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Planar dynamics and control of tethered satellite systems Pradhan, Satyabrata 1994

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PLANAR DYNAMICS AND CONTROL OFTETHERED SATELLITE SYSTEMSbySATYABRATA PRADHANB.Sc.(Engg., Honou’rs), Sambalpur University, India, 1987M.E. (Distinction), Indian Institute of Science, Bangalore, India, 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate StudiesDepartment of Mechanical EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1994© Satyabrata Pradhan, 1994In 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 mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)________________________Department of M€cRc.fliC.*I EreevThe University of British ColumbiaVancouver, CanadaDate 9 5DE-6 (2/88)ABSTRACTA mathematical model is developed for studying the inpiane dynamics andcontrol of tethered two-body systems in a Keplerian orbit. The formulation accountsfor:• elastic deformation of the tether in both the longitudinal and inplane transverse directions;• inplane libration of the flexible tether as well as the rigid platform;• time dependent variation of the tether attachment point at the platform end;• deployment and retrieval of the point mass subsatellite;• generalized force contributions due to various control actuators (e.g. momentum gyros, thrusters and passive dampers);• structural damping of the tether;• shift in the center of mass of the system due to the tether deployment andretrieval.The governing nonlinear, nonautonomous and coupled equations of motion are obtained using the Lagrange procedure. They are integrated numerically to assess thesystem response as affected by the design parameters and operational disturbances.Attitude dynamics of the system is regulated by two different types of actuators, thruster and tether attachment point offset, which have advantages at longerand shorter tether lengths, respectively. The attitude controller is designed usingthe Feedback Linearization Technique (FLT). It has advantages over other controlmethods, such as gain scheduling and adaptive control, for the class of time varyingsystems under consideration. It is shown that an FLT controller based on the rigidsystem model, can successfully regulate attitude dynamics of the original flexible11system. A hybrid scheme, using the thruster control at longer tether lengths andthe offset control for a shorter tether, is quite attractive, particularly during retrieval, as its practical implementation for attitude control is significantly improved.Introduction of passive dampers makes the hybrid scheme effective even for vibrationcontrol during the retrieval.For the stationkeeping phase, the offset control strategy is also used to regulate both the longitudinal as well as inpiane transverse vibrations of the tether. TheLQG/LTR based vibration controller using the offset strategy is implemented inconjunction with the FLT type attitude regulator utilizing thrusters as before. Thishybrid controller for simultaneous regulation of attitude and vibration dynamics isfound to be quite promising. The performance of the vibration controller is furtherimproved by introduction of passive dampers. The LQG based vibration controlleris found to be robust against the unmodelled dynamics of the flexible system.Finally, effectiveness of the FLT and LQG based offset controllers is assessedthrough a simple ground based experiment. The controllers successfully regulatedattitude dynamics of the tethered system during stationkeeping, deployment andretrieval phases.111TABLE OF CONTENTSABSTRACT .TABLE OF CONTENTSLIST OF SYMBOLS .LIST OF FIGURESLIST OF TABLESACKNOWLEDGEMENTINTRODUCTION1.1 Preliminary Remarks1.2 Review of the Relevant1.2.1 Dynamical modelling1.2.2 Control of the TSS1.2.3 Control algorithms2. FORMULATION OF THE PROBLEM2.1 Preliminary Remarks2.2 Kinematics of the System2.2.1 Domains and reference coordinates2.2.2 Position vectors2.2.3 Deformation vectors20202121232611ivixxvxxixxii. 11Literature 681013161.3 Scope of the Present Investigationiv• 28• 30• 32• 3234363839.40443. COMPUTER IMPLEMENTATION3.2 Preliminary Remarks3.2 Numerical Implementation .3.3 Formulation Verification3.3.1 Energy conservation3.3.2 Eigenvalue comparison3.4 Summary454545505254574. DYNAMIC SIMULATION . .4.2 Preliminary Remarks4.2 Deployment and Retrieval Schemes4.3 Simulation Results and Discussion• . . . 58• . .. 58• . . . 58• . . • 602.2.5 Librational generalized coordinates2.2.5 Angular velocity and direction cosine2.3 Kinetics2.3.1 Kinetic energy2.3.2 Gravitational potential energy2.3.3 Elastic potential energy .2.3.4 Dissipation energyEquations of MotionGeneralized ForcesSummary2.42.52.6v5.2.2 Control with the knowledge of complete dynamics5.2.3 Inverse control using simplified models5.2.4 Comments on the controller design models .5.3 Offset Control using the Feedback Linearization Technique5.3.1 Mathematical background5.3.2 Design of the controller5.3.3 Results and discussion5.4 Attitude Control using Hybrid Strategy5.5 Gain Scheduling Control of the Attitude Dynamics .4.3.1 Number of modes for discretization .4.3.2 Tether length4.3.3 Offset of the tether attachment point .4.3.4 Tether mass and elasticity4.3.5 Subsatellite mass4.3.6 Deployment and retrieval4.3.7 Shift in the center of mass4.4 Concluding Remarks5. ATTITUDE CONTROL5.1 Preliminary Remarks5.2 Thruster Control Using Dynamic Inversion .61647174788387899090935.2.1 Controller design algorithm 9397102• 110111• 112• 114• 119• 128• 130vi5.6 Concluding Remarks. 1321371371371391421451451461501577. EXPERIMENTAL VERIFICATION7.1 Preliminary Remarks7.2 Laboratory Setup7.2.1 Sensor7.2.3 Actuator7.2.3 Controller7.3 Controller Design and Implementation7.3.1 FLT design7.3.2 LQG design7.3.3 Controller implementation6. VIBRATION CONTROL OF THE TETHER .6.2 Preliminary Remarks6.2 Mathematical Background6.2.1 Model uncertainty and robustness conditions6.2.2 LQG\LTR design procedure6.3 Controller Design and Implementation6.3.1 Linear model of the flexible subsystem6.3.2 Design of the controller6.3.3 Results and discussion6.4 Concluding Remarks158158159159162162168168170171vii7.4 Results and Discussion 1737.4.1 FLT control 1757.4.2 LQG control 1837.5 Concluding Remarks 1898. CLOSING COMMENTS 1918.1 Concluding Remarks 1918.2 Recommendations for Future Work 194BIBLIOGRAPHY 196APPENDICESI. MATRICES USED IN THE FORMULATION 204II. ELEMENTS OF MATRICES ‘M’ AND ‘F’ 217III. NONLINEAR AND LINEARIZED EQUATIONS OF MOTIONFOR THE RIGID SUBSYSTEM 226IV. CONTROLLER DESIGN USING GRAPH THEORETIC• APPROACH 233V. LINEARIZED EQUATIONS OF MOTION AND CONTROLLERFOR THE FLEXIBLE SUBSYSTEM 242VI. LABORATORY TEST SETUP 249vii’LIST OF SYMBOLSJp offset of the tether attachment point; distance between theorigins of coordinate systems F and Fty—component of J,, D + Dd2 z—component of J, + DJs distance between the tether attachment point and the subsatellite centre of mass at the subsatellite endflexible deformation vector of the tether at a distance of yalong the axis Yunit vector along X, n=p, t, sJn unit vector along Y, n = p, t, sunit vector along the local verticalunit vector along Z, n=p, t, s{l} direction cosines of the local vertical (Yo-axis) with respect tothe body fixed framem1 m8L-I- ptL2/2m m8+pLma ptL+mo+msmc 1—maiMm0 mass of the offset mechanismmp mass of the platformm3 mass of the subsatelliterat mass of the deployed tether, ptL(t)mat msL2+pt3/3ixq vector of the generalized coordinates or parameter for the LQG/LTRdesignpositioi of an elemental mass on the th body with respect tothe th body fixed framet timeut(yt, L, t) deformation of the tether along the Xt—directionvt(yt, L, t) deformation of the tether along the Yt—directionwt(yj, L, t) deformation of the tether along the Zt—directiondistance measured along the Yt—axisA cross—sectional area of the tether or the system matrix in thelinear state equationth mode generalized coordinate for the out-of-plane transversetether vibration{B}{Be} equillibrium value of {B}B th mode generalized coordinate for the longitudinal tether vibrationNt{C} {c}.Cdl damping coefficient of the passive damper along the tetherCdt damping coefficient of the passive damper perpendicular to thetetherC mode generalized coordinate for inpiane transverse tethervibration{D} offset vector required by the controller{D} specified offset vectorspecified offset along the local vertical, y—component of {D}xspecified offset along the local horizontal, z—component of{D}offset required by the controller along the local vertical,y—component of {D}offset required by the controller along the local horizontal,z—component of {D}equivalent Young’s modulus of the tether material in the presence of structural damping, E + iE1E real component of E*; Young’s modulus of the tether materialin the absence of structural dampingimaginary part of E*F(q, , t) vector of nonlinear terms in the equations of motion due tocentrifugal, gravitational, and coriolis force contributionsFdl damping force along the undeformed tether due to passivedamperFdt damping force perpendicular to the undeformed tether due topassive damperF, i=p, t, s th body fixed frameF0 orbital frameF1 inertial frameF platform body—fixed frameFt tether body—fixed frameF3 subsatellite body—fixed frame{F}{F} {iYt L)}xiG universal gravitational constant[IJ, i= p, t, s inertia dyadic of the th body with respect to the frame F‘Pz’ Ipy, ‘PZ moments of inertia of the platform about the Xp, Y and Zpaxes, respectively‘Pyz product of inertia of the platform about Yp, Z axesL unstretched length of the tetherL deployment/retrieval velocityM total mass of the system, mp + m0 + ptL -I- m8Me mass of EarthM(q,t) mass matrixM torque applied by the control moment gyros about the Xp axisN1 number of admissible functions used to represent longitudinalvibration of the tetherNt number of admissible functions used to represent inpiane transverse vibration of the tetherNj Nt+NjQq vector of generalized forces corresponding to generalized coordinates qorbit radiusRdmj, i = p, t, s position vector of an elemental mass in the th body with respect to the inertial frame.j, i = p, t, s distance between the origin of the i12 body fixed frame (F)and the orbital frameRSM, {RSM } shift in the centre of mass of the system; also distance betweenthe origins of F and F0xiiRSM shift in the centre of mass along the local verticalRSMZ shift in the centre of mass along the local horizontalT kinetic energy of the th bodyT kinetic energy of the entire system[Tj] transformation matrix from the frame Fj to the frame F2control thrust along the undeformed tether applied at the sub-satellite endTat control thrust perpendicular to the undeformed tether appliedat the subsatellite endUG gravitational potential energy of the th bodygravitational potential energy of the entire systemU strain energy of the systemU potential energy of the system, U + UG14 volume of the unstretched tether, AL(t)j, 13,7,i = p, t, s Euler angles between the frames F2 and F06 true anamoly6 orbital angular velocity6 orbital angular acceleartion{w}, i=p, t, s angular velocity of the th body fixed frame with respect to theinertial framept mass per unit length of the tetherE strain in an elemental tether massIL GMe/2Rc3stress in an elemental tether mass without energy dissipation04 stress in the tether causing energy dissipationxlii77 E1/Ei(Yt, L) th shape function for the tether transverse vibration‘Ji(y, L) th shape function for the tether longitudinal vibrationDot above a character refers to differentiation with respect to timeSubscripts ‘p’, ‘t’,. and ‘s’ refer to platform, tether and subsatellite, respectivelyxivLIST OF FIGURES1-1 Schematic diagram of the Space Shuttle based Tethered SatelliteSystem (TSS) 21-2 Some applications of the tethered satellite systems 51-3 Dumbbell satellite and the associated forces 71-4 A schematic diagram showing the overview of the plan of study. . . 192-1 Domains and coordinate systems used in the formulation 222-2 Diagrams showing relative orientations of two different frames:(a) sequence of Eulerian rotations; (b) rotation about the X0-a.xis. 292-3 A schematic diagram of the Tethered Satellite System (TSS)showing different input variables 413-1 Flowchart showing the structure of the main program 473-2 Structure of the subroutine FCN to evaluate the first order differentialequations required by the integration subroutine 493-3 Flowchart for the subprogram CONTROL to compute actuationinputs 513-4 Figure showing variations of kinetic, potential (gravitational pluselastic) and total energies of the system during the stationkeepingphase: (a)L = 20 km, = = 0; (b) L = 5 km, = 1 m,d=0 534-1 Dynamic response of the system with the higher modes includedin the discretization of the system 634-2 System response for a shorter tether with higher modes includedin the flexibility modelling 654-3 Dynamical response of the tethered satellite system in the referencexvconfiguration with the offset set to zero 674-4 Response results showing the effect of decreasing the tether lengthon the system dynamics 684-5 Plots showing the effect of increasing the tether length on the systemdynamics 694-6 System response during stationkeeping showing the effect offlexibility on the tether attitude motion 704-7 Simulation results during offset motion along the local verticalwith the offset along the local horizontal fixed at zero 724-8 System response showing the effect of offset motion along the localhorizontal 734-9 Dynamical response of the system with fixed offsets along boththe local horizontal and local vertical directions 754-10 Plots showing the effect of decreasing the tether mass per unitlength on the system response 764-11 System response with an increase in the mass per unit lengthof the tether 774-12 Diagram showing the effect of decreasing the elastic stiffness 794-13 Response of the system with an increase in the elastic stiffness. . . 804-14 Response results showing the effect of decreasing the subsatellite mass. 814-15 System response with an increase in the subsatellite mass 824-16 Response of the system during deployment of the subsatelliteusing an exponential-constant-exponential velocity profile 844-17 System response during exponential retrieval of the subsatellite.Note instability of the system in rigid (aj) as well as flexible (B1)degrees of freedom 86xvi4-18 Plots showing shift in the center of mass during three different phasesof the system operation:(a) deployment; (b) stationkeeping;and (c) retrieval. Total deployed length is 20 km 885-1 Controlled response of the system during the stationkeeping phase. . 985-2 System response during controlled deployment with an exponential-constant-exponential velocity profile 1005-3 Controlled response during the exponential retrieval withouta passive damper 1015-4 System response during the controlled retrieval with passive dampers.Note, the flexible modes are controlled quite effectively keepingthe tether tension positive 1035-5 System response during the stationkeeping using the rigid nonlinearmodel for the controller design 1055-6 System behaviour during the stationkeeping with the outer PTcontrol loop and FLT controller using the rigid nonlinear model. . . 1075-7 Deployment dynamics with the controller based on the rigidnonlinear model 1085-8 Controlled response during exponential retrieval of the subsatellite.The controller is designed using the rigid nonlinear model andapplied in presence of passive dampers 1095-9 Schematic diagram of the closed-loop tether dynamics with the FLTbased controller 1195-10 Controlled response in presence of the modelling error introducedby neglecting the shift in the center of mass terms duringthe controller design 1215-11 System response with the shiftin the center of mass includedxviiin the controller design model 1225-12 Schematic diagram of the closed-loop system with the outer PTcontrol loop for the tether dynamics 1235-13 System response in presence of the outer PT control ioop 1245-14 Controlled response of the system during deployment. The offsetcontrol strategy in conjuction with the Feedback LinearizationTechnique (FLT) is used 1255-15 System response during controlled retrieval with an exponentialvelocity profile 1275-16 Controlled response during deployment using a hybrid strategy. 1295-17 System response during retrieval with a hybrid control scheme. 1315-18 Controlled stationkeeping dynamics using the graph theoreticapproach 1335-19 Gain scheduling control of the retrieval maneuver using thethruster augmented strategy 1345-20 Offset control of the retrieval dynamics using the gain schedulingapproach 1356-1 Standard feedback configuration 1396-2 Diagram showing different unstructured uncertainties: (a) additive;(b) output multiplicative; and (c) input multiplicative 1416-3 Closed-loop system with the LQG feedback controller 1436-4 Flow chart for the controller design 1516-5 Comparison of gain and phase of the return ratio at the plantoutput with those of the KBF loop tranfer function 1526-6 Robustness property of the vibration controller 1536-7 Three-level controller structure to regulate rigid as well asxviiitransverse and longitudinal flexible motions of the tether 1546-8 Response of the system using a three level controller in absenceof a passive damper 1556-9 Controlled response of the system in presence of a passive damper. 1567-1 A device to measure the tether swing through rotation of thepotentiometer shaft 1617-2 A schematic diagram of the experimental test—facility 1637-3 Photograph of the test—rig constructed to validate offset controlstrategy: (a) aluminum frame; (b) inplane motor; (c) carriage forout-of-plane motion; (d) wooden stand; (e) linear bearings;(f) tethered payload 1647-4 Carriage and sensor mechanism: (a) potentiometer on mountingbracket; (b) movable aluminum semi—ring mechanism withslots for tether; (c) tether; (d) inpiane traverse with linearbearings; (e) payload 1657-5 Digital hardware used in the experiment: (a) translator module,deployment and retrieval; (b) translator module, offsetmotions; (c) power supply; (d) function generator 1667-6 Photograph showing the recently constructed larger test—facilitywhich can accomodate tethers upto 5 m in length. The presentsmaller set—up on which the experiments reported herewere carried out can be seen in the foreground to the left 1677-7 Flow chart showing the real time implementation of the attitudecontroller 1747-8 Plots showing the comparison between numerical and experimentalresponse results during the stationkeeping phase: (a) uncontrolledand controlled system; (b) subsatellite and carriage positions 176xix7-9 A comparative study between numerical and experimental resultsfor the system during stationkeeping at 1 m: (a) uncontrolledand FLT controlled system; (b) subsatellite and carriage positions. . 1797-10 A comparative study between numerical and experimental resultsfor the system during stationkeeping at 2 m: (a) uncontrolledand FLT controlled system; (b) subsatellite and carriage positions. . 1817-11 Uncontrolled and controlled experimental results for retrieval of thesubsatellite: (a) retrieval time of 15s; (b) faster retrieval in 5s. . . . 1847-12 A comparative study with the LQG controller during stationkeepingat 1 m: (a) uncontrolled and controlled performance;(b) subsatellite and the tether attachment point positions 1867-13 Experimentally observed system response during retrievalwith the LQG controller 1887-14 Subsatellite and carriage positions during control of the sphericalpendulum using the LQG regulator 190xxLIST OF TABLES3.1 Inpiane dimensionless natural frequencies (w/Ô) of a 2-body system(L = 100 km, mp = i0 kg, m3 = 500 kg, Pt 5.76 kg/km,EA = 2.8 x 1O5 N) 553.2 Inpiane dimensionless natural frequencies (w/6) of a 2-body system(L = 20 km, mp = cc, m3 = 576 kg, p = 5.76 kg/km,.EA = 2.8 x N) 553.3 Comparison of natural frequencies of a tethered two-body systemwith different end masses (L = 20 km, p = 5.76 kg/km,EA = 2.8 x iO N) 565.1 Comparison of the time required (s) by the controllers using differentdesign models 1106.1 Comparison of the open-loop and closed-loop eigenvaluesof the system 149xxiACKNOWLEDGEMENTI am thankful to Prof. V. J. Modi and Prof. A. K. Misra (McGill University,Canada) for their guidance throughout the research project.A word of appreciation must be extended to my colleagues, Dr. C. A. Ng,Dr. A. K. Grewal, Dr. S. Hokamoto and Dr. F. Karray, for useful discussions andsuggestions as well as to Dr. I. Marom, Mr. S. R. Munshi, Mrs. M. Seto and Mr.Y. Chen for their advice in planning of the experimental test—program. Assistanceof Mr. John Richards in construction of the electronics hardware used during theexperimental study is gratefully acknowledged.The research project was supported by the Natural Sciences and Engineering Research Council’s grant (A—2181) held by Prof. V. J. Modi.2CXii1. INTRODUCTION1.1 Preliminary RemarksOver the past decades, a number of proposals have been made for the spaceexploration using Tethered Satellite Systems (TSS). The concept involves two ormore satellites connected by a tether upto 100km in length (Fig.1-1). In a typicalmission, the Space Shuttle carries the tether connected subsatellite payload. After adesired orbit is achieved, the subsatellite is deployed from the orbiter to the requiredaltitude. Intended scientific measurements are carried out at a fixed tether length.Subsequently the subsatellite is retrieved back to the Shuttle (Fig.1-1).The innovative idea of the TSS is attributed to the Russian scientist Tsiolkovsky, who explored the effect of gravity force on an individual climbing up atower extending upto geosynchronous altitude and beyond [1]. This analysis uncovered the principle that a string orbiting in a central force field always remains intension due to the gravity gradient, and led to the development of the TSS. So faras the actual application is concerned, interest in the system was initially associatedwith the retrieval of a stranded astronaut by throwing a buoy on a tether from arescue vehicle and reeling in the tether. Starly and Adihock [2] have shown that therotational motions of the tether grow continuously as it was reeled in. A proposalwas made to use tether for stationkeeping between two orbiting space vehicles [3],however, the idea was abandoned due to the difficulties involved in determining andcontrolling the required tether tension. On the other hand, Gemini XI and XIIflights have successfully demonstrated useful applications of a tethered system [4].The former used a rotating configuration aimed at artificial gravity generation while1TRAJECTORYORBITERFigure 1-1 Schematic diagram of the Space Shuttle based Tethered SatelliteSystem (TSS).BOOMTETHERSUBSATELLITE2the latter had a gravity-gradient stabilized configuration.After one and half decade, a joint U.S.A.—Japan space project called the TPE(Tethered Payload Experiment) used a sounding rocket based tether to conduct aseries of tests aimed at near earth environmental studies [5]. The technical andscientific data obtained in the TPE supported the electrodynamic tether missionTSS-i (Tethered Satellite System-i) launched in August 1992 [6]. The TSS missionis a collaborative project between NASA and the Italian space agency (ASI). Becauseof the mechanical failure, the mission objectives of the TSS-i were not met, but theinformation gathered during the short deployment upto 259 m, as compared to thedesired length of 20 km, demonstrated the fundamental concepts of orbital tetherflights in many ways. Another series of sounding rocket experiments called theOEDIPUS (Observation of Electrified Distributions in the Jonosheric Plasma — aUnique Strategy) are being carried out by the Canadian Space Agency (CSA) [7].The objective is to make passive observation of the auroral ionoshere and gain betterinsight into pane and sheath—wave propagation in plasmas using a radio transmitter.The first mission, called OEDIPUS-A, was successfully launched in January 1989which deployed a tether upto 958 m. The next mission, OEDIPUS-C, planned forJanuary 1995, will include a payload for tether dynamics measurements.The deployment of 20 km long tether was successfully completed in SEDS(Small Expendable Deployment System) missions [8]. SEDS-I and SEDS-Il missionsflew in March 1993 and March 1994, respectively. Each deployed a 26 kg instrumented probe to a distance of 20 km from the second stage of a DELTA II rocketwhile in orbit around the Earth. These tether missions have successfully demonstrated the feasibility of tether deployment. On the other hand, retrieval of a tetherstill remains to be realized in a real flight. With the advent of the Space Shuttle and3the proposed Space Station, several applications of the tethered subsatellite systemhas been suggested which can be summarised as:• sophisticated scientific experiments aimed at gravity gradient, magnetic, ionospheric, aero-thermodynamics and radio astronomy experiments (e.g. OEDIPUS-A and C);• use of tethered system as a flying wind tunnel [9];• deployment of payloads to new orbits [10] and retrieval of satellites for servicing;• provision of a desired controlled microgravity environment for scientific experiments and space manufacturing [11];• generation of electricity (electrodynamic tether, TSS-1);• power and cargo transfer between two orbiting bodies;• collection of atmospheric dust by a small probe tethered to a larger orbitingspacecraft thus avoiding landing [12];• expansion of the geostationary orbit resource by having tethered chain satellites [13, 14];• large antena reflector ( 1 km aperture) using tethers for gravity gradientstabilization and shape control [15];• orbiting optical astronomical interferometer consisting of three telescopes atthe corners and one at the center of a tethered triangle [16];• use of multiple tethers for attitude stabilization of satellites in elliptic orbit[17];and many others [10, 18]. Some of these applications are illustrated in Fig.