same as the direction of ETI.The station radiator response is insensitive to the PV array radiator disturbance; andis identical to the corresponding response in Figure 4-11 in absence of an externaldisturbance. The PV array is now vibrating with a peak amplitude of 0.1 cm. Theresponse trend is similar to that of the PV array radiator in Figure 4-14c where aninitial disturbance is now applied corresponding to the array tip deflection of 0.5 cm((H 16-1 ) 0 = 0.866 x 10 -4 ). Other observations are similar to those made during thediscussion of Figures 4-14a and 4-14b. Namely, the initial condition on PV array leadto only minor libration of the PMC. The distinct response trend of the power boomagain appears in the S. The stinger response, with its characteristic high frequencyharmonics, vibrates with a peak amplitude of about 0.005 cm in the local verticaldirection. Once again, the station radiator is insensitive to this particular initialcondition.126Ex40. 00 0.02^0.2 ^6T 0.0- 0.2 ^0.2^-^6f 0.0- 0.20.00 0 .02x10-yet0. cmCM-0.020.01xo0.00 -0.01111/30.0 0- 0.010.02(13°0.0 0(b) (1/24 ' 1 ). = 0.2485 x 10 - 3x10 -0.00^0.020, Orbit- 4-140, Orbit^ 0, OrbitEffect of station radiator, PV array, and radiator disturbance on thePMC dynamics: (b) PV array radiator subjected to a tip deflectionof 0.5 cm, (H 1 ' 1 ) 0 = 0.2485 x 10-3.1270.5 -^E2 0.0-0.50.2E4 0.0-0.20.1Es 0.0- 0.1x 10 cm0.00 0 02nn- 0 .010, Orbit0.5og 0.0-0.50.2Of 0.0-0.20.5oy 0.0-^-0.5 ^^0.5^6f o.o-^-0.5 0.00^0.02^0.00^0.020, Orbit^ 0, Orbit^Figure 4-14^Effect of station radiator, PV array, and radiator disturbance on thePMC dynamics: (c) PV array subjected to a tip deflection of 0.5 cm,(H16 4 ),, = 0.866 x 10-4.128Figures 4-12 to 4-14 show that, as in the case of the FEL, the rigid body motionis sensitive to the deformations of the flexible members. Even for the same flexiblemember, different directions of excitation results in different librational response. Dueto the relatively large mass and inertia of the power boom, even its small excitationinduces large deflections of the arrays and radiators. The stinger, which has thelargest stiffness among all the flexible components, excites librational and vibrationalresponses with high frequency modulations. Disturbances applied to the radiator orarray, in general, have little influence on the rigid body motion of the system as wellas flexible deformations of the power boom and stinger.4.3.2 Thermal deformation and eccentricityWith a large surface area of over 1,500 m 2 , it is important to investigate the effectof thermal deformation of the flexible members. The simulation here considers onlythe differential heating of the PV arrays, which have a total area of over 790 m 2 andare facing the sun at all times. Since no reliable data on the physical properties ofarrays are available, the value of L: is assumed to be 0.01 (i = 6, ... , 9) whereLZ = 1 i/17and /:, the thermal reference length, is as given by Eq. (2.19). Thermal referencelength can be visualized as the radius of curvature of the thermally deformed ele-ment. A large /7, and hence small L:, implies little thermal deformation effect. Anassumed value of Lz = 0.01 corresponds to /7 = 3,300 m, which represents littledifferential heating of the arrays. Even with this small L: value, the effect on thePMC response is not negligible as shown in Figure 4-15. Here, there is no initialdisturbance applied to the system, i.e. the excitation is purely due to deviation fromthe equilibrium configuration. A comparison with Figure 4-11 would be appropriate.129L: =0.01^..6,...,9)^10.0 ^^ifr° 0.0 -10.00.02A0,ky 0.000.01 4^0.5 ^EZ 0.0- 0.5 ^0.1 ^ETI 0.0 -- 0.1-0.2Es 0.0- 0.2x10 cmOf 0.0bl 0.0- 0.5-0.10.1X10 -3 Cbi 0.0-^X10-5 cm0 02^0.00 0 020, Orbit 0, OrbitFigure 4-15 Librational and vibrational responses of the PMC in absence of ex-ternal excitation showing the effect of thermally deformed PV arrays.130Since the PV arrays deform in planes parallel to the orbital plane, the pitch motion isstrongly affected. At 0.2 orbit, the pitch angle attains a magnitude of —10° and is stillincreasing. Most of the flexible generalized coordinates, except those correspondingto deformations in the orbit normal direction, are also affected. In general, thermaldeformations of the PV arrays induce larger response. The stinger and the PV arrayradiators experience the highest increase in deflections.As the space station will be orbiting in a near circular orbit (6 = 0.02), Figure 4-16studies the effect of eccentricity. With the system initially in the design configuration,a comparison with Figure 4-11 shows that both pitch and roll responses are affected.As for the flexible degrees of freedom, a small eccentricity does not produce anynoticeable change in their response.Figures 4-15 and 4-16 illustrate that both thermal deformation and eccentricityaffect the pitch response. However, thermal deformations, even of a small magnitude,also influence response of the flexible members.4.3.3 Velocities and accelerationsOne of the stated missions of the space station is to provide microgravity environ-ment for the purpose of scientific research. As stated in the NASA report [106], theobjective of the station is to provide a 1 — 10 pg environment in some portion of theLaboratory Module. Furthermore, a drift rate, apart from the orbital rate (0), below0.005 °/s (0.872 x 10 -4 rad/s) is desired. Using the data obtained for Figures 4-11 to4-16, variations of angular velocities (co x , coy , coz ) and angular accelartions (ax , ay ,az ) about the X,,Y,,Z,-axes and power boom accelerations (ag, an at the systemc.m. are plotted. Note, Eq. (3.11), where the orbital rate is assumed to be zero, ispurposely used to calculate angular velocities isolated from any effect of the orbitalrate. The angular accelerations expressed as pg/m can be interpreted as the pg level1310.2sc 0.0 X 10 - 5 cm-0.20.2bcz 0.0- 0.20.2sy 0.0- 0.0-0.2x 10- ° cmX 10 -5 cmIVE = 0.02^1.0 ^0 0.0-1.0 ^0.02,4, 0W 0.00 ^-0.02 ^0.01-^A ° 0.00-0.01 ,^0.00, Orbit0.2E2€  0.0E4 0.0-0.5- -^0.0^0.02^0.00^0.020, Orbit^0, OrbitFigure 4-16^Response of the PMC in eccentric orbit (€ = 0.02). Initial distur-bance is due to deviation from the equilibrium orientation.132at 1 m from 0,, the origin of the X,,Y,,Z,-axes. For instance, a x = 1 ii,g/m impliesthat a point at 1 m along the IT, or Zcaxis experiences an acceleration of 1 fig due tothe angular acceleration about the X,-axis. Assuming the microgravity experimentworkspace to be about 10 m from 0,, a 1 — 10 pg/m environment would thereforerequire the angular acceleration about any axis to be less than 1 ttg/m. Finally, thepower boom accelerations, expressed in terms of microgravity, are evaluated at thecentre of the boom using the following expressions:4( 9) = 1'c1 ( 9 )0( 0 ) ;4(8) = C(19)0RO) ; (4.3)since only one mode is considered in the boom vibration to illustrate the methodol-ogy. Of course, the formulation and the program are quite general and can readilyaccommodate desired number of modes.The dynamic response of the PMC in the design configuration was shown inFigure 4-11. The corresponding velocity and acceleration variations are presented inFigure 4-17. Note, the only disturbance being that corresponding to a small deviationfrom the equilibrium configuration (i,b, = 0, = 0.002°, and A, = 0), the resultingresponse is extremely small. Obviously, a word of caution is in order in the analysisof such virtually insignificant numbers. Stability of the numerical procedure andtruncation error may lead to noise. What is important is to recognize that all theresponses, even in the presence of 'noise', are within the permissible limits. Thus,this can serve as a reference while assessing influence of the initial disturbances.133xio- 7 vs--4411100-11000,0000.1100.wZ Ctswv CXY0. 1ci)x 0.00.5a 0.0- 0.5-0.1 ,^0.5Y 0.0 jx10-11- 0.50.5x10 -4 °/s1Wz 0. 0-0.50.0 0.20, Orbit0.5Agim x3.0 -1 jiga 0 . 00.50. 1^0.00^0 020. 1a 0.0•- 0.10.2x10-1az 0.0-0.2 -0.0X10 -7 pgim0, Orbit^ 0, OrbitFigure 4-17 Variation of angular velocities as well as and linear and angular ac-celeration fields for the PMC in absence of active disturbances.134Figure 4-18 illustrates velocity and acceleration variations for the PMC subjectedto an initial power boom or stinger disturbance. Figure 4-18a corresponds to the caseshown in Figure 4-12a, where the power boom is initially given 1 cm tip deflection inthe local vertical direction. Even with such a small disturbance, the resulting angu-lar velocities and accelerations approach or exceed the specified limit. For instance,maximum cox and ax attain values of 0.004 °/s and 500 pg/m, respectively. Themicrogravity level near the power boom centre is over 1,000 pg. When the initialdisturbance is applied in the local horizontal direction, the magnitudes of both ve-locities and accelerations are about one order smaller (Figure 4-18b); however, theaccelerations still exceed the design limit. Once again, angular velocity and acceler-ation about the Xc-axis are largest with magnitudes of 2 x 10 -4 °/s and 10 µg/m,respectively. The boom acceleration results in a microgravity level exceeding 1,000 µgin the local horizontal direction.Figure 4-13 showed that the influence of the stinger disturbance on the PMCresponse is small; hence, it is reasonable to expect the corresponding angular velocitiesand accelerations to be small. The prediction turned out to be correct; however, theyare not negligible when the disturbance is applied in the local vertical direction. Underthis condition, Figure 4-18c illustrates that the maximum drift rate is 0.0025 °/s aboutthe Xc-axis but the maximum angular acceleration has an amplitude of 300 pgimabout Y0-axis. The microgravity level due to the boom accelerations reaches a valueof 6 ag in the local vertical direction. When the same stinger disturbance is appliedin the out-of-plane direction, both the velocities and accelerations stay within theallowable limit (Figure 4-18d). Now the maximum angular velocity (0.3 x 10 -6 °/s)and acceleration (0.01 p,g/m) are about Z 0-axis, while the boom vibration results in0.02 µg acceleration.135(a)^(Eno = 0.4348 x 10-40.1x10 -1. °/s0.00.10 .1x10 -8 °/sWY 0.0—0. 10, Orbitco a11 )WZ)0, Orbit 9, Orbit0.1az 0 .0-0. 10.1ay 0. 0- 0.1x10 — AgimFigure 4-18 The effect of power boom and stinger disturbances on the PMC ve-locities and accelerations: (a) initial power boom tip displacementof 1 cm in the local vertical direction.136(b) (Q 1 )° = 0.4348 x 10 -40.2Wx 0.0- 0.20.1Wy 0.0- 0.1 ^^0.1 ^x10 -5^Wz 0.0^-0.1 ^0.00 0.026, Orbit0.1ay 0. 0—0. 1 x10 - pg/m0.2az 0.0- 0.2x10 -1 Ag/m0.00^ 0.026, Orbit 6, OrbitFigure 4-18 The effect of power boom and stinger disturbance on the PMC ve-locities and accelerations: (b) initial power boom tip displacementof 1 cm in the local horizontal direction.137x104 ihg/mx10 -5 °0.026, Orbit(AIX, CIXCI Zco , ayY)0.2Wz 0. 00.20. 0—0 2—0.2^0.00(c) (ED. = 0.1872 x 10 -30.5x10 -2 °/sWz 0.0—0.50. 1Y 0.0 xio -8 °/s-o. i ^0.2 0.2ay 0.0 aY o o-—0.20.5az 0. 0—0 50.00 0.026, Orbit^ 0, OrbitFigure 4-18 The effect of power boom and stinger disturbance on the PIVIC ve-locities and accelerations: (c) initial stinger tip displacement of 1 cmin the local vertical direction.138ihg/mx10 - 0.0--0 50.5ay 0.0-0.5(d) (Q 1 )0 = 0.1872 x 10 -30, Orbit0.2a c 0.0-0.20, Orbit 0, OrbitFigure 4-18 The effect of power boom and stinger disturbance on the PMC ve-locities and accelerations: (d) initial stinger tip displacement of 1 cmin the orbit normal direction.139Effects of station radiator, PV array and radiator disturbances on the PMC dy-namics were presented earlier in Figure 4-14. The corresponding velocity and accel-eration variations are shown in Figure 4-19 . Since the station radiator disturbancehas virtually no effect on the PNIC response (Figure 4-14a), the resulting velocitiesand accelerations are indeed small as shown in Figure 4-19a. Here, the maximumangular velocity, which is about the Ze-axis, is only 3 x 10 -7 °/s. The maximumangular acceleration, which is about the X e-axis, is only 0.02 µg/m. Note also thebeat type variations of velocity and acceleration about the Ye-axis. The maximumacceleration at the power boom centre is about 1µg and acts in the local horizontaldirection. From Figure 4-14b, an initially deformed PV array radiator results in verysmall amplitudes of libration and vibration. Yet, the corresponding angular veloci-ties and accelerations arising from such a disturbance barely meet the desired limit(Figure 4-194 Although the magnitude of power boom acceleration is only 0.5 jig,the maximum angular velocity and acceleration, which are about Xe-axis, have mag-nitudes of 1 x 10 -4 °/s and 10 itg/m, respectively. For the disturbance arising fromthe PV array, Figure 4-19c shows results to be relatively favorable compared to thosein Figure 4-19b. Angular velocities and accelerations about the IT, and Ze-axes areabout an order of magnitude smaller and the microgravity level at the boom centreis only 1µg. On the other hand, ax is about an order of magnitude higher than thatin Figure 4-19b, and exceeds the specification.As shown in Figure 4-15, the PV arrays subjected to even a small amount of ther-mal deformation can have a significant impact on the PMC response. The influenceof thermal deformation on velocities and accelerations is shown in Figure 4-20a. Acomparison with Figure 4-17 shows that the thermally deformed PV arrays have asignificant influence on the angular velocities and accelerations about the X, and Ye-axes. Without the thermal deformation, the maximum cox and ax are 1 x 10 -8 0 /s140(a) (H12 '1^0.2^x10 -6 °/8^co, 0.0^--\At-0.20.1x10-13 0/8WY 0.0--0.1^0.5Wz 0.0= 0.2485 x 10'0.020, Orbit6, Orbitw ay) Y0.1n Y•-•'c^^lx10az 0.0 0.00Aig0.020.2ax 0.0-0.20.1ay 0.0-0.1x].0 - Atg/rn0, OrbitFigure 4-19 Variations of PMC velocities and accelerations showing the effect ofstation radiator, PV array and radiator disturbances: (a) stationradiator with a tip deflection of 0.5 crn.141Cdx2) aZ6, Orbitt4; aY3^Y(b)^= 0.2485 x 10 -30.1CA.,y 0.00.1n Y-c 0.0- 0.10.00 0.020.026, Orbit 6, OrbitFigure 4-19 Variations of PMC velocities and accelerations showing the effect ofstation radiator, PV array and radiator disturbances: (b) PV arrayradiator with a tip deflection of 0.5 cm.1420.5 -^x10 -1 ice m0, Orbitaz 0 .0--0.50.00 0.02^0.00 0.02(c) (H26 ' 1 ) 0 = 0.866 x 10 -40.2—0.2ac 0.0- 0 2- 0.5x10Wx , axWY, aY0, Orbit^ 0, OrbitFigure 4-19 Variations of PMC velocities and accelerations showing the effect ofstation radiator, PV array and radiator disturbances: (c) PV arrayradiator with a tip deflection of 0.5 cm.143x102 pg/m—.-------00. 4.1••••■■■•Nrdl.M.411111111WzC4.;WY(a)^L: = 0:01^(i = 6, .^, 9)0.50.0—0.5 -i0.5x10-1 °/s0.0—0.5x10_ 6 °/s0.20, Orbit0.2ax 0.0UlmW.;tidy) all0.51^x10-4 °/sWz 0.0 ^-0.50.00, Orbit 0, Orbit^—0.2^^0.5 x10-3 ;hem^0.0 ^0.5 ^aYFigure 4-20 Influence of the thermal deformations and orbital eccentricity on thePMC velocities and accelerations: (a) thermally deformed PV arrays.144and 3 x 10 -4 p,g respectively. With inclusion of the thermal deformation, thecorresponding maximum magnitudes are 0.02 °/s and 10 p,g/m, an increase of over 5orders. The wz and a z remain essentially unchanged while the microgravity level atthe boom centre increases slightly.The velocity and acceleration variations in an eccentric orbit (€ = 0.02) are shownin Figure 4-20b. Recall from Figure 4-16 that eccentricity effects are limited to pitchand roll responses only. Indeed, Figure 4-20b shows that the power boom accelerationremains unchanged with a maximum value of 0.03 pg. As for angular velocities andaccelerations, those about X, and Ye-axes show a large increase as compared to thosein Figure 4-17 while w z registers only a small increase. Comparing with the thermaldeformation case (Figure 4-20a), the eccentricity influence is about an order smaller;the maximum wx and ax in eccentric orbit are 0.002 °/s and 0.3 p,g/m, which arestill within the permissible limit.The results of Figures 4-17 to 4-20 provide valuable information pertaining tothe velocity and microgravity environment of the PMC. The results indicate that therequirements on velocity and acceleration are stringent. With the system initially inthe design configuration, the velocity and microgravity level stay within the designlimit. However, even with a small disturbance applied to any flexible member inthe local vertical or local horizontal direction, the system velocities and accelerationseasily reach or exceed the acceptable value. A little thermal deformation of thePV arrays gives rise to angular velocities and accelerations above the specifications.On the other hand, a small orbit eccentricity does not adversely affect the systemperformance.145Wx axwz, az, ay(b)^E = 0.02Vswz0.2 1710 -20.0-0.20.5X10-6 0 /8W Y 0.0-0.50.5x 1 0 -4 ° /swz 0 0-0.5 r0.00, Orbit0.2ax050.0-0.5Ag/m11111,01i4+111.1111110.2 -p 0.2x10 - Ag/may ac 0.00.0-0.21 -0.20.2 0.5x10 -1 pg/maz 0.0 al■ ac^0.0-0.50.0 0.1 0.00 0 020, Orbit 0, OrbitFigure 4-20^Influence of the thermal deformations and orbital eccentricity on thePMC velocities and accelerations: (b) orbital eccentricity (€ = 0.02).1464.4 Mobile Servicing SystemOn board the space shuttle is a remote manipulator system designed mainly forsatellite retrieval and release. A similar system, known as Mobile Servicing System(MSS), is planned for the Space Station. The MSS is essentially a two link ma-nipulator attached to a mobile base which traverses along the station power boom.The functions of the MSS would not be limited to satellite retrieval and release. Infact, it is expected to be the workhorse for the station's construction, maintenance,operation, and future evolution.The MSS, being flexible in the links as well as the joints, is an extremely compli-cated system to study. Chan [15] has investigated in details the dynamics and controlof the MSS; however, the study is limited to the inplane maneuver case. The objec-tive here is twofold: (i) to demonstrate versatility of the computer program throughsimulation of systems with slewing Bij bodies; (ii) to explore system dynamics dur-ing general inplane (IP) and out-of-plane (OP) maneuvers of the manipulator. Thefollowing assumptions are used in the present study to have some appreciation of thecomplex system dynamics:(i) joint flexibility is assumed to be negligible as compared to the link flexibility;(ii) since the formulation does not consider translational motion, the MSS base isassumed to be stationary; however, the MSS can still be placed at an arbitraryposition on the power boom;(iii) in order to isolate effects of slewing maneuvers on the librational dynamics,the station is assumed to be a rigid body and in the stable gravity gradientorientation.The numerical values used in the simulation correspond to those employed byChan [15] and are listed in Table 4-3 .147= 1,800 kg (1-xx)1 = (ixx)2 = 101 kg m2= 7.5 m (iyy ) 1 = (iyy) 2 = 33,750 kg m 2= 0.03 Hz (Izz)]. = (./zz)2 -.