(1-2).The fundamental principle governing the tether dynamic can be illustrated4CargoCargo TransferRetrieval of Satellitefor ServicingManipulatorFigure 1-2 Some applications of the tethered satellite systems.Low Altitude Science ExperimentTether100 km- EarthTether5by a simple two-body tethered system as shown in Fig.1-3. Consider a ‘dumbbell’satellite in a circular orbit around the Earth. The top mass experiences a largercentrifugal force than the gravitational force, being in an orbit higher than that ofthe centre of mass. The reverse occurs at the lower mass. The resultant momentcauses The system to oscillate like a pendulum with the tether maintained taut undertension. Now, as the length of the tether decreases during retrieval, the conservation of angular momentum dictates the swinging oscillation of the tether to amplify.For a space platform supported tethered satellite system, the swing motion couldincrease to the point that the tether wraps itself around the platform. Furthermore,reduced gravity induced tension at smaller lengths together with large amplitude oscillations may render the tether slack. The presence of any offset between the tetherattachment point and the centre of mass of the end-bodies (platform and subsatellite) imposes an additional moment on the bodies involved. Flexibility of the tether,along with the above mentioned factors makes the system dynamics rather challenging. Obviously, a fundamental understanding of complex interactions betweenlibrational dynamics, flexibility, deployment and retrieval maneuvers, as well as development of appropriate control strategies is essential for successful completion ofthe proposed tether missions.1.2 Review of the Relevant LiteratureThe possible applications of the TSS being numerous and diverse, considerable amount of literature has developed, particularly during the past 30 years, whichwas reviewed quite effectively by Misra and Modi [19, 20]. The growing importanceof this new technology is reflected in a special issue published in The Journal of theAstronautical Sciences [21]. Objective here is to briefly touch upon more important6F,1F,g1F2Fg2Figure 1-3 Dumbbell satellite and the associated forces.CENTRIFUGAL FORCESGRAVITATIONAL FORCESFClC.M. FCl >FFC2FF>‘ I‘ I‘ I‘I‘IFC27contributions directly relevant to the present study.1.2.1 Dynamical modellingThe dynamics of a TSS consists of three major phases of operation dependingon whether the unstretched length of the tether is constant (stationkeeping phase) orvaries with time (if increasing, deployment; when decreasing, retrieval). The dynamics during stationkeeping is simpler than the other two phases and is quite similar tothat of other cable connected bodies, e.g. the space station-cable-counterweight system [22]. It has been investigated extensively by various researchers who concludedthat the gravity gradient can excite longitudinal as well as transverse vibrationsof the tether [22, 23]. However, a small amount of damping (1% of the criticaldamping) is quite effective in stabilizing the system [24].Perhaps the first systematic effort at understanding dynamics of the TSS wasmade by Rupp [25]. In his pioneering study, the librational motion in the orbitalplane was analyzed and the growth of pitch oscillations during retrieval phase noted.The system was further investigated by Baker et al. [26] taking into account thethree dimensional character of the dynamics as well as the aerodynamic drag inthe rotating atmosphere. During either deployment or retrieval, the out-of-planerotational motion cannot be neglected if the orbit is in a nonequatorial plane. Itis excited by the rotating atmosphere induced aerodynamic drag. Even when theorbit is in the equatorial plane, the initial out-of-plane disturbances may couple theinpiane and out-of-plane dynamics. Fortunately, in general, such disturbances arefound to be small. Furthermore, their effects can be minimized through the designof an appropriate active control system. Thus the coupling between the inpiane andout-of-plane motions can be neglected, for systems in the equatorial orbit, at least8in the preliminary design stage.The other important parameters, which make the system dynamics complicated, are mass and flexibility of the tether. The mass of the tether is expected tobe of the same order of magnitude as that of the subsatellite, particularly when thetether is long, and hence it can not be neglected. Because the tether diameter is expected to be small (around few millimeters) and the length quite long (upto 100km),the system shows considerable flexibility (the longitudinal stretch for a 100km longtether can be around hundred meters). The axial vibration has been representedin an approximate fashion by a single longitudinal displacement similar to that ofa spring mass system by several authors [26, 27, 28]. However, as the mass is distributed along the tether, a more accurate representation in terms of combination-of axial modes, similar to that of an elastic bar, would be more appropriate [29, 30].Furthermore, the tether can have transverse displacements causing it acquirecurvature. This happens mainly due to two reasons. The aerodynamic drag notonly causes the subsatellite to lag behind the platform but also forces the tether toassume a curved equilibrium position. Secondly during deployment or retrieval, theCoriolis force acts in the transverse direction. Since the tether has distributed massand elasticity, transverse vibrations are also excited. They can influence system dynamics substantially, particularly during retrieval [31]. The transverse oscillations ofthe tether have been studied using two distinct approaches. Quadrelli and Lorenzini[32] discretized the system using the lumped-mass approach. On the other hand,Kohler et al. [33], Modi and Misrà [27], and Xu, Misra and Modi [30] adopted thecontinuum model with admissible functions. To reduce the computational effortinvolved in simulation, a combination of lumped-mass and continuum model hasbeen proposed called the semi-bead model [34].9When the masses of the two end bodies are comparable, the system centre ofmass does not remain fixed with respect to any of the body fixed frames. The shiftin the centre of mass may become an important parameter in such situation and hasto be accounted for. The major conclusions based on the available literature maybe summarized as follows:• stationkeeping phase is normally stable;• deployment can be an unstable operation if velocity exceeds the critical rate;• retrieval is inherently unstable• particularly during the deployment and retrieval phases, the transverse vibrations can build up due to the Coriolis force induced excitation even thoughthere may not be any initial disturbance.The other aspects of practical importance are the damping in the tether [35],aerodynamics drag [36, 37], and orbital lifetime analysis [38]. The dynamics oftethered systems in different configurations, which has less relevance to the presentstudy, have been addressed by Misra and Diamond [39], and DeCou [40, 16]. In realsystems, there is always a chance of accident. The dynamics of the tether connectedto the orbiter after a catastrophic failure has been studied by Bergamaschi [41].1.2.2 Control of the TSSControl of the TSS, particularly during deployment and retrieval, is a challenging problem. Successful control of the unstable retrieval dynamics is of greatconcern since it is directly related to the success of the mission. Over the years,several control strategies have been proposed, which can be categorized into threetypes:(a) tension control and the law based on rotational rate of the tether reel (length10rate control or torque control);(b) thruster augmented active control;(c) offset control.(a) Tension and length rate controlAmong the control laws mentioned, the tension control law was developedfirst. Rupp [25] formulated this strategy to control the inplane rotation duringdeployment and stationkeeping. The retrieval problem was also touched upon briefly.Here the tension level is modulated as a function of the instantaneous length, lengthrate, and desired lengths of the tether. Several investigators have subsequentlymodified Rupp’s tension control procedure, however the inherent approach remainsthe same [26, 42—44]. For example, Bainum and Kumar [44] developed an optimalcontrol law, based on an application of the linear regulator theory, which modulatesthe tether tension to achieve acceptable level of the tether swing.As opposed to the tension control law, in the length rate scheme the ‘nominal’ unstretched length or its time derivative are modulated to achieve the desiredsystem response. The law corresponds to modulating the rotation of the reel of thedeployment mechanism. This can also be implemented by monitoring the torqueapplied to the reel mechanism. In principle, tension control, length rate control andtorque control affect the system dynamics by changing the tension in the tether. Thedifferences are in the mathematical modelling of the system used for the controllerdesign and its implementation. This type of control law was originally proposedby Kohier et al. [33], and used by Misra and Modi [45] and others to regulate theplanar longitudinal and transverse vibrations.11(b) Thruster augmented active controlDuring retrieval, as length of the tether reduces to a small value, the equilibrium tension in the tether due to gravity gradient approaches zero; and during adynamical situation, the tether may become slack. Thus a tension control law or itsmodification, such as a length rate law, becomes ineffective. To alleviate this difficulty, Banerjee and Kane [46], and Xu et al. [47] used a set of thrusters to controlthe retrieval dynamics. In this active control scheme, the thrusters are placed atthe subsatellite end to help reduce the motion and speed up the retrieval process.The thrusters provide control forces in both transverse and longitudinal directions.(c) Offset control strategyIt has been shown that, when the tether length is small the thruster augmented control is quite effective; however firing of thrusters in the vicinity of theshuttle or space platform is considered undesirable due to plume impingement, safetyand other considerations. To overcome this difficulty the “offset control strategy”was proposed by Modi, Lakshmanan and Misra [48]. Here, the point of attachmentof the tether at the platform is moved to control the tether swing. The coupling ofthe offset acceleration and tether swing makes this control strategy a success. Intheir study, Modi et al. considered a 3-dimensional rigid model without any shift inthe centre of mass. The results obtained from the numerical simulation were verifiedby a ground based experimental set-up. From the comparison between numericaland experimental results, they concluded the control strategy to be quite promisingfor shorter tethers. Because of relative advantages and disadvantages of differentschemes, a desirable solution would be to use a hybrid strategy, where the offsetcontrol is implemented at shorter lengths while the thruster, tension or length rate12control scheme is used for longer tethers [49].1.2.3 Control algorithmsAn important factor which needs careful attention is the methodology usedto obtain the control laws. In the pioneering work by Rupp [25], the feedbackgains were selected to achieve an appropriate stiffness and damping in the closed-loop system. This approach is feasible only when the system dynamics is relativelysimple, which is not the case in practice. Since then, linear and non-linear controllaws have been developed by many investigators (for example [26]), where the controlgains are obtained by trial and error.Among the wide variety of control methodologies, the Linear Quadratic Regulator (LQR) has received considerable attention [44, 48, 49]. On the other hand,a feedback control law using the second method of Liapunov was used by Vadaliand Kim [50, 51]. It was concluded that the tension control law is sufficient during fast deployment, however, thruster augmentation in the out-of-plane directionis required during the terminal phase of the retrieval [51]. The existence of limitcycles was avoided by the out-of-plane thrusting in conjunction with the tensioncontrol [50]. Monshi et al. [52] used the reel rate control law with nonlinear rollrate feedback, based on energy dissipation method. It was found to have better performance than that for the law developed using the Liapunov method. The studyshowed that the retrieval constant c (c = i’/l; ,, nondimensionalized retrieval rate;1, instantaneous tether length), in case of an exponential retrieval, does not have asignificant effect on limit cycle amplitude. As the retrieval rate increases the peakvalue of the pitch oscillation also increase. So the maximum retrieval rate is limitedby a maximum allowable pitch angle. The limit cycle amplitude found in [52] is13quite close to that obtained by Vadali and Kim [50].In the study by Liaw and Abed [53], tension control laws were establishedusing the Hopf bifurcation theorem, which guarantees stability of the system duringthe stationkeeping mode. They. also suggested a constant inplane angle controlmethod which results in stable deployment but unstable retrieval. In this method,the instantaneous tether length is treated as an external control input, and thelength rate is obtained as a function of some desired inpiane equilibrium angle. Thelength rate expression required to control the system during deployment does notstabilize the retrieval dynamics.Fleurisson et al, [54] designed an observer (Kalman filter) based controllerto follow a predetermined retrieval length history. The length history was obtainedto minimize a combination of final pitch angle, pitch rate at docking and totalretrieval time. The control strategy used feedforward trajectories motivated byoptimal control arguments. Optimal control procedure was employed to determinean invariant final approach path, or randezvous corridor, which the satellite mustfollow during the terminal phase to dock with the specified final conditions. Thestudy focused only on the last 2 km of the retrieval.In the study by Fujii et al. [55], the dynamics during deployment and retrievalis controlled by regulating the mission function, a positive definite quadratic functionof the mission states. The desired value of the system states are referred to as the‘mission states’. For the closed-loop system, the non-dimensional time derivativeof the mission function is set to be negative definite, guaranteeing stability of thesystem. Thus, the approach is quite similar to the Lyapuno’v’s second method.Onoda and Watanabe [56] as well as Fujii et aL [57] studied control of teth14ered systems in the presence of atmospheric drag. Attitude control of the tetheredsystems after blocking of the attachment point has been explored by Grassi, et al.[58]. The offset control, as discussed before, is for regulating the attitude motionof the tether. However, this strategy can also be used for precise control of thesubsatellite attitude, where the control moment can be generated by a combinedeffect of tether tension and the attachment point motion [59]. Numerous variationsof the control algorithms discussed here are also reported in the literature.Almost all the investigations reported so far deal with the control of attitudemotion of the tether or the endbodies. Little attention has been directed towardsthe control of structural vibrations of the tether. Xu et al. [30, 47) addressed theproblem of control of tether oscillations using length rate and thruster control. Onthe other hand, the study by Thornburg and Powell [60] considers the control ofonly transverse vibration in conjunction with the offset strategy.As seen in this literature review, the models developed so far do not includethe attachment point offset for a flexible tethered system capable of deployment andretrieval. The study in reference [60] incorporates the motion of the attachmentpoint, of a flexible tether, along a line parallel to the end body’s floor (i.e. motionalong one bodyfixed axis), however, it does not include the most critical operationalphases, deployment and retrieval. The tether is modelled as an arbitrary number ofpoint masses connected by elastic members. Use of the lumped mass model for acontinuum system may be appropriate for a preliminary study, however, any detailedanalysis of the system needs the more accurate continuum model. With the offsetmotion restricted to be parallel to the subsatellite floor, the strategy is limited onlyto the control of transverse oscillations. Furthermore, the analysis does not accountfor the shift in the center of mass, an important parameter for long tethers during15deployment and retrieval maneuvers.1.3 Scope of the Present InvestigationWith this as background, the thesis aims at studying dynamics and controlof two-body tethered systems, using a relatively general model, which is the fundamental requirement for their successful realization. Distinctive features of the modelmay be summarized as follows:(i) A flexible tether, with arbitrary mass distribution and finite dimensional rigidend bodies, is taken to be in a general Keplerian orbit. The tether is treatedas an elastic continuum during the system discretization.(ii) Offset of the tether attachment point from the end body’s center of massand its time dependent variations are included in the formulation. Thispermits development of several offset control strategies. Furthermore, theformulation is amenable to thruster augmented active control and its hybridsynthesis with the offset control procedure.(iii) The formulation accounts for time dependent variation of the tether length,thus permitting analysis of the tether performance during all the three operational phases of importance — deployment, stationkeeping and retrieval.The deployment/retrieval time histories are considered arbitrary.(iv) Kinetic and potential energy expressions are obtained for the general threedimensional (i.e. in the orbital plane and out-of the orbital plane) motionof the system. However, the governing equations of motion and hence thestudy based on them are purposely confined to the inplane dynamics. Thishelped focus on more important aspects of the system performance which aregoverned by a large number of variables. Furthermore, some appreciation of16the coupling between the inpiane and out-of-plane dynamics is already available through early studies (by the U.B.C. group and others) using simplermodels.(v) The formulation is based on the Lagrangian procedure which can account forboth holonomic and nonholonomic constraints.(vi) As can be expected the governing equations of motion are highly nonlinear, nonautonomous and coupled. Their decoupled, linearized forms are alsoobtained to facilitate the controller design.(vii) Numerical integration codes for nonlinear and linear systems are so structured as to help isolate effects of design parameters on the system performance.To begin with, kinematics and kinetics of the system, leading to a set of highlyCoupled, Nonlinear and Nonautonomous (CNN) equations of motion, are discussedin Chapter 2. This is followed by a brief account of the numerical integrationmethodology as applied to the nonlinear as well as decoupled linearized system(Chapter 3). Validation of the formulation by energy check and comparison offrequencies with reported results in the literature are also included here. Chapter 4focuses on the parametric dynamical study using the complete CNN set of equations.Objective is to assess effect of the system design parameters on its performance, andestablish critical conditions leading to unacceptable response or instability. This setsthe stage for an effective controller design.As pointed out before, the governing equations of motion are coupled, nonlinear and nonautonomous. To assist in the controller design, rigid and flexible partsof the system are often (but not always) decoupled due to their widely separated17frequencies. However, it must be emphasized that the coupling between the rigiddegrees of freedom, and among the flexible generalized coordinates, was retained.Two different control methodologies, linear eigenvalue assignment and the nonlinearFeedback Linearization Technique (FLT), in conjunction with the actuators in theform of offsets, thrusters and their combinations were used. Attitude controller wasdeveloped first (Chapter 5) followed by a composite attitude-vibration regulator design (Chapter 6). In all the cases, effectiveness of the controller was assessed throughits application to the original nonlinear, nonautonomous and coupled system.Finally, in Chapter 7, a ground based facility for studying dynamics of a tethered system as well as its control using the offset strategy is described. Experimentalresults are compared with numerically obtained simulation data.The thesis ends with a summary of results and recommendations for futurework. Figure 1-4 presents an overview of the research project.18IPLANARDYNAMICSANDCONTROLOFIITETHEREDSATELLITESYSTEMSPROBLEMFORMULATION*Kinematics*Kinetics*EquationsofMotion(CNN4,I PARAMETRICDYNAMICSICONTROLLERDESIGNI GROUNDBASEDUSINGTHECNNI4,EXPERIMENTAttitude(FLTorPolePlacement)*Offsetcontrol;DNN(Platform)andDLN2)andAttitude(FLTDLN(Tether)ercontrol,CNN,DNN,DLNtlledDynamics(CNN)controI;NN,DNN,DLNValidationofthe1CNN(CoupledNonlinearNonautonomous)offsetControl2DLN(DecoupledLinearNonautonomous)(FLT,LQG)3DNN(DecouplledNonlinearNonautonomous)Figure1-4Aschematicdiagramshowingtheoverviewof theplanofstudy.2. FORMULATION OF THE PROBLEM2.1 Preliminary RemarksConsiderable thought was directed in arriving at a model of the complextethered satellite system that would represent a logical step forward in understandingits dynamics and control. The model selected for study consists of a rigid platformconnected by a flexible cable (tether) of finite mass density to a rigid subsatellite.The entire system is in an arbitrary Keplerian orbit and free to undergo librationalas well as both longitudinal and transverse vibrations in the plane of the orbit.Furthermore, the tether can be deployed, retrieved or maintained at a constantlength (stationkeeping). Deviation of the tether attachment from the platform centerof mass is treated as a function of time permitting development of the offset controlstrategy. As in a practical situation, the platform based momentum wheels providecontrol torques for its attitude control. In addition, one may attempt to regulate thetether dynamics through an orthogonal set of thrusters at the subsatellite. Synthesisof the offset and thruster based strategies present interesting possibilities for a hybridcontrol design.As pointed out before, in a typical mission under consideration, the tethermay be deployed from a length of few meters to around 100 km. This results in alarge change in the system inertia as well as a shift in the system center of masswith respect to a body fixed reference frame. Thus, a realistic model must accountfor the effect of shift in the center of mass on the system dynamics. The structuraldamping of the flexible tether is also an important parameter and hence included inthe present formulation.20This chapter can be divided into three major sections: kinematics; kinetics;and equations of motion. System configuration and position of an elemental systemmass in the inertial space are established first. This is followed by evaluation ofthe kinetic and potential energies of the system. Finally the governing equations ofmotion are derived using the Lagrangian procedure. The generalized forces due toactive and passive control inputs are obtained using the principle of virtual work.2.2 Kinematics of the System2.2.1 Domains and reference coordinatesFour distinct domains can be established for the system described in Figure2-1. Domain ‘p’ represents the rigid space platform. The frame F Z) isattached to the centre of mass of the platform with its axes along any arbitrarydirections. Vectors described relative to this rotating frame are distinguished by thesubscript ‘p’. Mass of the platform is considered constant (rh = 0).The tether involves two major domains. The domain ‘o’ consists of the offsetmechanism and undeployed tether that is wrapped around a reel. In general, massof the offset mechanism is expected to be quite small compared to the mass of theplatform (Space Shuttle, Space Station). So, without any loss of generality, the offsetmechanism is treated as a point mass and its location is expressed with respect toF by the vector d. The mass of the offset mechanism is represented with subscript‘o’. As the tether is deployed or retrieved the mass of the spooi, around which thetether is wound, changes. The deployed portion of the tether is flexible and belongsto the domain ‘t’. The frame Ft (Xi, Y, Zt) has its origin at the point of attachmentof the tether at the platform end. The Yt-axis is along the undeformed length ofthe tether, i.e. the direction of tether in the absence of transverse deformation. The21Figure 2-1 Domains and coordinate systems used in the formulation.Ys xsSUBSATELIJTEdZsdmtyt iTETHERY rtZtYoOFFSET MECHANISM(MANIPULATOR) RdmpZ1xi22Xt-axis is so selected that, in absence of the out-of-plane motion, it is parallel tothe orbit normal. The Zt-axis completes the right-handed system. The mass ofthe deployed portion of the tether varies with time according to the deployment orretrieval rate. The total tether mass, composed of the deployed (domain ‘t’) andundeployed (domain ‘o’) portions, is constant (i.e. thj = —r’ao).Domain ‘s’ consists of the rigid subsatellite attached to the end of the tether.The frame F5 (X3,Y3,Z5) is used to describe its orientation in space. The originof this frame is at the centre of mass of the subsatellite. Vectors measured relativeto this rotating frame carry subscript ‘s’. Mass of the rigid subsatellite is constant(r = 0).Two more reference frames are required to completely define the kinematicsof a mass element in the system: the inertial frame F1 located at the centre of Earth;and the orbital frame F0 (X0,Y0,Z0). The origin of the orbital frame is located atthe instantaneous centre of mass of the system and follows a Keplerian orbit. Theorbital frame is so oriented that theY0-axis is along the local vertical (the line joiningthe centre of Earth and instantaneous centre of mass of the system) and points awayfrom Earth; the Z0-axis is along the local horizontal (the line perpendicular to Y0-axis and in the plane of the orbit) and points towards the direction of motion of thesystem; theX0-axis is along the orbit normal and completes the right-handed triad.2.2.2 Position vectorsWith the reference frames selected, the position vector of an elemental systemmass can be defined easily. As pointed out before, the instantaneous centre of massof the system follows a Keplerian orbit. This is based on the assumption that theorbital motion is not affected by the librational (attitude) and vibrational motions23of the system [61—63]. The position vector to the mass element with respect to theinertial frame can be obtained by a set of vectors added in sequence. First of all,the position vector is defined with respect to the frame attached to the domain inwhich the mass element is located (fj, i= p, t, s). For bodies like the platformand subsatellite, f and f consist of only rigid components. But for the flexibletether, the position vector has contribution from two sources: rigid (ytt) and elasticdeformation (ft(yt)), i.e.= yt3t + ft(yt),where 3t is the unit vector along Yt-axis, and y is the distance to the mass elementfrom the origin of F. The position of the origin of the body fixed frames, Fj, i =p, t, s, is defined with respect to the orbital frame F0 by the vector R, i = p, t, .s.Finally, the origin of the orbital frame is defined with respect to the inertial frameby Pc (i.e. radius vector of the Keplerian orbit). With these notations, position ofany elemental mass in the th domain with respect to inertial frame can be expressedasRdmj = Rc+R1+r. (2.1)The vector R, i= p, t, s, can be expressed in terms of other vectors defined in thebody fixed frames as:RPRSM; (2.2)= RSM + d; (2.3)R$—RsM+dp+L,t+ft(L)—ds; (2.4)where:4 distance between origins of frames F and Ft expressed in F;24L undeformed length of the tether;t unit vector along the fl-axis;ft(L) flexible deformation vector of the tether at a distance L from origin of Ft;J., offset of the tether attachment point at the subsatellite end and expressed inthe frame F3.In Eqs.(2.3 and 2.4) all the terms on the right hand side are not independent.The vector SM, the distance between origins of the frames F0 and F, representsthe shift in the centre of mass and can be expressed in terms of other system variables. By taking the first moment of the masses about the instantaneous centre ofmass and equating it to zero, it can be shown thatRSM = _{(mo + mt + ms)4, + ms(L3t + Jt(L) — J8) + J tdmt},where:M total mass of the system, mp + m0 + mt + m8;mp mass of the platform;m0 mass of the offset mechanism;mt mass of the tether, ptL;p mass of the tether per unit length;m3 mass of the subsatellite.It may be pointed out that the vectors defined above and in Eqs.