= 33,750 kg m 2m1^= mi.o.= /Lifl^= f1-4Table 4-3^Data of the space station based MSS used in the simulationSpace Station (Be ) me = 240,120 kg^ = 8 x 10 5 kg m2/,^= 115.35 m (40^2.67 x 10 8 kg m2(/„),^= 2.67 x 108 kg m2Upper and Lower Links (B 1 and BThe simulation is based on a 100 minute orbit using the coordinate systemsshown in Figure 4-21 . Considering only the first mode of vibration, the generalizedcoordinates are:q=0,0, A, Pi, Q 1. , /3 1 1 , and Q 1 hence, Ng = 7 .It is desirable to undertake a slewing maneuver with zero velocity and accelerationat the beginning and the end of the maneuver. The present simulation uses a sine-ramp function which has the required characteristic:{am,0 I Tsi — (am /27) sin(2701T,i) for 9 < Tsi ;asi =am^for > Tsi .(4.4)where a s / and am are the slew angle and its maximum value, respectively; T s/ isthe slewing period; and 0 is the true anomaly. Effect of the slewing period on thesystem response is shown in Figures 4-22 to 4-24. Here, the lower link is assumed toundergo a 180° maneuver in 5, 7.5, and 10 minutes. For each of the slewing periods,performance of the system is compared when subjected to two distinct maneuvers:the OP (out-of-plane) and IP (inplane) maneuvers. In the former case, the maneuver14811/741Z1, V x1,1/ThLocalVerticalLocalHorizontalOrbitNormalFigure 4-21^Coordinate systems and design configuration used in the simulationsof MSS slewing maneuvers.149is about the Y1 , 1 -axis resulting in the link traversing in a plane normal to the orbitalplane. In contrast, the link rotates about the Z 1 , 1 -axis and travels in the orbital planein the latter case.The system response when subjected to a 5-minute maneuver of the lower link isshown in Figure 4-22. For the OP maneuver, all the libration degrees are excited withthe yaw amplitude increasing most rapidly. With such a fast maneuver, the flexiblelinks experience large deflections during the maneuver. The maximum deflectionsfor the upper and lower links are 33.3 cm and 9.5 cm, respectively. With the IPmaneuver, Figure 4-22b shows that only the pitch librational motion is excited witha slightly higher amplitude than that of the OP case. In addition, the pitch responseshows high frequency modulations corresponding to the flexible member's frequency.The flexible members experience unacceptably high tip deflections of 62.4 cm and17.0 cm for the upper and lower links, respectively.As the maneuver time increases from 5 to 7.5 minutes, Figure 4-23 shows im-provements in the system response. With the OP maneuver (Figure 4-23a), the yawmotion is still significant; however, maximum tip deflections of the upper and lowerlinks have decreased to 11.4 cm and 3.4 cm, respectively. Similarly, the IP maneuvercase shows reductions in tip deflections to 19.2 cm and 4.6 cm, for the upper and lowerlinks, respectively. Notice also the disappearance of the high frequency modulationsin the pitch motion (Figure 4-23b).As the maneuver period is increased to 10 minutes (Figure 4-24), further reduc-tions in tip deflections is apparent. After 0.25 orbit, Figure 4-24a shows that the yawamplitude reaches 6.6°. This is lower than that observed before (8.2° for 5-minutemaneuver and 7.2° for 7.5-minute maneuver); however, it is still significantly higherthan the pitch or roll motion. The maximum tip deflections for the upper link is150AO 00, Orbit 6, Orbit70Si^0-70(b) Inplane maneuver40(a) Out—of—plane maneuver0.05 ^-0.050.05sp 0.00-40-0.05-10 -^ 201^-200.25 0.00-10-^0.00 0.250.1 cmal 0 . 06f.,1 01,vi 0.00-0.05^0.01 ^0° 0.00^-0.01^0.01^A ° 0.00^-0.01^0.00- 0.120r0- 20^0.1CM-0.10.25^0.000, Orbit 0, OrbitFigure 4-22 System response with the lower link of the MSS undergoing a 180°maneuver in 5 minutes: (a) out-of-plane maneuver; (b) inplane ma-neuver.15120- 0.05 -0.05-0.00 —Out-of-plane maneuvero ooI0-•^0-107 ^ 50.00^ 0 25^0.0O0, Orbit 0, Orbit0 25bi^0- 20- 0. 1106Y, 1^0- 100.05,oa,y 0.00- 0.050.010° 0.00-0.010.01A ° 0.00(b) Inplane maneuverCM0.161 , 1 0.0- 0 . 0 10.00^ 0 25Figure 4-230, Orbit^ 0, OrbitForced oscillations of the space station showing the effect of a 180°maneuver in 7.5 minutes of the lower link of the MSS: (a) out-of-planemaneuver; (b) inplane maneuver.- 0. 10.00 0.251526fgi 00.02,04,vi 0. 00-0.02Out—of—plane maneuverby(b) Inplane maneuver0.05y 0.00-0.050.0195 ° 0.00- 0 010. 1bi o o- 0. 1CM2CM1 0A°1 00 00- 2^0.25^0.006, Orbit 6, Orbit0 95CM0.01A ° 0.00- 0 010. 00^0.258, Orbit0. 115 T. ,1 0. 0- 0 .10.00 0 258, OrbitFigure 4-24 Response of the space station with the MSS subjected to a 180°maneuver of the lower link in 10 minutes: (a) out-of-plane maneuver;(b) inpla.ne maneuver.1536.3 cm and only 1.8 cm for the lower link. For the IP maneuver, there is no signif-icant reduction in the pitch motion (Figure 4-24b). In contrast, the maximum linkdeflections reduce to 7.9 cm (upper link) and 1.9 cm (lower link).Simulation results in Figures 4-22 to 4-24 indicate that the OP maneuver resultsin relatively smaller tip deflections of the links. However, it excites a large amplitudeyaw motion. Since the primary function of a manipulator is to position a payloadat a desired location, it would be useful to investigate the resulting pointing errorsdue to the maneuver. A suitable reference for measuring the error would be theorbital frame, X,,Y,,,Zs . With this as reference, errors due to flexible deflections aswell as rigid body libration must be accounted for. The 180° maneuver as before isconsidered. With respect to X,,Y,,Z, axes, the tip of the manipulator ideally (i.e.,in absence of librational and flexibility errors) travels from a point with coordinates(0; —7.5 m, 7.5 m) to the destination with coordinates (0, 7.5 m, 7.5 ni). Let thecoordinates of the manipulator tip be (s s , sy sz ) and define the errors in distance tobe Sx , Sy , and Sz in Xs , Y,, and Z, directions, respectively, then= Ss ;Sy = 7.5 — sy ;Sz = 7.5 — Sz ;and^St„ = .\/51^51^.^ (4.5)Using the results in Figures 4-22 to 4-24, Figure 4-25 shows the variation oferrors with maneuver time and path. The results of Figure 4-25a corresponds to themaneuver of Figure 4-22. Notice that Sx , Sy , and Sz are, in general, oscillatory due tovibratory character of the link response. For the rigid body motion, the mean valuesof Sx , Sy , and Sz are not necessarily constant. For the OP maneuver, effect of the154100Out—of—plane maneuver Inplane maneuverSm—1050Sy—5080—800, OrbitFigure 4-25^Effect of the maneuver time on the pointing error with the lower linkslewing through 180°: (a) 5-minute maneuver.155large yaw motion can be seen in the rapidly increasing amplitude of Sx . At the end of0.25 orbit, Sx has a mean value of over 1 m with an oscillatory contribution of 14.5 cmpeak-to-peak (p-p) superposed on it. Due to small amplitudes of pitch and yaw, Syand Sz variations remain fairly constant with p-p amplitudes of 16.4 cm and 25.6 cm,respectively. For the IP maneuver, since the yaw motion is not excited, Sx remainszero. However, the large link deflections result in prohibitively large magnitudes ofSy and Sz with p-p amplitudes of 82.3 cm and 132.6 cm, respectively. Although theOP maneuver results in smaller errors in the Y, and Z s directions, it leads to a largeerror in the Xs direction. At the end of 0.25 orbit, the manipulator is 103.4 cm off thetarget and the error is still increasing. On the other hand, the IP maneuver resultsin a mean error of 40.9 cm.As the maneuver time increases to 7.5 minutes (Figure 4-25b), there is a significantimprovement in Sy and Sz for both the maneuvers. The p-p amplitudes of Sy andSz have reduced to 1.3 cm and 2 cm, respectively whereas the corresponding valuesare 16 cm and 25.8 cm for the IP maneuver. However, the yaw motion for the OPmaneuver remains large resulting in a large error in the orbit normal direction; hence,Stot attains a magnitude of 94.4 cm after 0.25 orbit. In the IP maneuver case, themean value of the total error has dropped from 40.9 cm to 14.7 cm. As the maneuvertime further increases to 10 minutes (Figure 4-25c), the IP maneuver error showsfurther improvement. The difference in the p-p amplitudes of Sy and Sz between thetwo maneuvers is now relatively small. For instance the p-p amplitude of Sy is 0.5 cmfor the OP maneuver and 3.3 cm for the IP slew case. The major difference betweenthe two cases is the large amplitude of Sx resulting from the yaw motion in the OPmaneuver. Consequently, the mean pointing error in the IP case is only 9.5 cm at theend of 0.25 orbit compared to 86 cm for the OP slew.156cmcmOut—of—plane maneuver   Inplane maneuver0.15^ 0.250, OrbitFigure 4-25^Effect of the maneuver time on the pointing error with the lower linkslewing through 180°: (b) 7.5-minute maneuver,100sz— 101008tot— 10157Out—of—plane maneuver   Inplane maneuver100 ^CMSxsz—20100—^ cmstot—10-0.15Figure 4-250, OrbitEffect of the maneuver time on the pointing error with the lower linkslewing through 180°: (c) 10-minute maneuver.0.25158The manipulator is now moved 50 m in the X, direction from the central body'sc.m. The effect of this offset on the system performance is shown in Figures 4-26and 4-27 . Using a 10-minute maneuver, Figure 4-26 shows that the OP case resultsin increase in pitch and roll amplitudes; whereas the yaw response remains the sameas before. The maximum tip deflections also increase for both the links; from 6.3 cmto 10.4 cm for the upper link and from 1.8 cm to 3.1 cm for the lower link. Theseobservations remain valid, in general, for the IP maneuver; the pitch motion increasesin amplitude and so does the flexible links.Due to the increased librational motion, the pointing accuracy deteriorates forboth the IP and OP maneuver cases (Figure 4-27). For the OP maneuver, since theyaw motion is unaffected by the manipulator offset, the pointing error in X3 directionremains the same as before. With the increase in the pitch motion, the variation of Syand Sz is similar to Sx , i.e. the absolute mean values of Sy and Sz increase with time.Also, the increase in the link deflections results in larger p-p amplitudes of Sy and Sz .In the present case, they amount to 5 cm and 5.5 cm, respectively. Similar trends ofSy and Sz variations can be observed for the IP maneuver. It is interesting to notethat the p-p amplitudes for Sy and Sz , 3 cm and 2.7 cm, respectively, are smallerthan those for the OP maneuver. However, this is not enough to offset the effectof rigid body motion. At the end of 0.25 orbit, both maneuvers have poor pointingaccuracy; the IP maneuver is 95.9 cm off the target whereas the OP maneuver hasan error of 126.7 cm.Effect of increased link stiffness on the system performance is studied in Figures 4-28 and 4-29 . The links are assumed to have the stiffness doubled and the lower linkis undergoing a 180° maneuver in 5 minutes. With this fast maneuver, a comparisonbetween Figures 4-22 and 4-28 shows that there is no apparent improvement in the159Sy6f(a) Out-of-plane maneuveripo-1 ^0 11 ,^0.1-0. 0—0.15CM1,1^0 -5 10—A 0—10 ^0.00 0.258, Orbit^0, Orbit(b) Inplane maneuver1 1 0.16fcm0. 0—1 ■111•0, — 0.15-0.01 cm0° 0.00 6Y1 ,1^0—0.01 —50.01 0.1 cmA° 0.00z1 , 1 0.0—0.01 —0.10.00 0.25 0.00 0.25Figure 4-269, Orbit^ 0, OrbitDynamical response of the space station with the MSS located 50 mfrom the central body c.m.: (a) out-of-plane maneuver; (b) inpianemaneuver.160100—1 0Sz—80120CmStot80^0.15 0.25cm^ Out-of-plane maneuver   Inplane maneuver0, OrbitFigure 4-27 Pointing error of the MSS, located 50 m from the central body c.m.,with the lower link undergoing a 180° maneuver.161410X0 0-106, Orbit0, Orbit(a)0.05-le 0.00 -- 0.05 ^0.05 -0° 0.00- 0.057Out-of-plane maneuverOf.bi t i^0-4^0 25^0.00 0.250.00- 0.057^^0.01 ^dsyv o 0.00 ^- 0.01(b) Inplane maneuver0.050.00-0.10.01A° o oo0.25- 0 . 0 10.0046Y1,1^040.1 cm^0.0^-0 . 1^0.00 0.250, Orbit^ 6, OrbitFigure 4-28^Effect of the increased MSS stiffness on the response of the spacestation with the lower link undergoing a 180 ° maneuver in 5 minutes:(a) out-of-plane maneuver; (b) inplane maneuver.162librational response. The main advantage is the significant reduction in amplitudeof the flexible link vibrations. For the OP maneuver, the maximum deflection of theupper link is only 5.9 cm whereas it was 33.3 cm before. Even for the IP maneuver,the original deflection was 62.4 cm for the upper link but, with the stiffness increased,it is now reduced to only 8 cm. Thus the benefit of the increased stiffness is largerfor the IP maneuver. This point is further illustrated in the pointing accuracy plot(Figure 4-29). The errors in the local vertical and local horizontal directions havereduced significantly for both the maneuvers. Originally, for the OP case, the p-pamplitudes for Sy and Sz were 16.4 cm and 25.6 cm, respectively. With the stiffnessincreased, both these values have dropped to 0.5 cm. Similarly, there is a sharpreduction in the p-p amplitude of Sy and Sz for the IP maneuver. For instance, Szwas 132.6 cm before and is only 4.7 cm with the stiffness of the link doubled. Sincethe librational response is hardly changed, the OP maneuver still suffers from thelarge error in the orbit normal direction. At the end of 0.25 orbit, the OP maneuverhas a total error of 101.4 cm. The IP maneuver has now a reduced error of 9.5 cm,which was obtained before only with a 10-minute slew.A summary of the maximum tip deflections and pointing error at the end of 0.25orbit is presented in Table 4-4 .4.5 Space Flyer UnitThe Space Flyer Unit (SFU) is an unmanned, reusable and free-flying platformfor multipurpose use. The SFU is developed by a consortium of Japanese governmentagencies including the Institute of Space and Astronautical Science, the NationalSpace Development Agency, and the Ministry of International Trade and Industry[107]. The unit is scheduled to be launched in early 1993. The SFU consists of anoctagonal shaped central body which includes eight modules of scientific experiments.163CmOut—of—plane maneuver   Inplane maneuver100Sz—100Sy0sz—20100—^CmStot—10-1^^0.15 0.25Figure 4-290, OrbitPointing error of the MSS, of increased stiffness, with the lower linkundergoing a 180° maneuver.164Table 4-4^System performance vs. type of the maneuver and its periodManeuverPeriodManeuverTypeMaximumUpper LinkDeflections (cm)Lower LinkPointing Errorat 0.25 Orbit (cm)*5 minutes Out-of-plane 33.3 9.5 103.4In-plane 62.4 17.0 40.97.5 minutes Out-of-plane 11.4 3.4 94.4In-plane 19.2 4.6 14.710 minutes Out-of-plane 6.3 1.8 86.0In-plane 7.9 1.9 9.510 minutes** Out-of-plane 10.4 3.1 126.7In-plane 11.8 3.1 95.95 minutes*** Out-of-plane 5.9 1.8 101.4In-plane 8.0 1.9 9.5* Mean pointing error for the inplane case** Manipulator located at 50 m from Oc*** Stiffness of the links doubledTwo solar array pedals (SAPs), each 9.7 m x 2.4 m, are deployed at either end ofthe central body. The SAPs, besides generating power, are used for the High VoltageSolar Array (HVSA) experiment. The objectives of the experiment are to determine:(1) dynamical characteristics of the unit during deployment and retrieval of theflexible SAPs;(ii) the upper limit of the voltage generated which would be free from surfacebreakdown, power drain through space plasma, and enhancement of the aero-dynamic drag.Here, the present formulation is used to simulate the dynamics of the SFU duringdeployment and retrieval of the pedals. To this end, modifications to the presentformulation are necessary to account for (i) time varying component mass and stiffnessand (ii) rate of change of mode shape.165Recall that from the Lagrangian formulation, the governing equations of motionare given byd ( OT^UT ou r,dO^qr' = uin absence of any generalized forces. The terms, OT/aq and OU/Oq, remain thesame as before except that the mass and stiffness are functions of deployment andretrieval strategies. The term dIdO(OTIOV) can be written into the first and secondorder components, 01(d/c/9(OT/Oql) and 02(dIdO(OT 10V), as given by Eq. (3.10).To account for the rate of change of mass (m'), the first order component is nowrewritten as:n ( d OT) n ( d °Lys )^d OCDT rI s(.7) + Hays]—1 ydo00 — —1 c10 Oq' ) + dO Oq' l 'Y(9COT r dins^0+ 1 ( dHsys )i+ ^ [^Co- +1, ,s0i( dw )NI da^Y^dO^de Li+ 01 ( dCoT )0I-Isys + T d OHsys +^( OT)^(4.6)dO ) Oq'^dO Oq'^m^q'' )Note that, except for the last term, 0 1 is essentially the same as before except thatthe mass is a function of time. Similarly 02 term is given by Eq. (3.10b) with massas a variable.In the assumed mode method, flexural displacement (6) and velocity (S') are givenby6 =^q,(0)0,(C);r=1nr= E q7,(0)or«)r=1where qr and Or are the generalized coordinate and mode shape, respectively. Withthe inclusion of deployment/retrieval , Ibrahim [14] has shown that the velocity of166flexural displacement is modified to include the variation of mode shape, i.e.Thr8/^E( q 17.(0)0,(C) + q,(0)0 17.(()) ;^ (4.7)r=1where 4(0 is the rate of change of mode shape and is given by0,-(C) = 00,(C) 0(Incorporation of Eqs. (4.6) and (4.7) into the present formulation makes it pos-sible to study deployment/retrieval effects on the dynamics of the SFU. The unit isidealized as a rigid central body (B e ) with two deployable cantilevered plates (B 1 andB2 ) attached. Assuming a 90-minute orbit, the simulation is based on the data givenin Talbe 4-5 and using the coordinate system as shown in Figure 4-30. Consideringonly the first mode of the SAP vibrations, five generalized coordinates are requiredto describe the system dynamics:,^, A ,^ ,1The same data-set is used in the retrieval study except that the two columns of dataare interchanged. For instance, the length of the SAP before and after retrieval wouldbe 9.7 m and 3 m, respectively. Ideally, simulations should be based on SAP lengthfrom 0 m to 9.7 m. However, this would make the governing equations of motionextremely stiff initially and hence would require a lot of computational effort. It is,therefore, assumed that the pedal remains rigid until 3 m of its deployment. Evenwith this simplification, the natural frequency of the array would reach a high of1.93 Hz.The planned deployment/retrieval strategy of the SAPs can be separated intothree stages as summarized below:167x,LocalHorizontalYcLocalHorizontalBody B1Orientation I2^OrbitNormalABody ByOrientation IIOrbitNormal .4^\LocalVerticalLocalHorizontalOrbit^tNormal^ LocalVerticalOrientation IOrbitNormalOrientation II Figure 4-30 Coordinate systems and the two possible design configurations forthe SET central body and arrays.168Table 4-5^Data of the SFU used in the simulationMain Body (Body Bc )Before Deployment^After Deployment/,^=^4.9^ 4.9mc^3,964.8 3,886^kg(-rxx)c 7,839.2 7,524 kg-m2(-Tyy)c^ 4,410.0^4,410.0^kg-m2(1-zz)c 6,079.2 5,764 kg-m2Solar Array Pedal (Body Bi , i = 1, 2)Before Deployment After Deployment/i = 3 9.7 mmi = 17.6 57 kgfi = 1.934 0.185 Hz(rxx)i = 20 20 kg-m2(40i 68.7 2,321 kg-m2(izz)i = 69 2,331 kg-m2(1) The SAPs are deployed in 15 minutes with the longitudinal axis parallel tothe local vertical, i.e., one SAP points toward the earth while the other awayfrom the earth.