(2.1-2.4) are not withreference to the same coordinate system. So, appropriate transformation matricesare used during vector operations.252.2.3 Deformation vectorsDeformation at a point on the tether depends on its position and varies withtime. At a given instant, it can be expressed as three orthogonal components ut, vand wt along Xt, Yt and Zt directions, respectively. The assumed mode methodis used to discretize the deformations. The admissible functions are linearly independent and satisfy geometric boundary conditions [64]. In a system with constantlength, the admissible functions may represent the mode shapes. But, in the presentcase, where the tether length changes over time, the concept of mode shape doesnot apply. However, an admissible function can be chosen to satisfy the geometricboundary conditions. Since the diameter of the tether is very small (a few mm), theadmissible functions depend only on Y. So the tether deformations can be expressedas:ut(yt,L,t) =1mt,(t); (2.5)vt(yt, L, t) = n(yt, L)Bn(t); (2.6)Wt(yt, L, t)=n(yt, L)Cn(t); (2.7)where:4n(yt, L), ‘I’(yt, L) admissible functions for tether transverse and longitudinal deformations, respectively;A(t), B(t), C(t) generalized coordinates for out-of-plane transverse, longitudinal and inpiane transverse deformations, respectively.Theoretically, a complete set of admissible functions should include infinite26terms. Here the completeness implies that the energy of the discretized systemis the same as the energy of the continuous system [64]. For most engineeringsystems, finite number of functions are sufficient to represent the dynamics. In thepresent study, the first N1 and Nt functions from the complete set are consideredfor representing the longitudinal and transverse deformations, respectively. Here N1and Nt are arbitrary numbers. For the admissible functions, the kinematic boundaryconditions dictate that the transverse deformations at the supported ends be zero,i.e.u(0,L,t) = wt(O,L,t) ut(L,L,t) = wt(L,L,t) = 0; (2.8)and the longitudinal deformation at the boundary y = 0 is zero, i.e.vt(O,L,t) 0. (2.9)At the subsatellite end (y = L), a dynamic boundary condition relatingthe stretch and static tension in the tether can be obtained [65]. Having definedthe flexible deformations, elemental mass of the tether at any arbitrary unstretcheddistance y from the origin of Ft can be defined by the position vector= u(yt, L, t)j + (Yt + v(yt, L, t))3t + w(yt, L, t)k. (2.10)The admissible functions for transverse oscillations of the tether correspondto those of a flexible string [65],(y,L)= %ñsin(Tht), n 1,2,”,Nt. (2.11)For the longitudinal deformation, the admissible functions can be chosen as theeigenfunction of a elastic tether supporting a point mass [29],= sin(/3nyt/L), n = (2.12)27where 13 is governed by the equation/3ntan(i3n)=, n=1,2,•,Nj. (2.13)m3Alternatively, it can be taken as2n— 1= () , n = 1,2,•• ,N1 (2.14)which represents the vibration of a string with an end mass [36]. In the presentthesis the latter form of the admissible function (Eq.2.14) is used for the numericalsimulation.2.2.4 Librational generalized coordinatesThe generalized coordinates defining librational motion (rigid body motion)are identified here. The orientation of each frame, F (i p, t, s), is obtained relativeto the orbital frame through three modified Eulerian rotations starting from theorbital frame. The sequence consist of:• rotation ofF0(X,Y0,Z0) by angle a about theX0-axis resulting in (X1,Y1,Z1);• rotation of (X1,Y1,Z1) by angle ,t3 about the Y1-axis resulting in (X2,Y2,Z2);• rotation of (X2,Y2, 22) by angle about the Z2-axis resulting in (Xi, Y, Z).These sequences of Eulerian rotations are indicated in Figure 2-2. Figure2-2(b) shows a rotation about X0 axis in Yo, Z0—plane. In case of the tether, tworotations at and ‘it are sufficient to describe its orientation in space (i.e. /3t = 0).Therefore the generalized coordinates for the librational degrees of freedom are: ap;/3p ‘in; at; ‘it (assuming the subsatellite and offset mechanism to be point masses).28zi ZoZ,Z2/\/\i/\/L1 7x1, x0X2(a)0Yo(b)Figure 2-2 Diagrams showing relative orientations of two different frames: (a)sequence of Eulerian rotations; (b) rotation about the X0-axis.292.2.5 Angular velocity and direction cosineSince the gcneralized coordinates are defined with respect to the intermediateframes of the Eulerian rotations, their derivatives are not the same as the inertialangular velocities expressed in frame F. Before relating derivatives of the Eulerangles with the angular velocities, the following discussion on the vector transformation between the reference frames is appropriate.The matrix which transforms a vector from the frame Fj to the frame F2 canbe described by the equation{v} = [Tj]{vj}, (2.15)where:{v2} vector expressed in frame F;{v,} vector expressed in frame F;[T] transformation matrix relating (Xi, Y, Z) and (Xi, }, Zj).The transformation matrix [T] has the following properties [66]:[T} = [Tjk][Tk(k_1)} . [T}[Tij], V integer k; (2.16)[T][Tj]=[I] ; (2.17)[T] = [T]’ = [T]. (2.18)Using Eq.(2.16), the angular velocity vector of the frame F can be obtained as1wzi5 1+at{w} = Li..),j = [T2][T110 0(wzi) I 03010.1 10+ [T12][T21 èt + [T2] 0(oJI ( + at) cos(,6) cos(yj) + j sin(y) .1=—(Ô + àt)cos() sin(’yj) + cos(yi) (2.19)I (6+àt)sin(i3)+’y Jwhere: 6 is the orbital rotation rate; indices 1 and 2 in the transformation matricesrefer to the intermediate frames in the Eulerian rotation sequence; andcos(yj) sin(7j) 0T22 = —sin(7j) cos(7j) 00 0 1cos(f3j) 0 —sin(/3)T21= 0 1 0sin(16) 0 cos(/3)1 0 0T10 = 0 cos() sin(a)0 —sin(a) cos(a)It is necessary to obtain the direction cosines {l} of the frame F relativeto the local vertical as they are needed in evaluation of the gravitational potentialenergy later:I .1 1 {i} {i0}{l} = l = {j} . {j°}(l J ({k}.{j0}10= [T2][T21][Ti0] 1(0I sin(a) sin(i3j) cos(yj) + cos(a) sin(7j) 4=— sin(a) sin(/3) sin(7j) + cos(a) cos(’yj) , (2.20)( —sin(a)cos(6) Jwhere {i}, {jJ, {k:} are unit vectors along the X, Y, Z axes, respectively; and{j0} is the unit vector along the Y0-axis.312.3 Kinetics2.3.1 Kinetic energyThe inertial velocity of any elemental mass in the th domain can be obtainedby differentiating Eq.(2.1). In the following development, a bar over a character torepresent a vector quantity is dropped and replaced by brackets. The matrices areenclosed by square brackets. With these notations, the velocity of an elemental massin the domain becomes{dm}F1 = {ic}F1 + + {} + {w} x {r}. (2.21)Here, the terms with subscript F1 refer to velocities with respect to the inertialframe while those without any subscript are with reference to the local frames. Thekinetic energy of the elemental mass (Figure 2-1) can be expressed asTdmj = •{Rdm}F {Rdm.}Fdmj. (2.22)The energy of the th body (Ti) can be obtained by integrating the above equationover the mass of the body. The kinetic energy of the entire system (T) is the sumof the energies of the individual domains,T=Ti=p,t,s= J {kdmj} {dm}dimi, (2.23)i=p,t,s mwhere mj is the mass of the domain i. Substituting Eq.(2.21) in Eq.(2.23) and aftersome algebraic manipulations, the total kinetic energy of the system can be writtenasT=-{kc}F + M{RsM}F {SM}F32+ (m0+ mt + ms){4}F+— {it(L)}F1}— {it(L)}F1}+ > {w}T[I}{w}+ j{t} {t}dmt +J{t}. {{wt} x {rt}}dmt+ ({{Lt}} {{Lft}})_ms({{Lt}}_{ML)}F})+ {SM}F1 . {mo +mt +ms){4}F1+ {t} +{wt} x {rt} }dmt+ms{{Ljt}}_ms{{ds}F1- {it(L)}F}}{t}+{wt}x{rt} }dmt+ ms{{Ljt}}- ms{{s}F + {Jt(L)}FJ}}. (2.24)For conciseness, the kinetic energy is expressed as products of vectors and matrices.Note, the kinetic energy expression accounts for 3-dimensional rotation of the platform, tether and subsatellite, as well as flexibility of the tether. For the particularcase where the system dynamics is confined to the orbital plane, the kinetic energyexpression reduces to7’= M{ic}{Rc}F + {i?sM}{M{i?sM} + {i?m}}+ {w}T[I]{w} + (ms + pL).L2 + rnsL2w + ma{4}T{4}i=p,t+mp{cip}T[U]{dp}+ mawz{dp}T[UkjT[Uk]{dp}+ { }T[Tt ]T{K1}+ wpz{dp}T[UkjT[TtPJT{Kl}+{ã}[TjK2]{..k} + wpz{dp}[UkJTtp}K2]{X33+ {}T[T]T[K3]{X} + w{dp}’ [Uk]T [Tt]T[K3]{X}+ {iC}T[K4]{( +{(}T[K5]{X}-i- {X}T[K6]{ + {K7}{i(} + {K8}{X}, (2.25)where:ma = m0 + pjL + m3;{1m} = ma{{4} + {wp} x {d}} + J {{t} + {wt} x {rt}}dmt+ms{{{Lt}} + {Jt(L)F};{X}= { }, = { } , and are expressed with respect to Ft;{d} = velocity of the offset point in the frame Fr,;{B} = {B}1; {C} = {C}t1.Here, N1 and Nt are the number of generalized coordinates for longitudinal andinpiane transverse vibrations, respectively. {K1}, [K2], , {K8} are defined inAppendix-I. In the above energy expression (Eq.2.25), the first term represents theorbital kinetic energy; the second accounts for the shift in the centre of mass; whilethe rest of the terms arise from librational and vibrational motions of the system.2.3.2 Gravitational potential energyAs in the case of the kinetic energy, the total gravitational potential energyis the sum of the energies of the individual domains. The gravitational potentialenergy for an elemental mass dm can be written asdUG. =GMe dm. (2.26)IRdmjI34Using the binomial expansion with truncation of the series after the second degreeterms, the potential energy for the i’’ domain can be written asUc =—+ J ({R} {RJ —3 ({jo} {R}) 2) dm,where:G universal gravitational constant;Me mass of Earth;.Rc magnitude of the orbital radius;m mass of the th domain;{j°} unit vector along the Y0-axis (local vertical);and {R} as defined in Eqs.(2.3, 2.4). The gravitational potential energy for theentire system can now be written asU =— GMeM + GMe (M({RSM}. {RSM} — 3({jo} {RSM})2)+ ma(2{RSM} {d} + {d}. {d} — 6({j0} {RM})({jo} {dp})- 3({j0} {d})2)+ ms({Ljt} {Lj} + 2{Lj} {ft(L)} + {ft(L)} {ft(L)})+ 2ms{{RSM} + {d}} {{Ljt} + {ft(L)}}-3m ({io} {Ljt}) ({i0} {Ljt} + 2{j0} {RSM}+ 2{j0} {d} + 2{j0} {ft(L)})-3m ({i0}. {ft(L)}) (0 {ft(L)} + 2{j0} . {RSM} + 2{j0} . {d})+ 2{{RSM} + {d} - 3({jo} {d} + {jo} {RSM}){jo}} {J{rt}dmt}-(tr[zj_3{z}T[I]{l})). (2.27)i=p,t,s35Here the first term represents potential energy of the system treated as a point mass.The rest of the terms are due to the system’s finite dimension. The potential energyexpression when simplified and expressed in matrix notation for the planar case hasthe formUG =— GMeM+ ({RSM}T[Plj{RSM} + {d}T[P2]{d}+ 2{RsM}T[P] dp} + {X}T[P3]{RsM + d}+ {X}T{P4}+ {X}T[P5j{X} + {p}T{R + d}+ msL2(1— 3Tto) — (tr[i1 — (2.28)i=p,twhere[Ii] inertia dyadic of the domain i with respect to F (Appendix-I){l} direction cosine of the Y0-axis (local vertical) with respect to the frame F;and [F1], [F2],... , {P}, and Vto are defined in Appendix-I. In Eq.(2.28), termscontaining {RSM } account for the potential energy due to shift in the centre ofmass.2.3.3 Elastic potential energyIn the linear elastic theory of strings, it is generally assumed that the initial tension in the string is large enough and the transverse displacements causenegligible change in this tension. In a tethered orbiting system, the tension maybe reasonably large for long tethers (length of the order of kms). But at shorterlengths, the tension is very low due to the weak gravity gradient force. In the extreme case when the length approaches zero, the tension tends to zero. Thereforethe effect of transverse vibration on the tether tension, and hence on the elastic36oscillations can not be neglected.The strain energy of the tether is based on the theory of vibrating string[67]. In order to account for interactions between the longitudinal and transversemodes, transverse displacement terms upto the second order are retained in thestrain expressionövt 1 1 / ãut 2 ow 2\E=—+—( (—) +—) 1Oyt 2 c9yt f9yt /where is the total strain in an elemental tether mass of volume dVt. Total strainenergy of the tether can be written asusz_J cdVt,EAJ°( + 1((ôut)2 + (thvt)2))dY (2.29)where:u stress in an elemental mass of volume dVi, EE;A area of cross-section of the tether;L(t) unstretched tether length;T4 volume of the unstretched tether, AL(t);E young’s modulus of the tether material.Substituting from Eqs.(2.6, 2.7) in Eq.(2.29), strain energy of the planar system canbe obtained asU5JL(t) (d{F(Yt)}T{B}) dyt +JL(t) (d{Fb(vt)}T{c})4yt+JL(t) (d{F(t)}T ) (d{Fc15(Yt)}T{C}) dyt. (2.30)372.3.4 Dissipation energyThe energy dissipation during tether deformation can be accounted for throughstructural damping. Because of the complex mechanism of energy dissipation, thestress—strain relation does not correspond to the elastic case. The system exhibitsthe hysteresis phenomena during vibration. The area enclosed by the hysteresiscurve indicates the energy dissipation. In engineering applications, it is commonlyaccounted for by considering the structural damping coefficient determined experimentally [68]. The structural damping can be expressed as an equivalent complexYoung’s modulus,= E + iE1, (2.31)where the real part E is the Young’s modulus without structural damping; theimaginary part E1 contributes to the structural damping; and i = /1i. Now thetotal stress can be written asat E’c = (E + iE1)€ = E(l + ii)c,where= <<1. (2.32)If is harmonic with frequency ,i = — (2.33)woand the total stress becomes= E[e+ (_)E] = u+ad, (2.34)where a = Ee is the stress in absence of the structural damping; and ad =is the stress causing energy dissipation.38The dissipation of energy can be expressed in terms of the Rayleigh’s dissipation function [69],Wd= / OdfdVt= EAi71L2 (2.35)2.4 Equations of MotionUsing the Lagrangian procedure, the governing equations of motion can beobtained fromd ÔT ÔT c9U ãWd— ++Qq, (2.36)where:q vector of generalized coordinates;Q q vector of generalized forces corresponding to generalized coordinates q;U =UG+US.The governing equations of motion account for:(i) inpiane rotation of platform and tether negotiating any arbitrary trajectory;(ii) longitudinal and inpiane transverse vibrations of the tether;(iii) inpiane (two dimensional) offset of the tether attachment point from thespace platform’s centre of mass;(iv) effect of the controlled variation of the offset attachment point;(v) influence of deployment and retrieval on the system dynamics;(vi) effect of thrusters located at the subsatellite end and momentum gyros onthe platform;39(vii) shift in the centre of mass due to rigid body librations and elastic deformations of the tether.They would permit parametric response analysis of the system as well as aid in development of control strategies using thrusters, offsets and their hybrid combinations.In a compact form the equations of motion can be expressed as[M(q,t)]{ij} + {F(q,,t)} = {Qq}, (2.37)where:[M(q, t)} mass matrix (Appendix-IT);{ F(q, t, t)} nonlinear, nonautonomous terms accounting for gravitational, Coriolisand centrifugal forces (Appendix II);{q} vector of generalized coordinates, {ap, at, {B}T, {C}T}Tap platform pitch angle;at tether pitch angle;{B} vector of longitudinal elastic generalized coordinates;{ C} vector of transverse elastic generalized coordinates;{ Qq } nonconservative generalized force vector.2.5 Generalized ForcesThe generalized force vector (Qq) accounts for the effects of nonconservativeexternal forces and moments due to active and passive control, and environmentaldisturbances. As mentioned before, the platform based momentum gyros providecontrol torque (M,) about the platform axis Xp (Figure 2-3). As shown in the40TLA schematic diagram of the Tethered Satellite System (TSS) showing different input variables.SU BSATELLITETETHERTl:xtd’ORBITTaPLATFORMr aYiFigure 2-30Z1xi41figure, the tether swing can be controlled by manipulating an orthogonal set ofthrusters (Tat, TL) at the subsatellite, or the motion of the tether attachment pointd) at the platform end. As discussed in Chapter 4, the control influencematrix for the offset strategy can be obtained from the coefficient of the offsetacceleration in the equations of motion. However, for thruster control and platformpitch equation, it has to be obtained as the generalized force vector.The generalized force vector, Q, can be evaluated using the principle ofvirtual work. Let F1, F2, , Fm are m external forces acting on the systemat locations with position vectors r1, ‘2, , fm, respectively. The system hasn generalized coordinates (qi, q, qn). The position vectors are, in general,functions of qj’s. The virtual work, SW, by all the external forces can be expressed.asm m n -i=1 i=1(p. sqj = Qq38q, (2.38)j=1 := q3where(2.39)is the generalized force corresponding to the generalized coordinate qj• The externalforces and moments acting on the system are:M control moment acting on the platform about the X, axis;Tat thrust applied perpendicular to the tether line at the subsatellite end;TL control thrust along the undeformed tether applied at the subsatellite;Fdt damping force perpendicular to the undeformed tether;42Fdj damping force along the tether line.The transverse and longitudinal dampers are located at distances of dt anddl from the origin of Ft. Using Eq.(2.39), the generalized forces for the tether angleand elastic degrees of freedom can be obtained as:= Tat (L + {F(L)}T{B}) + Fdt (Ydt + {F(Ydt)}T{B}); (2.40){ Qn} = TL{F(L)} + Fdl{F(Ydl)}; (2.41){ Qc} = Fd{Fl(Yd)}. (2.42)Since the tether can not transmit any moment or transverse force, the computation of Qap needs special attention. To that end, rather than considering theexternal forces acting on the tether for virtual work computation, the equivalenttension (1”a) at the platform end of the tether is employed. The tension ta dependsOn Tat, TL, Fdt and Fdl, and attitude angles of the tether and platform. Tat andFdt are perpendicular to the undeformed tether and hence their contribution to t’acan be neglected. This results inThe generalized force for ap can now be obtained fromwhere a is the position vector of the tether attachment point on the platform.After appropriate transformation of vectors and expanding the dot product it canbe shown thatQap = M + (TL + Fdl) (Dt sin(a) — .Dt cos(a)), (2.43)43where:a = at — ap;= RSM + dp;= RSMZ + 4•2.6 SummaryA general set of kinetic and potential energy expressions are obtained fora platform based tethered satellite system undergoing three-dimensional dynamics.These expressions are used to obtain, through the Lagrangian procedure, governingequations of motion for the system dynamics confined to the plane of the orbit. Thehighly nonlinear, nonautonomous and coupled equations of motion are extremelylengthy even in the matrix notation. They account for a shift in the center of mass,time dependent variation of the tether attachment point at the platform, as well asdeployment and retrieval of the tether. The structural damping is modelled throughRayleigh’s dissipation function. The generalized force vector representing effects ofexternal forces is evaluated using the principle of virtual work. The relatively generalformulation can implement the offset and thruster control strategies to regulateboth the rigid and flexible dynamics of the tether with the platform attitude motioncontrolled by momentum gyros.443. COMPUTER IMPLEMENTATION3.1 Preliminary RemarksThe governing equations of motion developed in the last chapter were integrated numerically to study the dynamics and control of the tethered systems. Thecomputer code is quite lengthy ( 6200 lines), mainly due to the highly time varying nature of the system dynamics and moving tether attachment point. Moreover,the frequencies of attitude and elastic degrees of freedom are widely separated witha closely packed spectrum for the flexible system. The stiff characteristic of thesystem, which may present problems during numerical solution, demanded specialattention. It was desirable to explore several different approaches to arrive at anefficient controller design. To that end, it was necessary to evolve a flexible programstructure that would permit varied simulations with a few parameter changes in theinput file. Of course, the program should be easy to debug and efficient to run.This chapter begins with discussion on the numerical algorithm and programstructure developed. Next, validity of the equations of motion and the computercode are assessed by two different methods: conservation of total energy; and comparison of frequency spectra for a particular case of the linear system as reported inthe literature. The chapter ends with a summary of salient features of the numericalcode.3.2 Numerical ImplementationIntegration algorithmA computer program is written to numerically solve the governing equationsof motion of the tethered satellite system. The second order ordinary differential45equations representing the system dynamics are rearranged as first order equationsto this end. The system dynamics as represented in Eq.(2.37) can be expressed as= [M(q,t)]_1{Qq— F(q,i,t)}. (3.1)Definingthe above relation can be rearranged as{ [M(q,t)]1{Qq - F(q,,t)} }‘ (3.2)which is a set of 2N first order equations. Here N is the dimension of the generalizedcoordinate vector {q}. The IMSL:DGEAR subroutine is used to integrate the aboveequation. The main advantage of this method is the automatic adjustment of theiteration step—size for stiff systems with error check in each iteration cycle [70]. Thesubroutine uses Gear’s predictor-corrector algorithm [71].Program structureThe flowchart showing the structure of the program to simulate- both uncontrolled and controlled dynamics of the system is presented in Figure 3-1. Theprogram starts with initialization of the system parameters and generalized coordinates. Special program parameters are introduced to identify the type of controllerused in the simulation. The initialization is achieved by reading the input file. Ineach integration time-step, the generalized coordinates, length and offset variablesare written into the output files. The main program calls the integration subroutine DGEAR which in turn calls the subprogram FCN. The system dynamics asexpressed in Eq.(3.2) is computed by this subroutine. In case of controlled simulation, the actuator inputs are obtained from the subroutine CONTROL. Details of46I OrbitalI Elements_________________ControlL - - -MAINFigure 3-1 Flowchart showing the structure of the main program.INPUTMass and ElasticParametersDeployment andRetrieval ParametersInitialDisturbancesPROGRAMIMSL: DGEARParametersSubroutine DGEAR(Numerical Integration)47the subroutines are given below.Subroutine FCNThis subroutine computes the mass matrix and the nonlinear terms to formulate the vector ‘, Eq.(3.2), required by the integration subprogram (DGEAR).As shown in the flowchart (Figure 3-2), this subroutine calls a number of subprograms (such as KINETIC, POTENTIAL, etc.) which perform certain specificcomputation (e.g. computation of modal integrals used in Kinetic and potentialenergy expressions). This feature makes the program easy to debug. The velocityand acceleration of the tether deployment/retrieval maneuver are calculated by thesubprogram LENGTH. Based on the specified parameters, the subprogram OFFSET computes the velocity and acceleration profiles for the specified offset motionThese two subroutines define the system maneuvers. Next, the modal integrals andother matrices used in the formulation (Appendix—I) are evaluated using the subprograms KINETIC and POTENTIAL. As explained in Chapter 2, some intermediatematrices are defined for concise presentation of the governing equations of motion.The matrices appearing in the potential and kinetic energy expressions are evaluated through the subprograms SP and SK, respectively. The matrices used in thepotential energy equation are[F1], [F2], [F3], [P4], [F5], {P6},and those used in kinetic energy expression are{K1}, [K2], [K3], [K4], [K5], [K6], {K7}, {K8}.The contribution of the terms containing the inertia matrix (Appendix I) of thetether is computed by the subroutine INERTIA. The contribution of the strain48Calculate‘ H“FSubroutine IMSL: DGEARFigure 3-2 Structure of the subroutine FCN to evaluate the first order differential equations required by the integration subroutine.+SubroutineolNTROJNo49energy and structural damping are estimated in subprograms STRAIN and DAMPING, respectively.The above subprograms completely define all the terms required to evaluatethe governing equations of the system. Depending on the specified control parameters, the system inputs are established and the vector computed. This value isused by the integration subroutine DGEAR for the numerical solution.Subroutine CONTROLThe system inputs required to regulate the dynamics are computed by thesubroutine CONTROL (Figure 3-3). This subroutine selects the appropriate controller design subprogram for either offset or thruster strategy. The subprogramfor offset control obtains the system model and then designs the controller. In thethruster based regulator, three different models can be used for the controller design.Once the model is selected, the procedure is the same as in the offset strategy. Thecontrol inputs are computed and returned to the subroutine FCN.3.3 Formulation VerificationOnce the computer program is developed to integrate the equations of motion, the next logical step is to validate the formulation as well as the numericalcode. Two different approaches are used to this end. In the first place, total energyis computed for conservative configurations of the system. In the second approach,natural frequencies of the linear system are compared with those reported in theliterature. As can be expected, numerical results for the model selected for studyare not available. One is forced to be content with validation through comparisonof a few simplified cases.50Figure 3-3 Flowchart for the subprogram CONTROL to compute actuationinputs.Offset control513.3.1 Energy conservationIn the study of the attitude dynamics of spacecraft, the effect of attitudedynamics on the orbital motion is very small [61, 62, 63J. Therefore, it is a normalpractice to consider the satellite to be moving in a Keplerian orbit. This assumptiondoes not affect the results of an attitude dynamic study. However, it imposes an additional constraint and makes the system nonconservative. So, the attitude and theorbital motions are decoupled and the governing equations representing uncoupledattitude dynamics, which is a conservative system, is used for energy calculation.The inertia and elastic parameters considered in the analysis are the same as thoseused for the dynamic simulation (Chapter 4).With zero structural damping, the variations in kinetic, potential and totalenergies from the reference values are plotted in Figure 3-4. The energies at thebeginning of the simulation are considered as the reference values. Figure 3-4(a)shows results for the stationkeeping at a tether length of 20 km and zero offsetwhere as Figure 3-4(b) is for L = 5 km and an offset of 1 m along the the localhorizontal. In both the simulations, initial disturbances of 0.5° is given to the tetherattitude (at) and the platform pitch angle (ap). Initial conditions of 12 m and 0.8 mare given to the longitudinal oscillation (B1) of the tether for L = 20 km and 5 km,respectively. For the tether transverse vibration, the disturbance levels are taken as1 m and 0.01 m for L 20 and 5 km, respectively. As expected for a conservativesystem, change in the total energy remains zero and there is a continuous exchangebetween the potential and kinetic energies in both the simulations. Similar resultswere obtained for other lengths and offset positions of the tether. In all the casesconservative nature of the system was found to be preserved. It should be recognizedthat during deployment/retrieval and movement of the tether attachment point,5250.025.00.0-25.0-50.02.00.0-2.0Figure 3-4 Figure showing variations of kinetic, potential (gravitational pluselastic) and total energies of the system during the stationkeepingphase: (a)L = 20 km, d1,, = = 0; (b) L = 5 1cm, d = 1 m,= 0.Orbit 0.30.0 Orbit 0.353there is energy input to the system. So the system is no longer conservative and theenergy check can not be applied.3.3.2 Eigenvalue comparisonIn this case, linear models were obtained for decoupled rigid and flexible subsystems by neglecting the second and higher order terms and making appropriatesubstitution for the trigonometric terms. These linearized equations of motion forthe rigid and flexible subsystems are given in Appendices III and IV, respectively.The frequencies of the system are calculated and compared with the results reportedby Keshmiri and Misra [34], and Pasca and Pignataro [72]. In ref. [34], the number ofelastic modes of a tether in each direction is limited to two. The higher frequencies•of the system are obtained by using what authors call the semi-bead model, wherethe tether is divided into a number of smaller segments and negligible masses areplaced at the connection points. Ref. [72] analyzes a system consisting of an elastic continuum tether with mass, and orbiter and subsatellite represented as pointmasses. The linearized equations of motion are solved by means of a perturbationtechnique.-The present analysis, which models the tether as a flexible string using theassumed mode method, can include any arbitrary number of modes in both thelongitudinal and transverse directions. The eigenvalues are computed for the samemass and geometric parameters as in the references. The governing equations arelinearized about the static equilibrium position which is zero for the rigid and transverse flexible modes. The equilibrium value for the first longitudinal mode is 100 mand 3.6 m for tether lengths (L) of 100 km and 20 km, respectively. The normalizedfrequencies are compared in Tables 3.1 and 3.2. The mass and elastic parameters54considered for the frequency computation are indicated in the tables.Table 3.1 Inplane dimensionless natural frequencies (w/Ô) of a 2-body system(L 100 km, mp = i0 kg, m3 500 kg, pt = 5.76 kg/km,EA 2.8 x N).Mode Present Ref.[34] Ref.[72j TypeStudy1 1.732 1.731 1.794 Lib.2 5.958 6.388 6.905 Tran.3 11.917 12.059 12.457 Tran.4 17.876 17.879 18.269 Tran.5 23.835 23.742 24.142 Tran.6 29.794 29.627 — Tran.7 35.753 35.528 — Tran.8 41.712 41.445 — Tran.9 47.670 47.383 — Tran.10 53.627 53.344 53.807 Tran.11 54.266 54.541 54.559 Long.12 59.589 59.337 59.758 Tran.In the tables, w is the frequency of the system; 8, the orbital frequency; m3,the subsatellite mass; pt, the mass per unit length of the tether; E, the Young’smodulus and A, the cross sectional area of the tether. The first column representsthe mode number and the last column is the mode type, i.e. librational (Lib.),transverse (Tran.) or longitudinal (Long.).Table 3.2 Inpiane dimensionless natural frequencies (w/6) of a 2-body system(L = 20 km, mp = oo, m3 = 576 kg, p = 5.76 kg/km, EA =2.8 x N).Mode Present Ref.[34] Ref.[72] TypeStudy1 1.732 1.732 1.733 Lib.2 12.640 12.777 12.780 Tran.3 25.281 25.245 25.241 Tran.4 37.922 37.795 37.771 Tran.5 50.563 50.369 50.319 Tran.55The minor differences between the present results and those of ref. [34] areof the same order in both the cases. The discrepancies may be attributed to thedifferent formulation methods used. The larger differences for L = 100 km is due toa parameter 7 = pL/ms defined in ref.[72], which plays an important role. y has asmaller value in Table 3.2 than in the previous case. As pointed out by the authors,their results are more accurate for smaller -y, hence the correlation is better in Table3.2.Frequencies of the systems with different platform and subsatellite massesare shown in Table 3.3. The discrepancy between the present results and those ofRef. [34] follow the same trend as before. Since has a reasonable value, the resultsare in better agreement with those of Ref. [72]. Results for three cases are reportedhere with different end masses. One case is close to the TSS configuration where theplatform mass is iü kg and the subsatellite mass is 500 kg. Other two cases are forequal end masses. As shown in Table 3.3, the frequencies are the highest with thelargest end masses (mp = ms = iü kg) and the lowest for the smallest end masses(mp = m3 = 500 kg).Note, the results of the linearized system match quite well with the reporteddata although the three studies use different methods to determine frequencies ofthe system. Furthermore, the results are in close agreement for different tetherlengths as well as different masses of the endbodies. This with the conservation oftotal energ test provides considerable confidence in the validity of the governingequations of motion and the numerical code developed for their integration.56Table 3.3 Comparison of natural frequencies of a tethered two-body systemwith different end masses (L = 20 km, Pt = 5.76 kg/km, EA2.8 x iü N).Mode Present Ref.[34J Ref.[72] TypeStudymp = kg,m3 = 500 kg1 1.732 1.732 1.738 Lib.2 11.917 11.956 11.993 Tran.3 23.835 23.578 23.644 Tran.4 35.753 35.286 35.368 Tran.5 47.671 47.019 47.112 Tran.mp = ms = kg1 1.732 1.715 1.728 Lib.2 10.028 14.432— Tran.3 113.453 113.430 113,385 Tran.4 226.908 227.023 226.736 Tran.5 340.364 340.934 340.095 Tran.6 453.813 455.167 453.455 Tran.mp = ms = 500 kg1 1.732 1.732 1.742 Lib.2 8.294 8.486 8.341 Tran.3 16.588 16.507 16.191 Tran.4 24.883 24.637 24.155 Tran.5 33.178 32.799 32.142 Tran.3.4 SummaryThe governing equations of motion for the tethered satellite system are integrated numerically using the IMSL:DGEAR subroutine. The computer programis developed in a structured manner to reduce debugging and running time. Thenumerical code is validated by two methods: the total energy check for conservativesystems; and comparison of frequencies with those reported in the literature. Theexcellent correlation provide confidence in the simulation model and the numericalcode developed for its dynamical response and control studies.574. DYNAMIC SIMULATION4.1 Preliminary RemarksUnderstanding of the system dynamics is fundamental to the design anddevelopment of any engineering system. Furthermore, design of an appropriatecontroller requires appreciation of the system performance under a wide variety ofoperating conditions to guard against the possible instability. To that end, thegoverning equations of motion of the tethered satellite system are numerically integrated and the system’s dynamical response studied for different parameter valuesand operating conditions.A major concern in the operation of tethered systems is the dynamical behaviour during deployment and retrieval phases. In modelling of a flexible system,the number of modes used for discretization is important. The model developedhere can include an arbitrary number of modes for both the transverse and longitudinal vibration of the tether. An acceptable number of modes for the analysis ofthe system is arrived at through comparison of simulation results including highermodes. This chapter focuses on results of a parametric study carried out by systematic variations of the tether length (L), offset of the tether attachment point (dand Young’s modulus of the tether (E), density of the tether material (Pt) andmass of the payload (m3).4.2 Deployment and Retrieval SchemesBefore proceeding with the parametric analysis of the system dynamics,some remarks on the deployment and retrieval time histories used in the study58would be appropriate. The deployment is carried out with an exponential-constant-exponential velocity profile. Let L1 and L2 are the lengths where the velocity (L)profile changes from exponential to constant and from constant to exponential character, respectively. The deployment velocity profile is characterized by the followingrelations:LcdL, LaL<Li; (4.1)= CdL1, L1 L <L2; (4.2)—Cd(Ll+L2), L2 LLf; (4.3)where L0 and Lf are the initial and final tether lengths, and Cd is the proportionalityconstant. The above equations can be integrated to obtain the length expression:L = Loecd(t_t0), L0 L <L1; (4.4)= L1 + CdL1(t — t1), L1 L <L2; (4.5)= L1 + L2 — Lie_Cd(t_t2), L2 L Lf, (4.6)where and t are the time instants when the tether length is L1 and L2, respectively, starting from the beginning of deployment. Given the initial time t, finaltime tf, L0, Lf, L1 and L2, the proportionality constant can be obtained asCd (tftO){ln(t) +(L2i)1(Ll+L2_ f)} ()Similarly, expressions for the retrieval profile areL = crL; (4.8)and L =L0ed7(t_t0), (4.9)where L0 and t0 are the tether length and time, respectively, at the beginning of59the retrieval phase. The proportionality constant, c,, now has the formcr= ( ) in (t). (4.10)At times, particularly with the offset control in a hybrid scheme, only the exponentialvelocity profile is used. The corresponding length expressions for such deploymentscheme are similar to the retrieval expressions with appropriate initial and finalparameters.4.3 Simulation Results and DiscussionThe parametric study was rather comprehensive, however, for conciseness,only a few typical results are reported here to help establish the trends in thedynamic behaviour. The inertia and elastic parameters considered in the analysisare:I, = Inertia matrix of the platform,8,646,050 —8,135 328,108—8,135 1,091,430 27,116 kg-rn2;328,108 27,116 8,286,760rnp = mass of the platform, 90, 000 kg;rn0 = mass of the offset mechanism, 10 kg;Pt = mass of the tether per unit length, 4.9 x i0 kg/rn;rns = mass of the subsatellite, 500 kg;EA= 61,645 N.The system negotiates a circular trajectory with a period of 90.3 minutes. In thesimulation, the X-axis is oriented parallel to the orbit normal (X0). The platform60pitch,,,is the angle between the Yp-axis and the local vertical (i.e. Y0-axis).Similarly, X.t is parallel to X0 and the tether pitch is the angle between Y and thelocal vertical. The longitudinal and lateral elastic deformations of the tether aremeasured with respect to the frame Ft.The structural damping considered in the simulation corresponds to a damping ratio () of 0.5% based on the first natural frequency of the longitudinal oscillation of the tether. Computation of dissipative terms due to the structural dampingrequires the values of the parameters and w0 (Section 2.3.4). Here w0 is consideredas the frequency of the 1st longitudinal mode (B1) which can be computed from theequivalent stiffness and mass of the decoupled system,/--wo= 4! —,Vtm&where:= Ck3(1, 1) + Ck3(1, 1) + Wtl k4(l 1) — 2K6(1, 1)m = 2K4(1, 1).The structural damping parameter can be obtained from the damping ratio ,— 2Ewmb/LI ‘1’i \JoThe matrices used in the above expressions are defined in Appendix I.4.3.1 Number of modes for discretizationIn the modelling of a flexible system using the assumed mode method, the61system is complete in the sense that the energy of the mathematical model convergesto the true energy when the number of modes approaches infinite [64]. However,in reality, one can employ only a finite number of modes. Fortunately, for mostphysical systems, it has been observed that only a few modes can represent thesystem dynamics with considerable degree of accuracy. The model with the firstthree transverse and the first two longitudinal modes are considered to be reasonablyaccurate. The disturbance induced relative amplitudes of different modes were usedas a basis for selecting the acceptable number of modes.Simulations are carried out for two tether lengths. Figure 4-1 shows theresponse of the system with a tether length of 20 km and initial conditions as shownin the diagram. As expected, for the stationkeeping case, the rigid degrees of freedom(i.e. ap and at) exhibit pure oscillatory motion. The first and second longitudinalmodes (B1 and B2) show high frequency decaying oscillation due to the structuraldamping, which has a stronger effect on the second mode (B2). Modulation of theB2 response is due to its coupling with the first longitudinal mode. Note, an initialdisturbance of 10 m is given to the first transverse mode (C1). With zero specifiedoffset (D = = 0) of the tether attachment point, the tether dynamics isnot coupled with the platform motion. Therefore C1 has a pure oscillatory motionwith zero mean. As the structural damping has only the second order effect onthe transverse tether vibration, the responses in the higher modes, C2 and C3, arealso non-decaying. As apparent from the at response, there is very small couplingbetween the attitude and flexible motion of the tether even for, a length of 20 km.The modulation of the second and third transverse modes (C2 and C3) is due to thecoupling between the flexible degrees of freedom.The important aspect of this simulation is the relative magnitudes repre62B2010.00.0-10.0Figure 4-1OrbitDynamic response of the system with the higher modes included inthe discretization of the system.(O) 2°; a(0) = 20 C2(0) = C3(0) = 0B1(0) = 12 m; B2(0) = 0 Stationkeeping: L = 20 kmC.(0)=lOm D=0; D=O2.00.0-2.018.012.02.0cz 0.0-2.0B10.0 2.0 0.0 2.00.00.0-0.6C2C30.150.00-0.150.050.00-0.05o.5 2.063senting the energy content of the higher modes. A comparison for the longitudinalmodes shows that the amplitude of B2 is an order of magnitude lower than that ofB1. Moreover, the vibration of the second longitudinal mode decays in about 0.02orbit as compared to one orbit for the first mode. As mentioned before, the sustainedoscillation of B2 is due to its coupling with the B1 response. So the energy contentof the second longitudinal mode would be very small compared to that of the firstone. Similar conclusion can also be made for the transverse modes. The amplitudeof the first transverse mode (Ci) is higher by at least one order of magnitude fromthat of C2, and two orders of magnitude from that of the C3 response. Thus theenergy content in the higher modes of the tether vibration is minimal.Similar conclusion can be arrived at from the simulation results for a tetherlength of 10 km (Figure 4-2). Unlike the previous case, now the modulation ofthe third transverse mode is quite prominent, although the amplitude is rathersmall. The beat phenomenon is due to the weak coupling between the attitudeand flexible motions through a shift in the center of mass. Note, in this case thedifferences between the fundamental and the higher modes are more than two ordersof magnitude.From these two sample cases, it can be concluded that the energy contentin the higher modes is relatively small. Therefore, in the subsequent dynamicsand attitude control studies, only the first longitudinal and transverse modes areconsidered.4.3.2 Tether lengthTo facilitate comparison of results, a reference case is established for a tetherlength of 5 km and zero offset along both the local horizontal and local vertical direc64x(O) = 2°; cx.(0) =2° c2(o) = C3(0) =0B1(0) = 3m; B2(0) = 0 Stationkeeping: L = 10 kmC1(0) =2 m =0; =0B13 IWWWWVVvV.5Cl j0.02C20.01C3 0.00-0.010.0 Orbit 4.0Figure 4-2 System response for a shorter tether with higher modes included inthe flexibility modelling.65tions. In the reference case (Figure 4-3), the rigid body rotations of the tether (ct)and the platform (ap) are essentially unaffected by the tether flexibility. For zerooffset it is expected that the platform dynamics is not coupled with the tether motion and hence free from its dynamical behaviour. The tether rotation is negligiblyaffected by its flexibility dynamics. As apparent from the figure, the longitudinalflexible response (B1) consists of two frequencies. The lower frequency is due tothe coupling between the rigid body motion of the tether (ct) and the flexible dynamics, whereas the higher frequency corresponds to the flexibility itself. The highfrequency component of B1 decays due to the structural damping in the tether. Thestructural damping has a second order effect on the transverse tether vibration. Sothe C1 response has an essentially constant amplitude. For zero offset, as expected,the transverse vibration (C1) is not coupled with other degrees of freedom.The response of the system with a smaller tether length (L = 50 m) isshown in Figure 4-4. As in the reference case, the longitudinal mode containstwo frequencies. But the higher frequency of oscillation increases from 0.025 Hz(corresponding to the reference case) to 0.249 Hz when the length decreases to 50m. For L = 50 km (Figure 4-5), the higher frequency decreases to 0.0073 Hz. Therewas also a change in the frequency of transverse vibration (C1) from 0.0045 Hz forthe reference case to 0.0448 Hz for L = 50m and to 0.0014 Hz for L = 50 km. Thestiffness of the tether, which is inversely proportional to its length, increases as thelength decreases and hence results in a higher frequency of oscillation for a shortertether.Figure 4-6 shows the effect of the flexible tether on the rigid body dynamicsfor a length of 500 m. Simulation is carried out with zero initial condition forthe tether attitude motion. at response has a high frequency component due to it66cx(O) = 2°; cx(0) =2° Stationkeeping, L = 5 kmB1(0)=0.8mD =0; D=0C1(0)= 0.01 m2.00.0-2.0B10.0-2.00.00.90.80.0-0.000.02.02.02.01.0Figure 4-3 Dynamical response of the tethered satellite system in the referenceconfiguration with the offset set to zero.2.00.0OO2Orbit671.00.0-1.0Figure 4-4 Response results showing the effect of decreasing the tether lengthon the system dynamics.x(O) = 2°; x(0) =2° Stationkeeping, L =50 mB1(0)= 0.81 xlO4mC(0)=1.OxlO4m D=0; D=02.00cpo 0.0-2.00.02.00.0-2.01.0.ff4 IX IVTJB.,1.00.009_0.00 0.01.80.0 1.00.0 Orbit 0.168cxpCl2.00.0-2.0x(O) = 2°; tx(0) =2° Stationkeeping, L = 50 kmB1(0)=75.OmD=0m; D=0mC1(0) = 20.0 rn0.02.02.0Figure 4-5 Plots showing the effect of increasing the tether length on the system dyanmcis.2.02.00.0-2.00.0B1 80.00.0 Orbit69cxp°0.5-0.00.0x1.00.08.19B1 8.16C18.130.0Figure 4-61.0I—1.00.5System response during stationkeeping showing the effect of flexibility on the tether attitude motion.a(b) =0; a(0) =0 Station keeping, L = 500 mB1(0)=8.133x103mC(0)=1.OxlO2 D=0; D=00-1.0x103m0.170coupling with transverse tether oscillation (C1). But the amplitude of oscillation is ofthe order of i0 degree, which can be considered negligible in practical applications.4.3.3 Offset of the tether attachment point‘Effect of the tether attachment point offset was found to be quite significant.It couples the platform dynamics with the tether degrees of freedom. The offsetalong different directions, i.e. local horizontal and local vertical, was found to havedifferent coupling effects. Thus motion of the tether attachment point, which isrequired during the offset control, adds extra complexities to the system dynamics.Figure 4-7 shows the system response during the offset motion from the center ofmass of the platform by 2 m in 120 s along the local vertical while D2 is held fixedat zero. As shown in the figure, is varied according to a sinusoidal accelerationprofile. The tether pitch (c) and longitudinal elastic mode (B1) are unaffected bythe offset motion, while the platform pitch response (np) shows amplitude modulations. The amplitude of the transverse vibration (C1) reaches around 0.08 m duringthe offset motion and remains at that value subsequently. Note, here the responsereaches a higher value than the initial disturbance of 0.01 m.Response results were also obtained for an offset motion from zero to 2 m in120 s along the local horizontal (Figure 4-8). In this case there is strong couplingbetween the platform and tether dynamics. Particularly, the platform oscillatesat much lower frequency and about a mean orientation at _900 as compared tothe local horizontal position in Figure 4-7. The C1 response is modulated due tothe coupling with the platform dynamics. The other degrees of freedom behave asbefore.71Figure 4-7 Simulation result.s during offset motion along the local vertical withthe offset along the local horizontal fixed at zero.= 2°; c(O) =20B.(O)=0.8mC.(O)= 0.01 mStationkeeping, L=5 kmD: Oto2minl20sD=05.03.0a ° 0.0p-3.00.0B1C10.0 5.00.0 1.00.0 Orbit2.0-0.00.50.0 Time, s 200. 0.0 Time,s 200.72cz(O) = 2°; a(0) =2° Stationkeeping, L=5 kmB1(0)=0.8m D=0C1(0) = 0.01 m D: 0 to 2 m in 120 s0.020.01Cl-0.00-0.012.00.0 1.0bm..0.0 Time, SOrbit0.0cxp-180.00.0 0.0B 1 0.805.00.0-0.05.0D°•0pz200. 0.0 Time, s 200.Figure 4-8 System response showing the effect of offset motion along the localhorizontal.73When a fixed offset is given along both the local horizontal (D) and thelocal vertical (D2) directions (Figure 4-9), there is a strong modulation of thecrp response like that shown in Figure 4-7 but the frequency is not modulated asindicated in Figure 4-8. As shown in Figure 4-9, the transverse vibration (C1) has aregular amplitude modulations due to the coupling with the platform pitch motion,however, The tether attitude and the longitudinal flexible modes are unaffected bythe offsets.4.3.4 Tether mass and elasticityThe flexible motion of the tether is significantly dependent on the mass andelastic properties of the material. The two flexible characteristics which are mostaffected by the change in mass and elastic stiffness are: static deformation of thetether; and the frequency of both the transverse and the longitudinal vibrations.To understand these effects, studies were undertaken with different mass densitiesand elastic stiffnesses for the stationkeeping phase at a tether length of 5 km. Adecrease in the linear mass density (Pt) from 4.9 x i0 kg/rn (corresponding tothe reference case) to 1.0 x i0 kg/rn does not significantly affect the longitudinalresponse (Figure 4-10). However, the frequency of the transverse elastic mode isincreased from 0.0045 Hz, corresponding to the reference case (Figure 4-3), to 0.01Hz. The platform and tether attitude responses are not affected by the change inthe tether mass.There is an increase in the static elongation of the tether from around 0.81rn to 0.88 m when the linear mass density is increased to 25 x 10 kg/rn from thereference value of 4.9 x i0 kg/rn. As shown in Figure 4-11, the most significantchange is observed in the frequency of the transverse vibration mode (C1), which740.01-0.00-0.01Figure 4-9 Dynamical response of the system with fixed offsets along both thelocal horizontal and local vertical directions.x(O) = 2°; c(0) =2° Stationkeeping, L =5 kmB1(0)=0.8mD=2m D =2m, pzC1(0)= 0.01 m0.0-50.00.02.0cx° 0.0-2.05.00.0 5.0B 1 0.800.0 1.00.0 Orbit 5.075cx(O) = 2°; cx(O) =2° Stationkeeping, L =5 mB1(O)=O.BmD=O; D=O; p=1g/mC1(O)= 0.01 mFigure 4-10 Plots showing the effect of decreasing the tether mass per unitlength on the system response.2.0cxp° 0.0-2.02.00.0 2.00.0-2.00.0 2.00.85.B10.800.01Cl -0.00-0.010.0 0.50.0 Orbit 1.0762.00.0-2.00.00.01.00.90.80.00.02.02.00.51.0Figure 4-11 System response with an increase in the mass per unit length of thetether.x(O) = 2°; c(0) =2°Stationkeeping, L = 5 mB1(O)=O.BmD=0; D=0; p=25g/mC1(0) = 0.01 m2.00.0-2.0B1Cl -0.00Orbit77is decreased to 0.0021 Hz from the reference value of 0.0045 Hz. The rigid bodyresponses are essentially unchanged.The response of the system with a decrease in the the elastic stiffness (EA)to 40,000 N from the reference value of 61,645 N, is shown in Figure 4-12. Asexpected, with the reduced stiffness the static elongation of the tether increased to1.27 m compared to 0.81 m for the reference case (Figure 4-3). The frequency ofthe longitudinal vibration (B1) is also reduced to 0.02 Hz from the reference valueof 0.0249 Hz. The change in stiffness has only a small influence on the transverseoscillation (C1) of the tether. For a system with an increase in the elastic stiffnessof the tether (Figure 4-13), the trends are as expected. Now, the frequency of B1is increased to 0.0282 Hz from the reference value of 0.0249 Hz. As before, thetransverse vibrations exhibit insignificant change from the reference response.4.3.5 Subsatellite. massThe mass of the subsatellite (ms) is an important parameter affecting, particularly, flexible dynamics of the tethered satellite systems. The tether tension ismainly governed by the subsatellite mass and length profile during deployment/retrievalwhich, in turn, affects the elastic response of the system. The system behaviour witha decrease in the end mass to 50 kg and its increase to 5,000 kg from the referencevalue of 500 kg, is shown in Figures 4-14 and 4-15, respectively. For m3 = 50kg (Figure 4-14), frequency of the longitudinal oscillation increased to 0.0733 Hzfrom the reference value of 0.0249 Hz (Figure 4-3). This can be explained by modelling the longitudinal oscillation by a spring—mass system which has a frequencyinversely proportional to square root of the end mass. However, the frequency ofthe transverse vibration, which is proportional to square root of the tether tension78a(O) = 2°; c(0) =2°Stationkeeping, L =5 mB1(0)=1.2mD=O; D=0; EA=4OkNC1(0)= 0.01 m0.0 2.00.00.0 0.5Figure 4-12 Diagram showing the effect of decreasing the elastic stiffness.cx°2.00.0-2.02.00.0-2.01.4B12.01.31.20.010.00-0.01Cl0.0 1.0790.00.00.00.70.60.00.01-0.00-0.010.0Figure 4-132.00.51.0Response of the system with an increase in the elastic stiffness.a(O) = 2°; x(0) =2° Stationkeeping, L =5 mB1(0) = 0.6 m=0; =0; EA =80 kNC.(0)= 0.01 m2.0-2.02.00.0-2.0B12.0Orbit80cxpcx°B1C12.00.0-2.02.00.0-2.00.00.100.090.0(O) = 2°; x(0) =2° Stationi<eeping, L =5 mB1(o) = 0.09=0; =0; m =50 kgC.(0)= 0.01 rn0.11.0Figure 4-14 Response results showing the effect of decreasing the subsatellitemass.0.0 2.02.00.0 Orbit81x(O) = 2°; c(0) =2°Stationkeeping, L =5 mB1(o) = 7.5D=0; D=0; m=5,000 kgC.(0)= 0.01 m-2.00.0 2.02.00.02.00.0-2.00.0 2.00.0 1.0B10.01-0.001-0.010.0Figure 4-15 System response with an increase in the subsatellite mass.0.582[73] (tension is higher for a larger subsatellite mass), has a lower value of 0.00 16Hz as compared to 0.0045 Hz in the reference case. On the other hand, for an increase in the subsatellite mass to 5,000 kg, the frequency of longitudinal oscillationreduced to 0.0081 Hz and that of the transverse vibration increased to 0.014 Hz. Itis interesting to note that the frequencies of the transverse and longitudinal oscillations change in opposite directions (i.e. one decreases and the other increases) for achange in the subsatellite mass. As expected, the static elongation of the tether ismore for higher subsatellite mass.4.3.6 Deployment and retrievalFrom the dynamics point of view, deployment and retrieval are the criticalphases in a tether mission. As pointed out before, the retrieval dynamics is aninherently unstable operation where as the deployment can be unstable if the velocityexceeds a certain critical value. Figure 4-16 shows the dynamic response duringdeployment of the subsatellite from a tether length of 200 m to 20 km in 3 orbits. Anexponential- constant-exponential velocity profile is employed for the deployment.The deployment parameters are obtained from Eqs.(4.4-4.7). The velocity profileswitches from the exponential to constant at a tether length of 2.5 km and the secondswitch, from constant to exponential, occurs at 18 km (L1 2.5 km and L2 = 18km). As shown in the figure, even with a zero initial condition, the tether pitch(at) grows to —30° during the initial phase of the tether deployment. Subsequently,it slowly decreases and oscillates between ±2° after the deployment is over. Thisgrowth in the amplitude of at is mainly due to the Coriolis force caused by theinteraction between the orbital and deployment velocities. For zero offset, there isno coupling between the platform and the tether dynamics. Therefore, ap oscillatesbetween 0 and Q•50• The mean value of 0.25° is caused by the nonzero product83a(O) =0; cx(0) =0 Deployment:B1(0) = 0.1304 x 102 m 200 m to 20 km in 3 orbitsC1(O)=O D=O; D=OFigure 4-16 Response of the system during deployment of the subsatellite usingan exponential-constant-exponential velocity profile.0.0-30.01.5L0.00.5a°P 0.020.0L0.0B15.0C1 0.0-5.00.05.0 0.0 5.00.0 4.00.0 Orbit 4.084of inertia about the Y, Zp axes (‘P,z 0). As expected, the mean value of thelongitudinal oscillations, which is the static elongation of the tether, increases withthe length. As shown in the inset of the B1 plot, the deformation of the tether isgreater than zero implying that the tether tension is positive. Since tension in thetether is small at the beginning of the deployment, it is critical to use appropriateacceleration profile for deployment because higher acceleration may cause slackness(i.e. negative tension) in the tether. If the first switch in the velocity profile occurredat 800 m (L1 = 800 m) instead at 2.5 km, B1 became negative at the beginning ofthe deployment and the tether was slack (plot not shown). Because of the Coriolisforce, C1 increases to —7 m even without any initial disturbance. In the terminalexponential deployment phase, C1 decreases because of the decrease in the velocity(L). After the deployment is over C1 oscillates with a constant amplitude about thezero mean value.The system response for retrieval from L = 20 km to 200 m in 4 orbits withan exponential velocity profile is shown in Figure 4-17. The Coriolis force duringthe retrieval made the tether pitch () unstable. Since the offset in the simulationis taken to be zero, the platform attitude angle is not affected by the tether motion.As expected, the tether elongation (B1) decreased with the length. As the lengthdecreased, frequencies of the elastic modes (B1 and C1) increased as expected. Theinstability in the system excited the transverse mode (C1), which grew to ±50 m.The magnitude of the retrieval velocity, and hence the Coriolis force, decreases withthe tether length. This, along with the increase in the stiffness, led to the decreasein the amplitude of C1 towards the end of the retrieval. However, at the beginningof the retrieval the instability due to the Coriolis force is stronger resulting in anincrease in the transverse amplitude. The instability also excited the longitudinaloscillations resulting in negative B1, i.e. the slack tether.85250.00.00.00-4.050.020Ci 0.0-50.0OrbitFigure 4-17 System response during exponential retrieval of the subsatellite.Note instability of the system in rigid () as well as flexible (B1)degrees of freedom.a(O) = 0; c(O) =0 Retrieval:B1(O) = 13.0 m 20 km to 200 m in 4 orbitsC1(O)=O D=O; D=O0.5a°P 0.020.0L0.015.0B10.0L0.0 3.5 0.0 3.50.0 3.0864.3.7 Shift in the center of massIn a typical mission the length of the tether can vary from zero to 20 km.Experiments involving 100 km long tether are in the planning stage. Obviously, thiswould result in an extremely wide shift in the center of mass of the system. Figure4-18 shows plots for the shift in the center of mass of the system during deployment,stationkeeping and retrieval for a maximum tether length of 20 km. As expected,at the beginning of deployment (L 200 m), the shift in the center of mass is small(Figure 4-18 a). The shift along the local vertical (Rn) increases with the tetherlength and, at the end of the deployment, R becomes more than 120 m, whichis 0.6 % of the tether length. The shift along the local horizontal (R) is mainlydue to the tether libration. Though the tether pitch is large at the beginning ofthe deployment (Figure 4-16), because of a smaller length, variations remainessentially the same. As shown in Figure 4-16, there is a change in the mean valueof t as the deployment terminates. This effect can be easily seen in a change inthe amplitude of R2 at the end of deployment.Figure 4-18(b) shows the response during stationkeeping at L = 20 km. Thecoupling between the attitude motion of the tether and platform results in a beatresponse for both and variations are confined to one side of —121.26m, where as oscillates about the zero mean. As expected, decreases withthe tether length during retrieval (Figure 4-18 c). The retrieval is carried out from atether length of 20 km to 200 m in 4 orbits with an exponential velocity profile. Thetether pitch (c) being unstable, there is no specific pattern to the shift in the centerof mass. Though at becomes very large during the retrieval, does not increasebeyond —25 m. This is due to the fact that an increase in the tether attitude angleis associated with a decrease in the tether length. The combined effect of these two87Figure 4-18 Plots showing shift in the center of mass during three differentphases of the system operation:(a) deployment; (b) stationkeeping;and (c) retrieval. Total deployed length is 20 km.0.0 Orbit 5.0 Orbit 5.0(a)0.0-120.0(b)-120.9-121.2(c)0.0-120.015.0R°.