(ii) The SFU then undergoes a 90° roll so that the longitudinal axes of the SAPsare aligned with the orbit normal.(iii) At the end of the mission, the SAPs are retrieved with the SFU in the sameorientation as in (ii). The retrieval time is also 15 minutes.Contingency plans are also drafted in the event that either SAP fails to deployor retrieve.The parametric analysis presented here aims at studying the dynamical response169of the SFU during the deployment/retrieval of the SAPs. Since it is assumed that thepedal length varies from 3 to 9.7 m instead of 0 to 9.7 m, the deployment/retrievaltime is adjusted to 10 minutes. The deployment velocity is taken to be constant.Figure 4-31 shows the system response during the deployment of the SAP. Withthe SAPs deployed in the nominal orientation, the SFU response is shown in Figure 4-31a. Since the SAPs are deployed in the inplane direction, only pitch motion isexcited. Although the SFU remains stable, it undergoes a large amplitude pitchmotion reaching a minimum of —63°. As the SAP becomes more flexible duringdeployment, the tip deflection increases too; however, the SAPs hardly oscillate. Assoon as the deployment terminates, the pedals start to vibrate with a peak-to-peakamplitude of about 0.0025 cm. Figure 4-31b studies the feasibility of an alternativedeployment strategy. Here, the SFU undergoes a 90° roll before the deployment, i.e.,the SAPs are deployed in the out-of-plane direction. Advantages of this strategy areobvious. The pitch libration has a considerably smaller amplitude with a minimumof —5.8°. At the end of the deployment, the SAPs vibrate at amplitudes about twoorders smaller than the corresponding ones in Figure 4-31a. The only disadvantagewith this strategy is that both roll and yaw are also excited; however, their amplitudesare of the order 10 -5 even after 0.5 orbit.Even if one SAP fails to deploy, the SFU remains stable under asymmetric deploy-ment (Figure 4-32). Here, only B2 is assumed to deploy successfully. In Figure 4-32a,B2 is deployed in the direction towards the earth. The figure shows that both thelibrational and vibrational responses remain essentially the same as if both pedalswere deployed. In contrast, when only B2 is deployed in the out-of-plane direction,the SFU response is different from that in Figure 4-31b. Although the pitch motionhas reduced its minimum amplitude to —2.6° from —5.8°, the roll and yaw ampli-17010-900.5-0.1 ^0.00.1 -^X00 Orbit0.5Ex1-0.50.5End of Deploymentx10 -2 cm ^I, .11^14^thi^IA /1'10111111j^11^di Ili^11141' 1 ,14/II^11.1x10 -2 cm,1 1,11 14;11 0111i "i ll^II II "11 ,11 141 II I ill^I g 1 1 , d ill-0.5 7^0.00, OrbitFigure 4-31 Dynamical response of the SFU during deployment of the solar arraypedals: (a) inplane deployment.0.2171A0. 10. 1- 0. 1950X 0- 0.10.50 .0-100, Orbit0.5End of Deploymentx10 -4 cmC lz0. 00.5 -^C220. 00 .0 0.20, OrbitFigure 4-31 Dynamical response of the SFU during deployment of the solar arraypedals: (b) out-of-plane deployment1721 0\o)-90 -0. 1100-0.10.100. 1 ^0.0 0.50.5Ex2-0.50, OrbitEnd of Deployment0, OrbitFigure 4-32^Response characteristics of the SET during deployment of the solararray pedal B2 : (a .) inplane deployment.173tudes increase significantly (Figure 4-32b). With both the pedals deployed, the orderof magnitude of roll and yaw is about 10 -5 at the end of 0.5 orbit. With only B2deployed, roll and yaw attain values of —12.9° and —2.7°, respectively, over the sameperiod. The response trend of the SAP vibration also changes. At the end of thedeployment, the SAP oscillates with a peak-to-peak amplitude of 0.0002 cm. Thisvalue, though small, is about one order higher than that in the case with both theSAPs deployed.Figure 4-33 shows the system response during the SAPs retrieval. The effect ofretrieval time is studied here. In Figure 4-33a, a 5-minute retrieval period is assumed.Even with this fast retrieval, the SFU remains stable with a maximum pitch angle of6°, which is about the same in magnitude as the deployment case. The roll and yawangles remain small but are considerably larger than those during the deployment. Atthe end of 0.5 orbit, their amplitudes are —0.0013° and —0.0005°, respectively. As theSAPs are becoming more and more rigid with the progress of retrieval, it is reasonablethat they attain lower deflection (-0.5 x 10 -5 cm) during retrieval and oscillate witha peak-to-peak amplitude of less than 10 -7 cm afterwards. With the retrieval timeincreased to 10 minutes, Figure 4-33b shows that the libration response remainsunchanged. Furthermore, the SAP vibration retains the same response trend andhas about the same minimum deflection (-0.64 x 10 -5 cm). Even with the retrievaltime further increased to 20 minutes, the pitch and SAP responses remain the same;however, the roll and yaw degrees of freedom are no longer excited (Figure 4-33c).As seen before, the SFU remained stable for inplane or out-of-plane deployment ofthe SAPs. This is no longer true for the SAP retrieval. Although out-of-plane retrievalof the SAPs has no adverse effect on the SFU dynamics, the inplane retrieval tendsto destabilize the system. Figure 4-34a shows that with a 5-minute inplane retrieval,1740-200.50 .0510xo-10A0, Orbit0.2x10 - cm-0.2 ^0.0.1, End of Deployment0, Orbit0.2Figure 4-32^Response characteristics of the SFU during deployment of the solararray pedal B2: (b) out-of-plane deployment.17500.21 0i^x10-O—0.2 I^^0.1 ^xi0 -2xo—0.1 ^0.0 0.50, Orbit. End of Retrievalx1 0- 4 cmx10 -4 cmEx20. 1El- 0.1 7^0.0 0. 10, OrbitFigure 4-33^Librational and vibrational responses of the SFU during the out-of-plane retrieval maneuver of the solar array pedals: (a) 5-minuteretrieval period.17610c50— 0.2^0.1 ^x10 -2xo— 0.1 ^0.0 0.50, Orbit, End of Retrieval0.1Elx10 -4 cm—0 . 1 -1^0.1X 10 4 cmEx2—0 .1 ^0.0 0.20, OrbitFigure 4-33 Librational and vibrational responses of the SFU during the out-of-plane retrieval maneuver of the solar array pedals: (b) 10-minuteretrieval period.1771 0 0^^0.2xio -200—0.20.11^xi0 -2xo—0.1^0.00, Orbit0.5End of Retrieval0.1x10' cmE l—0.1 ^0.1x10' cmEx— 0.1 I0.0^ 0.30, OrbitFigure 4-33 Librational and vibrational responses of the SFU during the out-of-plane retrieval maneuver of the solar array pedals: (c) 20-minuteretrieval period.17850E 21290]^vio0 ^0.1^c5° ^—0. 10.1]^X 0 ^—0. 1 ^0. 0 5.06, Orbit *to 3CM—50 r^cm0e l0.0^ 5.06, Orbit^1. 0- 3Figure 4-34 System response of the SFU shoving unstable motion induced bythe inplane retrieval of the solar array pedals: (a) 5-minute retrievalperiod.179the SFU starts to tumble almost immediately (after 0.0004 orbit). Over the sameperiod, the tip deflections of the SAPs are over 50 cm. As the retrieval time increasesto 10 minutes, Figure 4-34b illustrates that the system remains unstable although theonset of tumbling motion has been delayed to 0.008 orbit. Recall from Figure 4-31athat in the inplane deployment, both SAPs vibrate with the same magnitude but inopposite directions. Now both the SAPs deflect with different amplitudes. B 1 , whichpoints away from the earth, has a tip deflection of over 50 cm at the end of 0.01 orbit.B2 , which points towards the earth, deflects only 10 cm at 0.01 orbit. As can be seenfrom Figure 4-33, the appendages softens during retrieval. It is apparent that duringthe inplane retrieval case, the degree of SAP softening is higher and different for thetwo SAPs due to difference in the gravitational torque. This softening effect coupledwith higher pitch rate result in large deflections of the SAPs; however, further studiesare needed to confirm this observation. As the retrieval time is further increased to20 minutes, the system remains stable until 0.13 orbit (Figure 4-34c). Once again,B1 experiences a considerably larger deflection than B2. At the end of 0.02 orbit, thetip deflections for B1 and B2 are 40 cm and 6 cm, respectively.When only one of the SAPs is deployed, its retrieval is found to be a difficulttask. Whether the maneuver is performed out-of-plane or inplane, the system remainsunstable for either B1 or B2 retrieval. A typical response plot is shown in Figure 4-35.Here, B2 is retrieved in the out-of-plane direction. Figure 4-35a corresponds to a 5-minute retrieval which clearly indicates that the system experiences large amplitudesof pitch, roll, and yaw motions. In less than 0.02 orbit, the SFU becomes unstableand starts tumbling. Note also the large deflection of B2. As the retrieval time isincreased to 10 minutes, the SFU manages to maintain its stability until about 0.03orbit (Figure 4-35b). The duration of stable motion further increases to 0.05 orbitwhen the retrieval time is increased to 20 minutes (Figure 4-35c).18090 0-i0.1 1.-00^- 0.1 ^^0.1^X00.1^0.00r0.010, OrbitE lC2cm—50500.00 0 01Figure 4-340, OrbitSystem response of the SFU showing unstable motion induced by theinplane retrieval of the solar array pedals: (b) 10-minute retrievalperiod.181900—0.10.000, Orbit0.020.1X 00Ex2cm—50500 -1^^0.00 0 020, OrbitFigure 4-34 System response of the SFU showing unstable motion induced by theinplane retrieval of the solar array pedals: (c) 20-minute retrievalperiod.18290vo —9090—90t90xo—90 -0.000, Orbit0.0240—^Ex—40—)^0.000, OrbitFigure 4-35^Out-of-plane retrieval of one of the solar array pedals (B 2 ) showinginstability of the system: (a) 5-minute retrieval period.0 021830.040 .0090ripo—9090cbo—9090X°—900, Orbit40cmEx2—400.00 0 040, OrbitFigure 4-35^Out-of-plane retrieval of one of the solar array pedals (B 2 ) showinginstability of the system: (b) 10-minute retrieval period.18490190cbo—9090X°— 90 ^0.000, Orbitvfo— 900.0740—400.00€210.070, OrbitFigure 4-35^Out-of-plane retrieval of one of the solar array pedals (B 2 ) showinginstability of the system: (c) 20-minute retrieval period.185A summary of the SFU stability subjected to various deployment/retrieval con-dition is presented in Table 4-6 . The results suggest that inplane or out-of-planeretrieval of the single SAP and inplane retrieval of both the SAPs require some con-trol effort. Without that, it is apparent from Figures 4-34 and 4-35 that the retrievalcan be successfully performed only if it is carried out at a very slow rate.Table 4-6^Summary of SFU stability with respect to deployment/retrieval pe-riod and orientationDEPLOYMENTDeploymentDirection*Array(s)DeployedDeploymentPeriod (min.)StabilityIP B1 & B2 1 0 StableOOP B1 & B2 10 StableIP B1 or B2 1 0 StableOOP B1 or B2 10 StableRETRIEVALRetrievalDirection*Array(s)RetrievedRetrievalPeriod (min.)StabilityIP B1 & B2 5 UnstableOOP B1 & B2 5 StableIP B1 & B2 10 UnstableOOP B1 & B2 10 StableIP B1 & B2 20 UnstableOOP B1 & B2 20 StableIP B1 or B2 5 UnstableOOP B1 or B2 5 UnstableIP B1 or B2 10 UnstableOOP B1 or B2 10 UnstableIP B1 or B2 20 Unstable00P B1 or B2 20 Unstable* OOP — out-of-plane deployment/retrievalIP — inplane deployment/retrieval1864.6 Closing CommentsSince there are numerous combinations of system parameters and initial condi-tions of interest, the dynamical study presented in this chapter is in no way complete.The objective here is to demonstrate the versatility of the general formulation throughthe parametric analysis of four different spacecraft models. They represent a widevariety of situations: (i) arbitrary orientations in circular or elliptic orbit; (ii) en-tirely flexible structure; (iii) spacecraft with a large number of interconnected bodies;(iv) thermally deformed members; (v) slewing members; (vi) deployable/retrievableelements; and their combinations. Although the present formulation has been modi-fied to simulate the SFU deployment/retrieval only, its implementation for a generalcase is rather straightforward.1875. NONLINEAR CONTROL5.1 Preliminary RemarksVersatility of the general formulation in studying dynamics of a variety of space-craft configurations was demonstrated in Chapter 4. The next logical step is toimplement a control algorithm suitable for the general equations of motion which arenonlinear, coupled and nonautonomous.Nonlinear control has received considerable attention in the robotics research,particularly during the past decade. Control strategies based on linearized systemmodels have been found to be inadequate. The working conditions of robot armsoften deviated from those predicted by linearized approaches. One possible solutionwas put forward by Freund [107]. The idea is to use the state feedback to decouple thenonlinear system in such a way that an arbitrary placement of poles is possible. Thetechnique, however, was found to be difficult to apply to systems with more than threedegrees of freedom. Freund [108] subsequently showed that by careful partitioning ofthe equations of motion, the procedure can be extended to systems with more thanthree degrees of freedom. However, the approach did involve simplification of theequations of motion.Slotine and Sastry [109] applied the sliding mode theory to the control of robotmanipulators. Consider a differential equation with the right-hand side discontinuousaround a hypersurface. If the trajectory of the solution points toward the discon-tinuity, it is plausible that the trajectory eventually slides along the hypersurface.By a suitable choice of sliding surfaces, control laws can be formulated to force themanipulator to travel along a specified trajectory defined by the surfaces. However,unmodelled dynamics usually results in high frequency oscillations of the manipulator188as it slides along the surface. Slotine [110,111] improved the performance by using afiltering process with a high bandwidth for the sliding variable. Slotine and Li [112]also incorporated the sliding mode control in an adaptive PD feedback controller. Theidea is to utilize the PD controller to give zero velocity error. The nonzero positionerrors are then eliminated through the sliding mode controller.Inverse control, based on the Feedback Linearization Technique (FLT), was firstinvestigated by Beijczy [113] and used by Singh and Schy [114] for rigid arm control.Spong and Vidyasagar [115,116] also used the FLT to formulate a robust controlprocedure for rigid manipulators. Using the FLT and given the dynamics modelof the manipulator, the controller first utilizes the feedback to linearize the systemfollowed by a linear compensator to achieve the desired system output. At times,the method is also referred to as the Computed Torque Technique which is, to beprecise, is a particular case of the FLT. Spong [117] later extended the method to thecontrol of robots with elastic joints. Advantages of this approach are twofold: (i) thecontrol algorithm based on the FLT is simple; and (ii) the compensator design, basedon a feedback linearized model, is straightforward. Recently, Modi et al. [118,119]extended the technique to include structural flexibility for a model of an orbitingmanipulator system studied by Chan [15]. The technique is found to provide adequatecontrol for both rigid as well as flexible manipulator.The study in this chapter is based on the FLT as applied to the INdian SATelliteII (INSAT II) and the MSS. The chapter begins with an introduction to the FLT.To begin with, the method is utilized to control the rigid INSAT II. Next, the morerealistic situation of flexibility is tackled. Both, the quasi-open loop and quasi-closedloop control strategies are discussed. This is followed by the application of the quasi-closed loop control to the flexible INSAT II subjected to disturbances from librationalmotion, flexibility and thermal deformation of the appendages. Finally, the controlstrategy is implemented to improve pointing accuracy of the Out-of-Plane (OP) ma-189neuver of the MSS.5.2 Feedback Linearization TechniqueConsider a rigid system given by1\4 (qr, t)g;^F(qr, 4r, t) = Q(qr, 4r, t)^(5.1)where qr denotes rigid generalized coordinates and (qr , , qr, is the nonlinear control.Let the control have the formQ(qr , 4r , t) = M(qr ,t)V F(qr , qr , t) ,^ (5.2a)where v = (qr)d Kv((4r)d — 4r) + KP((gr)d qr)^(5.2b)with (qr)d, (qr)d, and (qr )d representing the desired displacement, velocity and ac-celeration, respectively. The nonlinear control when substituted into (5.1) results ina linear closed-loop system,gr = ;^ (5.3a)or (47-)d — 4.7. +Kv((gr)d — qr) + Kp((qr)d qr) = 0 .^(5.3b)Since e (qr ) d — qr denotes the displacement error, Eq. (5.3) can be rewritten as.e+ K,,Ce +Kp e= O.