°-15.00.010.0R°°pz-10.00.00.0 Orbit 10.0 Orbit 10.025.0R°0pz-25.00.0 Orbit 3.0 0.0 Orbit 3.088factors limits any increase in beyond a certain value.4.4 Concluding RemarksResults of a comprehensive parametric study suggest the following:(i) Higher modes carry insignificant amount of energy.(ii) In absence of the tether attachment point offset, tether flexibility has littleeffect on the rigid body dynamics.(iii) Offset of the tether attachment point couples the platform dynamics withthe tether degrees of freedom.(iv) Flexible dynamics of the tether is substantially affected by the mass andelastic properties of the tether material. In particular, they affect staticdeflection of the tether, and frequency of both transverse and longitudinaloscillations.(v) The tension of the tether is primarily governed by the subsatellite mass andlength of the tether and, in turn, affects its natural frequencies.(vi) Deployment and retrieval represent critical phases in a tethered mission. Depending on the deployment/retrieval rate, the system is susceptible to instability and the tether may become slack.(vii) There can be significant shift in the system center of mass during deployment,stationkeeping and retrieval.895. ATTITUDE CONTROL51 Preliminary RemarksThe instability during retrieval, large amplitude swinging motion at the timeof deployment and undesirable performance in the stationkeeping phase would demand some active control strategy for a successful tether mission. Depending onthe mission requirements, the objective of the controller is to regulate the attitudeand/or flexible motion of the system. The design and implementation of the attitudecontroller is addressed in this chapter.As mentioned in Chapter 1, in the past, control of tethered systems hasbeen approached through three different strategies: tension or length rate control;thruster control; and offset strategy. The tension or length rate control schemedepends on the differential gravity field, and hence is ineffective at shorter tetherlengths. Although the thruster based control procedure is unaffected by the lengthof the tether, it is not advisable to use thrusters in the vicinity of the space platformdue to plume impingement and safety considerations. Therefore, the offset controlstrategy is the most effective choice for shorter tethers. The offset motion requiredfor controlling the tether attitude dynamics increases with the length and hence theapplicability of this strategy is limited by the space available to move the tetherattachment point on the platform.The advantages associated with the three control strategies over differentlengths lead to a hybrid scheme where the offset method can be used for shortertethers and thruster or tension based approaches can be applied at longer lengths.Such a hybrid strategy was originally used by Modi et al.[49j for a rigid tethered90system. In this thesis, the hybrid strategy is extended to a system with flexibletether. The thruster augmented active control is used here to regulate the attitudedynamics for longer tethers.The dominant feature, which governs the system characteristics, is the timevarying length of the tether. If the time dependent variation of the system parameters is relatively small, the Linear Time Invariant (LTI) robust control techniquescan be used to regulate the dynamics with the parameter variations treated as uncertainties. But, in the present situation, the length of the tether varies from fewmeters to 20 km, and hence the robust LTI controller will not be effective. Thereare mainly four different approaches to control such highly time varying systems:• gain scheduling control;• adaptive/self-tuning control;• simultaneous control of multiple plants;• control based on the Feedback Linearization Technique (FLT).In the gain scheduling approach, a number of LTI controllers are designed fordifferent operating conditions and are stored on-board. In the real time operation,the controller selects and implements the appropriate gains corresponding to anoperating condition. The major difficulties with these controllers are the design andstorage of the controller gains for a number of operating conditions. Any change inthe mission objective may need redesign of the controllers for new operating points.In the adaptive control, a simple system model is estimated from the knowledge of the system inputs and outputs. Based on the estimated model, the controlleris designed and implemented in real time [74]. This approach has advantages forsystems whose dynamics can not be modelled accurately.91Another alternative to these approaches is the simultaneous control of anumber of LTI plants, over the entire range of parameter variation, by a singlecontroller. Here, the time varying system is considered as a collection of controllableand observable LTI plants described by:th = Ax + Bu; (5.1)y=Cx, i=l,2,...,np; (5.2)where: x, yj and u are the state, output and control input vectors, respectively;and A, B and C are the LTI matrices of appropriate dimensions. The index ‘i’corresponds to the th plant and Tip is the total number of plants. Consideringthe output feedback, the objective is to design a single controller, u = —Ky, i =1,2, -. , Tip, so that all the plants are simultaneously controlled. The control can alsobe achieved by state or dynamic output feedback. Several algorithms to accomplishthis objective have been reported in the literature [75—81].In the FLT, the nonlinear and time varying dynamic equations are transformed into an LTI system by the nonlinear, time-varying feedback. Thus a singlecontroller structure can regulate the highly time varying and nonlinear dynamics ofthe system. References [82-88] discuss detail mathematical background and designprocedures for the feedback linearization control.The FLT based controller is quite effective for systems which can be modelledaccurately. The controller performance is sensitive to the modelling errors [89]. Inthe present study, attitude dynamics of the system can be modelled quite accuratelyand hence the FLT is selected for the controller design. In a general problem withfeedback linearization, the control algorithm consists of three steps: transformationof the state space; control variable transformation making the system linear in the92new coordinate system; and control of the linear system in the new state space. Thedimension of the transformed linear controllable system depends on its propertiesand type of the linearization procedure applied, such as the input-state or the input-output feedback linearization [83].The attitude controller for shorter and longer tethers are designed using theoffset and the thruster strategies, respectively. The following section discusses thedesign algorithm and implementation of the thruster based controller. The designand implementation issues related to the offset strategy are addressed in the nextsection. Finally, a hybrid control scheme is presented followed by some concludingremarks.5.2 Thruster Control Using Dynamic InversionSelection of an acceptable design model is important in the development ofa controller using the dynamic inversion. From the implementation point of view,it is advantageous to use a simpler model for the controller design. However, anapproximate model may affect the system stability [89] and performance. In thepresent study an acceptable dynamic model for the controller design is obtained vianumerical simulation. Three different models — complete nonlinear flexible, rigidnonlinear, and rigid linear — are selected for the controller design. Simulation resultsfor each case are compared to arrive at an acceptable model.5.2.1 Controller design algorithmAs pointed out before, the thruster control strategy is used to regulate theattitude motion of the longer tether, and momentum gyros control the platformpitch motion. The rigid body modes, i.e. ap and aj, are controlled actively through93feedback and the flexible degrees of freedom are regulated by passive dampers. Thecontrol of the flexible generalized coordinates becomes important particularly duringretrieval when the elastic modes are unstable. The controller for the rigid degreesof freedom is designed using the inverse control procedure, which is a special caseof the Feedback Linearization Technique (FLT). In case of the thruster control, thesystem is already in the canonical form. So only control transformation is necessaryfor linearization of the system.The system model for the controller design can be expressed in the formM3 1j8 + F8 = Q, (5.3)where: M3 is the mass matrix; F3 is the vector of nonlinear terms; and q3 andare the vectors of generalized coordinates and forces, respectively. If the numberof independent control inputs in Q3 is the same as the dimension of q3, the abovesystem can be transformed into a linear time invariant form by the generalized forcevector= M3v + F3. (5.4)This transforms Eq.(5.3) into a linear form of= v. (5.5)Here v is the new control input required to regulate the transformed decoupledlinear system. A linear control theory can be used to design the control input. Inthe present study, v is chosen in such a way that the error (e = q3 — q3) dynamicshas poles at desired locations in the s-plane. This is accomplished by= ‘1d + Kv(sd — s) + Kp(q$ — qs), (5.6)94which results in the error equationë+Kvé+Kpe=O. (5.7)Here K and K are the diagonal matrices of velocity and position feedback gains,respectively, and q, sd and sd represent the desired trajectory. The controllerexpression in Eq.(5.4) is referred to as the primary controller while that in Eq.(5.6)is called the secondary controller.The next step in the controller design is to decide an acceptable structure forM8, F8 and Q for the system model in Eq.(5.3). As mentioned before: nonlinearflexible; nonlinear rigid; and linear rigid models are used for the controller design.The linear and nonlinear rigid models are in a form which can directly be used for thecontroller design. But the nonlinear flexible model has a different form. However,it can be transformed to have the required structure as follows.The system under consideration has the formM(q, t) + F(q, , t) Qq, (5.8)where: q and Qq are vectors of the generalized coordinates and forces, respectively;M(q, t) is the nonlinear time varying mass matrix; and F(q, , t) is the vector containing nonlinear gravitational, Coriolis and centrifugal terms. Here the dimensionof q is more than the number of independent control inputs in Qq. Let q, anddenote the components of q; Fr and Ff the components of F(q, , t); and Qq,. andQqf the components of Qq corresponding to the rigid and flexible modes, respectively. Here Qq,. represents the generalized forces due to the control inputs, andQq corresponds to the generalized forces due to the passive dampers in the system.95Partitioning the mass matrix appropriately, the above equation can be rewritten as[M,.,. Mrf1f4ri> fFrfQqr1 ff J 1q J l f J 1.. qHere: q, = {crp, ct}T; qf = {B1, B2,• • , B1 C1, C2,• • , CNt}T; and Nt andN represent the number of flexible modes in transverse and longitudinal directions,respectively. If the controller is to be designed based on the rigid model of thesystem, Mrf and the flexibility terms from Mrr, F,. and Qq,. are neglected and theresulting rigid body (qr) equation can be used to design the controller using Eqs.(5.4)and (5.6). But if the effect of flexibility on the rigid modes is to be accounted forby the controller, the equations of motion for the rigid degrees of freedom (q,.) canbe obtained as [84]M8 &+Fs—Qqr, (5.10)where:—1M8 = M,.,.—MrfMff Mf,.;and F F,. + MrfMJ(Ff + Qqf)’Eqs.(5.4) and (5.6) can be used to design the controller for this system.In the implementation of this control strategy, the submatrices of M and Fare to be computed, which need the knowledge of the flexible system model and theflexible generalized coordinates. Using the assumed mode method to discretize theelastic deformations, the dynamic model can be obtained with a considerable degreeof accuracy. The problems associated with the observation of the flexible modes andtheir resolution is a subject in itself. Here, it is assumed that the flexible generalizedcoordinates are available for the control purpose.965.2.2 Control with the knowledge of complete dynamicsThe system model considered for the controller design includes all the nonlinear time varying terms including the effect of tether flexibility (Eq. 5.10). Thecontroller is designed using the dynamic inversion (Eqs. 5.4 and 5.6) and implemented on the complete nonlinear flexible model of the system (Eq. 5.8). Thenumerical values for the system parameters considered in the simulation are thesame as those used for the dynamic study in Chapter 4. Only the first longitudinal(Bi) and lateral (Ci) modes are considered in the simulation, however, the programdeveloped can incorporate an arbitrary number of modes in both directions.Figure 5-1 shows controlled response of the system during the stationkeepingphase at a tether length of 20km. Since the dynamics of the rigid system is exactlycancelled by the controller, the platform pitch (ap) and tether swing (as) behaveas required. The desired performance specifications for the closed ioop system arecharacterized by the rise and settling times, which are 0.1’r and 0.4T, respectively,for both the platform and tether librations. Here r is the orbital period. The thrust(Tat) requirement is quite small for an initial disturbance of 2° in the tether pitch(at). The nonzero mean value of M is required to regulate ap about zero which isnot its equilibrium value, and to cancel the extra moment on the platform due to anonzero offset. The control inputs (M and Tat) cancel the effect of flexible dynamicson the rigid degrees of freedom. This introduces the high frequency component inthe time histories of M and Tat. The coupling between the rigid and flexiblegeneralized coordinates excites the transverse vibration (C1) even in absence of anyinitial disturbance. As expected, the longitudinal elastic mode decays due to thestructural damping.The controlled response during deployment from L = 200m to 20km in 397M 45.0x40.00.0B1 14.012.00.010.0C1 0.0-10.00.02.01.02.0Figure 5-1. Controlled response of the system during the stationkeeping phase..ct(O) = 2°; ci(0) =2°Stationkeeping, L =20 kmB1(0)=13mD=0; D=1mC(0)=O2.00.02.00.0.IIIIlIIIIIIjIIIIIIIIiIlI,1lvlIIIIiIIIIIIIIIIlIuiIIIlIOrbit98orbits is shown in Figure 5-2. An exponential-constant-exponential velocity profileis used for the deployment. The first switch from exponential to constant velocityoccurs at L = 2.5 km and the second switch takes place at L = 18 km. Thebehaviour of the rigid modes, c, and,is similar to the stationkeeping case.It is interesting to note the similarity between the deployment velocity (L) andcontrol thrust (Tat) profile governed by the Coriolis force. The moment M is verysmall due to zero offset of the tether attachment point from the platform centerof mass (D = = 0). As expected, elongation of the tether (B1) increaseswith the length. With the present deployment profile, inset within the B1-plotshows the tether elongation at the beginning of deployment to be greater than zero.Sometimes, if the deployment acceleration is high, tether may become slack. Theresponse of the transverse flexible mode (C1) is governed by two effects: change inthe tether tension due to the deployment acceleration profile; and the Coriolis forceeffects due to the deployment velocity.The rigid body dynamics during the retrieval is controlled quite successfully(Figure 5-3). However, it is important to recognize the unstable dynamics of theflexible subsystem. The uncontrolled longitudinal elastic mode (B1) becomes negative implying that the tether is slack. The amplitude of C1 response also becomesvery high (±80m) due to the Coriolis force. The retrieval is carried out with anexponential velocity profile (Chapter 4). As the velocity decreases the effect of theCoriolis force becomes small. Furthermore, frequencies of the flexible degrees offreedom increase with a decrease in the tether length. The decay in the amplitudesof B1 and C1 responses after certain time during retrieval is due to the combinedeffect of the tether length, changing Coriolis effect and structural damping. The99OrbitFigure 5-2 System response during controlled deployment with an exponential-constant-exponential velocity profile.cx(O) = 2°; cs(O) =2° Deployment:B1(O) = 0.1304 x 10.2 m 200 m to 20 km in 3 orbitsC(O)=O D=O ; D=O2.0-0.020.00.0xto2.0-0.0ap°LMB1C10.00.010.00.0-10.04.00.0 4.0100Figure 5-3 Controlled response during the exponential retrieval without a passive damper.cz(O) = 2°; O) =2° Retrieval:B1(O) = 13.0 m 20 km to 200 m in 4 orbitsC1(O)=O D=O ; D=O2.0-0.02.0-0.0LMB1C120.0 0.0L0.0 -4.05.0 5.013.0 Eii10.00.0 5.080.00.0-80.00.0 Orbit 5.0101control thrust cancels effects of the flexible dynamics on the rigid modes resultingin the high frequency oscillations of Tat.In the present study, the unstable dynamics of the flexible degrees of freedomis controlled only by passive dampers. Passive dampers are provided for both thetransverse and longitudinal oscillations. The transverse damper (damping coefficient = 0.03 Ns/m) is placed at the middle point of the tether and the longitudinal damper (damping coefficient = 1.5 Ns/m) is located at the subsatellite. Themagnitude of the damping coefficients considered in the simulation are within thepractically achievable range [90]. The results for controlled retrieval with passivedampers are presented in Figure 5-4. The rigid body responses are the same asthe previous case without damper. However, the flexible responses (B1 and C1)now settle to their equilibrium values rather quickly. Note, the magnitude of B1 isalways greater than zero implying that the tether has positive tension.5.2.3 Inverse control using simplified modelsThe controller design in the last section is based on the complete nonlinearmodel of the system. The implementation of this controller needs knowledge of allthe flexible modes, which are difficult to obtain, and requires considerable amountof the real time computational effort. These limitations may make the controllerdifficult to implement. Therefore, to obtain a readily implementable controller, twosimplified models are considered here. The results of the uncontrolled dynamicsimulation (Chapter 4) serves as the guideline in selecting the simplified model. Asseen in Figure 4-6, the rigid body responses are not significantly affected by thetether flexibility. Since objective of the attitude controller is to regulate the rigidbody modes, the nonlinear rigid body model is considered for the controller design.102Figure 5-4-0.00.0-4.0-0.0Tat-6.05.0 0.0= 2°; cx(0) =2°B1(0) = 13.0 mC1(0)=0Retrieval:20 km to 200 m in 4 orbitsD=0; D=02.0cx°L(:xp°LMB1C12.0-0.020.00.02.00.0-2.013.00.070.00.00.0 5.00.0 Orbit 5.0System response during the controlled retrieval with passive dampers.Note, the flexible modes are controlled quite effectively keeping thetether tension positive.103The governing equations of motion can now be expressed as[Mn M12 f&pj+JF1_f M j 511)[M21 M22 &tJ 1F2J1LTatJ’ (.where the coefficients are defined in Appendix III. This system is in the canonicalform and hence Eqs.(5.4 and 5.6) can be used directly to design the controller.The other simplified model considered for the design corresponds to the rigid linearsystem, which can be expressed as[ML11 ML12 fz’l+ 1ii CL12 I a1)[ML21 ML22j 1 & J [GL21 CL22] at+[KL11 KL12 I 1 + I FL1 — f M (5 12KL21 KL22j 1cJ FL2J— 1LTatJ’ ‘with the coefficients defined in Appendix III. This controller is also designed usingEqs.(5.4 and 5.6). The controllers, designed by dynamic inversion using simplifiedmodels, are implemented on the complete nonlinear flexible system. Figure 5-5shows the system response during stationkeeping with the controller based on thenonlinear rigid model. As expected, ap and at settle to the desired value in about0.5 orbit. Note, in the present case, the control inputs do not cancel the effect offlexibility on the rigid modes. Therefore the high frequency components of M andTat have much lower amplitudes than those in Figure 5-1. As can be seen from theinset of at-plot, due to the coupling between C1 and at, the tether pitch has a highfrequency component with very small amplitude. These oscillations slowly decaydue to active control and, in turn, decease the amplitude of C1 through couplingeffects.As shown in Figure 5-5, the controller based on the rigid nonlinear modelresults in a steady state error in the platform pitch (ap). An outer ProportionalIntegral (PT) controller loop is used to take care of this error. With this, the primary104Figure 5-5 System response during stationkeeping using the rigid nonlinearmodel for the controller design.a(O) = 2°; a(O) =2° Stationkeeping, L =20 kmB1(O)=13mD=O; D=1mC. (0) = 02.00.02.00.02.0T°°at-2.044.042.00.0 2.0MB1C10.014.012.02.00.010.00.0-10.01.00.0 Orbit 2.0105controller of Eq.(5.4) becomesI KpT(ap—pd)+KITj(ap_apd)dt= M8v + F5 + , (5.13)( 0 Jwhere KPT and KIT are the proportional and integral gains, 2.0 and 0.05, respectively. The response of the system with the P1-controller is presented in Figure 5-6.Note, the steady state error in the platform pitch response is reduced to zero. Theother responses remain essentially unchanged.The response of the controlled system during deployment, with dynamicinversion, using rigid nonlinear model is shown in Figure 5-7. The deploymentvelocity profile is the same as in Figure 5-2. Comparing Figures 5-2 and 5-7, it canbe concluded that the system dynamics is virtually identical except for two effects:amplitude of the steady state ocillations of the control thrust (Tat) is very small;and the C1 response decays slowly with the controller based on the rigid nonlinearmodel. In Figure 5-7, the controller does not cancel the effect of flexibility on therigid model. It leads to a smaller fluctuation of Tat as compared to that in Figure5-2. The tether transverse oscillation (Ci) decays slowly due to its coupling withthe a response which has a small diminishing amplitude. Similarly, response of thesystem with the controller based on the rigid linear model (Eq. 5.12) was found tobe almost identical to that in Figure 5-7. Thus the controllers based on the rigidnonlinear and rigid linear models lead to essentially the same performance duringdeployment.The simplified linear and nonlinear models of the rigid system were also usedto design the controller for regulating the attitude motion during retrieval. Responseof the system with the controller based on the rigid nonlinear model and in presenceof passive dampers is shown in Figure 5-8. The dampers used here are the same106M 44.042.00.0B1 14.0C112.010.0-10.0Figure 5-62.0Tat2.00.0-2.00.0 2.01.02.0System behaviour during stationkeeping with the outer PT controlloop and the FLT employing the rigid nonlinear model.a(O) = 2°; x(O) =2° Stationkeeping, L =20 kmB1(0)= 13mD =0; D= 1 mC1(O)=O2.00.02.00.00.0 Orbit107B11.50.01.50.00.0 4.0Figure 5-7 Deployment dynamics with the controller based on the rigid nonlinear model.a(O) = 2°; x(O) =2° Deployment:B1(0) = 0.1304 x 102 m 200 m to 20 km in 3 orbitsC1(0)=0 D=O ; D=O2.0-0.0xpoLM2.0-0.020.00.02.00.0-2.013.0ILT{N-m0.0 4.010.00.0-10.0Cl0.0 Orbit 4.01082.0-0.020.0 0.0L -0.0 -4.02.00.0M Tat-2.0-8.0 —0.0 5.0 0.0 5.0B1 13.00.070.0C10.0Figure 5-8cx(O) = 2°; cx(0) =2°B1(0)=13mC.