^ (5.4)The function of Kv and Kp is now obvious; they are position and velocity gains toinsure asymptotic behaviour of the closed-loop system. A suitable candidate for Kpand Kv would be diagonal matrices of the form-2Kp = [2CoiKv =2DnW2(5.5)190leading to a globally decoupled system with each generalized coordinate behaving asa critically damped oscillator. For attitude control of a rigid spacecraft, Kp and Kvare 3 x 3 matrices for pitch, roll, and yaw degrees of freedom. In general, a largervalue of c:;"),,, gives rise to a faster response of the n-th generalized coordinate.As an example, the FLT is applied to the INSAT II. The satellite, designed by theIndian Space Research Organization (ISRO), is scheduled to be launched this year.INSAT II is a telecommunications satellite orbiting at the geosynchronous orbit. Themain body itself has the dimensions of 1.7 m x 1.8 m x 1.9 m. It has two flexiblecomponents attached to the main body. An array extending to 9 m collects solarenergy to power the electronic components onboard. A 15 m solar boom attached onthe opposite end is used to counterbalance the torque produced by the solar radiationpressure exerted on the array and the main body. The simulation carried out hereis based an the data presented in Table 5-1 and employs the coordinate systems ofFigure 5-1 .In the simulation, the main body is taken to be rigid whereas the solar arrayand boom are considered to be either rigid or flexible cantilevered plates and beam,respectively. The equilibrium configuration coincides with the design configuration,i.e., e = qe = A e = O. To begin with, the array and boom are considered to berigid. Figure 5-2 shows the libration response for initial displacement and velocitydisturbances. In Figure 5-2a, the satellite is initially given a displacement of 1° inpitch, roll, and yaw. The simulation results show that the satellite is in an inherentlyunstable orientation. At the end of 1/2 orbit, the amplitudes are 5.4 ° , —1.9°, and26.1° for pitch, roll, and yaw, respectively. In Figure 5-2b, the satellite is given aninitial velocity of 1 rad/rad in the direction of pitch, roll, and yaw axes. Once again,the initial condition leads to unstable motion of the satellite with pitch, roll and yaw191Z1.X1x,Y2^►X2LocalLocal^VerticalHorizontal k^A ^ OrbitNo.NormalOrbit^AiNormalFigure 5- 1^Coordinate systems and design configuration used in simulations ofthe INSAT II.192Main Satellite (Body Bc)1.7 m1,035 kg718 kg-m 21,960 kg-m21,810 kg-m2Solar Array (Body Bi)^Solar Boom (Body B2)9 m /2 15 m60 kg 7n2 5 kg0.29 Hz 1W2 0.32 Hz50 kg-m2 (ixx)2 2207 kg-m2 (Iyy)2 300 kg-m2257 kg-m2 (Izz)2 300 kg-m 2Table 5-1^The INSAT II dataexceeding 90° in less than 1/2 orbit. The controlled response of the satellite subjectedto the same initial disturbances is shown in Figure 5-3 . For simplicity, letKp = Kp I ,^Kv = Kv I ,where I is the identity matrix. It is required that(4;-)d = (qr)d = (qr) = 0 •Since the satellite's main purpose is for telecommunications, the pointing error shouldnot be greater than 0.1 ° . Three sets of control gains which would give criticallydamped response, are compared here: (i) Kp = 36 x 104 , /-C, = 1.2 x 103 ; (ii) Kp =64 x 104 , K, = 1.6 x 10 3 ; and (iii) Kp = 100 x 104 , I-C, = 2.0 x 10 3 . With aninitial displacement of 1 ° in each direction, these gains are capable to reduce theerror to about 0.1° in less than 2 minute (or 0.001 orbit). The critically damped193(a) Yb0 =^= A0 = 1010 ^(b) ?PO = ClO = ao = 190—900.50, OrbitA °—907^0.00, Orbit0.5—90-)^^90—5 ^—2^30 ^A°0^0.0Figure 5-2^Uncontrolled response of the rigid INSAT II showing instability ofthe system: (a) 0 0 = 00 =^= 1°; (b)^= 0 1. =^= 1.194response is shown in Figure 5-3a. Since the same set of gains are applied to thelibrational degrees of freedom, the response is identical in pitch, roll, and yaw. Ingeneral, the larger the Kp , and hence Kv , , the faster the response. The errors atthe end of 0.001 orbit are 0.11°, 0.04°, and 0.01°, for Kp = 36 x 104 , 64 x 104 , and100 x 104 , respectively. Figure 5-3b shows the critically damped response with aninitial condition of 1 rad/rad in the pitch, roll, and yaw. With Kp = 36 x 104 , thelibrational amplitude increases to a mere 0.04 ° and then decreases to 0.009 ° at theend of 0.001 orbit. The corresponding values for Kp = 64 x 104 are 0.03° and 0.002°whereas for Kp = 100 x 104 , they are only 0.02° and 0.001°, respectively.Even though the response for each degree of freedom is identical, the controleffort for each generalized coordinate is different due to differences in inertia. Forthe controlled response of Figure 5-3a, variations in the effort about the three axes,denoted by Q0 , Q0, and QA, are shown in Figure 5-4 a. Since the inertia about thepitch axis is considerably smaller than that for the roll or yaw, the control torque Q.is the smallest among the three. For Kp = 36 x 104 , an initial torque of —0.06 Nm isrequired which reaches a peak of 0.01 Nm before the magnitude finally decreases withthe pitch attitude. A similar trend can be observed in the Qo variation: an initial—0.24 Nm effort gradually reaches a peak value of 0.04 Nm before the magnitudefinally decreases. At the end of 0.001 orbit, the pitch and roll efforts required are neg-ligible. Since the inertias about roll and yaw axes differ by about 5%, the magnitudeand time history of control torque Q, are similar to those of Qo.The control effort variation corresponding to the response in Figure 5-3b is shownin Figure 5-4b. Although the controlled response in Figures 5-3a and 5-3b are dif-ferent, the corresponding effort time histories for the two cases are similar. ForKp = 36 x 104 , initial torque values of —0.007, —0.027, and —0.028 Nm are required195100 .00, Orbit..............................0-i.....^... ... . ...^...........0-i(a)^=^= A o = 1 °^ Kp = 36 x 10 4, K, = 1.2 x 103Kp = 64 x 10 4, K, = 1.6 x 103Kp = 100 x 104, K, = 2.0 x 103Figure 5-3^Controlled librational response of the INSAT II for three differentsets of gains: (a) 00 =^=^= 1°.196(b)^=^= 1^ Kp = 36 x 104, K, = 1.2 x 103_ _ _ Kp = 64 x 104, K, = 1.6 x 103^ Kp = 100 x 104, K, = 2.0 x 1030.04-- -- - ................^ 7 7 7 . ^0.00 ^^0.04 .........0.00 -1^0.04. .............................................1.00. 00 -1^0.00, Orbit *io-3Figure 5 - 3^Controlled librational response of the INS AT II for three differentsets of gains: (b) 0 10 = Olo = A t° = 1.1970.01QA.........•••■•••••■•■••■Nm -----------*10 30, Orbit•......................•••• - 1.•^...—0.30—0.10 ^Nm••^0.05 i e.'Q0^..•/•0.05 --^ Nm ..^ ..........1.0—0.30 ^0.0c2x(a) /Po =^= A ° = 1°^ Kp = 36 x 10 4, Kt, = 1.2 x 103Kp = 64 x 104, Ku = 1.6 x 10 3^ Kp = 100 x 104 , Kt, = 2.0 x 103Figure 5-4^Comparison of control efforts for three different sets of gains used inthe INSAT II attitude control: (a) 7,/)0 = 40 = Ao = 1°.198Qa(b) o =^= ao = 1^ Kp = 36 x 10 4, Kt, = 1.2 x 103Kp = 64 x 10 4, Kt, = 1.6 x 103^ Kp = 100 x 10 4, K, = 2.0 x 1030, OrbitFigure 5-4^Comparison of control efforts for three different sets of gains used inthe INSAT II attitude control: (b)^= 0'0 = ao = 1.199for pitch, roll, and yaw control, respectively. Subsequently, the torque decreasessteadily such that at the end of 0.001 orbit, the effort required is negligible.One disadvantage of the FLT is the difficulty in predicting the torque requirement.From Eq. (5.2), it is apparent that the control torque has two components with onea linear function of the gain and states. The second component, which depends onthe equations of motion, is a highly nonlinear function of the states. In general,the larger the gain, the faster the response; however, this does not necessarily implyhigher control effort. For instance, for Kp = 36 x 104 , the direction of QA is clockwise;however, the direction changes when Kp is reduced by four orders (not shown).5.3 Control Implementation ProceduresIn general, dynamics of a flexible spacecraft with qr and qf corresponding tolibrational and vibrational generalized coordinates, respectively, is governed byMrr FrMrfl {qr {Qr(5.6)Mfr mff qf Ff QfHere Mrr (qr ) is a 3 x 3 submatrix for the librational degrees of freedom; M rf(qr , qf),of dimension 3 x Nq — 3, represents the coupling between the rigid and flexible gen-eralized coordinates; Mfr = MrfT ; Mff(qf ) is a Nq — 3 x Nq — 3 submatrix forthe flexible degrees of freedom only. Fr and Pf are 3 x 1 and Nq — 3 x 1 vectors,respectively, representing first and second order coupling terms. Assuming only thelibrational degrees of freedom to be observable, the control force Q f is not applicableand hence set to zero. The objective is to determine Qr such that the closed-loopsystem is linearized. Rewriting Eq. (5.6) into two sets of equations and with Q f = 0gives:Mrr6r +Mrf6f + Pr = Qr;^(5.7a)200(5.7b)1\44 + 1\ 4ff 4' ,f‘ Ff = 0 ;which can be solved for qr and qf :f\46, + = (2 , ;= _mff-1mfr6. - mff-lpfwhere:^= Mrr MrfMff iMfr= Fr — MrfMiC iFf(5.8a)(5.8b)As in the case of Eq. (5.2), a suitable choice of Qr would beC2r(qr,g1,4r,4f,t)=M(qr,41,01) +P(qr,g1,4r,4f,t),with U = (41.)d + Kv((dr)d 4r) + Kp((qr) d — qr)Now the controlled equations of motion become:gr = ;^ (5.9a)mff-lpf^ (5.9b)Note thatQr = (6.)d^+ fq(Kv 6 + Kp e)which can be visualized as a combination of two controllers: primary (0 rp ) andsecondary (Q r,^ ), whereQr,p = M(gr) d + F ;^ (5.10a)Qr,s = M(Kve + Kp e) . (5.10b)The function of the primary controller is to offset the nonlinear effects inherent in therigid degrees of freedom; whereas the secondary controller ensures robust behaviouragainst the error.201To evaluate the control effort (2r required, a priori information of M and F isneeded. In turn, calculation of M and F requires the knowledge of qf and 4.f. Tothis end, two schemes, Quasi-Open Loop Control (QOLC) and Quasi-Closed LoopControl (QCLC), are suggested by Modi et al. [118].5.3.1 Quasi-open loop controlIn this scheme, the flexible coordinates are evaluated off-line, i.e. integration ofEq. (5.9b) is performed independently and with q r substituted with (4r )d. The mainadvantage of the scheme is a reduced computation effort. Under this scheme, discrep-ancies between the calculated and actual flexible coordinates would exist. Hence, thesuccess of the scheme depends on the robustness of the controller.5.3.2 Quasi-closed loop controlUnder this scheme, both the rigid and flexible coordinates are evaluated simulta-neously, i.e. Eqs. (5.9a) and (5.9b) are integrated concurrently. The disadvantage ofthe scheme is a relatively large computational effort as compared to the QOLC. Onthe other hand, the QCLC is less sensitive to system uncertainties.Block diagrams for both the control schemes are shown in Figures 5-5a and 5-5b. Implementation of the QCLC into the present formulation is straightforward.The program can remain as it is with the inclusion of the subprogram CNTROL.The function of the subprogram is to partition the M matrix and then evaluate theM and E. In contrast, the QOLC scheme requires considerable modification of theprogram codes. With this in mind, the present study is limited to the QCLC of theflexible INSAT II only.202SecondaryControllerStateOutputDesiredTrajectoryIm■^110(qr)ch^(fr)dOffline VibrationalDynamics Evaluation INSAT II1kPrimaryControllerr- 2(4;) dOP, (If(if, (IfFigure 5-5^Block diagram for control of the flexible INSAT II: (a) quasi-openloop control.203 DesiredTrajectory(Ma, (ir)d, (dr)dSecondaryControllerStateOutputINSATgf)PrimaryControllerM 'flr, f firFigure 5-5^Block diagram for control of the flexible INSAT II: (b) quasi-closedloop control.2045.4 Application of the Quasi-closed Loop Control to INSAT IIWith only the first mode of vibration in the system discretization process, thegeneralized coordinates for simulation are:qr = P2u1,1AN = 6 .Q12- ,Figure 5-6 shows system response of the INSAT II subjected to the same initialdisturbance as in Figure 5-2a. With this initial condition, flexibility does not appearto have any influence on the librational motion as pitch, roll and yaw responses remainthe same as in Figure 5-2a. Similarly, the librational disturbance hardly excites theflexible motion  of the appendages. Maximum tip deflection of the solar array or boomis of the order of 10 -6 cm.Control is now applied to the system using the same three sets of gains as inFigure 5-3. For the librational response, integration of Eq. (5.9a) is essentially thesame as that of Eq. (5.3a). For the same initial conditions, the controlled librationalresponse would be identical to that in Figure 5-3a. The controlled response accountingfor appendage flexibility is shown in Figure 5-7 . Note, each control gain setting affectsthe vibrational behaviour differently. In general, application of the control results inhigher initial vibration amplitudes. For K p = 100 x 104 , the peak amplitude of thearray and boom are 0.054 cm and 0.073 cm, respectively, an increase of over fourorders. As the control effort diminishes with attitude, the amplitude of vibrationdecreases; however, the rate of decay is minimal.Figure 5-8 shows the control torque variation to give the desired rigid body re-sponse. Comparing with Figure 5-4a, the time history and the magnitude of the2052 f /I3fl= O o = A o = 1°100 595 0—230NO00.0 0, Orbit 0.50, Orbit^ *to 3Figure 5-6^Librational and vibrational responses of the uncontrolled INSAT IIwith initial condition of O. =^= A c, = 1°.2060, Orbit *1 0-3Vo =^= A° _10^ Kp = 36 x 10 4, K„ = 1.2 x 103Kp = 64 x 10 4, K, 1.6 x 103^ Kp = 100 x 104, K, = 2.0 x 103Figure 5-7^Variation of tip deflections of the INSAT II flexible appendages show-ing the effect of control gains.207= C60 = Ao = 1°^ Kp = 36 x 104, K, = 1.2 x 103Kp = 64 x 10 4 ,^= 1.6 x 103^ Kp = 100 x 10 4, K, = 2.0 x 103QA 0.02In01.17m.....■••■••Q0—0. 100. 1Q00, OrbitFigure 5-8^Control effort variations for the INSAT II subjected to three differentcombinations of control gains.208control torque qo is almost the same as before. For Kp = 100 x 104 , an initialtorque of —0.067 Nm is required as compared to —0.061 Nm for the rigid spacecraft.However, flexibility has some influence on Q cb and QA. Since the beam vibration inthe local vertical direction (4) is about the roll axis, Q o can be expected to be af-fected. Similarly, both the array and beam vibrations in the local horizontal direction(ET and SD are expected to influence QA. It is, therefore, reasonable that the controltorques Qo and QA both oscillate at high frequencies to compensate for the vibration.The general trend for Q tp is the same for both rigid and flexible configurations of thesatellite. In fact, in the present case, flexibility actually aids in reducing the maxi-mum torque. For Kp = 100 x 104 , — 0.235 Nm is needed for a rigid satellite but only—0.151 Nm is required when flexibility of the appendages is accounted for. On theother hand, the appendage flexibility results in higher QA. For Kp = 36 x 10 4 , initialtorque for yaw control is 0.329 Nm in the present case as compared to -11236 Nm forthe rigid spacecraft. Note that flexibility results in the reversal of the torque direc-tion. As the vibration magnitude is gradually damped through control, the torquerequired also decreases. Again, as the vibration amplitude diminishes at a slow rate,so do the control torques Qo and QA.The effect of reduced stiffness of the flexible members is shown in Figure 5-9. Here,both the array and boom stiffnesses have been reduced to 10% of the design value. Forthe same initial librational disturbance, the rigid body response remains essentiallyunaffected by the flexibility. The array and boom vibration, however, have increasedby about two orders. Note also that, with the beam natural frequency reduced tentimes, it is now close to that of gravitational disturbance resulting in the beat responsein 4. For the same three sets of control gains, the librational response was againobserved to be the same as in Figure 5-3a. The corresponding vibrational behaviouris as shown in Figure 5-10. The controllers again have a significant influence on the209IP 0 = (i)0 = 0 = 1010 ^O 050 02 ^30X 000.0^ 0.56) , Orbit0.4 —^0.00.1Sy2x 1 0- ' cmFigure 5-90, OrbitResponse of the uncontrolled INSAT II, with reduced stiffness of theappendages, to the initial conditions of *0 =^=^= 1°.210= O a, = A t, = 1°Kp = 36 x 104 , K,, = 1.2 x 10 3Kp = 64 x 104 , K,, = 1.6 x 103 ^Kp = 100 x 104 , K = 2.0 x 10320. 0 1.052ExCMcm0, OrbitFigure 5-10 Vibrational response of the INSAT II with flexible appendages ofreduced stiffness.211appendage motion leading to a significant increase (over four orders) in vibrationamplitude. For Kp = 100 x 10 4 , the peak amplitude of ET, (I and S2 are 3.1 cm,4.5 cm and 4.3 cm, respectively. The values are about two orders of magnitudehigher than the corresponding ones with the design stiffnesses (0.054 cm, 0.073 cm,and 0.071 cm). The time histories of Qv, and Q cy are similar to those observed forthe design case (i.e. without any reduction in stiffness), except for the amplitudemodulation which now occur at lower frequencies (Figure 5-11 ). However, QA nowoscillates at constant amplitude because ET and S2 do not exhibit any sign of decay.For Kp = 100 x 10 4 , QA varies in the range of ±0.207 Nm.The performance of the controller under thermal disturbances on the array andboom is next investigated. Figure 5-12 shows the response of the satellite induced bythe thermal deformations of the array and boom. The rigidity of the array and boomis again taken to be 10% of the design value. Thermal reference length parameters, LI(for the array) and 14' (for the boom), are chosen to be 0.22% and 0.13%, respectively.With these values, maximum tip deflections of the array and boom due to thermaleffect would be 1 cm. As can be seen, thermal deformations totally change thelibrational response of the satellite, with the pitch and roll most severely affected.In less than 0.5 orbit, the satellite starts to tumble (the pitch angle reaches —90°).The fast deterioration of the attitude is due to large amplitudes of array and boomvibrations. Without thermal deformation, the maximum displacements are 0.2 x 10 -5and 0.1 x 10 -4 cm for the array and boom, respectively. These values increase to 1.7and 3.6 cm in the presence of thermal deformations (from the deformed equilibriumstate). With the inclusion of an initial libration disturbance of 1° in pitch, roll,and yaw, Figure 5-13 shows that the system response is almost identical to that ofFigure 5-12.212Nm ........... 0.0•• \•= /50 =^= 1°^ Kp = 36 x 104, K„ = 1.2 x 103Kp = 64 x 10 4, K„ = 1.6 x 103^ Kp = 100 X 104, K„ = 2.0 x 103Q).0.02Q9511.00, Orbit^*10 3Figure 5-11^Comparison of the control effort variations for the INSAT II withappendages of reduced stiffness.21390,tpo—90 ^1 0 ^0°—9090X°00.0^0, Orbit^0.5—4 ^0.000, OrbitFigure 5-12 Librational and vibrational responses of the INSAT II showing theeffect of thermally deformed appendages.0.052 . 