(0)=02.0Retrieval:20 km to 200 m in 4 orbitsD=O; D=O-0.0Orbit 5.0Controlled response during exponential retrieval of the subsatellite. The controller is designed using the rigid nonlinear model andapplied in presence of passive dampers.109as those for the simulation in Figure 5-4. A comparison of results in Figures 5-4and 5-8 shows similar behaviour except for reduced Tat fluctuations in the presentcase. Similar trends were observed with the controller based on the rigid linearmodel. Identical observations can be made for the controller design aimed at thestationkeeping case.5.2.4 Comments on the controller design modelsAs seen in the previous section, the controllers based on the complete flexiblenonlinear, rigid nonlinear, and rigid linear models result in very similar systemperformance. Hence, the computational time may serve as an important criterionin selection of the model. To that end, the controlled dynamics was assessed overfive orbits with a time—step of 1 s. This corresponds to the situation where thecontroller can be implemented as a sampled data system with a step—size of 1 s.The computational time with the controller design based on different models arecompared in Table 5-1 for stationkeeping, deployment and retrieval phases.Table 5.1 Comparison of the time (s) required by the controllers using differentdesign models.Nonlinear Nonlinear LinearFlexible Model Rigid Model Rigid ModelStationkeeping 402.6 67.6 70.4Deployment 5195.1 68.1 71.6Retrieval 5332.2 68.4 71.7Each computation loop involves calculation of the control inputs, and recording of the system output as well as input values in a file. Calculations for the stationkeeping case correspond to L = 20 km. Deployment is carried out from a tether110length of 200 m to 20 km in 3 orbits using an exponential-constant-exponentialvelocity profile with the switch over at tether lengths of 2.5 km and 18 km. Duringretrieval, the tether length decreases from 20 km to 200 m in 4 orbits with an exponential velocity profile. The study was carried out on a SUN Sparc-2 workstation.As expected, in all the three modes of tether operation, the controller basedon the nonlinear flexible model demands the maximum computational effort. Thecontroller based on the rigid nonlinear model takes the minimum time. In the linearrigid model, the equations are linearized about an arbitrary reference trajectory.So the governing equations involve more algebraic and trigonometric operationsthan the rigid nonlinear case. This results in a longer computational effort forthe controller based on the rigid linear model as compared to the riid nonlinear.case. As seen in the previous section, the system performance with the differentcontroller is essentially similar. So the controller based on the rigid nonlinear systemis preferred. Of course, it should be recognized that irrespective of the system modelchosen for the controller design, its effectiveness is assessed through application tothe nonlinear, nonautonomous and coupled flexible system.5.3 Offset Control using the Feedback Linearization TechniqueAs mentioned before, because of practical limitations, it is advantageous touse the offset strategy for attitude control when the tether length is small. Thissection presents the design procedure and simulation results for the offset control ofa class of tethered systems. The Feedback Linearization Technique (FLT) is usedto design the controller for this highly time varying tether dynamics.1115.3.1 Mathematical backgroundFor two vector fields f and g on R”, the Lie bracket [f, g] is a vector fielddefined by [83]ôg of[f,g]f—g,where Of/Ox and ag/ax are the Jacobians. It is also denoted by ad}(g), and byinductionad(g)= [f, ad1(g)],with ad°f(g) = g. A set of vector fields (x) {x1,..., xm} is said to be involutiveif there are scalar fields ajjJ such thatj,=or in other words [xi, Xj] E s(X). It is called completely integrable if for everypoint, assuming x1, , x are linearly independent, there exists an rn-dimensionalmanifold M in R such that at each point of M the tangent space of M is spannedbyx,•,xm.Let h R” —‘ R be a scalar field. The gradient of.h, denoted by dh, is arow vector fieldfOh OhOx1’ ‘8on R”. A set of scalar fields are linearly independent if their gradients are a linearlyindependent set of row vector fields. For a scalar field h and a vector field f =(fi, •• , f)T, the dual product of dh and f, denoted by (dh, f),is a scalar fielddefined byOh OhThe feedback equivalence for control systems is based on three operations112[86]: coordinate transformation in the state space; coordinate transformation in thecontrol space; and feedback. For linear systems, with the operations defined in alinear fashion, it is well known that every single input system which is time invariantand controllable is feedback equivalent to the canonical formZ[ Z2 0= + V. (5.14)dt Z_1 Z 0Zn 0 1Any nonlinear system which is feedback equivalent to the above form is called linearequivalent.Theorem 1 (Su, 1982 [86)): A system = f(x, u) is a linear equivalent if and onlyif(i) J(x,u)—_ f(z)+g(x)4’(z,u) where f(O)= 0, 4(0,O)=0 and 84/ôu 0;(ii) the vectors g, ad}(g),. , ad)(g) span R about the origin;(iii) the set of vector fields {g, ad}(g),... , ad?)(g)} is involutive. DFor systems satisfying the above theorem, the state transformation z =T(x) {T1, . , Tn}T required for linearization can be obtained from the followingconditions:(dTi,adjc(g)) =0, i = 0,1,,n—2; (5.15)(dT1,ad’g)) 0; (5.16)T2 0= +. (5.17)dt TnEqs. (5.15) and (5.16) can be used to get the function T1. The complete transfor113mation, z = T(x), can be obtained from Eq.(5.17). Ifq5(x,u) is selected asq(x,u) v —P(x) (5.18)where:nP(x) =nandj1Eq.(5.17) acquires the same canonical form as that of Eq.(5.14).The expression for (x,u), Eq.(5.18), which transforms the system into thecanonical form, is referred to as the primary controller. Now the task is to design afeedback controller (called the secondary controller) to generate the control input,v, so that the transformed states z1,•• , Z1’ and hence the original states x1,• ,follow the desired trajectory. This can be achieved by using the linear time invariantcontrol procedure.5.3.2 Design of the controllerThe purpose of the attitude controller is to regulate the rigid body rotationsof the platform and tether. The dominant characteristic, which governs the systemdynamics, is the time varying length of the tether during deployment and retrieval.As discussed in the previous section, the controller based on the rigid model ofthe system gives almost the same performance as that obtained using the flexiblenonlinear model. Furthermore, as shown in Chapter 4, the tether response is notsignificantly coupled with the platform dynamics; however, for a nonzero offset, theplatform attitude is strongly affected by the tether motion. Therefore the controllerfor the tether dynamics is designed based on the rigid model decoupled from theplatform motion. On the other hand, the model for the platform controller includes114the effect of tether libration.As shown in the thruster control case (Section 5.2), performance of the systemwith the controllers based on linear and nonlinear rigid models is almost identical.Therefore, it is sufficient to consider only the linear model for design of the tethercontroller. The rigid body equation for the tether dynamics, decoupled from theplatform motion, is linearized about the quasi-equilibrium trajectory (at) and thespecified offset motion. The total offset position, {d}, is the sum of the specifiedvalue, {D}, and the controller coordinate {D}. The resulting dynamic equationsfor the controller design are:ma(ap, x, t)& + f(ap, p, x, , t) = (5.19)= alat +a2D + a3at +a4Dz + b + cDi, (5.20)wherer iTx =jat, a, -‘jHere D2 is the z-component of {D} which is the offset along the local horizontalrequired by the controller. The coefficients of Eqs.(5.19) and (5.20) are defined inAppendix III.The equilibrium value for the platform pitch angle (p) is specified by themission requirement, and the quasi-equilibrium angle for the tether pitch rotation(at) is given by{miDpz + m1(Ô2 — 2ji)D +61imi(Dp — Da) += mi(—Dpy + Da)— mcrtS—that8+ mi(Ô2 — 2u)(Dpyäp + D2). (5.21)115Here Dpv and represent the ‘y’ and ‘z’ components, respectively, of the vector{D} and other coefficients are defined in Appendix III. In Eq.(5.20), the acceleration of the tether attachment point along the local horizontal (Di) is used to controlthe tether swing. Since it is necessary to regulate the position and velocity of thetether attachment point, the dynamic model for the tether pitch is augmented bythe identity= ut. (5.22)The alternate method to control the offset motion is to include the physical behaviour of the offset mechanism into the system dynamics. It should be pointed outthat the time constant of the offset mechanism is usually much smaller than that ofthe tether libration. Hence the use of Eq.(5.22) to regulate the offset motion doesnot affect the implementation of the control strategy.The governing equation for the platform dynamics (Eq. 5.19) is already inthe canonical form. Therefore, only the control variable transformation is requiredto linearize the system. The structure of the primary controller which accomplishesthe linearization isQcrp = m(cp, x, t)v + fap(p, &p, Z,,t). (5.23)The secondary control input, Vp, which asymptotically drives the error (e= p—apd)dynamics to zero, can be expressed asVp k1 (Pd — + kp2(à d — p) + &pd, (5.24)where: and k2 are coefficients of the desired closed loop polynomial for theerror dynamics; and cpd, &pd, cpd represent the desired trajectory. Now the error116equation becomesë+kp2ê1e= 0.The model used for the tether controller design, Eqs.(5.20) and (5.22), can be represented by= f(x, t) + g(x, t)ut, (5.25)where:I X3X4‘ ‘ — ax1 + a2x + a3x + a4x + b0(UJoand g(x,t)=IiIn general, the coefficients in Eq.(5.25) are time varying. The transformation ofthis system to time invariant canonical form can be obtained using the algorithmof Hunt and Su {91j. However, in the present case, the coefficients of the tetherdynamics are slowly time varying parameters and hence a quasi-static approach issufficient to achieve the objective. This results in a quasi-static controller structurewhich transforms the system into the canonical form. The functions f(x) and g(x)satisfy the conditions in Theorem 1, and hence the system is feedback equivalent.The state transformation, which maps the system into the canonical form, isz = T(x) = [t]x, (5.26)where:= a + a1c + a3(a4 + a3c);t12 =—c(a2 + aic) + a4(a4 + a3c);= a4— a3c;117= c(a4 + a3c);and for i = 2,3,4til =tj2 =tj3 = t(_1)+a3t(_l);+ a4t(_l)3.The structure of the transformed system is similar to Eq.(5.14), where the primarycontroller has the form— P(x)Ut=, (5.27)with:P(z) =a1t43x—i- at43z+ (t41 +a3t4)z + (t42 +a4t3)x;Q(x) = Ct43 + t44.The secondary controller for the system is designed to place the closed ioop eigenvalues at some desired locations. This leads to the following expression for thetransformed control input Vt,Vt =—zld) + kt2(z — Z2d) + kt3(z — Z3d) + kt4(z — Z4d), (5.28)where kt and Zid = 1, , 4, are the coefficients of the desired closed loop polynomial and the desired values for z, respectively. Zid can be computed from Eq.(5.26)with the knowledge of the desired x. The structure of the tether pitch controller isshown in Figure 5-9.118Figure 5-9 Schematic diagram of the closed-loop tether dynamics with the FLTbased controller.5.3.3 Results and discussionThe controller designed for the platform and tether pitch angles are implemented on the complete flexible and nonlinear model. As mentioned before, theoffset control strategy has advantages at shorter tether lengths. In this section, controlled response results are presented for operations with a maximum tether lengthof 200 m. The inertia and elastic parameters are the same as those used in thedynamic simulation of Chapter 4. One possible drawback of the inverse control procedure may be its lack of robustness against the model uncertainty [89]. To assessthis aspect, intentional modelling error was introduced by neglecting the shift inthe center of mass terms from the controller model, but retaining the correspondingterms in the simulation model.The desired system performance is characterized by the settling and risetimes. The settling times for platform, tether and offset motion are 0.4 ‘r, 0.8r and0.82T, respectively. The rise times are O.lr and 0.15r for the platform and offset1 IFLT controller119motions, respectively. The rise times for the tether libration are 0.2r, 0.16r and0.18r during deployment, stationkeeping and retrieval, respectively. Here r is theorbital period. The maximum allowable offset is taken as ±15m.Figure 5-10 shows the controlled performance of the system for stationkeepingat L = 200 m with the modelling error in the controller design. Though the rigidmodes are stabilized to some steady state values, the performance of the systemis not satisfactory. The desired steady state values for ap and are 0 and 1 m,respectively, which are not achieved by the controller. Moreover, the at and d2response is quite oscillatory. The moment required to control a1,, i.e. M, is alsooscillatory because of the coupling between the platform and the tether dynamicscaused by a nonzero offset.When the shift in the center of mass terms are included in the controllermodel, the oscillatory nature of a and d1,2 responses is reduced substantially andthe steady state value of a1, become zero (Figure 5-11). The steady state value ofd1, is still less than the required magnitude of 1 m. An outer Proportional-Integral(P1) loop was introduced to reduce this steady state error. With this, the structureof the tether primary controller becomesUt=Q(r +K1,0(d — D1,2)+K10j (d1, — D1,2)dt, (5.29)where K1,0 and K10 are the proportional and integral gains, respectively. The secondary control input Vt IS obtained as before using Eq.(5.28). In the present simulation, the gains are: K1,0 = 1.0 x 10—6 and K10 = —1.5 x i0. The schematicdiagram of the controllers with the integral loop is shown in Figure 5-12 and response of the system is presented in Figure 5-13. Now, the steady state offset is 1.0m. Note, the offset requirement is a little higher than the previous case, however120xp°MB12.0-0.00.02.00.0-2.00.010.02.02.00.1Figure 5-10 Controlled response in presence of the modelling error introducedby neglecting the shift in the center of mass terms during the controller design.a(O) = 2°; cx(0) =2° Stationkeeping, L = 200 mB1(0)=0.1304x10mC.(O)=0 D=O ; D=1.0m2.0 0.00.0Cl -0.00-0.010.0 Orbit 0.5121a(O) = 2°; c(O) =2° Stationkeeping, L = 200 mB1(0)=0.1304x10mC1(0) =0 = 0 ; D = 1.0 mFigure 5-11 System response with the shift in the center of mass included in the2.0-0.00.0 2.0 0.0 2.0xp°M2.25B11.250.00.01C1 -0.00-0.01 —0.00.1Orbit 0.5controller design model.122Figure 5-12 Schematic diagram of the closed-loop system with the outer P1control loop for the tether dynamics.the response is substantially improved. The longitudinal oscillation (B1) decays dueto the structural damping in the tether. Although the transverse vibrational degreeof freedom is not subjected to any initial disturbance, the offset motion during thecontrol excites the C1 response resulting in a small amplitude oscillation.Response results were also obtained for the controlled deployment of thesubsatellite from a tether length (L) of 50 m to 200 m in one orbit (Figure 5-14). Anexponential-constant-exponential velocity profile was used for the deployment. Thefirst switching of the velocity from exponential to constant profile occurs at L 80m and the second switching is at L = 180 m. The controller used in this case includesthe outer P1 loop for the tether dynamics with the gains as mentioned before. Asshown in Figure 5-14, the platform pitch angle is controlled quite successfully. TheD+PlantCLp123Figure 5-13 System response in presence of the outer PT control ioop.a(O) = 2°; a(O) =2°Stationkeeping, L = 200 mB1(0)= 0.1304 x 10.2 mC,(0)=0 D=0 ; D=1.0m2.0-0.00.02.00.0-2.00.02.02.00.0xp°MB1C1 -0.00-0.010.0-2.02.0 0.0 2.00.00.010.1LOrbit 0.5124cx(O) = 2°; a(0) =00 Deployment:B1(0) = 8.14 x i0 m 50 m to 200 m in 1 orbitC(O)=O D=O ; D=OOrbitFigure 5-14 Controlled response of the system during deployment. The offsetcontrol strategy in conjunction with the Feedback LinearizationTechnique (FLT) is used.0.02.0-0.00.20.0-10.0L0.040.00xp°LMB15.0‘Cl 0.0-5.0x100.0 1.5125tether pitch (at) is controlled about the quasi-equilibrium trajectory as defined byEq.(5.21). After deployment, at settles to zero. The total offset motion requiredis within ±5 m, which is much less than the limiting value of ±15 m. The controlmoment M cancels the coupling between the tether and platform dynamics, anddrives ap towards its desired value. The coupling between the tether and platformdynamics is due to the offset (d2). This leads to similarity between the moment(Mi) and d2 time histories. The nonzero steady state value of M is needed tostabilize a, about zero, which is not its equilibrium state.As expected, the mean value of the longitudinal oscillations increase withthe tether length. Note, frequency of the transverse vibrations decrease as thetether length increases. Effect of the Coriolis force during deployment is stabilizingand keeps amplitude of the transverse oscillations quite small, however the meanvalue of C1 is not zero. In the post—deployment phase, amplitude of the transverseoscillations increase with zero mean. Simulations were also carried out for severalother deployment rates (0.7 and 1.5 orbit, plots not shown). As can be expected,the maximum offset requirement was found to be higher (±7 m) for the fasterdeployment in 0.7 orbit and lower (±3 m) for the slower deployment rate.The controlled response of the system during the exponential retrieval froma tether length of 200 m to 50 m in 1 orbit is shown in Figure 5-15. The generalizedcoordinate for the transverse vibration (C1) grows to around 0.02 m even when it isnot excited initially, i.e. C1(0) = 0. The initial disturbance in the first longitudinalmode (B1) decays quite rapidly due to the structural damping, however a smallamplitude oscillation persists due to coupling with the transverse vibration (inset inB1 plot). In the present case the destabilizing Coriolis force is not enough to makethe tether slack (i.e. B1 < 0). The platform pitch (ap) response settles to zero126Figure 5-15velocity profile.d pzO.O3.0 0.0a(O) = 2°; cx(0) = 0° Retrieval:B1(0) = 0.1304 x 102 m 200 m to 50 m in 1 orbitC1(O)=O D=O ; D2=O2.0-0.010.00.0-0.00-0.0515.0L::4.00.0LMB1Cl0.0 3.0x103m1.500 1.5050.1-0.0-0.11.5—-0.0 Orbit 1.5System response during controlled retrieval with an exponential127within 0.4 orbit while the tether pitch (at) is stabilized about the quasi-equilibriumtrajectory, which is zero after the retrieval. The moment (Mi) required to controlap is only 4 Nm. The offset position required to control the tether swing is between+12 m and —5 m, which is within the limit of ±15 m used in the present study. Asin the case of deployment, simulation results were also obtained for several retrievaltimes (results not shown). As anticipated, the maximum offset required to regulatethe tether swing was larger for a faster retrieval.5.4 Attitude Control using Hybrid StrategyAs discussed earlier, thruster and tension control schemes have disadvantagesat shorter tether lengths which leaves the offset strategy as an efficient alternative.This section presents simulation results of a tethered system implementing a hybridcontrol scheme. The controller design procedures are the same as those describedearlier. In the study, the offset strategy is used for tether lengths less than 200 mand the thruster control law is applied for longer (> 200 m) tethers. The thrustercontrol strategy is based on the rigid nonlinear model of the system.Response of the system during controlled deployment from 50 m to 20 kmusing this hybrid strategy is shown in Figure 5-16. Deployment from 50 m to 200m is carried out in 1 orbit with an exponential velocity profile while the rest of thedeployment (200 m to 20 km) is completed in 3 orbits with an exponential-constantexponential velocity profile. In the second deployment stage, the first switch (fromexponential to constant velocity profile) takes place at a tether length of 2.5 kmand the second switch occurs at 18 km. The tether pitch is controlled about itsquasi-equilibrium value during the first orbit. In this period, the maximum offsetrequired along the local horizontal is within ±3 m from the steady state value of 11280.010.0Figure 5-16 Controlled response during deployment using a hybrid strategy.(O) = 2°; cx(O) =2° Deployment:B1(0) = 0.814 x m 50 m to 20 km in 4 orbitsC.(O)=0 D=O; D=1m2.0-0.020.0LL0.0-8.01.50.02.0-0.020.00.00.060.00.03.00.0M TB1Cl 0.0-10.06.0 0.0 6.00.0 Orbit 6.0129m. The nonzero steady state value of the offset leads to coupling between the tetherand platform dynamics. This coupling exhibits through the small fluctuations inthe cp response. As observed earlier in the thruster control section (Sec. 5.2), theM and Tct profiles have similarity with the L and L responses, respectively.The simulation results for retrieval from a tether length of 20 km to 50 m in5 orbits is shown in Figure 5-17. An exponential velocity profile is used for the retrieval. The initial retrieval from 20 km to 200 m is carried out in 4 orbits. To limitthe offset motion, the final stage of the retrieval from 200 m to 50 m is completedin 1 orbit. The instability of the flexible modes is controlled by passive dampers.As in the case of the thruster control, the damper for the longitudinal oscillationis located at the subsatellite and for the transverse vibration at the center of thetether. The damping coefficients for the longitudinal and transverse dampers are1.5 and 0.03 Ns/m, respectively. The thruster control is used for retrieval upto 200m, and the offset scheme for the rest of the retrieval as well as the subsequent stationkeeping. During the offset control, the tether pitch angle is regulated about thequasi-equilibrium trajectory (Eq.5.21). As in the case of deployment, M and Tatprofiles are similar to the L and L trajectories, respectively. The offset excursionsto control the unstable tether attitude during the retrieval range from —4 m to +13m, a significantly large distance than that required during deployment. Of course,this is expected due to unstable character of the retrieval maneuver.5.5 Gain Scheduling Control of the Attitude DynamicsFor a comparison between the FLT based regulator and a linear controllerwith gain scheduling, design of an attitude controller using the eigenvalue assignmentalgorithm was undertaken. This linear time invariant controllers were implemented130Figure 5-17 System response during retrieval with a hybrid control scheme.a(O) = 2°; x(0) =2° Retrieval:B1(O) = 13 m 20 km to 50 m in 5 orbitsC1(O) =0 =0; = 1 m8.0cx°0.00.0L-4.00.0Tat-6.0N-mj2.00.020.00.050.00.012.00.070.00.0LMCl1B10.0 6.0 0.0 6.00.0 Orbit 6.0131on the time varying nonlinear plant in conjunction with the gain scheduling. Thecontrollers were designed using the Graph Theoretic Approach. The detail mathematical background and the controller expressions are given in Appendix IV. Only afew typical results are presented here for comparison. Figure 5-l8shows the station-keeping dynamics in presence of the thruster augmented active control. Note, theattitude degrees of freedom are controlled quite successfully. The control effort (Mand Tat) requirements are rather modest. However, the system performance duringretrieval with a fixed specified offset of 1 m along the local horizontal (D2=1 m) isnot satisfactory (Figure 5-19). Particularly, the platform pitch response has a largeovershoot of more than —5° at the beginning of the retrieval. This is attributed tocoupling, caused by a nonzero offset, between the tether and platform dynamics.The linear controller (with gain scheduling) takes some time to compensate for thecoupling effect. Similar response was also observed during the retrieval maneuverof shorter tethers using the offset strategy (Figure 5-20). The performance of thesystem during a hybrid control is essentially the same. Comparison of these resultswith those presented in Figure 5-17, where the FLT based hybrid control is usedto regulate the retrieval dynamics, clearly shows better performance of the FLTcontroller compared to the gain scheduling linear regulator.5.6 Concluding RemarksA controller based on the FLT is found to be adequate in regulating thehighly time varying attitude dynamics of the TSS with a flexible tether. Two different strategies, thruster and offset control, are used for longer and shorter tethers,respectively. Three different system models (nonlinear flexible, nonlinear rigid, andlinear rigid) are considered to arrive at an efficient FLT controller design. Results132cc(O) = 2°; a(O) =2° Stationkeeping, L =5 kmB1(0)=0.8mD=o D =0, pzC(0)=0.1 mFigure 5-18 Controlled stationkeeping dynamics using the graph theoretic approach.xtoTat2.0-0.00.20.02.0cx°-0.0M0.5-0.00.85B1 0.800.750.0 2.0 0.0 2.00.0 1.0Cl -0.0-0.30.30.0 Orbit 2.0133cz(O) =0; a(0) =0 Retrieval:B1(0) = 13.0 m 20 km to 200 m in 4 orbitsC.(0)=0 D=0; D=1m0.0 6.0 0.0 6.00.00.0-5.020.0L0.040.0M0.013.0B10.060.030.00.0-4.0-0.0-3.030Qo OrbitFigure 5-19 Gain scheduling control of the retrieval maneuver using the thrusteraugmented strategy.6.01345.00.0-5.00.20.03.0M -0.0B1-0.00-0.0510.0d0.0-10.00.0Cl3.00.0-3.00.0Figure 5-20 Offset control of the retrieval dynamics using the gain schedulingapproach.x(O) = 0; a(0) =0 Retrieval:B1(0) = 1.304 x m 200 m to 50 m in 1 orbitC(0)=0 D=0; D=014.00.0L-3.0 —0.0 3.0 3.0x103m0.0 2.0x102m1Orbit 1.5135suggests that the rigid body model (either nonlinear or linear) is sufficient to develop an effective thruster based regulator with a PT control ioop. The FLT basedcontroller with an PT loop is also shown to be effective during the offset control.Both the offset and thruster strategies are also implemented in a hybrid fashion.The hybrid control strategy using the thruster control at longer tether lengths andthe offset control for shorter tethers appears quite promising.Linear time invariant regulators, designed using the graph theoretic approachand implemented through gain scheduling, though effective are not as efficient asthe FLT based control system.1366. VIBRATION CONTROL OF THE TETHER6.1 Preliminary RemarksTether missions involving controlled gravity environment or precise positioning of the subsatellite require regulation of the tether’s vibratory motion. Thetether may oscillate in both the longitudinal and transverse directions. The designand implementation of the controller to suppress tether vibrations is addressed inthis chapter. The results presented here correspond to the stationkeeping situationwhere most of the mission objectives are carried out. Only the first longitudinaland transverse modes are controlled actively here as their energy content is dominant. The higher transverse modes constitute critically stable degrees of freedom.The tether vibrations can be controlled either by some active methods (if controllable and observable) or using passive dampers at appropriate locations. In thepresent study, a passive damper is used to control the higher transverse modes. Therigid degrees of freedom are regulated by thrusters and momentum gyros, and thetransverse and longitudinal modes are governed by the offset strategy.This chapter begins with some mathematical background for the controllerdesign. This is followed by the system linearization. Finally design of the controlleris undertaken which includes the choice of system inputs and outputs as well assome typical results showing its performance.6.2 Mathematical BackgroundThe nominal design model of a plant in the linear state space form can be137expressed as:th=A0x+Bu 1’; (6.la)y=C0x+q; (6.lb)where: x E R1; E Rm; y E R’; E R1; and A0, B0, C0 and F are constantreal matrices of appropriate dimensions. x, u and are the state, control inputand measured output vectors, respectively. is the state noise vector and i is themeasurement noise vector. These are white noises uncorrelated in time (but maybe correlated with each other) with covariancesE[T]= E 0, E[T] = 0 > 0, and E[T] = 0,where 0 is the null matrix of appropriate dimension. In the transfer function notation the system can be represented by(6.2)where:C(s) =C04(s)Bo;d =and 4(s) = [sU — A0J1.Here, C(s) is an m x r transfer function matrix and U is the unit matrix. Thestandard feedback configuration of the system is illustrated in Figure 6-1. It consistsof an interconnected plant C(s) and a compensator F(s) forced by the commandinput r, measurement noise and disturbance d.138d6.2.1 Model uncertainty and robustness conditionsIn reality, the true model of the system, G’(s), is not the same as the nominaldesign model, G(s). Hence, no nominal model can be considered complete withoutsome assessment of its error, normally referred to as model uncertainty. The representation of these uncertainties varies primarily in terms of the amount of structureit contains. For example, the uncertainty caused by variation of certain parameters in the governing equations of motion is a highly structured representation. Ittypically arises from the use of linear incremental models, e.g. error in the momentof inertia of a spacecraft, variation in the satellite mass due to firing of thrusters,changes in the aerodynamic coefficients of an aircraft with flight environment andconfiguration, etc. In these cases, the extent of variation and any known relationshipbetween the parameters can be expressed by confining them to appropriately definedsubsets in the parameter space. An example of the less structured representationof uncertainty is the direct statement for the transfer function matrix of the modelFigure 6-1 Standard feedback configuration.139such asG’(jw) = G(jw) + G(jw) (6.3a)withã[G(jw)] <la(w), Yw 0,where: la() is a positive scalar function confining the matrix C’ to the neighborhoodof C with magnitude la(w); and a(.) represents the maximum singular value of amatrix. The statement does not imply a mechanism or structure that gives rise toG(jw). The uncertainty may be caused by parameter changes as above, or byneglected dynamics, or by some other unspecified effects. This is also referred to asadditive uncertainty. The alternative statement has the multiplicative forms:G’(jw) = [U + Lo(jw)]G(jw); (6.3b)orG’(jw) = G(jw)[U + L(jw)], (6.3c)withö[L0(jw)} <lmo(w), U[L(jw)] < lm(w), Vw 0,where lmo() and lm() are positive scalar functions; U is the unit matrix of appropriate dimensions with Lo(jw) and L(jw) representing output and input multiplicativeuncertainties, respectively. The structure of the system with these uncertainties isshown in Figure 6-2.The objective of the feedback design problem is to find a compensator F(s)such that:(i) the nominal feedback system, GF[U + GF]1,is stable;(ii) the perturbed system, G’F[U + G’F]1,is stable for all possible C’; and140(a)(b)True Plant G’(s)rFigure 6-2 Diagram showing different unstructured uncertainties: (a) additive;(b) output multiplicative; and (c) input multiplicative.True Plant G’(s)(c) True Plant G’(s)141(iii) performance objectives are satisfied for all possible G’.The requirement (i) demands that the encirclement count of the map det[U+GF(s)], valuated on the standard Nyquist D-counter, be equal to the (negative)number of unstable open loop modes of GF(s) [92]. As shown by Yuan and Stieber[93], for the additive uncertainty used in the present study, the closed loop systemsatisfies the stability robustness requirement (ii) if1la(w)< ã[R(jw)]’ (6.4)where [R(jw)] = F(j)[U + G(jw)F(jw)]1.Simillar conditions for other uncertainties can also be obatained [94].6.2.2 LQG\LTR design procedureLinear Quadratic Gaussian (LQG) procedure is a widely used approach forfeedback design [95, 96]. The LQG controller is an ordinary finite dimensional LTIcompensator with the internal structure as shown in Figure 6-3. It consists of aKalman-Bucy Filter (KBF) which provides an estimate of the state, &. The KBFgain, Kf, is given by [94]T-1Kf PfC0 0 , (6.5where Pf is the solution of the algebraic Riccati equationPfA + AOPf — PfC’O1COPf + F E FT =0, (6.6)and Pf = PJ 0. In general there are several solutions to Eq.