1490]^,tpo—90 ^10—9090X004bZ20.00 0.05= Cbo =^= 1°0.0^8, Orbit^0.58, OrbitFigure 5-13 Dynamics of the INSAT II with thermally deformed appendages andinitial conditions of 0 0 = o^A c, = 1°.215Control is next applied to the case of Figure 5-13, i.e. thermal effect coupled withinitial disturbance. Application of the control would once again result in the samelibrational response as in Figure 5-3a. The vibrational dynamics of the controlledsatellite is shown in Figure 5-14 . Note, the response trends of the appendages areessentially the same as in Figure 5-10 indicating dominant influence of the controllersto offset the effect of thermal deformations. A typical control torque variation isshown in Figure 5-15 . As the attitude of the satellite increases rapidly withoutcontrol, it is reasonable that larger control torques are needed to force the satelliteinto the desired trajectory. The increase is approximately 5% for Qo and Q0 , andabout 15% for QA. For instance, QA increases from ±0.207 Nm to ±0.237 Nm forKp = 100 X 104 .The maximum control effort required for the cases studied is summarized in Ta-ble 5-2 .5.5 Application of the Quasi-closed Loop Control to MSSIn Section 4.4, the performance of the Mobile Servicing System (MSS) during therobotic arm maneuver was discussed. The pitch, roll, and yaw motion excited bythe Out-of-Plane (OP) maneuvers, resulted in poor pointing accuracy of the robotarm as compared to the InPlane (IP) maneuver. It is apparent that to improve theperformance of the OP maneuver, the attitude of the MSS has to be first controlled.The objective here is to assess effectiveness of the QCLC as applied to the attitudecontrol of the MSS.The control gains selected in the present study are Kp = 1.0 and Kv = 2.0.As will be pointed out later, even such small gains are capable of improving thesystem performance significantly. To assess effectiveness of the controller, the erroris purposely allowed to build up over an arbitrary duration before implementing the216cm5ez1cm• S.o =^= A o = 10^ Kp = 36 x 10 4, Kt, = 1.2 x 103Kp = 64 x 104, K, = 1.6 x 103^ Kp = 100 x 10 4, K, = 2.0 x 103 Cm2—5 752—3 ^^0.0 1.00, OrbitFigure 5-14 Tip deflections of the thermally deformed INSAT II appendagesshowing the effect of control gains.2170.02Qip—0.100, Orbit7P0 = 00 = Ao = 1°^ Kp = 36 x 104, Kt, = 1.2 x 103Kp = 64 x 104, Kt, = 1.6 x 10 3^ Kp = 100 x 10 4, Kt, = 2.0 x 103 QaFigure 5-15^Plots of control efforts required for the INSAT II with thermallydeformed appendages.218Table 5-2^Summary of maximum Q0, Q0, and QA required for the cases studiedGain Qo Nm* 1^Qo Nm* 1^QA Nm*Rigid array & boom (0 0 = 00 = A o = 1°)Kp = 36 x 10 4 , Kv = 1.2 x 103 -0.023 -0.088 -0.089Kp = 64 x 10 4 , Kv = 1.6 x 103 -0.040 -0.154 -0.155Kp = 100 x 104 , Kv = 2.0 x 103 -0.061 -0.235 -0.236Rigid array & boom (0 10 = Of) = A ic, = 1)Kp = 36 x 10 4 , Kv = 1.2 x 103 -0.005 -0.017 -0.018Kp = 64 x 104 , Kv = 1.6 x 10 3 -0.006 -0.022 -0.023Kp = 100 x 104 , Kv = 2.0 x 10 3 -0.007 -0.027 -0.028Flexible array & boom (00 = cho = Ao = 1°)Iip = 36 x 10 4 , Kv = 1.2 x 10 3 -0.025 -0.061 +0.124Kp = 64 x 10 4 , Kv = 1.6 x 10 3 -0.044 -0.102 +0.215Kp = 100 x 104 , Kv = 2.0 x 10 3 -0.067 -0.151 +0.329Flexible array & boom (10% rigidity & 00 = o = A = 1 ° )Kp = 3.6 x 104 , Kv = 1.2 x 103 -0.024 -0.075 ±0.080Kp = 64 x 10 4 , Kv = 1.6 x 103 -0.042 -0.130 ±0.138Kp = 100 x 104 , .Kv = 2.0 x 103 -0.064 -0.198 ±0.207Flexible array & boom (10% rigidity, thermal effect & 00 = 00 = A o = 1°)Kp = 36 x 104 , Kv = 1.2 x 103 -0.025 -0.077 ±0.117Kp = 64 x 104 , Kv = 1.6 x 103 -0.044 -0.133 ±0.173Kp = 100 x 104 , Kv = 2.0 x 103 -0.067 -0.202 ±0.237* + and - signs represent counterclockwise and clockwise directions,respectively; ± denotes equal magnitudes for both directions.control. In the present case, the operation of the controllers is assumed to begin athalf-way of the maneuver. For instance, during a 10-minute maneuver, the controlleris turned on 5 minutes after the commencement of the maneuver.Figure 5-16 shows the controlled system performance for a 5-minute OP ma-neuvers. The controller is turned on 2.5 minutes after the robotic arm begins the219NO0.255 -^05 -^0.00(a) libration response0.‘P^0.00 .1(b) vibration response40 cmal 0-40 ,^40z0(c) control effort time historiesQ4, o-2030QA 0300, Orbit^ 0, Orbit(d) pointing error variation0.00^025^015^0.250, Orbit^ 0, OrbitFigure 5-16 Controlled system performance of the MSS undergoing a 5-minuteOP maneuver: (a) libration response; (b) vibration response; (c) con-trol effort time histories; (d) pointing error variation.220maneuver. The controlled attitude response is shown in Figure 5-16a. A comparisonwith the uncontrolled case (Figure 4-22) is appropriate. With application of the con-trol, there is no noticeable improvement in the pitch and roll response because theirmagnitudes are small already. On the other hand, the relatively large yaw angle isdamped quite effectively. At the end of 0.25 orbit, the yaw angle is only —1.1° ascompared to 8.2° for the uncontrolled case. Note, influence of the controller on theappendage vibration is rather small (Figure 5-16b). The peak deflections of the upperand lower links are 35.9 and 9.4 cm, respectively. The corresponding values are 33.3and 9.5 cm for the uncontrolled system. Since the inertias of the MSS about the pitchand roll axes are of the order 108 kg m 2 , even a small error in pitch or roll requires alarge control effort. Similarly, although the inertia about the yaw axis is about threeorders of magnitude smaller, the relatively large error in the yaw angle demands a sig-nificant control torque. The peak torques Qo (4, QA are —34.5, 12.1, and 25.5 Nm,respectively (Figure 5-16c). The pointing accuracy has improved remarkably with theapplication of the control (Figure 5-16d). With the yaw attitude damped, the errorin the orbit normal direction (Sx ) no longer increases progressively as in Figure 4-25.Instead, Sx oscillates about a mean value of —15 cm with 30 cm p-p amplitude. At0.25 orbit, the mean error is 25.7 cm with 9.8 cm p-p superposed. This is a significantimprovement compared to the errors for the uncontrolled system which are 103.4 and40.9 cm for OP and IP maneuvers, respectively.With the maneuver time increased to 10 minutes, the performance of the MSSimproves significantly (Figure 5-17). Here, the controller is turned on 5 minutesafter the slewing maneuver begins. The attitude errors at 0.25 orbit are only 0.054°,—0.027°, and —0.862 ° for pitch, roll, and yaw, respectively (Figure 5-17a). Thesevalues are smaller than those in a 5-minute maneuver (0.103°, —0.058°, —1.05° forpitch, roll and yaw). Comparing with Figure 4-24, the control effort results in a221response(b) vibration response10- 1010Of 0- 10cm0.0020 25B, Orbit-5100^1 Nm0 0.00QA0.15 0.25(c) control effort time histories10QV) 0105Q qs o0 25Nm0sx-100Sy- 300sz-1030stot0CMA A A cm0.05 1tp^0 00 6Y1-0 10525NO 0 61,10.00^ 0 256, Orbit(d) pointing error variation6, Orbit 0, OrbitFigure 5-17 Controlled system performance of the MSS undergoing a 10-minuteOP maneuver: (a) libration response; (b) vibration response; (c) con-trol effort time histories; (d) pointing error variation.222slight increase in p-p amplitude of the robot arm vibration. For instance, without theattitude control, the maximum deflections for the upper and lower links were 6.3 and1.8 cm, respectively. These values increase to 6.4 and 2.0 cm with the application ofthe control (Figure 5-17b). For a longer maneuver period, the peak control torquevalues are smaller (Figure 5-17c). With a 10-minute maneuver, the maximum valuesof the torques Q,p, Q cb, and Qx (-7, 3, and 5.7 Nm) are about 20% of those for a5-minute maneuver (-34.5, 12.1, and 25.5 Nm). Obviously, these savings are dueto smaller excitation in attitude. Similarly, reduced attitude errors are sufficientto improve the pointing accuracy and offset the minor increase in link deflections.Once again, the control of the yaw angle helps stabilize the error in the orbit normaldirection, Sx . At 0.25 orbit, the arm is only 14 cm from the target as compared to86 cm for the uncontrolled case (Figure 5-17d).Figure 4-26 shows that the MSS performs poorly when the robotic arm is located50 m from the station centre of mass. Even with a 10-minute maneuver, the systemattains a —0.55° pitch, 0.03° roll and 6.5° yaw in 0.25 orbit. Consequently, the armis already 126.7 cm from the target in 0.25 orbit (Figure 4-27). Figure 5-18 clearlydemonstrates effectiveness of the controller, even for a 5-minute maneuver. Here, theattitude error is reduced to only 0.013° in pitch, —0.016° in roll and —0.8° in yawat 0.25 orbit (Figure 5-18a). Again, the flexible motion of the arm is not adverselyaffected by the control torques. For instance, without attitude control, the maximumdeflections for the upper and lower links are 33.4 and 9.6 cm, respectively. Withattitude control, the corresponding values are 34.5 and 9.2 cm (Figure 5-18b). Sincethe controlled response for pitch and roll angles is different from that without anyarm offset, the controlled torques Q,0 and Qo are expected to be different for thetwo cases. In contrast, the controlled yaw response is the same, with or withoutoffset implying that Qx should be similar in both the cases. This is substantiated223-510ho 0.00-0.050.10° 0.05-^0X05-T^bi 0cmbi, 1 0-100.00 0.25 0.00 0.250, Orbit 0, Orbitsz(c) control effort time historiesQ8 iNmo 0 0200-1Q0"Scot30 ^70 ,^0.00^ 0.25 0.150, Orbit^ 0, Orbit80-2000.25-40 -1-60--^300(2). INTFigure 5-18 The performance of the MSS controller in the presence of offset anda 5-minute OP maneuver: (a) libration response; (b) vibration re-sponse; (c) control effort time histories; (d) pointing error variation.224by the control effort profiles in Figure 5-18c. Peak torques for Q0 , Q0, and QA are—74.3, —175.8 and 24.5 Nm as compared to —34.5, 12.1 and 25.5 Nm when thereis no offset. Without the attitude control, the errors Sx Sy, and Sz increase withtime (Figure 4-27). This is no longer true in the presence of control. The errorsare periodic with mean values of —10.6, —78.5, and —14.6 cm for Sx , Sy , and Sz,respectively. At the end of 0.25 orbit, the mean pointing error of the arms is 82.6 cmwith a p-p fluctuation of 8.3 cm. This error is larger than that for the case withoutthe manipulator offset (25.7 ± 4.9 cm). This is due to the relatively large error in thelocal vertical direction, Sy in the present case. The robot arm points towards andaway the earth before and after the maneuver, respectively. Consequently, there is ashift in centre of mass in the local vertical direction. Without the manipulator offset,this shift is negligible. However, this is no longer true when there is a manipulatoroffset. Thus, it is important to recognize that, even in presence of the attitude control,influence of the manipulator offset is still significant.To determine advantages, if any, of longer maneuver period in the presence ofarm offset, the maneuver is now increased to 10 minutes in Figure 5-19. Note that,the controlled pitch and roll responses in Figure 5-19a are no longer similar to thosein the 5-minute maneuver (Figure 5-18a). At the end of 0.25 orbit, the pitch, roll,and yaw errors are —0.085°, 0.023°, and —1.43°, respectively. These relatively largeerrors do not imply the ineffectiveness of the controller. Rather, it suggest lack ofsufficient time to damp the attitude response in 0.25 orbit. Comparing the resultswith those in Figure 4-26, it is apparent that the controller once again does not havea significant influence on the arm vibrations (Figure 5-19b). In fact, the maximumdeflection of the lower link reduced to 2.5 cm with attitude control (3.1 cm withoutattitude control). The biggest advantage of a longer maneuver period is the reduceddemand on the control torques (Figure 5-19c). Maximum torques, though higher225100.1—5,fro 0.0—0.1 ^50—5 -1(a) libration response —20 ,^0.10.0—0.15 c^m5^xo6f 0—105m—3050Q0 030Qv, 0^— 50^^10 N^m^QA 0^—10 ,0.00 0.25N0.00 0.25 0.00 0.250, Orbit 9, Orbit(d) pointing error variationszC121(c) control effort time histories —4010, Orbit 9, OrbitFigure 5-19 The performance of the MSS controller in the presence of offset anda 10-minute OP maneuver: (a) libration response; (b) vibration re-sponse; (c) control effort time histories; (d) pointing error variation.226than those without the offset case, are significantly lower than those for a 5-minutemaneuver. Here, peak Q v) , Q0, and QA are —22.6, —47.9 and 7.5 Nm, respectively.Since the attitude errors remain large at 0.25 orbit, the pointing accuracy (Figure 5-19d) is still poor compared to the case of a 10-minute maneuver without link offset.However, compared with Figure 5-18, the smaller p-p error together with a smallercontrol effort justify the use of a longer maneuver period.Of course, implementing control at the beginning of the maneuver would restrictany librational motion as shown in Figure 5-20a. Here, a 5-minute maneuver of therobot arm is assumed. Comparing Figures 5-16b and 5-20b, the control strategyhas little effect on the flexible motion of the links. When the libration error is notallowed to build up, the effect on the control torques is two-fold. On the one hand,less control effort is needed to drive a large error to zero. On the other hand, a largercontrol torque is necessary to maintain zero libration error. Consequently, dependingon the dominance of these two effects, the peak control torques required might belarger or smaller. A comparison between Figures 5-16c and 5-20c show that withthe control strategy implemented early, smaller Q,0 and Q0 are needed whereas Qxincreased from 25.5 Nm to 32.1 Nm. With the elimination of the libration error,the positioning accuracy would definitely improve as shown in Figure 5-20d. At theend of 0.25 orbit, the mean error is only 18.9 cm as compared to 25.7 cm when thecontroller was implemented half-way through the maneuver.With the inclusion of a 50 m offset of the manipulator, the controller imple-mented at the beginning of the maneuver once again helps eliminate the libration er-ror (Figure5-21a) and improve the positioning error (Figure 5-21d). Once again, thecontrol strategy has negligible influence on the flexible motion of the links (Figure 5-21b). Compared with the case when the control was implemented half-way through227crricmcm0.250.000, Orbit -0.25^0.00Orbit—8080 Nmo0.150.250.008, Orbit 0, Orbit(c) control effort time histories200Qo o—200QAsz—10-1^—60--^by(a) libration response —4030Of0.01sv^0.00—30—0.010.01 58Y1,1kp^0.00—5—0.01100.01 -X° f,i^00.00—0.01 - —10Figure 5-20 The performance of the MSS undergoing a 5-minute OP maneu-ver with the controller implemented at the beginning of the slew:(a) libration response; (b) vibration response; (c) control effort timehistories; (d) pointing error variation.2280.00 0.25 0.15 0.256, Orbit 6, Orbit(c) control effort time histories30Qa o—30Stat.(a) libration response —40400.01ar° 0It/^0.0 0—40-0.0110691,1 00.01]tp^0.00-0.01^ —10^0.01^A ° 0.001-0.01 ^0.006, Orbit20bl l o—200.25 0.00^0.256, Orbit(d) pointing error variationFigure 5-21 MSS performance in the presence oOP maneuver with the controllerthe slew: (a) libration response; (effort time histories; (d) pointing229f offset and undergoing a 5-minuteimplemented at the beginning ofb) vibration response; (c) controlerror variation.Table 5-3^Summary of the system performance for the controlled MSS with out-of-plane maneuverPeak Control Effort (Nm) Maximum Deflections (cm) Pointing Errorat 0.25 Orbit (cm)*Qo Qo Upper Link Lower LinkControl Implemented Half-way through the Maneuver5-minute maneuver-34.5 12.1 25.5 35.9 9.4 25.7 ± 4.9 (103.4)10-minute maneuver-7.0 3.0 5.7 7.8 2.0 14.0 ± 1.6 (86.0)5-minute maneuver & 50 m offset-74.3 -175.8 24.5 37.1 10.1 82.6 ± 8.3 (139.7)10-minute maneuver & 50 m offset-22.6 -47.9 7.5 11.7 2.6 83.8 ± 1.2 (126.7)Control Implemented at the Beginning of the Maneuver5-minute maneuver-3-1.8 10.7 32.1 36.1 9.5 18.9-± 11.3 (103.4)5-minute maneuver & 50 m offset-61.4 -213.0 31.5 37.3 9.5 81.7 ± 7.4 (139.7)* Pointing error for the uncontrolled case are indicated in parenthesesthe maneuver, the peak torque Qo is now reduced to -61.4 Nm from -74.3 Nm. Onthe other hand, peak Qo and QA increase by about 20% to -213 Nm and 31.5 Nm,respectively (Figure 5-21c).The maximum control effort required and the pointing errors for the six casesstudied are summarized in Table 5-3 .5.6 Closing CommentsThe FLT has been found to be effective in controlling the attitude motion of therigid INSAT II. The performance with three sets of control gains is compared. Withthe satellite initially disturbed by 1 ° in pitch, roll, and yaw, the gains are found to230be adequate in reducing the attitude error to the design limit in about 2 minutes. Ingeneral, the rate of response is directly proportional to the magnitude of the controlgains. With the inclusion of flexibility in the satellite model, the QCLC was foundto be effective. Even in presence of relatively large initial attitude disturbances, thecontrol scheme works remarkably well. The inclusion of thermal deformations of thearray and beam does not adversely affect the controller's performance. As expected,although larger control torques are needed, the equilibrium attitude of the satellite isagain restored in less than 2 minutes.The QCLC is also effective in controlling the attitude of the space station andhence the accuracy of the MSS. Even with a 5-minute maneuver and an offset locationof the manipulator (from c.m. of the station), the controller continues to be effectivein damping the slew excited attitude motion. In turn, the pointing error of the armdiminishes significantly. Considering the large inertia of the space station, the controleffort required is reasonable. In the presence of an offset, the controllers performanceremains essentially unaffected except for larger control torques. In general, a longerthe maneuver period results in smaller control effort and p-p amplitude of the armvibration.From practical considerations, it would be useful to study effectiveness of thecontroller subjected to constraints and system uncertainties. The gains and desiredtrajectory are selected arbitrarily in the present simulation. A more systematic ap-proach in making these selections is needed. Also, as seen before, the control effortmay oscillate at high frequencies implying a need for high bandwidth. Although theFLT is promising, these issues are among many which remain to be resolved.2316. CONCLUDING REMARKS6.1 ConclusionsA relatively general Lagrangian formulation for studying the dynamics of space-craft with interconnected flexible bodies forming a tree-type topology is presented.The formulation is applicable to a large class of systems negotiating arbitrary orbitsand having any desired orientation in space. Each member is free to undergo trans-verse vibration, slewing maneuvers and thermal deformations. The governing equa-tions are highly nonlinear, nonautonomous, and coupled. In general, implementationof these equations into a comprehensive computer code requires enormous amount ofeffort This is minimized by defining three new operators. The computer programis structured in a modular form permitting assessment of the effects of shifting c.m.,flexibility, thermal deformations, shape functions and higher modes. Validity of thecomputer program is established through extensive comparison with particular casesstudied by Ng [90] and Chan [15]. Convergence of the beam shape functions has alsobeen established.Versatility of the formulation is then demonstrated through its application tofour spacecraft models of contemporary interest: the First Element Launch (FEL)and Permanently Manned Configuration (PMC) of the proposed U.S. space stationFreedom; Mobile Servicing System (MSS) under development by Canada; INSAT--II, a multipurpose communications satellites of India; and Japan's Space Flyer Unit(SFU). Simulation results provide insight into the interactions between librational andvibrational dynamics, slewing maneuvers, thermal deformations, orbit eccentricity,deployment and retrieval, etc. The focus is on results which help establish trends.The concept of nonlinear control using the Feedback Linearization Technique (FLT)232is discussed next. Effectiveness of the procedure to control rigid as well as flexiblemodes of the INSAT II is explored. Control of the satellite in the presence of thermaldeformation is also demonstrated. The technique is then applied to the control ofspace station attitude during MSS maneuvers. Once again, the FLT has proved to beeffective in controlling the attitude of the station. More important conclusions basedon the parametric and control studies are summarized below:(i) The FEL's nominal orientation is neither in equilibrium nor stable. Evenwith a very small initial pitch, roll, or yaw disturbance, a rigid FEL startsto tumble in less than 1 orbit. With the inclusion of member flexibility,the FEL's librational and vibrational responses become highly coupled. Ingeneral, even a small disturbance to a flexible member results in significantrigid body motion. Initial condition an the power boom is undesirable asit leads to pitch and roll responses with high amplitudes and frequencies.Furthermore, response results show the power boom and stinger dynamics tobe quite sensitive to the disturbance direction.