(6.6),.but only oneof them is positive-semidefinite.The state estimate, I, is multiplied by the full-state Linear Quadratic Regulator (LQR) gain, K, to produce the control command which drives the plant and142Figure 6-3also feed—backs internally to the KBF. The LQR gain K is obtained to minimizethe cost functionJ= j (zTQz + uTRu)dt,where: z = Mx is some linear combination of the states; and Q = QT 0,R = RT > 0 are weighting matrices. The solution to this problem isI’ D1 1-lTD.LkC — c, 6.7where P satisfies the algebraic Riccati equation+ PA0 - PCBOR BQTPC + MTQM =0, (6.8)and P=P’>0.Both the LQR and KBF loops have good robustness properties [97, 98].Therefore, it may be expected that the LQG compensator would generally displayacceptable robustness and performance. Unfortunately, it has been shown that theLQG designs can exhibit arbitrarily poor stability margins [99]. However, there areCompensator F(s)IClosed-loop system with the LQG feedback controller.143procedures to design either LQR controller or KBF so that the full state-feedbackproperties are recovered at the output or input, respectively, of the plant [100]. Fora square minimum phase plant, design for the Loop Transfer (function) Recovery(LTR) at the plant output consists of two steps:(i) Design a KBF by manipulating the covariance matrices and 0 until a returnratio C0(sU — Ao)’Kf (i.e. the KBF loop transfer function) is obtainedwhich would be satisfactory at the plant output (i.e. at the break point (i)in Figure 6-3).(ii) The ioop transfer function obtained by breaking the loop at point (ii) in.Figure 6-3 is OF(s), where F(s) is the compensator transfer function. It canbe made to approach Co(sU — Ao)1Kf pointwise in s by designing the LQRin accordance with a sensitivity recover’y procedure due to Kwakernaak andSivan [101]. To achieve this, synthesize an optimal state-feedback regulatorby setting Q = Q0H- qU and R = R0 (or Q = Qo and R = R0 + pU), andincrease q (reduce p) until the return ratio at the output of the compensatedplant converges sufficiently close to Co(sU — Aa)’Kf over a large range ofoperational frequencies.For non-square plants, the inputs and/or outputs can be redefined to makethe system square and the LQG/LTR procedure can be applied to the modified plant(if it is observable and controllable). For non-minimum phase plants the recoverymay be achievable at those frequencies at which the plant’s response is very close tothat of a minimum-phase plant, i.e. at frequencies which are small compared to thedistance from the origin to any of the right-half plane zeros [94]. A simple strategyto use with the non—minimum phase plant is to follow the usual LTR procedure. Ifthe right half-plane zeros lie well out side the required bandwidth, then adequate144recovery of the ideal characteristics would be achieved at all significant frequencies.6.3 Controller Design and ImplementationThe prime objective of the controller is to regulate longitudinal and transverse tether vibrations using the offset strategy with the attitude dynamics regulatedby the thruster augmented active control. As explained earlier, only the first longitudinal and transverse modes are considered for the controller design. The LQG/LTRbased procedure is used which can account for the model uncertainty due to theneglected dynamics.6.3.1 Linear model of the flexible subsystemThe model is obtained by linearizing the decoupled equations of motion forthe flexible subsystem about the equilibrium positions. The equilibrium position forthe transverse vibrations is zero and for the longitudinal oscillations corresponds tothe static deflection value. The linearized equations can be written asM2Z + G Z + K Z + Mdy + Gddpy + Kdydpy+ Md2dtpz + Gdzdpz + Kdzdpz + P2 = QZTL, (6.9)where: Z {{B — Beq}T, CT}T is the flexible generalized coordinate vector;TL, the control thrust along the undeformed tether line; d, (D + D) and cL2(D2 + D2), the offsets of the tether attachment point along the local vertical andlocal horizontal, respectively; and the specified offsets; and D and D2,the offsets required by the controller. The expressions and numerical values of thecoefficient matrices (for a system with 10 transverse and 2 longitudinal modes) aredefined in Appendix V. The numerical values are for the system without any passivedamping. Depending on the choice and controllability, the variables TL, D, D,145D and D2 can be used as control inputs, either separately or in combinations.The selection of outputs completes system characterization in the linear state-space form. It should be recognized that placement of sensors on the tether tomeasure vibrations is impractical due to small diameter (one to two mm) of thetether as well as its deployment and retrieval maneuvers. The objective of thecontroller is to suppress longitudinal as well as transverse vibrations using the offsetcontrol strategy. The output vector consists of: longitudinal deformation from theequilibrium value of the tether at yt = L; slope of the tether due to transversedeformation at y = 0; and offset positions along the local vertical and horizontal;i.e.{ i(L), ..., %(L), 0, ..., 0}{Z}{ 0, •.., 0, (aia) ••, (aN1/aYt)}{Z} (6.10)6.3.2 Design of the controllerIn the nondimesional form, the lowest elastic and maximum attitude frequencies (w/O) are 12.6 and 1.732, respectively. This separation of frequencies allows thecontrollers for the rigid and flexible subsystems to be designed based on the decoupled equations of motion. The rigid body controller was designed using the FeedbackLinearization Technique (FLT) discussed in Section 5.2.3. Now, the controller forthe flexible subsystem is designed by the LQG based approach. Here, the objectiveis to regulate both the longitudinal and transverse vibrations by the offset controlstrategy. As mentioned earlier, the 1st transverse (C1) and 1st longitudinal (B1)146modes are controlled. The nondimensional frequencies of C1 and B1 degrees offreedom are 12.6 and 65.5, respectively. Again, the frequency separation permitsthe design to be based on the decoupled C1 and B1 models. The control input forthe B1 model is the offset acceleration along the local vertical (.b) and that forC1 model is D (offset acceleration along the local horizontal). The outputs areselected from Eq.(6.10). For the B1 subsystem, the output vector isI ,b1(L)Z’ (6.11)D, Jwhere Z1 = B1 — Beg1, and Beg1 is the equilibrium (static deflection) value of the1st longitudinal mode (B1). For the C1 equation, the outputs areI (ôi/ôt) c1 ‘1= Yt0.(6.12))Controller for the B1 model, which is a two output and a single input system, isdesigned using the LQG/LTR method. The design approach is due to Doyle andStein [100]. The controller matrices are obtained using the subroutines available inthe Robust Control Toolbox of MATLAB [102]. This procedure requires the systemto be square. The rectangular linear model for the B1 dynamics is rendered squareby defining the output asYb =b1(L)Z + D. (6.13)The system and control influence matrices are obtained consistent with the firstmode assumption. With this modified output, the system is found to be controllableand observable. The controller design equations for the C1 dynamics is obtained ina similar way. In this case, the LQG/LTR controller, obtained after redefining theoutput, does not give satisfactory performance. The degradation was attributed tothe modified output and can be avoided by using the original input-output structure147in conjunction with the LQG design procedure. To regulate the offset motion, thedesign equations are augmented by the following identities:= u;and jjz = u,for B1 and C1 dynamics, respectively. The controller design models can be represented by:Xb = Abxb + Bbub; (6.14a)Yb = Cbxb, (6.14b)andth Ax + Bcuc; (6.15a)Yc (6.15b)where:Z z{(B1— Beq1), E1, D, ]3}T;z={C1, O1, D2, .Oz}T;with Yb and Yc given by Eqs.(6.13) and (6.12), respectively.Here, the subscripts ‘b’ and ‘c’ refer to B1 and C1 dynamics, respectively. Thedynamic feedback controllers have the form:Xfb = Afb Zfb + Bfb Yb; (6.16a)Ub = Cfb Xfb, (6.16b)148andXfc = Af Xfc + Bfc ye; (6.17a)U = Cf Xfc, (6.17b)where Xfb and Xfc are the state vectors for the B1 and C1 controllers, respectively.The numerical values of the design model are obtained from the higher order model.The complete linear model, controller and weighting matrices are given in AppendixV. The data are for a tethered system during stationkeeping at L = 20 km withmass and elastic properties as given in Chapter 4. Next, the closed-loop eigenvaluesfor the linear flexible system were obtained to have some appreciation as to thecontroller’s effectiveness. The eigenvalues of the open-loop and closed-loop systemsare shown in Table 6.1.Table 6.1 Comparison of the open-loop and closed-loop eigenvalues of the system.Mode Open-loop Closed-loopTrans. 1 0.0 ± 1.468e-2— 9.880e-4 ± 1.463e-2Trans. 2 0.0 ± 2.937e-2 0.0 ± 2.937e-2Trans. 3 0.0 ± 4.405e-2 0.0 ± 4.406e-2Trans. 4 0.0 ± 5.874e-2 0.0 ± 5.874e-2Trans. 5 0.0 ± 7.342e-2 0.0 ± 7.343e-2Long. 1 — 3.797e-4 ± 7.601e-2— 1.144e-2 ± 7.614e-2Trans. 6 0.0 ± 8.811e-2 0.0 ± 8.811e-2Trans. 7 0.0 ± 1.028e-1 0.0 ± l.028e-1Trans. 8 0.0 ± 1.174e-1 0.0 ± 1.174e-1Trans. 9 0.0 ± 1.321e-1 0.0 ± l.321e-1Trans. 10 0.0 ± 1.468e-1 0.0 ± 1.468e-1Long. 2 — 2.267e-2 ± 5.870e-1— 2.183e-2 ± 5.912e-1Additional controller was designed for the model with a passive damper hay149ing a damping coefficient (Cdt) of 0.2 N.s/m and located on the tether at a distanceof 9.8 km from the platform. The damper was introduced to reduce the transversevibrations. The controller design procedure is given in Figure 6-4. As before, thecontroller for theB1-dynamics is designed by the LQG/LTR method and the C1-controller is designed using the LQG algorithm. The attitude controller is obtainedthrough the FLT procedure. The recovery of the return ratio at the plant outputwith the LQG/LTR controller for the B1 dynamics is shown in Figure 6-5. The gainand phase of the return ratio approaches those of the KBF loop transfer functionwith an increase in the design parameter q. In the present case, the B1 dynamicsis a non-minimum phase system with zeros at +0.045 and —0.042. As mentionedbefore, the recovery in this class of plants is not guaranteed as apparent in Figure6-5.Finally, the robustness property of both the B1 and C1 controllers is checkedagainst the additive model uncertainty G(j). Figure 6-6 compares the boundfor G(jw), i.e. la(w), and the inverse of the maximum singular value of R(jw)F(jw) (u + L(3w)) , where: L(jw) = G(jw)F(2w is the ioop transfer function;F(jw) is the controller transfer function (both the controllers); G(jw) is the openlooptransfer function; and U is the unit matrix. As shown in the figure, the stabilitycondition of Eq.(6.4) is satisfied. The structure of the closed-loop system, withcontrollers for rigid body dynamics as well as longitudinal and transverse vibrations,is shown in Figure 6-7.6.3.3 Results and discussionThe controllers were implemented on the complete nonlinear system withtwo longitudinal and three transverse modes. Figure 6-8 shows the response of the15028 states4 outputs___________2 inputsTruncated flexible modelfor controller design8 states4 outputs2 inputsThree level controllerfor rigid and flexible modelFigure 6-4 Flow chart for the controller design.Complete (rigid and flexible)nonlinear modelDecoupled linearflexible modelNonlinear rigid-body modelController design(LQG and LQG/LTR)8 dimensionalcontroller 4,Is controlleracceptaIe?Improve damping FLT basedcontroller designNoIIB1 - Controller C.ControllerAttitudeController151100.0Gain100.0q = 0 .q=lx-200.0 .. .... ..1 102 10.1 10° 101 102 io3degree!180.0q=1 \Phas:0____- .01 0 10.2 10.1 10° 01 02 1Frequency, rad/sFigure 6-5 Comparison of gain and phase of the return ratio at the plant outputwith those of the KBF loop transfer function.152100.00.0.S..-...-— __/\-100.0 V1 1Frequency, rad/sFigure 6-6 Robustness property of the vibration controller.system without a passive damper and using a three level control structure for simultaneous regulation of flexible and rigid modes. The total longitudinal deformation,v(L), is controlled about its equilibrium value of 13.896 m. As shown in Figure6-8, it returns to the equilibrium value in less than 0.05 orbit. The initial specifiedoffsets for this simulation was set at zero (D = = 0.0). The offset motionrequirement for control of the flexible modes is much lower than ± 15 m limit setfor regulation of the rigid degrees of freedom. The total offset motion along thelocal vertical (d) is less than ± 2 m. The first transverse mode settles to zero inaround one orbit. The offset motion along the local horizontal (d), required bytheC1-controller, is around ± 0.6 m. As discussed earlier, the platform motion isstrongly coupled with the tether dynamics through Since the FLT controllerfor the platform pitch (ap) dynamics cancels all the coupling effects, the shape of153Input I I OutputM(q,t)’ + f(q,,t) = QqLQG/LTR Controller______for Long. VibrationLQG Controllerfor Trans. VibrationDesired ITrajectory FLT Controller for[e}Figure 6-7 Three-level controller structure to regulate rigid as well as transverse and longitudinal flexible motions of the tether.the moment (Me) time history required to regulate p is similar to that of Themagnitude of M is well within the acceptable limit. The thrust (Tat) required tocontrol the tether pitch (at) is also very low.Figure 6-9 shows the response of the controlled system with a passive damper.The rigid body and longitudinal vibration responses are similar to those observedin the previous case. However, there is a substantial improvement in the transversevibration responses, particularly in the C1 and C3 modes. There is a room forfurther improvement through optimum location of the damper.From these simulations, it can be concluded that the tether vibrations inboth longitudinal and transverse directions can be controlled effectively by the offsetstrategy. It is important to note that the offsets required to control the flexible modes154Figure 6-8 Response of the system using a three level controller in absence ofa passive damper.(O) = 10; Ex(0) = 10 c2o = C3(O) =0B1(O) = 12 m; B2(O) =0 Stationkeeping: L = 20 kmC(0)=lOm D=0; D=01.00.00.01.00.02.018.012.0czCl0.00.50.0 2.013.0 Eii10.0-13.00.00.10.0v(L)C2M0.02.01.0C3Tat40.020.00.0_on A0.0 1.0::2.00.03.00.0 Orbit 0.12.00.0 Orbit 2.0155cz(O) = 10; (0) = 10 C2(0) = C3(0) = 0B1(0) = 12 m; B2(0) = 0 Stationkeeping: L = 20 kmC(0)=lOm D=0; D=01.0 1.00.0 0.00.0 1.0 0.0 1.018.0 13.0v(L) C1 0.012.0___________________-13.0___________________0.0 0.1 0.0 1.00.50.5C2 C3 0.00.0-0.50.0 1.0 0.0 1.0M8°0.0 T0.6at0.0-8.0___ ____ ___ ___ ___ ___0.0 1.0 0.0 1.00.22.00.0 0.0-2.0__ ___-0.2__ _0.0 Orbit 0.1 0.0 Orbit 1.0Figure 6-9 Controlled response of the system in presence of a passive damper.156are less than ±2 m, which is small compared to the limit (±15 m) set for regulatingthe attitude motion.6.4 Concluding RemarksIssues involving the control of elastic motion of the tether using offset strategywere addressed in this chapter. To begin with a linear model of the flexible subsystemwas obtained. Next, three controller ioops were designed for simultaneous regulationof attitude as well as longitudinal and transverse flexible modes. The controllers forflexible modes were obtained using the LQG based approaches while the attitudecontroller employed the FLT algorithm. Offset accelerations along local horizontaland vertical were used as inputs for transverse and longitudinal modes, respectively,while the attitude motion was controlled through thrusters. Effectiveness of thecontrollers was assessed through its application to the complete nonlinear coupledsystem. From the simulation results it can be concluded that the procedure is quiteeffective in regulating, during stationkeeping, both the rigid and flexible degrees offreedom.1577. EXPERIMENTAL VERIFICATION7.1 Preliminary RemarksSince the first practical application of a tethered system in 1966 during theGemini flight, several missions have been carried out to demonstrate the concept.As mentioned in Chapter 1, they include: a joint U.S.—Japan sounding rocket basedTethered Payload Experiment [TPE, 5]; U.S.A.—Italy TSS—i (Tethered SatelliteSystem—i) mission in August 1992 [6]; sounding rocket based OEDIPUS (Observation of Electrified Distributions in the lonosperic Plasma—a Unique Strategy) experiment launched by the Canadian Space Agency in January 1989 [7]; and themost recent studies called SEDS—I and II [Small Expendable Deployment System,8], launched in 1993 and 1994, respectively by NASA. Several ground based laboratory experiments have also been carried out to verify the tether dynamics andeffectiveness of different attitude control strategies. Dynamical aspects of a spinning tethered system were explored by Jablonski et al. [103] and Tyc et al. [104]particularly with reference to the OEDIPUS—C experiment scheduled for lunch inDecember 1995.Gwaltney and Greene [105] implemented a tension scheme by converting thecontrol algorithm to a length rate law, for both inpiane and out-of-plane dynamics,as well as accounting for the deployment and retrieval phases. The control scheme,with measured tension feedback, was investigated by Shoichi and Osamu [106]. Alaboratory set—up, for studying the attitude control of the subsatellite using theoffset strategy, has been developed by Kline -Schoder and Powell [59]. Effectivenessof the offset control strategy was verified by Modi et al. [48] using a ground based158experimental facility. An LQR type offset control algorithm was used to regulatetether librations during deployment, stationkeeping and retrieval. Using the modifled facility to improve the sensor accuracy, the present study assesses effectivenessof the FLT and LQG control methodologies.To begin with, the experimental set—up is described followed by details of thecontroller design and implementation procedure. Only a sample of typical results ispresented to attest effectiveness of the two controllers. The chapter ends with someconcluding remarks.7.2 Laboratory SetupThe objective of the present experiment is to have some appreciation as tothe effectiveness of the FLT and LQG based offset control schemes for regulating theattitude motion of the tether. The experimental setup consists of a spherical mass,representing the subsatellite, which can be deployed or retrieved from a carriage.The carriage, depicting the offset mechanism translating on a platform, can bemoved in a horizontal plane. This permits time dependent displacement of thetether attachment point to implement the offset control strategy. A Nylon thread,1 mm in diameter, connecting the subsatellite with the carriage, serves as tether.The tether can have a maximum length of 2.25 m. A larger platform with a heightof 5 m to accommodate longer tethers has also been designed and constructed.Details of the dimensions and mass properties of the setup are given in AppendixVI. The experimental apparatus consists of three main parts: the sensor; actuator;and controller.7.2.1 SensorRole of the sensor is to measure the angular deviation of the tether from the159vertical position. To that end, a pair of optical potentiometers (Si series Softpot,U.S. Digital Corp.) were used. The Softpot optical shaft encoder is a noncontactingrotary to digital converter. Useful for position feedback, it converts the real—timeshaft angle, speed and direction into the Transistor—Transistor Logic (TTL) compatible two channel quadrature outputs plus a third channel index output. It utilizesa mylar disk, a metal shaft with bushing, and an LED. The unit operates from asingle +5 V supply. Low friction ball bearing makes it suitable for motion controlapplications.The output pulses are monitored by a counter circuit, which generates aneight bit digital signal. The counter output can directly be read by the data acquisition system (a commercially available card [107]). The resolution of the sensor depends on whether one or two potentiometer output channels are used in the counter.Two different resolutions, 0.25° and 10, can be obtained with the present countercircuit. The index pulses are used to reset the counter to a reference value whenthe tether swings through the vertical position. This feature eliminates sensor driftthat is normally associated with a resistance potentiometer. Attractive features ofthe system are : low friction in the bearing; no sensor drift; and accurate resolutionto 0.25°.An important step in the sensor development was the design of a suitablemechanism to rotate the potentiometer shaft with the tether swing. The mechanismhas to perform its task without interfering with the deployment/retrieval maneuversand yet maintain the unavoidable friction at a low level. The design consists of alight, slotted, aluminum semi—ring as shown in Figure 7-i. The ring is so designedas to maintain dynamic balance during acceleration of the carriage. Any dynamicunbalance may introduce an inertia torque on the ring leading. to an error in the160CarriageFront View Side ViewFigure 7-1 A device to measure the tether swing through rotation of the potentiometer shaft./PotentiometerSlotted Ring —‘BearingTetherSubsatetlite161potentiometer reading. One side of the ring is connected to the potentiometer shaftwhile the other is supported by the bearing thus permitting it to rotate about afixed axis. The tether passes through the slot in the ring permitting detection of theswinging motion even during deployment and retrieval. A pair of such semi—ringsare used to measure the angles in two orthogonal planes.7.2.2 ActuatorThe actuation mechanism can be visualized as a large x-y table, where a carriage representing the tether attachment point traverses a horizontal plane (Figure7-2). The motion of the carriage is controlled by a pair of stepper motors. Thecarriage carried a reel mechanism, driven by another stepper motor, to deploy orretrieve the tethered payload. All the three motors were commanded by a digitalcomputer that implements the control strategy.7.2.3 ControllerThe controller was a 486/33 MHz IBM compatible personal computer. A typical control loop consists of sensing the tether angles, computation of the correctivecontrol effort, transmitting actuation commands for moving the tether attachmentpoint, and saving the required information into an output file. The stepper motor commands are sent through translators, which supply required voltage to themotors for each pulse received from the controller. The pulse train for the deployment/retrieval motor is generated by a commercially available card [108]. Atrapezoidal velocity profile is selected for this maneuver. The pulses for the carriage motors are generated through the digital output port of the PCLabCard [107].A linear velocity profile is chosen to implement the constant acceleration of thetether attachment point during each time—step. Photographs of the test facility arepresented in .Figures 7-3 to 7-6.162mCD I?3 C) CD S C) oqI-.C-) I z Q mH 0io>coOQm0-iQCoCl)-rr-<-A.,,m—1:imQC-IH-o z mFigure7.3Photographofthetest-rigconstructedtovalidateoffset controlstrategy:(a)aluminumframe;(b)inpianemotor;(c)carriagefor out-of-planemotion;(d)woodenstand;(e)linearbearings;(f)tetheredpayload.(.11Figure7.4Carriageandsensormechanism:(a)potentiometeronmountingbracket;(b)moveablealuminumsemi—ringmechanismwithslotsfortether;(c)tether;(d)inpianetraversewithlinearbearings;(e)payload.II•4Pd&,.dDigital hardware used in the experiment: (a) translator module,deployment and retrieval; (b) translator module, offset motions;(c) power supply; (d) function generator.Figure 7-5166I.Figure7.6Photographshowingtherecentlyconstructedlargertest—facilitywhichcanaccomodatetethersupto5minlength.Thepresentsmallerset—uponwhichtheexperimentsreportedherewerecarriedoutcanbeseenintheforegroundtotheleft.7.3 Controller Design and ImplementationThe equations governing dynamics of the laboratory model of the tetheredsatellite system were obtained. The linearized equations of motion can be writtenas:a=1a+a3& cd; (7.1)7= 7+a-y+c, (7.2)where:ai = —g/L; a = —2L/L; c = —ilL.Here: L is the instantaneous tether length; g is the acceleration due to gravity; aand ‘y are the tether angles in two orthogonal vertical planes; and da and th are theaccelerations of the offset point for controlling a and y, respectively. The a andequations are independent of each other and hence the controllers can be designedseparately. The controller design is based on the FLT and LQG procedures whichare explained below.7.3.1 FLT designThe procedure for the FLT controller design was outlined in Chapter 5. Forthe experimental model, it consists of state and control transformations:z = T() = [tjj]x; (7.3)—P(x)Uj=; (7.4)where:tj1= aic+ac;1682= —arc + a3c,= —a3c;=and for i= 2,3,4ui =t2O= t(_l)l +a3t(_l);P(z) =a1t43x+a2t43x+ (t4i +a3t43)x + (u42 1-a4t3);Q(x) = ct43 + t44.The secondary controller, Vt, for the system is designed to place the closed ioopeigenvalues at some desired locations. This leads to the following expression for v= kt1(zi zid)+kt2(z2 _Z2d)+ kt3(z _Z3d)+kt4(z4_z4d), (7.5)where: and Zid i = 1, , 4, are the coefficients of the required closed looppolynomial and the desired values for zj, respectively.Controllers for a and 7 degrees of freedom are designed by considering x as{a, cl, ix, d}T and {, J, , q7}T, with u as and d.y, respectively. Thecoefficients of the desired closed loop polynomials correspond to rise—times of 0.5and 0.6 s for tether and offset motions, respectively, and settling—times of 3.0 s forboth the tether and offset motions.1697.3.2 LQG designConsidering the dynamics of , the linear equations for the LQG controllerdesign can be written as:= Acxc + brua; (7.6a)= Ccxc, (7.6b)where:= {cr da a da}T;ua =o 0 1 0 (0o 0 0 1 JoAaai 0 0o 0 0 0 Ii_r T.ya_1a (hJ1000Ccx 0 1 0 0The controller design (i.e. LQR gains and KBF matrices) is arrived at using thesubprograms available in MATLAB. The weighting matrices considered in the designare:50.0 0 0 0o 2.0 0 00 0 1.0 0 IL—LOS0 0 0 0.130.0 0 0 001.0 0 0 11.0 0—= 0 100.0 0and 0= L 0 1.0o o 0 100.0The continuous time LQG controller is implemented as a sampled data system Tothat end, the dynamic controller equation is transformed into the discrete model,170which can be expressed as:Xfa(k + 1) AfaXf(k) + Bfaya(k); (7.7a)u.