(ii) As in the case of the FEL, the PMC of the space station is also in a nominallyunstable orientation. Unlike the FEL, due to the relatively large inertia of thepower boom, the system is less susceptible to disturbances from the arraysand radiators. However, even a small disturbance applied initially to thepower boom can result in significant librational and vibrational motions.(iii) In general, both eccentricity and thermal deformation of the PV arrays canaffect the PMC pitch motion. Furthermore, thermal deformations influenceflexible response of the system.(iv) The study of the PMC's velocity and acceleration profiles indicate that inabsence of any disturbance, the system velocity and microgravity distribution233stay within the design limits. However, even with a small disturbance in thelocal vertical or local horizontal direction, the design limits are easily violated.The presence of a small orbit eccentricity does not adversely affect the systemperformance; however, even small thermal deformation of the PV arrays resultin velocity and acceleration beyond the acceptable values.(v) As expected, the MSS study shows that longer the maneuver period, smallerthe response amplitude both for rigid and flexible degrees of freedom. Ingeneral, an out-of-plane (OP) maneuver excites a smaller flexible motion thanthe corresponding inplane (IP) case. However, this advantage is offset by the3 dimensional rigid body motion excited. Increasing the link stiffness resultsin only a small reduction in the librational response. On the other hand, thevibration response diminishes significantly. Offset of the MSS location fromthe system c.m. is detrimental to the spacecraft performance.(vi) The excitation of roll and yaw responses during the OP maneuver results in apoor pointing accuracy. Even increasing the maneuver period or link stiffnessdoes improve the situation significantly. The OP maneuver is feasible only ifthe system libration is controlled.(vii) The SFU study shows that the spacecraft remains stable under both sym-metric and asymmetric deployment of the Solar Array Pedals (SAPs) Theparametric study also shows that the OP deployment has the advantage overthe IP extension as the excitation is smaller.(viii) The retrieval of the SAPs may pose some problems. The symmetric OPretrieval poses no difficulty. However, if only one SAP is retrieved, the space-craft becomes unstable even when the retrieval time is increased. The SAPretrieval in the IP direction is found to be undesirable. Regardless of symme-234try and period of retrieval, the spacecraft starts to tumble in a short time.(ix) Feedback Linearization Technique (FLT) for attitude control of highly non-linear systems appears promising. The method is straightforward and thecontrol algorithm is simple. For the control of the rigid INSAT II, the effortrequired is minimal.(x) For the control of flexible spacecraft, Quasi-Closed Loop Control (QCLC)based on FLT, performs well. In general, flexibility of the solar array andboom does not have significant influence on the control effort. Flexibilityimplies that controllers of higher bandwidth are needed especially for the rolland yaw control of the INSAT II. Even if appendages are thermally deformed,the controllers remain effective. In general, larger control torques are neededfor the satellite with thermally deformed appendages.(xi) The control of the space station during MSS maneuvers using the QCLCis again proved to be feasible. The controllers are effective in damping outattitude motion induced by OP maneuvers of the robotic arms. In turn, thepointing accuracy of the robotic arms improve significantly. Even the offsetof robotic arms does not adversely affect the performance of the controllersexcept that larger control torques are required. In general, the longer themaneuver period implies smaller control effort required and p-p fluctuationsof the pointing errors.6.2 Recommendations for Future WorkThe formulation presented in this thesis is relatively general; however, there is ascope for improvement. A few features that would enhance its general character andimplementation efficiency include the following.235(i) As pointed out in Chapter 3, there are two areas in subroutine FCN whichcan be executed in parallel resulting in a substantial saving in computationaltime and cost. If parallel processing facility is available, the computer codeshould be modified to take its advantage.(ii) The convergence of the beam shape functions is shown in Figure 3-11. Itwould be useful to study the convergence of the shape functions of cantileverplates. The formulation accounts for transverse vibration, thermal deforma-tions, and slewing maneuvers of the members. Other possible member motionnot accounted for includes longitudinal vibration, translational motion, andtorsional oscillations. As pointed out in Chapter 4, the present formulationwas tailored to facilitate the deployment/retrieval study of the SFU. A for-mulation that accounts for deployment/retrieval of arbitrary members wouldbe useful.(iii) As shown in Table 2-1, shape functions based on Warburton [94] are inad-equate for modeling free-free plates. Incorporation of Gorman [96] shapefunctions in the general formulation is one alternative. Another option wouldbe to use quasi-comparison functions put forward by Meirovitch and Kwak[88].(iv) Thermal deformation of flexible members is the only environmental distur-bance considered in the thesis. This is because it affects both high as wellas low altitude spacecraft. However, for spacecraft at high altitude, solarradiation pressure effect may also be significant whereas at lower altitudes,aerodynamic drag may become significant. Inclusion of these two environ-mental disturbances would add to make the versatility of the formulation.The dynamics and control studies of the five spacecraft configurations, though236comprehensive, are not complete. Obviously, there are numerous system parametersand countless possibilities for initial conditions (disturbances) that can be studied.Some of the simulations which are likely to be enlightening are indicated below.(i) As the nominal orientations of the FEL and PMC are unstable, active controlis necessary. It would be useful to investigate the feasibility of alternativeorientations with respect to the control effort. The effect of slewing maneu-vers of PV arrays on the dynamics of the FEL and PMC is likely to be ofimportance.(ii) The study of the MSS presented in thesis is based on a gravity stabilized andrigid space station. Performance of the MSS under different orientations ofthe flexible space station is not easy to predict. Only a detailed investigationinto orientation and flexibility effects can provide the insight needed.(iii) The deployment/retrieval profile of the SFU assumed constant velocity. Thestability study of the SFU under the influence of more efficient profiles wouldbe a logical extension to the present study.(iv) Control based on the FLT is promising; however, it needs to be further ex-plored. In particular, a more systematic approach to select the controllergains and desired trajectory is needed. Furthermore, performance of the con-troller in the presence of logical constraints needs attention. The problems ofrobustness and bandwidth will have to be addressed. Investigations aimed atthe FEL, PMC, and SFU would result in important and timely contributionsto the field. It would be particularly useful to investigate performance of theQCLC algorithm in the presence of deploying or retrieving members.237BIBLIOGRAPHY[1] Taylor, L., "The SCOLE Design Challenge - An Overview," Proceedings of the2nd Annual SCOLE Workshop, Hampton, Virginia, December 1985, pp. 1-13.[2] Singh, R.P., VanderVoort, R.J., and Likins, P.W., "Dynamics of Flexible Bod-ies in Tree-Topology - A Computer-Oriented Approach," Journal of Guidance,Control and Dynamics, Vol. 8, No. 5, September - October 1985, pp. 584-590.[3] Ho, J.Y.L., and Herber, D.R., "Development of Dynamics and Control Sim-ulation of Large Flexible Space Systems," Journal of Guidance, Control andDynamics, Vol. 8, No. 3, May - June 1985, pp. 374-383.[4] Ho, J.Y.L., Herber, D.R., Clapp, B.R., and Schultz, R.J., "ALLFLEX Pro-gram - Simulation Methodology," Proceedings of the Workshop on MultibodySimulation, Eds: G. Man and R. Laskin, April 1988, pp. 811-856.[5] Ho, J.Y.L., "Direct Path Method for Flexible Multibody Spacecraft Dynam-ics," Journal of Spacecraft and Rocket, Vol. 14, No. 2, February 1977, pp. 102-110.[6] Chun, H.M., Turner, J.D., and Frisch, H.P., "Experimental Validation of Or-der (N) DISCOS," AAS/AIAA Astrodynamics Specialist Conference, Stowe,Vermont, August 1989, Paper No. AAS-89-457.[7] Likins, P., "Spacecraft Attitude Dynamics and Control - A Personal Perspec-tive on Early Developments," Journal of Guidance, Control and Dynamics,Vol. 9, No. 2, March - April 1986, pp. 129-134.[8] Modi, V.J., "Attitude Dynamics of Satellites with Flexible Appendages - ABrief Review," Journal of Spacecraft and Rocket, Vol. 11, No. 11, November1974, pp. 743-751.[9] Shrivastava, S.K., "Effects of Solar Radiation Pressure and Aerodynamic Forceson Satellite - Attitude Dynamics and Their Utilization for Control: A Sur-vey," Journal of the Indian Institute of Science, Vol. 58, No. 9, September1976, pp. 391-411.[10] Shrivastava, S.K., Tschann, C., and Modi, V.J., "Librational Dynamics ofEarth Oriented Satellites - A Brief Review," Proceedings of 1.4th Congress onTheoretical and Applied Mechanics, Kurukshetra, India, 1969, pp. 284-306.238[11] Roberson, R.E., "Two Decades of Spacecraft Attitude Control," Journal ofGuidance and Control, Vol. 2, No. 1, January —February 1979, pp. 3-8.[12] Markland, C.A., "A Review of the Attitude Control of Communications Satel-lites," 32nd Congress of the International Astronautical Federation, Rome,Italy, 1981, Paper No. IAF-81-344.[13] Lips, K.W., Dynamics of a Large Class of Satellites with Deploying Flexi-ble Appendage, Ph.D. Thesis, The University of British Columbia, September1980.[14] Ibrahim, A.M., Mathematical Modelling of Flexible Multibody Dynamics withApplications to Orbiting Systems, Ph.D. Thesis, The University of BritishColumbia, April 1988.[15] Chan, J.K., Dynamics and Control of an Orbiting Space Platform Based MobileFlexible Manipulator, M.A.Sc. Thesis, The University of British Columbia,April 1990.[16] Hooker, W.W., and Margulies, G., "The Dynamical Attitude Equations for anu-Body Satellite," The Journal of the Astronautical Sciences, Vol. 12, No. 4,1965, pp. 123-128.[17] Roberson, R.E., and Wittenburg, J., "A Dynamical Formalism for an Arbi-trary Number of Interconnected Rigid Bodies, with Reference to the Problemof Satellite Attitude Control," Proceedings of the 3rd International Federationon Automatic Control, London, 1966, Paper No. 46D.[18] Roberson, R.E., "A Form of the Translational Dynamical Equations for Rela-tive Motion in Systems of Many Non-Rigid Bodies," Acta Mechanica, Vol. 14,1972, pp. 297-308.[19] Hooker, W.W., "Equations of Motion for Interconnected Rigid and ElasticBodies: A Derivation Independent of Angular Momentum," Celestial Mechan-ics, Vol. 11, No. 1,1975, pp. 337-359.[20] Jerkovsky, W., "The Structure of Multibody Dynamics Equations," Journalof Guidance and Control, Vol. 1, No. 3, May —June 1978, pp. 173-182.[21] Hooker, W.W., "A Set of r Dynamical Attitude Equations for an Arbitraryn-Body Satellite Having r Rotational Degrees of Freedom," AIAA Journal,Vol. 8, No. 7, July 1970, pp. 1205-1213.239[22] Hughes, P.C., "Dynamics of a Chain of Flexible Bodies," The Journal of theAstronautical Sciences, Vol. 27, No. 4, October — December 1979, pp. 359-380.[23] Kane, T.R., and Levinson, D.A., "Formulation of Equations of Motion forComplex Spacecraft," Journal of Guidance and Control, Vol. 3, No. 2, March —April 1980, pp. 99-112.[24] Modi, V.J., and Ibrahim, A.M., "A General Formulation for Librational Dy-namics of Spacecraft with Deploying Appendages," Journal of Guidance, Con-trol and Dynamics, Vol. 7, No. 5, September — October 1984, pp. 563-569.[25] Meirovitch, L., and Quinn, R.D., "Equations of Motion for Maneuvring Flex-ible Spacecraft," Journal of Guidance, Control and Dynamics, Vol. 10, No. 5,September — October 1987, pp. 453-465.[26] Vu-Quoc, L., and Simo, J.C., "Dynamics of Earth-Orbiting Flexible Satelliteswith Multibody Components," Journal of Guidance, Control and Dynamics,Vol. 10, No. 6, November — December 1987, pp. 549-558.[27] Keat, J.E., "Multibody System Order n Dynamics Formulation Based onVelocity Transform Method," Journal of Guidance, Control and Dynamics,Vol. 13, No. 2, March — April 1990, pp. 207-212.[28] Huston, R.L., "Multibody Dynamics Formulations Via Kane's Equations,"AIAA Dynamics Specialist Conference, Long Beach, California, April 1990,Paper No. AIAA-90-1248—CP.[29] Kurdila, A.J., "Multibody Dynamics Formulations Using Maggi's Approach,"AIAA Dynamics Specialist Conference, Long Beach, California, April 1990,Paper No. AIAA-90-1248—CP.[30] Balas, M.J., "Some Trends in Large Space Structure Control Theory: FondestHopes; Wildest Dreams," Proceedings of the 1979 Joint Automatic Control,Denver, Colorado, June 1979, pp. 42-53.[31] Meirovitch, L., and Oz, H., "An Assessment of Methods for the Control ofLarge Space Structures," Proceedings of the 1979 Joint Automatic Control,Denver, Colorado, June 1979, pp. 34-41.[32] Meirovitch, L., and Oz, H., "Modal-Space Control of Distributed GyroscopicSystems," Ignorable Coordinates," Journal of Guidance and Control, Vol. 3,No. 2, March — April 1980, pp. 140-150.240[33] Meirovitch, L., and Oz, H., "Modal-Space Control of Large Flexible SpacecraftPossessing Ignorable Coordinates," Journal of Guidance and Control, Vol. 3,No. 6, November — December 1980, pp. 569-577.[34] Meirovitch, L., Baruh, H., and Oz, H., "A Comparison of Control Techniquesfor Large Flexible Systems," Journal of Guidance, Control and Dynamics,Vol. 6, No. 4, July — August 1983, pp. 302-310.[35] Oz, H., and Meirovitch, L., "Optimal Modal-Space Control of Flexible Gyro-scopic Systems," Journal of Guidance and Control, Vol. 3, No. 3, May —June1980, pp. 218-226.[36] Balas, M.J., "Active Control of Flexible Systems," Journal of OptimizationTheory and Applications, Vol. 25, No. 3, July 1978, pp. 415-436.[37] Balas, M.J., "Feedback Control of Flexible Systems," IEEE Transactions onAutomatic Control, Vol. AC-23, August 1978, pp. 673-679.[38] Balas, M.J., "Modal Control of Certain Flexible Dynamic Systems," SIAMJournal of Control and Optimization, Vol. 16, May 1978, pp. 450-462.[39] Balas, M.J., and Ginter, S., "Attitude Stabilization of Large Flexible Space-craft," Journal of Guidance and Control, Vol. 4, No. 5, September — October1981, pp. 561-564.[40] Balas, M.J., "Direct Velocity Feedback Control of Large Space Structures,"Journal of Guidance and Control, Vol. 2, No. 3, May — June 1979, pp. 252—253.[41] Wie, B., and Bryson, A.E. Jr., "Pole-Zero Modeling of Flexible Space Struc-tures," Journal of Guidance, Control and Dynamics, Vol. 11, No. 6, Novem-ber — December 1988, pp. 554-561.[42] Wie, B., "Active Vibration Control Synthesis for the Control of FlexibleStructures Mast Flight System," Journal of Guidance, Control and Dynamics,Vol. 11, No. 3, May — June 1988, pp. 271-277.[43] Chu, P.Y., Wie, B., Gretz, B., and Plescia, C., "Space Station Attitude Con-trol: Modeling and Design," AIAA Guidance, Navigation and Control Con-ference, August 1988, Minneapolis, Minnesota, Paper No. AIAA-88-4133.[44] Kida, T., Ohkami, Y., and Sambongi, S., "Poles and Transmission Zeros ofFlexible Spacecraft Control Systems," Journal of Guidance, Control and Dy-241namics, Vol. 8, No. 2, March — April 1985, pp. 208-213.[45] Goh, C.J., and Caughey, T.K., "A Quasi-Linear Vibration Suppression Tech-nique for Large Space Structures via Stiffness Modification," InternationalJournal of Control, Vol. 41, No. 3,1985, pp. 803-811.[46] Reddy, A.S.S.R., Bainum, P.M., Krishna, R., and Hamer, H.A., "Control of aLarge Flexible Platform in Orbit," Journal of Guidance and Control, Vol. 4,No. 6, November — December 1981, pp. 642-649.[47] Yedavalli, R.K., "Critical Parameter Selection in the Vibration Suppression ofLarge Space Structures," Journal of Guidance, Control and Dynamics, Vol. 7,No. 3, May — June 1984, pp. 274-278.[48] Sundararajan, N., Joshi, S.M., and Armstrong, E.S., "Robust Controller Syn-thesis for a Large Flexible Space Antenna," Journal of Guidance, Control andDynamics, Vol. 10, No. 2, March — April 1987, pp. 201-208.[49] Modi, V.J., and Brereton, R.C., "Planar Librational Stability of a FlexibleSatellites," AIAA Journal, Vol. 6, No. 3, March 1968, pp. 511-517.[50] Modi, V.J., and Flanagan, R.C., "Effect of Environmental Forces on the Atti-tude of Gravity Oriented Satellites, Part I: High Altitude Orbits," The Aero-nautical Journal of the Royal Aeronautical Society, Vol. 75, November 1971,pp. 783-793.[51] Modi, V.J., and Flanagan, R.C., "Effect of Environmental Forces on the At-titude of Gravity Oriented Satellites, Part II: Intermediate Altitude OrbitsAccounting for Earth Raidation," The Aeronautical Journal of the Royal Aero-nautical Society, Vol. 75, December 1971, pp. 846-849.[52] Flanagan, R.C., and Modi, V.J., "Effect of Environmental Forces on the At-titude Dynamics of Gravity Oriented Satellites, Part III: Close Earth OrbitsAccounting for Aerodynamic Forces," The Aeronautical Journal of the RoyalAeronautical Society, Vol. 76, January 1972, pp. 34-40.[53] Modi, V.J., and Flanagan, R.C., "Librational Damping of a Gravity OrientedSystem Using Solar Radiation Pressure," The Aeronautical Journal of theRoyal Aeronautical Society, Vol. 75, August 1971, pp. 560-564.[54] Modi, V.J., and Kumar, K., "Librational Dynamics of a Satellite with Ther-mally Flexed Appendages," The Journal of the Astronautical Sciences, Vol. 25,No. 1, January — March 1977, pp. 3-20.242[55] Beletsky, V.V., and Starostin, E.L., "Planar Librations of a Gravity OrientedSatellite Under the Influence of Solar Radiation Pressure," 38th Congress ofthe International Astronautical Federation, Brighton, U.K., October 10-17,1987, Paper No. IAF-87-352.[56] Goldman, R.L., "Influence of Thermal Distortion on Gravity Gradient Stabi-lization," Journal of Spacecraft and Rocket, Vol. 12, No. 7, July 1975, pp. 406—413.[57] Yu, Y.Y., "Thermally Induced Vibration and Flutter of a Flexible Boom,"Journal of Spacecraft and Rocket, Vol. 6, No. 8, August 1969, pp. 902-910.[58] Frisch, H.P., "Thermally Induced Vibrations of Long Thin-Walled Cylindersof Open Section," Journal of Spacecraft and Rocket, Vol. 7, No. 8, August1970, pp. 897-905.[59] Frisch, H.P., "Thermally Induced Response of Flexible Structures: A Methodfor Analysis," Journal of Guidance and Control, Vol. 3, No. 1, January —February 1980, pp. 92-94.[60] Tsuchiya, K., "Thermally Induced Nutational Body Motion of a SpinningSpacecraft with Flexible Appendages," AIAA Journal, Vol. 13, No. 4, April1975, pp. 448-453.[61] Tsuchiya, K., "Thermally Induced Vibrations of a Flexible Appendage At-tached to a Spacecraft," AIAA Journal, Vol. 15, No. 4, April 1977, pp. 505—510.[62] Krishna, R., and Bainum, P.M., "Effect of Solar Radiation Disturbance ona Flexible Beam in Orbit," AIAA 21st Aerospace Sciences Meeting, Reno,Nevada, January 1981, Paper No. AIAA-83-0431.[63] Bainum, P.M., and Krishna, R., "Control of an Orbiting Flexible Square Plat-form in the Presence of Solar Radiation," 1.4th International Symposium onSpace Technology and Science, Tokyo, Japan, May — June 1984, Paper No.i-2-1.[64] Krishna, R., and Bainum, P.M., "Orientation and Shape Control of an Orbit-ing Flexible Beam under the Influence of Solar Radiation Pressure," AAS/AIAAAstrodynamics Conference, Lake Placid, N.Y., August 1983, Paper No. AAS-83-325.[65] Krishna, R., and Bainum, P.M., "Environmental Effects on the Dynamics243and Control of an Orbiting Large Flexible Antenna System," 35th Interna-tional Astronautical Congress, Lausanne, Switzerland, October 1984, PaperNo. IAF-84-358.[66] Krishna, R., and Bainum, P.M., "Dynamics and Control of Orbiting FlexibleStructures Exposed to Solar Radiation," Journal of Guidance, Control andDynamics, Vol. 8, No. 5, September — October 1985, pp. 591-596.[67] Jordan, P.F., "Comment on 'Thermally Induced Vibration and Flutter of Flex-ible Booms'," Journal of Spacecraft and Rocket, Vol. 8, No. 2, February 1971,pp. 204-205.[68] Augusti, G., "Comment on 'Thermally Induced Vibration and Flutter of Flex-ible Booms'," Journal of Spacecraft and Rocket, Vol. 8, No. 2, February 1971,pp. 202-204.[69] Sikka, S., Iqbal, M., and Aggarwala, B.D., "Temperature Distribution andCurvature Produced in Long Solid Cylinders in Space," Journal of Spacecraftand Rocket, Vol. 6, No. 8, August 1969, pp. 911-916.[70] Lutz, J.D., Allen, D.H., and Haisler, W.E., "Finite Element Model for theThermoelastic Analysis of Large Composite Space Structures," Journal ofSpacecraft and Rocket, Vol. 24, No. 5, September — October 1987, pp. 430-436.[71] Ng, A.C., Dynamics of Gravity Oriented Axi-Symmetric Satellites with Ther-mally Flexed Appendages, M.A.Sc. Thesis, The University of British Columbia,November 1986.[72] Silver, W.M., "On the Equivalence of Lagrangian and Newton-Euler Dynamicsfor Manipulators," The International Journal of Robotics, Vol. 1, No. 2,1982,pp. 60-70.[73] Hurty, W.C., "Dynamic Analysis of Structural Systems Using ComponentModes," AIAA Journal, Vol. 3, No. 4,1965, pp. 678-685.[74] Craig, R.R. Jr., and Bampton, M.C.C., "Coupling of Substructures for Dy-namic Analysis," AIAA Journal, Vol. 6, No. 7,1968, pp. 1313-1319.[75] Benfield, W.A., and Hruda, R.F., "Vibration Analysis of Structures by Com-ponent Mode Substitution," AIAA Journal, Vol. 9, No. 7,1971, pp. 1255-1261.[76] MacNeal, R.H., "A Hybrid Method of Component Mode Synthesis," Comput-244ers and Structures, Vol. 1, No. 4, December 1971, pp. 581-601.[77] Rubin, S., "Improved Component-Mode Representation for Structural Dy-namic Analysis," AIAA Journal, Vol. 13, No. 8,1975, pp. 995-1006.[78] Hughes, P.C., "Dynamics of Flexible Space Vehicles with Active Attitude Con-trol," Celestial Mechanics, Vol. 9, No. 1,1974, pp. 22-39.[79] Hughes, P.C., "Modal Identities for Elastic Bodies, with Application to VehicleDynamics and Control," Transaction of ASME, Journal of Applied Mechan-ics„ Vol. 47, No. 1, March 1980, pp. 177-184.[80] Hablani, H.B., "Constrained and Unconstrained Modes: Some Modeling As-pects of Flexible Spacecraft," Journal of Guidance and Control, Vol. 5, No. 2,March — April 1982, pp. 164-173.[81] Hughes, P.C., "Space Structure Vibration Modes: How Many Exist? WhichOnes Are Important?" IEEE Control Systems Magazine„ Vol. 7, No. 1, Febru-ary 1987, pp. 22-28.[82] Hughes, P.C., and Skelton, R.E., "Modal Truncation for Flexible Spacecraft,"Journal of Guidance and Control, Vol. 4, No. 3, May — June 1981, pp. 291-297.[83] Gregory, C.Z. Jr., "Reduction of Large Flexible Spacecraft Models Using In-ternal Balancing Theory," Journal of Guidance, Control and Dynamics, Vol. 7,No. 6, November — December 1984, pp. 725-732.[84] Moore, B.C., "Principal Component Analysis in Linear Systems: Controllabil-ity, Observability, and Model Reduction," IEEE Transactions on AutomaticControl„ Vol. AC-26, No. 1, February 1981, pp. 17-32.[85] Spanos, J.T., and Tsuha, W.S., "Selection of Component Modes for the Simu-lation of Flexible Multibody Spacecraft," AAS/AIAA Astrodynamics Special-ist Conference, Stowe, Vermont, August 1989, Paper No. AAS-89-438.[86] Hale, A.L., and Meirovitch, L., "A General Substructure Synthesis Methodfor the Dynamic Simulation of Complex Structures," Journal of Sound andVibration, Vol. 69, No. 2, 1980, pp. 309-326.[87] Meirovitch, L., and Hale, A.L., "On the Substructure Synthesis Method,"AIAA Journal, Vol. 19, No. 7, 1981, pp. 940-947.[88] Meirovitch, L., and Kwak, M.K., "On the Modeling of Flexible Multi-Body245Systems by the Rayleigh-Ritz Method," AIAA Dynamics Specialist Confer-ence, Long Beach, California, April 1990, Paper No. AIAA-90-1245-CP.[89] Johns, D.J., Thermal Stress Analysis, Pergamon Press, Oxford, U.K., 1965,pp. 48-55.[90] Ng, A.C., and Modi, V.J., "Dynamics of Gravity Oriented Satellites with Ther-mally Flexed Appendages," AAS/AIAA Astrodynamics Specialist Conference,Kalispell, Montana, August 1987, Paper No. AAS-87-432.[91] Blevins, R.D., Formulas for Natural Frequencies and Mode Shapes, Van Nos-trand Reinhold Co., New York, N.Y., 1979, pp. 108-109.[92] Timoshenko, S., and Woinowsky-Kreiger, S., Theory of Plates and Shells,McGraw-Hill Book Co., New York, N.Y., 1959, pp. 79-88.[93] Leissa, A.W., "The Free Vibration of Rectangular Plates," Journal of Soundand Vibration, Vol. 31, No. 3,1973, pp. 257-293.[94] Warburton, G.B., "The Vibration of Rectangular Plates," Proceedings of theInstitution of Mechanical Engineers, Vol. 168, No. 12,1954, pp. 371-384.[95] Gorman, D.J., and Sharma, R.K., "A Comprehensive Approach to the FreeVibration Analysis of Rectangular Plates by Use of the Method of Superposi-tion," Journal of Sound and Vibration, Vol. 47, No. 1,1976, pp. 126-128.[96] Gorman, D.J., Free Vibration Analysis of Rectangular Plates, Elsevier NorthHolland, Inc., New York, New York, 1982, pp. 112-145.[97] Hughes, P.C., Spacecraft Attitude Dynamics, John Wiley & Sons, New York,N.Y., 1986, pp. 18-22.[98] Moran, J.P., "Effects of Planar Libration on the Orbital Motion of a DumbbellSatellite," ARS Journal, Vol. 31, No. 8, August 1961, pp. 1089-1096.[99] Yu, E.Y., "Long-term Coupling Effects Between the Librational and OrbitalMotions of a Satellite," AIAA Journal, Vol. 2, No. 3, March 1964, pp. 553-555.[100] IMSL Library Reference Manual, Vol. 1, IMSL Inc., Houston, Texas, June1980, pp. DGEAR-1 - DGEAR 9.[101] Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equa-tions, Prentice-Hall, Englewood Cliffs, New Jersey, 1971, pp. 158-166.246[102] Space Station Engineering Data Book, NASA SSE-E-87-R1, NASA Space Sta-tion Program Office, November 1987.[103] Modal Analysis of Selected Space Station Configurations, NASA SSE-E-88-R8,NASA Space Station Program Office, Washington D.C., June 1988.[104] Kane, T.R., Likins, P.W., and Levinson, D.A., Spacecraft Dynamics, McGraw—Hill Book Co., New York, N.Y., 1983, pp. 199-211.[105] Modi, V.J., and Ng, A.C., "Dynamics of Axisymmetric Satellite Under theInfluence of Solar Radiation: Analytical and Numerical Approaches," TheJournal of the Astronautical Sciences, Vol. 37, No. 1, January — March 1989,pp. 17-40.[106] Space Station Microgravity Environment Status, NASA SSE-E-88-R12, NASASpace Station Program Office, Washington, D.C., June 1988.[107] Small Space Platform Working Group, Advanced Technology Experiment On-board Space Flyer Unit (SFU), Institute of Space and Astronautical Scienceand Space Station Task Team, Japan, March 1986, pp. 115-202.[107] Freund, E., "The Structure of Decoupled Nonlinear Systems," InternationalJournal of Control, Vol. 21, No. 3,1975, pp. 443-450.[108] Freund, E., "Fast Nonlinear Control with Arbitrary Pole-Placement for In-dustrial Robots and Manipulators," The International Journal of RoboticsResearch, Vol. 1, No. 1,1982, pp. 65-78.[109] Slotine, J.E. and Sastry, S.S., "Tracking Control of Non-linear Systems us-ing Sliding Surfaces with Application to Robot Manipulators," InternationalJournal of Control, Vol. 38, No. 2,1983, pp. 465-492.[110] Slotine, J.E., "Sliding Controller Design for Non-linear Systems," InternationalJournal of Control, Vol. 40, No. 2,1984, pp. 421-434.[111] Slotine, J.E., "The Robust Control of Robot Manipulators," InternationalJournal of Robotics Research, Vol. 4, No. 2,1985, pp. 49-64.[112] Slotine, J.E., and Li, W., "On the Adaptive Control of Robot Manipulators,"International Journal of Robotics Research, Vol. 6, No. 3,1987, pp. 49-59.[113] Beijczy, A.K., Robot Arm Dynamics and Control, JPL TM 33-669, CaliforniaInstitute of Technology, Pasadena, California, 1974.247[114] Singh, S.N., and Schy, A.A., "Invertibility and Robust Nonlinear Control ofRobotic Systems," Proceedings of 23rd Conference on Decision and Control,Las Vegas, Nevada, December 1984, pp. 1058-1063.[115] Spong, M.W., and Vidyasagar, M., "Robust Linear Compensator Design forNonlinear Robotic Control," Proceedings of IEEE Conference on Robotics andAutomation, St. Louis, Missouri, March 1985, pp. 954-959.[116] Spong, M.W., and Vidyasagar, M., "Robust Nonlinear Control of Robot Ma-nipulator," Proceedings of the 24th IEEE Conference on Decision and Control,Fort Lauderdale, Florida, December 1985, pp. 1767-1772.[117] Spong, M.W., "Modelling and Control of Elastic Joint Robots," Journal of Dy-namic Systems, Measurement and Control, Vol. 109, December 1987, pp. 310—319.[118] Karray, F., Modi, V.J., and Chan, J.K., "Inverse Control of Flexible Orbit-ing Manipulators," Proceedings of the American Control Conference, Boston,Mass., June 1991, Editor: A.G. Ulsoy, pp. 1909-1912.[119] Modi, V.J., Karray, F., and_ Chan,. J.K., "On the. Control of a Class of Flexi-ble Manipulators Using Feedback Linearization Approach," 42nd Congress ofthe International Astronautical Federation, October, 1991, Montreal, Canada,Paper No. IAF-91-324.248Appendix I:^Details of 1-18y„ I , and fisysThe details of the various components that make up Tsy, of Eq. (2.35) are givenbelow:1^•^•Torb = —2 M Rcm, • Rcm1^n. c.,f•^c.,f•Tern = —1V-L L' CM • L'2Th = —1E Li . ai dmi2 i=1ni3= fm2 ,3 ](ai + Clii) • ( i + C lij) dmi ,j ;1 NTir = ...E I d( pi + T-i + 6.0 • q(pi + T-i + 60 dmii=1 mi+ E f { qdii + ^+ ^+ +j=1 m'i,j{^ ]qClii +^+^+ ij +(q JAL j CZ JILL j)(15ij T-^6i,j)} dmi , iN• (C ipist- ii) dmi j1N= O c • Oc drnc f (C) • (C .Si) dminlcnii=1 mij,(C4U,i,j6ij) • (CZpi,j6ij) drni ,jj 1Th,jr = E [f iii • qcoi +  + 6i ) dm,i + E I. { iii + q}1 mi= i^ni3=1 ri. '2,3{qClii + (OZilii,i + CZ jiti, j)(pii + Tip + oij)} drni ,ji249T= — f 4.--c • ,t-c cn, + —E21 1mc 2 i=1 [19ni (q4-- i ) • (q't- )dm,Th ,ti=1[1:^i) dmi E^(di + qdii) • (C4aist- ij) dm inij=1 mijTh,v = • (Cf^dmif . •^i 3^1^J•(di^CF di •) • (C jF lc. • -) dm' •Ni=if i q(iji + Ti + Si) • (Cm) dmi + E f . .{dfdi,[j=1 mz,3+^+ CULL' ) (Pi.; + Tii^ 1(Cf,i ,aisiii) dmi ,jniTjr ,v = E [f q(pi + Ti +Si) • (q6i ) dmi E f {qdi,i=1 Tii SO}^dmic • Sc Cintc^f^. (q i ) dmii=17Tt = Lcnimi,i (q,i,ai,j+ii) • (q, j,Lti,;(50^.^ (I — 1 )The system inertia, Lys , is given by Eq. (2.36) and is the sum of the followingcomponents:'cm = M [Ccm • Ccrn U CCM CCM] ;^Ih = E [f {di diU —^dmi + E^{ (di + qdii ) • (di +i=i ma^— (di + qdij )(di^dmi,j]Ir =meN+ E [j {(crpi) • (qiox - (crpi)(crpi)} dmini+ E^pii) • (qpi, ipii )uj=1i= 17Pc • PcU — Pc -Pc} dmc250]— (q j it i , i pii )(CUL i jpii ) } dm i ,i ;It= f { 7f, • i-c U —^dm,+E [f {(qT-i) • (qi-ou - (qi-i)(qT-i )} dm ii=1ni f• Emi,i— dmij] ;Iv = f {6, • SOU — ScSc} dm c911c+ [f { (cm) • (q4i )u - (cm)(q6i)} dmii=1ni+ E^{(cuLi jsij ) • (cuL igij)uj=1 rni,jjOij)(q,^jOij)} d'ati ,j];= E [./ {2di • (CF pi)U — di(Crfii) — (qpi)dil dmii=1 mini+E f {2(di + qdii ) • (culi,Jpii)u - (di + qdii )(cu.tigii )j=1 rni,j— jpij)(di qdij)} Chni ,j]= E [f {2di • (Crfi)U — di(qT-i) —^dmii=ini+E f {2(di + qdij) • (q, i tti,i'T-ii)U — (di + C ic. dii )(CU.twij )j=i rni,j—(CFstisrij)(di qdji) dmi,j] ;Th,t, = E[I {2di • (C icA)U — di(q6i) — (CP5i)dil dmii=1Mc251f { 2(di qdij) • (C4tti,j6ii)U . — (di +^jOii)j=i— qdji)} dmi,j] ;{2p, TcU — pcTc — tcpc } dm,+ [fm {2(qpi) • (m-ou - (cpi )(qm - (qT-i)(criii) dmii=1^ini+E f {2(c? jitupii) •^-- dmi,j] ;Ir,v = Lc {2fic • 6,15 — pc s, — scp-c } dm,+ E [f {2(q fii) • (Ci 6i)U — (C i, pi ) (q6i ) — (q6i)(C ilii)} dmini+E f {2(q, Ji-oii ) • (cu,i joix -j=1. rni,j—(CULL^jpii)} dmij ]{21-, • 6,15 — Tcsc — Oc'f, dm,+ [I {2( Ti) • (qoi )u - (q'T-i)(q6,) — (q6i)(q1-i)} dm ii=1ni+E^{2(cu.ti^• (CU.ti,j6ii)U —j=1— j p i, iSii) ( C41.1i , j Tii ) dMi,j] •^ (I — 2)Here U is the unit matrix.The components that contribute to Lys , the system angular momentum vector,= Lc= Lc252Ht = f (7t, X +,)dmc E [f (C; Ti xme ni+ E f x^ •,j=i mi,jare shown in Eq. (2.37) and their details are as follows:Hcm = —M[Ccm x cm] ;niHh = EL/ (di x ;:/i)dmi^f (di +^x (di +i=i mi^j=iAir =E[1 (Pi "Ti + Si) x q(pi^+ Si) dmii=1 mini+E f {qdii +^+ 6ii )}J=1x^ CZjiti,j)(fiii^6ij)}H„ = f(8, X (5. c) dm, + [f (qSi6  x^6i)• dmi771c^ i=172,i+E f^xJ=1 mi,jHh,jr = E[ J. di X^(pi +^(52; ) dmii=1 mini+ E f di x+Tij 6ij)}dmi,iij=1 mi,jniHh,r =E[f^x dmi + f fczpigii x + qdii )} dmi ,j]E mi mi,Jh,t = E [Ii=1 mi(Crfi x di + di xni+E f ^x +J=1 mi,J253= [f (qai x + di x q6i) dmii=1 mini+ E^{q, ;Auk.; x (di+ qaii )j=1 Mi,j+ (di + qdii) X (q j ij)} drnij] ;= f (pc x dm, + E [f (qpi x cF-9;) dmim,^i=i rnini+Ei x j=1 rni,jHroi = f (pc x dritc E [f (qpi x cF;50 dmim,^i=ini+Ei (CicA itupij x^dmi,-];j j=i(Tc X Sc + 6, x c) dm, + E [f (qti x q&; + q(59; x^dmii=i mini+Ei^xj=1+ CICAlli, j6ij X (q, JAI, j+ii)} drnij 1^ (I — 3)Ht,v fmc254Appendix II: Representation of Tsy „ Lys , and Lys in termsof the New OperatorsIn terms of operators T and 0, the components of T,y, in Eq. (I-1) can be writtenas follows:1T,,, = — 2MT(U,U) o e(Ccni ,^;Th =^[f T(U, o e(di , di ) dmii=1ni+ {T(U, U) o e(cri,dD 2T(U,^o 0(d2,j=iT(q, CD 0 e(dii ,Tjr = 12^T(Cr, Cr) 0 e((fii + Ti + Si ) / ( Pi + Ti+ Si)) dmii=1 Li ni+E f {T(cr, Ci') 0 coii , dii )j=1 f+ 2T(Cr, (q,itti,j)') 0 e(dii , (p-ii + ,7=ii + Sii ))]+ T((Cfjiti,iY, (q,illi,i) i ) o e((pii + -iii + Oii), (Ai +i-ii + Oii )) dmi ,j ;Tt = fmc1^T(U^ NU) o ^7f- ) dm, + E j T(q, CD o 001,11) dmi1^[i=1 miT(Cf jp,i , j, C4Lti , j) 0 O( , Tti) drni ,i ;1T„ = 2 fmc T(U , U) 00(6'c , S'c ) dm, +1^[f T(q, CD o^dmii=1ni+E f^citti,J) 0 e(sij ,^dmi,i ;ni+Ej=1255NTh,jr = E f T(U, Cr) o e(di , (pi + Ti + Si )) dm ii=1^nnini+E i {T(U, Cr) o e(d i , di,)j=1 mj,j+ T(U, (C4u,i, .0') 0 Co(d i , (fiij + •ii + Sij))]+ T(q, Cr) 0 e(d i i , (fiii + 7fij + Sii ))} dmi ,iN=Th,t — E f T(U, q) o 0(d'i, Ti) dmii=1 miniTh,]+ T(q, CULi , j) 0 e(Cii, Ili)} d'ini,i[ rnii T(U, CD o e(di , OD dmii=1  {T(U,^o 0(di ,Tir,t =+ T(q, C ip,i,j) 0 0(crii ,6ii )} dm i ,i1N [f r(cr,q) 0 sa((pi + 1 i + 8i), Ti) dmii=1 mjni+E^{T(Cr, CUti,j) o e(d ij ,T1i )j= 1 mi,i + T((q,j1-11,J)', q, j ,Lii,j) 0 eqpii +Tij+^dmi,i[f T(Cr,^°^+^Si) dmiJ= 3. rnini+E f {T(Cr, C4u,i,j) oj=1Tjr,v3= frni,3{T(U, qpi,j) o (di,]+ T ( (qj iti, j Y, CUt i, i ) o 6 ( (Pij +i-Z? + Sij ) , Pii ) 1 dmi,jNTt,v =^T(U, U) 0 0( 7f- ,S1c ) dm c + E [f T(q, C) o 0(72 7 O') dm imcf i=1 mini+ E f^T(qii,,,, j , qJ p,i, j ) 0 0 ('T-tj , 6ij ) dm i ,j .j=1 mi,i(II — 1)The components of inertia Lys in Eq. (I-2) can be written in terms of the oper-ators T, 1-1 , and 0:'cm = —m- [r(u, u) 0 e(ccm , c cm )u—r(u, u) 0^Gem)] ;Th, = E [f {T(U,U) 0 g(di, di)U — r(u, o e(di , di)} dmiimni+ { [T(u, U) 0 g(di , di) + 2T(U, CD 0 g(di , dii )j=1+ T(q, CD 0 e(dij, dii)]U — [c(u, u) 0 e(di, di)+ 21-1 (U, CD 0 0(di, dij) + r(q, CD 0 o(dii , dii)] dm i ,j ]Ir = f {T(U,U) o 0(pc , fic )U — r(u, u) 0 coc , pc )} dm cmc• E[ f {T ,^0 (10i, 100U — r(q, CD 0 e(pi, pi)} dm ii=1 rnini+ E j {T(culi,J, q,p,i, J ) 0 e( fiii , fiii )uj=1—r(cuLi c4u, i , j ) 0 c(Pii , fiij )} dmi ,j] ;f {• TM, U) 0 0(7f,, T-c )U — r(u,u) 0 Cl(i-c ,T-c )} dm c+ E [fm^CD o^-fiyu - r(q, cD 0 ecti ,^dmii=1257f j, CULL j) 0 0(Tii, Tii)Uj=i- ,j) 0 o(Tii , Tij )} drni ,d ;Iv = T(U, U) 00(6c , 6,)U — r(u,^o 0(6,, .5 c)} dm,me+E [ f {T(Cf,^o0(&, &)U — r(q, cp 0 co((5i , Si)} dmii=1.ni+E f^cf,pi,j) 0 e(k,62i ) uj=1- r(cut i^J) 0 o(sii , sij )} dm i ,j] ;= E [f {2T(U, CD 0(di , pi)Ui=1- [r(U, CD o 0(di, pi) 11 (g, U) o e(pi , di)] dm ini+E f {2 [T(U, CU-4, j) 0 e(di,^+ T(q, c4u, i , j ) 0 o(dii , p-ii )]uj=i- [r(u, cutij ) 0 so(di,^+ r(q, cuL i j ) 0 so(dii , p-ii )+ o 0(pii, di) + r(cut i , j , cp 0 e(pii, dii)]} dmi,j] ;N^t = E [f {2T(U,^o 0(di ,i=1- [r(u,^o e(di, Ti) r(q,u) 0^di)] dm ini+E f {2 [T(U, CUti,j) 0 e(di, ti) T(q, CUzi,j) o e(dii,i=1o^Tij,^_- [r(U, cuz i^r (e , CF^) o A(d^)r(CUti,j,U) 0 e(fij, di) r(cuL i J , q) 0^dij)]}^= E [f { 2T(U,^0 0(di, 6i)Ui=1- CD o e(di, 6i) r(q,u) o 0(6i, di)] dm i258Ir,v =ni+E f {2 [T(U, CZ pui,j) e(di,^+ T(q,^e(dij, sii )juj= 1- [11 (U, CULL j) 0 e(di)6ij) r(q, czpi j ) 0 c(dii ,sij )+ r(cuLii ,u) o 0(6.ii, di) + r(q JAL j , cp 0 e(sii , dii)] dmi,j] ;= f {2T(u, u) coc, 'fc )U — r(u,u) [e(fic,tc) + (fc Pc)]} dmcmcN+E [f {2T(q, CD o 0(5i, t-i)Ui=1 mi- r(c, cn 0^+ cl(f, pi)] dmini+ E f {2T(CUzi, j , C4tti,j) o^tii)Uj=1 mi,j- r(C4tti,j,^.1) o [(-3(pij,Tij) +^Aid} dmi ,j];{2T(U, U) 0 e(fic , c)U — r(U, U) 0 [e(fic , o) + 00c , pc)] dmc+ {2T(q, CD o 0C/3i, SOUi=1 mi- r(q, cn 0 [0 (fii, Si) + 0020pi) ] } dmini+E f {2T(C4tti,j, CUti,j) o e(fijj, Sii )Uj=1- j, cul i , j) [e(pij,Oij) + e(Oij, pij)i} dm i ,j];= f {• 2T(U , U) o (tc , 6 c )U — r(u, u) o [e(tc , Sc) +^c,tcd} dmcmcE [f+ {2T(q,^SOUi=1- r(q, cn [eo-i , Si) + e(Oi, ti)] dmini+E f {2T(q j, CZ j) oj=1 259— r(q^\ 0^+ 0(8^d^j p,i , j / o [( ,^_ij, Yid^. (II — 2)Similarly, the angular momentum components of fi ns as given in Eq (I-3) canbe represented by using operators A and 0:Hcm, = —M[A(U,U) o CI(C,,,,C,,,)];171h E[fi=1ni+E {A(U,U) o e(di, di) A(U,^o 0(di ,j=1 rnij▪ A.(C,U) 0 e(dij,^A(q^o g(dii,^dmiHST= i[j A(q, Cr) 0 g(Cfii^+ Si), (pi + + 6i)) dmii=1ni+E^{A(g, cr) o cof ii, did)i▪ A(q jp,i,j, Cr') 0^62i), dii )A(q,^o^(pad +Tad +▪ A(q^(qiiti,in o 13((pii Tij^(pii Tii bidHt =^A(U,U) o 0(1-,,77D dm,meA(U, U) o e(di, di ) dmi+E A(q, o^dmimini+E A(q^o (Tij,j=1 9fli,j—77-1, A(U,U) o 0(6,, 6',) dm,c+E [f A(q,^o 0(6i, 64j ) dmii=1 rni260ni+E f^0(6ii, S'j ) dm i ,j ]j=iA(U, Cr) o e(di, (Pi + + Oi )) dmini+E f {A(u, Cr') 0 e(di, (Pi^+ Si)) dmij=1 mi,i• A(U, (CZ ipi,j)') o 0(4^+ 6ii))} drni,j]Hh,r = E [fi=1 miA(q , U) 0 ( -pi, di ) dm ini+ f {A(C4ai,j,U) 0 e(fiii, di )j=3.A(CZ^j,^o^crii)} dmi ,j]Hh ,t = ^[fm {A(q,u.^di) + A(U,^o e(di ,^dmii=1^ini+E f {A(cuii, ; ,-u) 0^di)A(q^o^crij)J=1 + A(U, CULi,j) o e(di,^A(q, CULi,j) o e(dii,^dmi,j] ;^A(q,U) 0 e(pi, di) + A(U,^0 e(di , 6";)} dmini+E f {A(cu.ii, j ,u) 0 c(Sii , di ) + A(CZ i pi,j, CD 0 0(6ii, crij )j=1 mij+ A(U, CU.ti,j) 0 0(di, blij) A(q, C4ai,j) 0 e(dii,^dmi,j]NHr , t = f A(U, U) 0 e(fic , ;7--D dmc E [f A(q,^0 e(pi , ;fp dmimc^ i=1ni+E f J) 0 e( fiii , f1j ) dmij=1261Lc+ ^[f A(q, q) o [0(T-i, Si) + 0(6i ,^dmii=1 minii=^i,jA(U, U) o [0(1-c , 6 /c ) + 0(6,, 1-- )] dm,A(C i^Cup,,,j) o [0(;fii, 6 1ii ) 0(6ii,2j )] dmi,j]Hr,v = I A(U, o e(fic , 6 1c ) dm, + E [f A(q,o ^e(pi, dmirn,^ i=1ni+ E^A(C4ai,j, C ijp,i,j) o e(pii, 6 1ij ) dmi,j]j=1t,v( II — 3)262Appendix III: A Sample of the Computer Program for theEvaluation of a^llTsys Ia Isys and BrsysIt is impractical to list the whole computer program here as it is over 5,000 lines.Instead, the objective is to illustrate simple algorithms required to assemble aTsys /aq,Isys and _ans . In so doing, only arrays that are used are declared in the DIMENSIONstatements. The actual program would require additional arrays to accomodate theevaluation of other derivatives of Lys , Isys and Hays . The array names are the sameas those in Tables 3-1 and 3-2. A few points that may help understand the notationsused are as follows:(i) The R91, R99, and R93 arrays have the following structures:R99: R99c(m, n, p), R99i(m, n,,p,i), R99ij (mi n, p, AR99j(m,n,,p, j);R93: R93c(m, n, p), R93i(m, n, p, i), R93ij(m,n,p, j),R93j(m,n,p, j);where m (for R91) and m, n (for R99 and R93) refer to U rn , y n n , and Anz,n)respectively, in Eq. (3.4); p is the subscript of T, F, and A in Table 3-1; i andj denote body Bi and Bi ,j , respectively.(ii) In order to save the array space, each B i ,j body is identified by the numbersk and Jcon(k). For instance, consider a system with two Bi bodies (B 1and B2 ) such that B 1 has three Bi ,j bodies attached (B1,1, B1,2, and B1,3)while B2 has only B2,1 attached. Without any identification numbers, theminimum array size for, say R91j, would be R91j(9, 5, 2, 3) where the thirdand fourth dimensions are for identifying Bi and its associated Bi J. Usingthe identification number so that k = 1, 2, 3, 4 for B1,1, B1,2, B1,3, and B1,4,respectively; Jcon(1) = Jcon(2) = Jcon(3) = 1 and Jcon(4) = 2, each Bi ,3remains unambiguously identified but the minimum array for R91j would be263R91j(9, 5, 4) resulting in a saving in the array size.(iii) The arrays for 0 operation have the similar structure as T operation. UsingDH as an example,DH: DHi(m,p,i); DHij(m,p,j); DHj(m,p,j);where m identifies Om of Eq. (3.4d); p is the subscript of 0 in Table 3-2; iand j denote body Bi and Bi, j , respectively.(iv) The generalized coordinates, q, are each assigned a number mq (mq = 1,... ,Nq). The first three, mq = 1,2, 3, are for the librational degrees of freedom,IP, 0, and A, respectively. The rest (mq = 4, ... , Nq) are for vibrationaldegrees of freedom.(v) The prefix Q in the array name represents the (0Iaq) of the array; hence,one more dimension is required to identify the differentiation with respectto the generalized coordinate qmq . For instance, QR91i(m, 3, i, mq ) refers to&um/aging of T3 (C i l , C ie ).(vi) It should be pointed out that, for simplicity, derivative arrays have the addi-tional dimension of length Nq in the sample program here. This length canbe shortened by recognizing the fact that not all arrays are functions of allgeneralized coordinates.264Sample subroutine to calculate OTsys /OqCC^The array QKe represents the T{sys} derivativeCSubroutine KINENE(QKe)Implicit Real*8(a-h,o-z)CC^First declare the common blocksCCommon /No/^N, Nj, Nq, Jcon(Nj)CC^Rmc, Rmi, Rmj are the masses of body Bc, Bi, and Bij,.C^respectivelyCCommon /Ratio/ Rmi(N), Rmj(Nj)CCommon /CM/ CMc(9,4),^QCMc(9,4,Nq)CCommon /DD/ DDi(9,4,N),^QDDi(9,4,Nq,N),2^DDij(9,4,Nj), QDDij(9,4,Nq,Nj),3 DDj(9,4,Nj), QDDj(9,4,Nq,Nj)Common /DF/ DFi(9,2,N),^QDFi(9,2,Nq,N),2^DFij(9,2,Nj), QDFij(9,2,Nq,Nj),3 DFj(9,2,Nj), QDFj(9,2,Nq,Nj)Common /DG/ DGi(9,4,N),^QDGi(9,4,Nq,N),2^DGij(9,4,Nj), QDGij(9,4,Nq,Nj),3 DGj(9,4,Nj), QDGj(9,4,Nq,Nj)Common /DH/ DHi(9,4,N),^QDHi(9,4,Nq,N),2^DHij(9,4,Nj), QDHij(9,4,Nq,Nj),3 DHj(9,4,Nj), QDHj(9,4,Nq,Nj)Common /RR/ RRc(9,1),2^RRi(9,1,6),265C3 RRj(9,1,4)2Common /RT/ RTc(9,3),^QRTc(9,3,Nq),RTi(9,3,N),^QRTi(9,3,Nq,N),3 RTj(9,3,Nj),^QRTj(9,3,Nq,Nj)2Common /RV/ RVc(9,3),^QRVc(9,3,Nq),RVi(9,3,N),^QRVi(9,3,Nq,N),3 RVj(9,3,Nj),^QRVj(9,3,Nq,Nj)C2Common /TT/ TTc(9,4),^QTTc(9,4,Nq),TTi(9,4,N),^QTTi(9,4,Nq,N),3 TTj(9,4,Nj),^QTTj(9,4,Nq,Nj)C2Common /VV/ VVc(9,4),^QVVc(9,4,Nq),VVi(9,4,N),^QVVi(9,4,Nq,N),3 VVj(9,4,Nj),^QVVj(9,4,Nq,Nj)2Common /TV/ TVc(9,4),^QTVc(9,4,Nq),TVi(9,4,N),^QTVi(9,4,Nq,N),3 TVj(9,4,Nj),^QTVj(9,4,Nq,Nj)C2Common /R91/ R91c(9),R91i(9,5,N),^QR91i(9,5,Nq,N),3 R9lij(9,4,Nj),^QR9lij(9,4,Nq,Nj),4 R91j(9,5,Nj),^QR91j(9,5,Nq,Nj)CC^Declare matrices QKe and the temporary matrices QKC^QK(m,n) (m=1,...11; n=1,Nq), refers to the m-th termC^derivative in Eq. (I-1)For instance, QK(1,iq) for T{cm}', QK(2,iq) for T{h}',C^and so onReal*8 QKe(Nq), QK(11,Nq)CData Hf/0.5d0/266CC^Initialize the arrays QKe and QKCDo 7010 iq=1,NqQKe(iq)=0.d07010 ContinueCDo 7012 iq=1,NqDo 7012 in=1,11QK(in,iq)=0.d07012 ContinueCC^Evaluate all terms for body BcCDo 710 iq=1,MqiDo 710 i1=1,9QK(1,iq) =QK(1,iq)-Hf*R91c(i1)*QCMc(i1,1,iq)QK(4,iq) =QK(4,iq)+Hf*R9 -1c(i1)*QTTc(i1,1,iq)QK(11,iq)=QK(11,1q)+Hf*R91c(i1)*QTVc(i1,1,iq)710 ContinueDo 714 i =1,NDo 714 iq=1,NqDo 714 i1=1,9CC^Proceed to Bi bodiesCQK(2,iq) =QK(2,iq)+Hf*R91c(i1)*QDDi(i1,1,iq,i)QK(3,iq) =QK(3,iq)+Hf*QR91i(i1,1,iq,i)*(RRi(i1,1,i)2^+RTi(i1,3,i)+RVi(i1,3,i)3 +TTi(i1,4,i)+VVi(i1,4,i)+TVi(i1,4,i))4^+Hf*R91i(i1,1,i)*(QRTi(i1,3,iq,i)5 +QRVi(i1,3,iq,i)+QTTi(i1,4,iq,i)6^+QVVi(i1,4,iq,i)+QTVi(i1,4,iq,i))QK(4,iq) =QK(4,iq)+Hf*(QR91i(i1,4,iq,i)*TTi(i1,1,i)2^+R91i(i1,4,i)*QTTi(i1,1,iq,i))267QK(5,iq) =QK(5,iq)+Hf*QR91i(i1,4,iq,i)*VVi(i1,1,i)QK(6,iq) =QK(6,iq)+QR91i(i1,3,iq,i)*(2^DFi(i1,1,i)+DGi(i1,2,i)+DHi(i1,2,i))3 +R91i(i1,3,i)*(QDFi(i1,1,iq,i)4^+QDGi(i1,2,iq,i)+QDHi(i1,2,iq,i))QK(7,iq) =QK(7,iq)+QR91i(i1,5,iq,i)*DGi(i1,1,1)2^+R91i(i1,5,i)*QDGi(i1,1,iq,i)QK(8,iq) =QK(8,iq)+QR91i(i1,5,iq,i)*DHi(i1,1,i)2^+R91i(i1,5,i)*QDHi(i1,1,iq,i)QK(9,iq)=QK(9,iq)+QR91i(i1,2,iq,i)*(2^RTi(i1,1,i)+TTi(i1,3,i)+TVi(i1,3,0)3 +R91i(i1,2,i)*(QRTi(i1,1,iq,i)4^+QTTi(i1,3,iq,i)+QTVi(i1,3,iq,i))QK(10,iq)=QK(10,iq)+QR91i(i1,2,iq,i)*(RVi(i1,1,i)+VVi(i1,3,i))2^+R91i(i1,2,i)*QVVi(i1,3,iq,i)QK(11,iq)=QK(11,iq)+Hf*(QR91i(i1,4,iq,i)*TVi(i1,1,i)2^+R91i(i1,4,i)*QTVi(i1,1,iq,i))714 ContinueCC^QKe for all the generalized coordinates are evaluated if BijC^body does not existCDo 718 iq=1,NqDo 718 in=1,11718 QKe(iq)=QKe(iq)+QK(in,iq)CC^Now execute the same procedures for Bij bodies providedC^they existCIf(Nj .eq. 0)Go to 810CC^Initialize the temporary arraysCDo 8020 jq=1,NqDo 8020 jn=1,11QK(jn,jq)=0.d02688020 ContinueCDo 822 j =1,NjDo 822 jq=1,NqDo 822 j1=1,9i=Jcon(j)QK(2,jq) =QK(2,jq)+Hf*(2^(Rmj(j)/Rmi(i))*R91c(j1)*QDDi(j1,1,jq,i)3 +2.d0*(QR91i(j1,5,jq,i)*DDij(j1,1,j)4^+R91i(j1,5,1)*QDDij(j1,1,jq,j))5 +QR91i(j1,4,jq,i)*DDj(j1,1,j)6^+R91i(j1,4,i)*QDDj(j1,1,jq,j))QK(3,jq) =QK(3,jq)+Hf*(QR91i(j1,1,jq,i)*DDj(j1,4,j)2^+R91i(j1,1,0*QDDj(j1,4,jq,j)3 +2.d0*(QR9lij(j1,1,jq,j)*(DFj(j1,2,j)4^+DGj(j1,4,j)+DHj(j1,4,j))+R91ij(j1,1,j)*(QDFj(j1,2 i jq,j)6^+QDGj(j1,4,jq,j)+QDHj(j1,4,jq,j)))7 +QR91j(j1,1,jq,j)*(8^RRj(j1,1,j)+RTj(j1,3,j)+RVj(j1,3,j)9^+TTj(j1,4,j)+VVj(j1,4,j)+TVj(j1,4,j))+R91j(j1,1,j)*(QRTj(j1,3,jq,j)+QRVj(j1,3,jq,j)1^+QTTj(j1,4,jq,j)+QVVj(j1,4,jq,j)+QTVj(j1,4,jq,j)))QK(4,jq) =QK(4,jq)+Hf*(QR91j(j1,4,jq,j)*TTj(j1,1,j)2^+R91j(j1,4,j)*QTTj(j1,1,jq,j))QK(5,jq) =QK(5,jq)+Hf*(QR91j(j1,4,jq,j)*VVj(j1,1,j)2^+R91j(j1,4,j)*QVVj(j1,1,jq,j))QK(6,jq) =QK(6,jq)+QR91i(j1,3,jq,i)*DDij(j1,2,j)2^+R91i(j1,3,0*QDDij(j1,2,jq,j)3 +QR91j(j1,3,jq,j)*(DFij(j1,1,j)4^+DGij(j1,2,j)+DHij(j1,2,j))5 +R91j(j1,3,j)*(QDFij(j1,1,jq,j)6^+QDGij(j1,2,jq,j)+QDHij(j1,2,jq,j))7 +QR91i(j1,2,jq,i)*DDj(j1,3,j)8^+R91i(j1,2,j)*QDDj(j1,3,jq,j)2699^+QR9lij(j1,3,jq,j)*(DFj(j1,1,j)+DGj(j1,2,j)+DHj(j1,2,j))1^+R91ij(j1,3,j)*(QDFj(j1,1,jq,j)2 +QDGj(j1,2,jq,j)+QDHj(j1,2,jq,j))QK(7,jq) =QK(7,jq)+QR91j(j1,5,jq,j)*DGij(j1,1,j)2^+R91j(j1,5,j)*QDGij(j1,1,jq,j)3 +QR91ij(j1,4,jq,j)*DGj(j1,1,j)4^+R91ij(j1,4,j)*QDGj(j1,1,jq,j)QK(8,jq) =QK(8,jq)+QR91j(j1,5,jq,j)*DHij(j1,1,j)2^+R91j(j1,5,j)*QDHij(j1,1,jq,j)3 +QR91ij(j1,4,jq,j)*DHj(j1,1,j)4^+R91ij(j1,4,j)*QDHj(j1,1,jq,j)QK(9,jq) =QK(9,jq)+QR9lij(j1,2,jq,j)*DGj(j1,3,j)2^+R91ij(j1,2,j)*QDGj(j1,3,jq,j)3 +QR91j(j1,2,jq,j)*(4^RTj(j1,1,j)+TTj(j1,3,j)+TVj(j1,3,j))5 +R91j(j1,2,j)*(QRTj(j1,1,jq,j)6^+QTTj(j1,3,jq,j)+QTVS(j1,3,jq,j))QK(10,jq)=QK(10,jq)+QR9lij(j1,2,jq,j)*DHj(j1,3,j)2^+R91ij(j1,2,j)*QDHj(j1,3,jq,j)3 +QR91j(j1,2,jq,j)*(RVj(j1,1,j)+VVj(j1,3,j))4^+R91j(j1,2,j)*QVVj(j1,3,jq,j)QK(11,jq)=QK(11,jq)+Hf*(QR91j(j1,4,jq,j)*TVj(j1„j)2^+R91j(j1,4,j)*QTVj(j1,1,jq,j))822 ContinueCC^Hence QKe for all the generalized coordinates are evaluatedCDo 828 jq=1,NqDo 828 jn=1,11828 QKe(jq)=QKe(jq)+QK(jn,jq)C810 ContinueStopEnd270Subroutine to calculate the Lys of the systemCC^The I{sys} of the system is denoted by the 3X3 array ImatC^The common blocks, except for R99, are the same as KINENEC^and are omitted hereCSubroutine INEMAT(Imat)Implicit Real*8(a-h, o-z)CC^Common /R99/ R99c(9,9),^R99i(9,9,2,N),2^R99ij(9,9,1,Nj), R99j(9,9,2,Nj)CReal*8^Imat(3,3)CC^Set up the temporary matricesC^In Eq. (I-2), each component has two termsC^The first one is a scalar product times the unit matrixC^The second one is the product of two 3X3 matricesC^The scalar product is stored in Sc(m)C^(m=1,...,11) for the m-th term in Eq. (I-2)C^Am(m,n,p)=Sc(m)X the unit matrixC^Bm(m,n,p) is the second term of each componentC^Dm(m,n,p) (m=1,...,4) are the dummy matricesCReal*8 Am(11,3,3), Bm(11,3,3), Dm(4,3,3), Sc(11)CC^First, initialize the matricesCDo 5000 i1=1,115000 Sc(i1,1)=0.d0CDo 5002 i2=1,3Do 5002 i1=1,35002 Imat(i1,i2)=0.d0CDo 5004 i2=1,3271Do 5004 il=1,3Do 5004 in=1,11Am(in,i1,i2)=0.d05004 Bm(in,i1,i2)=0.d0Do 5006 i2=1,3Do 5006 i1=1,3Do 5006 in=1,35006 Dm(in,i1,i2)=0.d0CC^First start with body Bc to calculate the first termCDo 500 il=1,9Sc(1) =Sc(1)-R91c(i1)*CMc(i1,4)Sc(3)=Sc(3)+R91c(i1)*RRc(i1,1)Sc(4)=Sc(4)+R91c(i1)*TTc(i1,4)Sc(5) =Sc(5)+1191c(i1)*VVc(i1,4)Sc(9) =Sc(9)+R91c(i1)*RTc(i1,3)Sc(10)=Sc(10)+R91c(i1)*RVc(i1,3)Sc(11)=Sc(11)+R91c(i1)*TVc(i1,4)500 ContinueCC^Now proceed the calculation for Bi bodiesCDo 502 i =1,NiDo 502 i1=1,9Sc(2) =Sc(2)+R91c(i1)*DDi(i1,4,i)Sc(3)=Sc(3)+R91i(i1,4,0*RRi(i1,1,i)Sc(4)=Sc(4)+R91i(i1,4,i)*TTi(i1,4,i)Sc(5)=Sc(5)+R91i(i1,4,i)*VVi(i1,4,i)Sc(6)=Sc(6)+2.dO*R91i(i1,5,0*DFi(i1,2,i)Sc(7)=Sc(7)+2.dO*R91i(i1,5,i)*DGi(i1,4,i)Sc(8)=Sc(8)+2.dO*R91i(i1,5,i)*DHi(i1,4,i)Sc(9) =Sc(9)+R91i(i1,4,i)*RTi(i1,3,i)272Sc(10)=Sc(10)+R91i(i1,4,i)*RVi(i1,3,i)Sc(11)=Sc(11)+R91i(i1,4,i)*TVi(i1,4,i)502 ContinueCDo 504 i1=1,3Do 504 in=1,11504 Am(in,i1,i1)=Sc(in)CC^Second, the calculation of the second term for BcCDo 506 i2=1,3Do 506 i1=1,3Do 506 i3=1,9Cia=(i1-1)*3+i2Bm(1,i1,i2) =Bm(1,i1,i2)-R99c(ia,i3)*CMc(i3,4)Bm(3,i1,i2) =Bm(3,i1,i2)+R99c(ia,i3)*RRc(i3,1)=Bm(4,i1,i2)+11.99c(ia,i3)*TTc(i3,4)Bm(5,i1,i2) =Bm(5,i1,i2)+R99c(ia,i3)*VVc(i3,4)Bm(9,i1,i2) =Bm(9,i1,i2)+R99c(ia,i3)*RTc(i3,3)Bm(10,i1,i2)=Bm(10,i1,i2)+R99c(ia,i3)*RVc(i3,3)Bm(11,i1,i2)=Bm(11,i1,i2)+R99c(ia,i3)*TVc(i3,4)506 ContinueCC^Repeat the calculation of the second term for BiCDo 508 i =1,NDo 508 i2=1,3Do 508 i1=1,3Do 508 i3=1,9ia=(i1-1)*3+i2Bm(2,i1,i2) =Bm(2,i1,i2)+R99c(ia,i3)*DDi(i3,4,i)Bm(3,i1,i2) =Bm(3,i1,i2)+R99i(ia,i3,1,0*RRi(i3,1,i)Bm(4,i1,i2) =Bm(4,i1,i2)+R99i(ia,i3,1,i)*TTi(i3,4,i)Bm(5,i1,i2) =Bm(5,i1,i2)+R99i(ia,i3,1,i)*VVi(i3,4,i)273Dm(1,i1,i2) =Dm(1,i1,i2)+R99i(ia,i3,2,i)*DFi(i3,2,i)Dm(2,i1,i2) =Dm(2,i1,i2)+R99i(ia,i3,2,i)*DGi(i3,4,i)Dm(3,i1,i2) =Dm(3,i1,i2)+R99i(ia,i3,2,i)*DHi(13,4,1)Bm(9,i1,i2) =Bm(9,i1,i2)+R99i(ia,i3,1,i)*RTi(i3,3,i)Bm(10,i1,i2)=Bm(10,i1,i2)+R99i(ia,i3,1,i)*RVi(i3,3,i)Bm(11,11,i2)=Bm(11,i1,i2)+R99i(ia,i3,1,i)*TVi(i3,4,i)508 ContinueDo 510 i =1,NDo 510 i2=1,3Do 510 i1=1,3Bm(6,i1,i2)=Dm(1,i1,i2)+Dm(1,i2,i1)Bm(7,i1,i2)=Dm(2,i1,i2)+Dm(2,i2,i1)Bm(8,i1,i2)=Dm(3,i1,i2)+Dm(3,i2,i1)510 ContinueCC^Evaluate I{sys} for the system Bc and Bi bodiesCDo 512 12=1,3Do 512 i1=1,3Do 512 in=1,11Imat(i1,i2)=Imat(i1,i2)+Am(in,i1,i2)-Bm(in,i1,i2)512 ContinueCC^Proceed if Bij is not zeroIf(Nj .eq. 0)Go to 600CC^Initialize the matricesCDo 6000 j1=1,116000 Sc(j1)=0.d0CDo 6002 j2=1,3Do 6002 j1=1,3274Do 6002 jn=1,11Am(jn,j1,j2)=0.d0Bm(jn,j1,j2)=0.d06002 ContinueCDo 6004 j2=1,3Do 6004 j1=1,3Do 6004 jn=1,46004 Dm(jn,j1,j2)=0.d0CC^Begin the first term calculation for Bij bodiesCDo 602 j =1,NjDo 602 j1=1,9i=Jcon(j)Sc(2) =Sc(2)+(Rmj(j)/Rmi(i))*R91c(j1)*DDi(j1,4,i)2^+2.d0*R91i(j1,5,i)-*DDij(j1,4,j)3^+R91i(j1,4,0*DDj(j1,4,j)Sc(3) =Sc(3)+R91j(j1,4,j)*RRj(j1,1,j)Sc(4)=Sc(4)+R91j(j1,4,j)*TTj(j1,4,j)Sc(5)=Sc(5)+R91j(j1,4,j)*VVj(j1,4,j)Sc(6)=Sc(6)+2.d0*(R91j(j1,5,j)*DFij(j1,2,j)2^+R91ij(j1,4,j)*DFj(j1,2,j))Sc(7)=Sc(7)+2.d0*(R91j(j1,5,j)*DGij(j1,4,j)2^+R9lij(j1,4,j)*DGM1,4,j))Sc(8)=Sc(8)+2.d0*(R91j(j1,5,j)*DHij(j1,4,j)2^+R91ij(j1,4,j)*DHj(j1,4,j))Sc(9)=Sc(9)+R91j(j1,4,j)*RTj(j1,3,j)Sc(10)=Sc(10)+R91j(j1,4,j)*RVj(j1,3,j)Sc(11)=Sc(11)+R91j(j1,4,j)*TVj(j1,4,j)602 ContinueDo 604 j1=1,3Do 604 jn=1,11604 Am(jn,j1,j1)=Sc(jn)CC275CC^Then, calculate the second term for BijCDo 606 j =1,NjDo 606 j2=1,3Do 606 j1=1,3Do 606 j3=1,9i =Jcon(j)ja=(j1-1)*3+j2Bm(2,j1,j2) =Bm(2,j1,j2)2^+(Rmj(j)/Rmi(i))*R99c(ja,j3)*DDi(j3,4,i)3 +R99i(ja,j3,1,i)*DDj(j3,4,j)Dm(1,j1,j2) =Dm(1,j1,j2)+R99i(ja,j3,2,i)*DDij(j3,4,j)Bm(3,j1,j2) =Bm(3,j1,j2)+R99j(ja,j3,1,j)*RRj(j3,1,j)Bm(4,j1,j2) =Bm(4,j1,j2)+R99j(ja,j3,1,j)*TTj(j3,4,j)Bm(5,j1,j2) =Bm(5,j1,j2)+R99j(ja,j3,1,j)*VVj(j3,4,j)Dm(2,j1,j2)_ =Dm(2,j1,j2)+R99j(ja,j3,2,j)*DFij(j3,2,j)2 +R99ij(ja,j3,1,j)*DFj(j3,2,j)Dm(3,j1,j2) =Dm(3,j1,j2)+R99j(ja,j3,2,j)*DGij(j3,4,j)2^+R99ij(ja,j3,1,j)*DGj(j3,4,j)Dm(4,j1,j2) =Dm(4,j1,j2)+R99j(ja,j3,2,j)*DHij(j3,4,j)2^+R99ij(ja,j3,1,j)*DHj(j3,4,j)Bm(9,j1,j2) =Bm(9,j1,j2)+R99j(ja,j3,1,j)*RTj(j3,3,j)Bm(10,j1,j2)=Bm(10,j1,j2)+R99j(ja,j3,1,j)*RVj(j3,3,j)Bm(11,j1,j2)=Bm(11,j1,j2)+R99j(ja,j3,1,j)*TVj(j3,4,j)606 ContinueDo 608 j2=1,3Do 608 j1=1,3Bm(2,j1,j2)=Bm(2,j1,j2)+Dm(1,j1,j2)+Dm(1,j2,j1)Bm(6,j1,j2)=Dm(2,j1,j2)+Dm(2,j2,j1)Bm(7,j1,j2)=Dm(3,j1,j2)+Dm(3,j2,j1)Bm(8,j1,j2)=Dm(4,j1,j2)+Dm(4,j2,j1)608 Continue276CC^Hence, the matrix Imat is calculatedCDo 610 j2=1,3Do 610 j1=1,3Do 610 jn=1,11Imat(j1,j2)=Imat(j1,j2)+Am(jn,j1,j2)-Bm(jn,j1,j2)610 ContinueC600 ContinueStopEndSample subroutine to calculate H,y, of the systemCC^The H{sys} of the system is represented by the array HvecC^The common blocks, except for R93, are the same as KINENEC^and are omitted hereCSubroutine ANGMOM(Hvec)Implicit Real*8(a-h, o-z)CCommon /R93/ R93c(3,9),^R93i(3,9,5,N),2^R93ij(3,9,4,Nj), R93j(3,9,5,Nj)CC^Set up the matrix Hvec and temporary matrix HvC^Hvec(n) (n=1,2,3) refer to the X, Y, and Z components,C^respectivelyC^Hv(m,n) (m=1,...,12; n=1,3) refers to the contributionC^from the m-th term in Eq. (I-3)C^For instance, Hv(m,n) for H{cm}, Hv(2,n) for 11{11} and so onCReal*8 Hvec(3), Hv(12,3)CC^Initialize the matricesCDo 3000 i1=1,3Hvec(i1)=0.d03000 ContinueCDo 3002 i1=1,3Do 3002 in=1,12Hv(in,i1)=0.d03002 ContinueCC^Start with Body BcCDo 300 i2=1,9Do 300 il=1,3278Hv(1,i1) =Hv(1,i1)-R39c(il,i2)*CMc(i2,3)Hv(4,i1) =Hv(4,i1)+R39c(il,i2)*TTc(i2,3)Hv(5,i1) =Hv(5,i1)+R39c(i1,i2)*VVc(i2,3)Hv(10,i1)=Hv(10,i1)+R39c(i1,i2)*RTc(i2,1)Hv(11,i1)=Hv(11,i1)+R39c(il,i2)*RVc(i2,1)Hv(12,i1)=Hv(12,i1)+R39c(il,i2)*TVc(i2,3)300 ContinueCC^Proceed to the Bi bodiesCDo 302 i =1,NDo 302 i2=1,9Do 302 i1=1,3Hv(2,i1) =Hv(2,i1)+R39c(il,i2)*DDi(i2,3,i)Hv(3,i1) =Hv(3,i1)-R39i(il,i2,2,i)*(RRi(i2,1,i)2^+RTi(i2,3,i)+RVi(i2,3,i)3 +TTi(i2,4,i)+VVi(i2,4,i)+TVi(i2,4,0)Hv(4,i1) =Hv(4,i1)+R39i(il,i2,4,i)*TTi(i2,3,i)Hv(5,i1) =Hv(5,i1)+R39i(il,i2,4,i)*VVi(i2,3,i)Hv(6,i1) =Hv(6,i1)+R39i(il,i2,3,i)*(2^DFi(i2,2,i)+DGi(i2,4,i)+DHi(i2,4,0)Hv(7,i1) =Hv(7,11)-R39i(i1,i2,5,i)*DFi(i2,1,i)Hv(8,i1) =Hv(8,i1)+R39i(il,i2,5,i)*(-DGi(i2,2,i)+DGi(i2,3,i))Hv(9,i1) =Hv(9,i1)+R39i(il,i2,5,0*(-DHi(i2,2,i)+DHi(i2,3,i))Hv(10,i1)=Hv(10,i1)+R39i(i1,i2,4,0*RTi(i2,1,i)Hv(11,i1)=Hv(11,i1)+R39i(il,i2,4,0*RVi(i2,1,i)Hv(12,i1)=Hv(12,i1)+R39i(il,i2,4,i)*TVi(i2,3,i)302 ContinueCC^Hvec calculated if Bij does not existCDo 304 i1=1,3Do 304 in=1,12Hvec(i1)=Hvec(i1)+Hv(in,i1)279304 ContinueCC^Proceed if Nj is not zeroCIf (Nj .eq. 0)Go to 400Do 4000 in=1,12Do 4000 11=1,34000 Hv(in,11)=0.d0Do 402 j =1,NjDo 402 j2=1,9Do 402 j1=1,31=Jcon(j)Hv(2,j1) =Hv(2,j1)+R39c(j1,j2)*(Rmj(j)/Rmi(i))*DDi(j2,3,1)2^+R391(j1,j2,5,1)*(DDij(j2,3,j)-DDij(j2,2,j))3 +R391_(j1,j2,4,i)*DDj(j2,3,j}Hv(3,j1) =Hv(3,j1) -R391(j1,j2,2,1)*DDj(j2,4,j)2^+(R391j(j1,j2,3,j)-R391j(j1,j2,2,j))*(3 DFj(j2,2,j)+DGj(j2,4,j)+DHj(j2,4,j))4^-R39j(j1,j2,2,j)*(RRj(j2,1,j)+RTj(j2,3,j)+RVj(j2,3,j)5 +TTj(j2,4,j)+VVj(j2,4,j)+TVj(j2,4,j))Hv(4,j1) =Hv(4,j1)+R39j(j1,j2,4,j)*TTj(j2,3,j)Hv(5,j1) =Hv(5,j1)+R39j(j1,j2,4,j)*VVj(j2,3,j)Hv(6,j1) =Hv(6,j1)+R391(j1,j2,3,1)*DDij(j2,4,j)2^+R39j(j1,j2,3,j)*(3 DF1j(j2,2,j)+DGij(j2,4,j)+DH1j(j2,4,j))Hv(7,j1) =Hv(7,j1) -R39j(j1,j2,5,j)*DFij(j2,1,j)2^-R391j(j1,j2,4,j)*DFj(j2,1,j)Hv(8,j1) =1-117(8,j1)+R39j(j1,j2,5,j)*(DGij(j2,3,j)-DGij(j2,2,j))2^+R391j(j1,j2,4,j)*(DGj(j2,3,j)-DGj(j2,2,j))Hv(9,j1) =Hv(9,j1)+R39j(j1,j2,5,j)*(DHij(j2,3,j)-DHij(j2,2,j))2^+R391j(j1,j2,4,j)*(DHj(j2,3,j)-DHj(j2,2,j))Hv(10,j1)=Hv(10,j1)+R39j(j1,j2,4,j)*RTj(j2,1,j)Hv(11,j1)=Hv(11,j1)+R39j(j1,j2,4,j)*RVj(j2,1,j)CC280Hv(12,j1)=Hv(12,j1)+R39j(j1,j2,4,j)*TVj(j2,3,j)402 ContinueCC^Calculate Hvec for the whole systemCDo 404 j1=1,3Do 404 jn=1,12Hvec(j1)=Hvec(j1)+Hv(jn,j1)404 ContinueC400 ContinueStopEnd


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items