(k)=CfXfy(k). (7.7b)For L = 1 m; L = 0; and a sample time of 0.09 s, the controller matrices are:0.52214 0.00177 0.06212 0.00639A — 0.01194 0.62945 0.00585 0.06604fcr— —1.17623 0.01768 0.81021 0.159770.22942 —0.74557 0.14336 0.804240.43029 0.002430.00341 0.36614Bf= —0.03394 0.08718 ; and0.11417 0.64020Cfa [5.49377 —1.41421 1.73788 —2.12424].The controller for regulating ‘y is designed in a simillar way and leads to exactly thesame design.7.3.3 Controller implementationThe two controllers were implemented as sampled data systems with the zeroorder hold. The FLT compensator is described by algebraic equations. Therefore,the control inputs can be computed by Eqs.(7.3-7.5) at the sampling point. TheLQG control inputs are calculated by the discrete time model given in Eq.(7.7).As pointed out before, in the present experimental setup, stepper motors areused to move the tether attachment point. These motors are commanded to moveto a specified position by providing required number of pulses in each time—step.Therefore, the acceleration requirement, i.e. the control input, is converted into adisplacement requirement. The displacement can easily be converted into the num171ber of pulses required, from the knowledge of pulses per revolution and parametersof the transmission mechanism (chain drive in the present case). The conversion ofacceleration requirement to displacement is accomplished by integrating the acceleration profile, which is considered constant over a time—step. This leads to:dcz(k + 1) = o!a(k) + Ja(k)zt; (7.8a)da(k + 1) = dcx(k) + dcx(k)L?&t + cicr(k)(t)2, (7.8b)where: da(k) and da(k) are the magnitudes of cia and da at the kt1’ sampling point,respectively; and t is the sampling period.In case of the LQG controller, the control input da(k) is directly computedfrom the knowledge of the system outputs a(k) and da(k). In the present study, a(k)is measured directly by the potentiometers and da(k) is obtained from Eq.(7.8b).For the FLT controller, in addition to these outputs, magnitudes of à(k) and da(k)are required. da(k) can be computed using Eq.(7.8a). Since no sensor is used tomeasure a, backward finite difference method is used for its evaluation, i.e.a(k) = a(k) — a(k —1) (7.9)As the equations of motion for a and‘degrees of freedom are identical in form, the7-controller can be implemented using exactly the same procedure.The next important step in the controller implementation is the selection ofthe sampling time t. It has to be sufficiently large so that the computations required in the control ioop can be completed in the time—step. In the present setup,the time required to complete a control loop, i.e. to perform sensing, computation,actuation and data storage is less than 5 ms. The sampling rate is based on the requirement that the sampling frequency should be higher than the Nyquist frequency.172For better performance of the system, the sampling time should be less than onesixth of the minimum period of the system [109], which is 1.45 .s in the present case.From these considerations, the sampling time (st) is taken to be 90 ms.The computer program to implement the controller was written in MicroSoftC language. The real time implementation starts with initialization of the systemparameters and special purpose computer cards, and calibration of the potentiometers for reference vertical position. Application of a disturbance activates the controlloop. Each loop consists of measurement of the angles; computation of the offsetposition, velocity and acceleration; checking the steady state and safety conditions;commanding the motors to move; writing the data into the output file; and updatingthe variables for the next time—step. The controller operation can be stopped bypressing an arbitrary key at any time. This can be used for emergency exit of thecontroller. The fow chart for the controller operation is shown in Figure 7-7.7.4 Results and DiscussionThe experimental validation of the offset strategy in conjunction with theFLT and LQG controllers provided considerable insight into the feasibility of theapproach. The concept of controlling the attitude motion of a tether, using offset acceleration as the input, augmenting the tether dynamical equations with theidentity cia u, and providing both angle and offset variable feedback proved tobe feasible. Small discrepancies observed between the experimental and numericalsimulation results may be attributed to the following factors:• Damping effects can not be completely eliminated in a real system. Herethe contributions come from friction in the sensing mechanism (Figure 7-1),linear bearings of the carriage and aerodynamic effects.173Figure 7-7 Flow chart showing the real time implementation of the attitudecontroller.IN ITIALIZATION:Define constants and deployment/retrieval parametersInitialize port addresses and system variablesChoose controller (FLT or LOGMeasure angles and compute angular velocities ICompute offset parameters at the end of AtSend pulses to inplaneand out-of-plane motorsMeasure timeNo174• The stepper motors are controlled in the open ioop fashion. Therefore, anymissing pulse can not be compensated by the controller.• The step—size for the motors is 1.8°. The required angular rotation of themotor in each time—step may not be a multiple of 1.8°. This introduces errorin the implementation of the controller.Considerable amount of information was obtained through a series of carefully planned experiments. Only a sample of results is presented here. The systemwas purposely subjected to a very large disturbance of 10° in both the directions(a and ‘y) to assess the controller’s effectiveness under demanding situations. Inpractice, an external excitation would seldom result in motions larger than a fewdegrees. Performance of the FLT and LQG controllers is discussed in the followingtwo subsections. The acceptable steady state error limits are set at ± 1 cm for offsetpositions and ± 0.5° for the tether angles. The maximum allowable offset positionsare limited to ± 20 cm.7.4.1 FLT controlFigure 7-8(a) compares numerical and experimental results during the stationkeeping phase at a tether length of 0.5 m. The uncontrolled response is fairlywell predicted by the numerical simulations, however, small attenuation is observedin the amplitude of the experimental response. As pointed out before, this may beattributed to friction in the ring mechanism used to measure the angles. Comparison of results during the controlled phase shows acceptable correlation. Note, thesteady state errors are within the limits. The initial disturbance is damped withina. very short time (< 4 s). The offset motions required to regulate a and 7 are alsowithin the specified limits.Subsatellite and carriage positions during the controlled experiment are shown175Numerical10.00.0-10.010.00.0-10.0cx°10.00.0-10.0Experimental10.00.0-10.08.0da o.o-8.08.0d 0.0-8.00.0Figure 7-8Time, s 10.0Plots showing the comparison between numerical and experimentalresponse results during the stationkeeping phase: (a) uncontrolledand controlled system.(0) = 100, (O) = 10Stationkeeping: L=0.5 mFLT ControllerI U ncont rol ledIUncontrollediIControlledlVControl led)Th\/j176Numerical ExperimentalFigure 7-8 Plots showing the comparison between numerical and experimentalresponse results during the stationkeeping phase: (b) subsatelliteand carriage positions.a(O)=1O°, y(O)=1O°FLT Controller Stationkeeping: L=O.5 m10.05.00.0-5.0-10.05.00.0-5.0-10.0yOd7-5.0 0.0 5.0 10.0xo[cj I I, I Icm[-5.0 0.0 .5.0da177in Fig.7-8(b). The projection of the subsatellite position in a horizontal plane hastwo orthogonal components, L sin a cos and L sin y COS a. For small angles theymay be represented, approximately, as La and L’y. Therefore, vs. a plot represents the scaled position of the subsatellite projected on a horizontal plane. Inthe present simulations, disturbance is given to the angular positions only. Initially,librational velocities, as well as position and velocity of the carriage (tether attachment point)are zero. Under this situation, the subsatellite should oscillate in oneplane during both the uncontrolled and controlled conditions. The numerical simulation (solid line in -a plot) substantiates this observation. The small discrepancyin the experimental results is again due to the sensor noise mentioned before. Theplots for carriage position show similar trends.Figure 7-9(a) shows response of the system during stationkeeping at a longertether length of 1 m. The numerical and experimental responses for the uncontrolledas well as controlled system show excellent agreement. As expected, the frequency ofangular motion is lower compared to that for L = 0.5 m. The offset motion requiredto regulate the inpiane and out-of-plane librations is rather modest (within +6 cmand —4.5 cm). Experimental results for the subsatellite and carriage positions arequite close to the straight line predicted by the numerical simulation (Figure 7-9b).Similar trends continued even at a higher length of 2 m (Figure 7-10).As emphasized earlier, retrieval is the most critical phase in the tether operation. To show effectiveness of the controller, two different retrieval rates areconsidered with maneuver completed in 15 s and 5 s. In each case, the tether lengthwas reduced from 2 m to 0.5 m in accordance with a trapezoidal velocity profile.The velocity was linearly increased (constant acceleration) from zero to the maximum value in O.ltr, held constant (zero acceleration) at the maximum value for178Numerical Experimental-8.00.0= 100, y(O) = 100FLT Controller Stationkeeping: L=1 mlUncontrolled!10.00cx 0.0-10.010.000.0-10.010.0a° 0.0-10.010.00.0-10.08.0da 0.0-8.08.0d 0.0ControllediControlled\\/‘•i1..==========..=.===.====Figure 7-9Time,s 10.0A comparative study between numerical and experimental resultsfor the system during stationkeeping at 1 m: (a). uncontrolled andFLT controlled system.179Numerical Experimental10.05.00.0-5.0-10.05.0d 0.0-5.0-5.0 0.0d5.0Figure 7-9 A comparative study between numerical and experimental resultsfor the system during stationkeeping at 1 m: (b) subsatellite andcarriage positions.cx(O) = 100, ‘y(O) = 100FLT Controller Stationkeeping: L=1 m-10.0 -5.0 0.0 5.0 10.0a°cm.1....180Numerical Experimentala°cx°dad110.00.0-10.010.00.0-10.010.00.0-10.015.00.0-15.015.00.0-15.00.0 10.0(0) = 100, y(0) = 10Stationkeeping: L=2 mFLT Controller I10.00.0-10.0\\\_IControllediControllediFigure 7-10Time, sA comparative study between numerical and experimental resultsfor the system during stationkeeping at 2 m: (a) uncontrolled andFLT controlled system.181Numerical Experimentala(O)=1O°, y(O)=100FLT Controller Stationkeeping: L=2 m, I • I I I. I I I-5.0 0.0 5.0 10.0x°cm] I II Iicr0.0dFigure 7-10 A comparative study between numerical and experimental resultsfor the system during stationkeeping at 2 m: (b) subsatellite andcarriage positions.10.05.00.0-5.010.0d1 0.0-10.0-10.0 10.0182O.8tr, and finally linearly reduced to zero in 0.lt,.. Here tr is the specified time tocomplete the retrieval. The maximum velocities are 0.11 rn/s and 0.33 rn/s for theretrieval times of 15 s and 5 s, respectively.Figure 7-11(a) shows experimental response results for the retrieval completed in 15 s. As expected, with the tether length becoming smaller, inertia ofthe system reduces causing the tether oscillations to grow to conserve the systemangular momentum. During the uncontrolled operation, the system response (c and7) grows to a maximum amplitude of ± 290 as the retrieval ends. Subsequently,the frictional damping causes the amplitudes to slightly diminish as expected. Theoffset procedure effectively controls the motion within 8 s, i.e. long before the retrieval is completed. Note, excursion of the tether attachment point is maintainedwithin the specified limit of ± 20 cm. Even at a faster retrieval rate (t = 5 s), theoffset strategy continues to be effective (Figure 7-llb).7.4.2 LQG controlThe offset controller, designed using the LQG algorithm (Section 7.3.2), wasalso implemented on the same tethered system. The uncontrolled and controlledresponses, for the stationkeeping case at L = lm, are compared in Figure 7-12(a).As before, the results show good agreement with the numerically predicated performance. The initial disturbance is damped within 5 s. Considering the frictionaleffects, the subsatellite and carriage positions also show good correlation with thenumerically predicted results (Figure 7-12b).The experimental results for the retrieval phase are presented in Figure 7-13.As before, a trapezoidal velocity profile was employed with the retrieval from 2 rnto 0.5 rn completed in 10 s. Instability of the uncontrolled system is shown ratherdramatically. Gain scheduling was used during the retrieval phase with the gains183Figure 7-11 Uncontrolled and controlled experimental results for retrieval of thesubsatellite: (a) retrieval time of 15s.= 110,‘y(O) = 110 Retrieval:FLT Controller 2.0 m to 0.5 m in 15 sdegrees_________ ________rolIed’”11VV\/\.30.00.0-30.010.00.0-10.010.00.0-10.020.00.0-20.020.00.0-20.02.0. [ptrolldJ1c°dadL0.Qoo Time, s 20.018430.00.0-30.010.0cx° 0.0-10.010.00.0-10.020.0d 0.0-20.0d20.0‘‘ 0.0-20.02.0L0.00.0 Time, s 20.0Figure 7-11 Uncontrolled and controlled experimental results for retrieval of thesubsatellite: (b) faster retrieval in 5s.x(0) = 110, y(0) = 110 Retrieval:FLT Controller 2.0 rn to 0.5 m in 5 s185Numerical ExperimentalIControlledi1Time, sFigure 7-12 A comparative study with the LQG controller during stationkeeping at 1 m: (a) uncontrolled and controlled performance.x(O)=1O°, y(O)=1O°LQG Controller Stationkeeping: L=1 mUncontroIled10.00.0-10.0 -10.000.0-10.010.0a° 0.0-10.010.00.0-10.08.0d 0.0cx-8.08.0d 0.0-8.00.0Control led10.018610.05.00.0-5.0-10.010.0ct(0) = 1 00, ‘y(O) = 100LQG Controller Stationkeeping: L=1 m.—.—..--;.‘._._4.__ I,s...— ,.c •,•-.-..-./ .—..-,.—-— -—-10.0 -5.0 0.0 5.0 id.0c°cm. I I crn[0.0dFigure 7-12 A comparative study with the LQG controller during stationkeeping at 1 m: (b) subsatellite and the tether attachment point posiNumerical-—•-••-••--••- Experimentald 0.0-10.0-10.0 10.0tions.1871i1—----0jTime, sFigure 7-13 Experimentally observed system response during retrieval with the30.00.0-30.010.00a 0.0-10.0a(O) = 110, ‘y(O) = 110 Retrieval:LQG Controller 2.0 m to 0.5 m in lOsdegrees_________---•-••-‘‘c011JVVVVVV\Controlled10.00.0-10.020.0da 0.0-20.020.0d 0.0-20.02.0LLQG controller.20.0188adjusted at 0.75 m and 1.5 m. The LQG controller is quite successful in regulatingsuch a large initial disturbance within 12 .s. The steady state errors are less than5 % of the initial disturbance for the angles and less than ± 1 cm for the offsetposition.Finally, performance of the LQG controller in regulating the spherical pendulum type motion was evaluated (Figure 7-14). This required application of a rathersevere set of initial disturbances both to the angular velocity and position. The controller continues to be remarkably effective as shown by the inward spiral motion.The corresponding carriage position also converges to the steady state value. Thetime to damp the disturbance was found to be around 15 .s.A video of the experimental set—up, presenting in some detail the sensor,actuator and the controller, was taken. It shows, rather dramatically, effectivenessof the offset control strategy in damping a variety of severe disturbances.7.5 Concluding RemarksExperiments carried out employing a unique ground based test—facility suggests that the offset control strategy, using the FLT as well as LQG algorithms, caneffectively damp rather severe disturbances during both stationkeeping and retrievalphases. The controllers continue to be effective even during a faster retrieval rate of0.33 rn/s. Note, the above mentioned performance is attained within the specifiedlimit of the offset motion. Correlation between the experimental and the numericallypredicted results is also good considering the frictional effects encountered in thereal-life situation. A video captures, quite effectively, the remarkable performanceof the offset control procedure.189Spherical Pendulum MotionStationkeeping: L=1 .5 mLQG Controller-10.0 -5.0 0.0 5.0cm10.010.05.00.0-5.010.0d o.o-10.0___-10.0 10.0dFigure 7-14 Subsatellite and carriage positions during control of the sphericalpendulum using the LQG regulator.0.0cm[1908. CLOSING COMMENTS8.1 Concluding RemarksUsing a relatively general model, the thesis develops a methodology and associated computational tools for studying planar dynamics and control of tetheredsatellite systems. Versatility of the model is illustrated through its application during all the three phases of a typical mission involving deployment, stationkeeping,and retrieval. The focus is on the system control using two distinctly different typesof actuators: thrusters located at the subsatellite; and movement of tether attachment at the platform. Both linear as well as nonlinear controllers, using the actuatorsand their hybrid combinations, are developed applying the Linear Quadratic Gaussian (LQG) regulator and the Feedback Linearization Technique (FLT). It may berecalled that the FLT accounts for the complete nonlinear dynamics of the system.As can be expected, the governing equations of motion are highly nonlinear,nonautonomous and coupled. They were used to assess uncontrolled dynamical performance of the system as affected by the important system parameters. However,widely spaced frequency for the rigid and flexible degrees of freedom was often takenadvantage of through decoupling during the controller design. Thus controllers forthe rigid and flexible parts of the system were designed separately using the coupledsets of linearized equations. Of course, effectiveness of the designed controllers wasassessed through their application to the original nonlinear and coupled system.The thesis presents innovations in several areas. More important contributionsof the thesis, which have not been reported in the literature, include the following:(i) the system model that accounts for the tether flexibility and motion of thetether attachment point at the platform end;191(ii) control of the system’s attitude dynamics using the offset scheme in presenceof tether flexibility;(iii) application of the Feedback Linearization Technique, which accounts for thecomplete nonlinear dynamics, to tethered systems;(iv) vibration suppression along both the longitudinal and transverse directionsusing the offset strategy;(v) simultaneous attitude and vibration control of the tethered systems;(vi) ground based experiment to substantiate effectiveness of the FLT and LQGbased offset control synthesis.It should be emphasized that the objective here was to establish a methodologyto understand dynamics and control of such a complex system. It was not intendedto acquire large amount of information useful in the system design. Of course, suchinformation can be generated quite readily as the dynamics and control programsare operational. Even then the amount of information obtained is rather extensive.The thesis presents only some typical results useful in establishing trends. Basedon the analysis following general conclusions can be made:(a) Offset of the tether attachment point leads to coupling between the platformand tether dynamics. The tether pitch libration significantly affects the platform dynamics, however, the effect of platform motion on the tether dynamicsis relatively unsignificant.(b) As can be expected, static deflection of the tether as well as frequency ofboth transverse and longitudinal oscillations are affected by the mass densityand elastic properties of the tether material. Higher modes of the longitudinalvibrations have rapidly decaying characteristics due to the structural damping,which has only the second order effect on the transverse tether vibrations.192(c) The tether dynamics is susceptible to instability when the deployment/retrievalrate exceeds the critical value.(d) The FLT based controller using the rigid nonlinear model of the system isquite successful in regulating attitude dynamics of the system with a flexibletether. The controller structure is relatively easy to implement. A single control algorithm is applicable to all the three operational phases of deployment,stationkeeping and retrieval.(e) The FLT based controller is found to be better than linear, time invariant regulators designed using the graph theoretic approach and implemented throughgain scheduling.(f) A hybrid strategy, relying on the thruster control at longer tether lengths andthe offset control for shorter tethers, appears quite promising for regulatingthe tether pitch libration. Results show that a pair of passive dampers can beused to control the unstable elastic degrees of freedom.(g) Besides controlling the tether pitch motion, the offset strategy can be usedto regulate, simultaneously, both longitudinal and transverse oscillations ofthe tether duing the stationkeeping. Results for a 20 km tether showed thecontroller to be remarkably effective in damping the motions with the tetheroffset maintained much below the specified ± 20 m limit.(h) Substantiation as to the effectiveness of the FLT and LQG based offset controlstrategies using a rather unique test facility represents a significant contribution to the field. Results show that the concept of controlling the motion ofa tethered payload through specification of the acceleration at the point ofattachment is not only effective but can also be implemented in practice.1938.2 Recommendations for Future WorkThe present thesis represents a modest contribution to the challenging fieldof tether system dynamics and control. There are several avenues open for futureexploration which are likely to improve our understanding of the field. A few ofthem, more directly related to the present study, are indicated below:(i) Extension of the present investigation to three dimensions, i.e. generalizationof the model to account for inplane as well as out-of-plane dynamics representsthe logical next step.(ii) Offset control of the tether vibrations during a retrieval maneuver needs tobe explored. If successful, it will make the offset strategy more attractive andversatile.(iii) In presence of an offset along the local horizontal, the tether dynamics significantly affects the platform motion. This presents an exciting possibilityof controlling the platform libration through an offset strategy. Simultaneouscontrol of the platform and tether dynamics through offset would represent animportant contribution to the field.(iv) It would be useful to assess effect of the free molecular environment forces onthe tether dynamics and control.(v) Multibody tethers have been proposed for several scientific experiments, including monitoring of Earth’s environment as in the case of Mission to PlanetEarth. The present model can be extended, through a recursive formulation,to account for such configurations.(vi) Extension of the ground based experiment to attest effectiveness of the offsetstrategy in controlling transverse and longitudinal oscillations of the tetherwould represent an important step forward.194/(vii) Validation of various offset control strategies using a longer tether is desirable particularly during retrieval. It would also help study the concept oftethered elevator system. 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C., Robust Control Toolbox User’s Guide, TheMath Works Inc., 1992, pp. 2.72-2.74.[103] Jablonski, A. M., et al., “The Tether Laboratory Demonstration System (TELAB) for Dynamics of OEDIPUS,” Advances in Astronautical Sciences, Editors:A. K. Misra, et al., Vol. 85, Part I, 1993, pp. 711-731.[104] Tyc, G., et al.,”Ground Experiments of a Hanging Spinning Tethered Body,”Proceedings of the AIAA/AAS Astrodynamics Conference, Scottsdale, AZ, U.S.A.,August 1994, pp. 341-350.[105] Gwaltney, D. A., and Greene, M. E.,”Ground-Based Implementation and Verification of Control Laws for Tethered Satellite,” Journal of Guidance, Control,and Dynamics, Vol. 15, No. 1, 1992, pp. 271-273.[106] Shoichi, Y. and Osamu, O.,”An Experimental Study of Tether Reel System - ALaboratory Model,” TR-1176T, National Aerospace Laboratory, Chofu, Tokyo,Japan, August 1992.[107] User’s Manual: PCL-812PG Enhanced Multi-Lab Card, Advantech Co., 1990.[108] Model 5030, Software Guide, Manual No. 40041—009—bA, Industrial ComputerSource, San Diego, California, U.S.A., 1992[109] deSilva, C. W., Control Sensors and Actuators. Prentice Hall, Englewood Cliffs,New Jersey, U.S.A., 1989, pp.5-30.[110] Lakshmanan, P. K., Dynamics and Control of an Orbiting Space Platform BasedTethered Satellite System, Ph.D. Thesis, Department of Mechanical Engineering, University of British Columbia, Vancouver, Canada, 1989, pp. 216-220.203APPENDIX I: MATRICES USED IN THE FORMULATIONFor concise derivation and efficient computer implementation of the governingequations of motion, a number of matrices are defined to express the system energies.The matrices, dependent on modal integrals and attitude angles, are also reportedhere.Matrices used in the Kinetic Energy (Eq. 2.25){K1} = LL{Ak0}+L2wt{Bk0}+ msL{Dk1}+m8Lwt{Dk2} E R3;[K2] = [Ak2] + ms[Ap1] E R3)<Ntl;[K3] = L[Ak1]+ wt[Bk1]+ msct[Uk][Ap1] E R3tl;[K4] = -ms[Ap1]T[ + -[Ck2] E RNtltl;[K5] = mswt[Ap1]T[Uk][j+ L[Ck3]+ wt[Hk4] E RNtltl;[K6] = mswz[ApljT [Uk]T[Uk] [A1 + j,2[ck] + Lwt[Hk3] E RNtZtl;{K7} + + msL{Dk1}[Ap]+ msLwt{Dj1,2}[Ap E RNtZ;204{K8} = 1L2{ck}T + Lwt{Hk1}T+ msLwt{Dk1}T[Uk][Apij+ mgLw{Dk2[Ukj[Apl1 E RI.Matrices used in the Potential Energy (Eq. 2.28)[F1] = M[U] — 3M{V] E R3<;[F2] = ma[U] — 3ma[Vpo] E2p—6pt cos(at){f{F.p(yt)}dyt}{I/0}T31— 2p{f{F(yt)}dyt}{Utp}T + 6pt sin(at){f{F(yt)}dyt}{17po}T+2m8[Api]T[Ttp] — 6ms[Api]T{17to}{Wpo}T E RNt1X3;{F4} = 2m8L { {F(L)} } — GmsLcos(crt)[Ap1]T{Wto} E RNtZ;[F5] = ms[Api]T [Ap1] — 3ms[Api]T{Wto}{Wto}T[A1j E RNtlTtl;{P6} = (2m3 + pL)L{Wtp} — 3(2m + pL)L cos(crt){T’Vpa} E= cos2(at) E R.205Inertia Dyadics and Their DerivativesThe terms associated. with the inertia matrices and their derivatives, used inderivation of the governing equations of motion, are presented here.‘Pz ‘Pzy ‘Pzz[4] = 1Pzy ‘Py ‘Pyz4zz 4yz ‘Pz1t 0 0 Ib+Ic 0 0[1t] = 0 jt 1ty2 = 0 Ic ‘bc- 0 ‘ty ‘tz 0 ‘bc ‘bwhere:= + {B}T[Iblj{B} + 2{1b} B};Ic {C}T[Ici]{C};1bc = —{Ib1}{C} — {B}T[1oc2]{C};[1b] = Pt f{Fb}{Fb}TdYt;{Ib2} ptJyt{Fp}Tdyt;[Icr] = Pt j {Fq}{F,}Tdyt;{1&ci} =jLvt{F}Tdyt;[‘bc2l = pt jL{i}{i}Tdyt.The time derivative of [Ip] is zero. The derivatives of the elements of [‘] used inthe equations of motion, are as follows:= + 2{B}T[Ibl){B} + {B}T[.ibl]{B}+2{1b}{B} + 2{lb} B} + 2{C}T[I1J C}+{C}T[ici]{C};206o{X}(tT[1t]) 2f{ItzB}1I sin(2czt){ItyZB } +sin2(ct){ItZB } %ô{X} t}Ttjt}) t cos(t){Itz} — Sifl(2t){Ityz}ã{X} ({w}T[I]{w}) = { }where:WiT.{‘tZB}=2[Ib1]{B}+2{Ib{ 1tzc} = 2[1c1J{C};{ ItYZ } = —[IbC2]{C};{ = {J& }T_ [Ibc2lT{B};[1b] = F fL{{DF}{F}T -f {F}{DF1p}T}dy + {F}{F}T];LoUb1} = pt{JL{+ +[i1 = F fL{{DF}{F}T + {F}{DF}T}dy + {F}{F}T].[JOModal Integral and other Constant Matrices{Ako}={Pt}ER3;207o 0[Ak1]= pJoL{DF}dy 0 ER3tl;o ptJ{DF#}dyto 0[Ak2] = Pt f{F(yt)}Tdyt 0 e R3tl;o p f{F(yt)}Tdyto 0[A1] {F(L)}T 0 Eo {F1,(L)}T(01{Bk0}z 0(pt/2Jo o[Bk1]= 0 _ptf{F(yt)}Tdyt ER3>tl;Pt fJ{F(ut)}Tdyt 0Ptf{DF}{DF}Tdut o1 — RNtZXNt1I k1i— 0 Pt fj’{DF,}{DF}Tdytpt ,f{F1p(yt)}{F(yt)}Tdyt 0[Ck2] = E RNt1)<Ntl;0 Ptff{F(vt)}{F(vt)}Tdyt2080 ] ER’tltl;[Ck3] = 0 2pt f{F(yt)}{DF}Tdyt{Ck4}= { J’ó’{DF}dyt } E RNtZ;{Ck5}= { 2Pt f{Fi,(yt)} } € RNt1;{Dk1} { } € R3;0{oi{Dk2}= O €R3;1{Hk1}= { foL{yt{DF#} — {F}}dyt } E RNtZ;{Hk2}= { f yt{F}dyt } €0 pt f{F(yj)}{DF,}Tdyt[Hk3]= 0 ] € RNtltl;2090—Pt[Hk41 = e RNtltl;pt 01 0 0[T] = 0 cos(at — Qp) sin(at — crp)0— sin(czt — ap) cos(at — czp)100[Uj= 0 1 0001000[Uk] = 0 0 —1 e0101 0{U} —sin(at—ap) hcos(at — ap) J0 0 0[V0] 0 cos2(ap) —sin(ap)cos(ap)0 — sin(ap) cos(ap) sin2(ap)1 0{W} cos(at — cxp) ;I sin(at — cxp) J(0{1i70}= cos(ap) ;(—sin(ap) J10 ‘I{1’t,} = cos(at) ;I —sin(at) JNt{F(yt)} ={.(y,L)} e RNt;210{F,,1,(y)} ={‘I’ct,L)}’1 E Rl;OF ôF{DF}+ ERNt;8F ôF,b{DFti}=-ã---+--- eRl.Here:N1 = number of modes used to represent the longitudinal tether vibration;Nt = number of modes for transverse oscillation of the tether;Nti= Nt+Nj.Time Derivative of Matrices{Ak0} ={O};o 0[Ak1] f{D2F}Tdyt + {DF(L)}T 0o - JL{D2F}Tdy {DF(L)}T[Ak2] [Ak1];{E} ={0};211- 0 0-[E] = 0 —[Ak2(3,Nl + 1: N)][Ak2(2,1 N1)] 0f{{D2F,,1,} F}T +{DF1p}{ 2F}T}dyt[Ck1 =ptL00,f{{D2F}{DF}T + {DF}{D2F}T}dyt+{DF4,(L)}{ F(L)}T+ {F}{DF,,11}T}dy[Oh2] =L00+ {F}{DF#}T}dyt+ {F,}{D2F}T}dy[Oh3] =2pL02120J’{{DF}{DF}T +I f{D2F}dyt + {DF(L)}{Ck4}=2pL1 0 ){Ok5} .L{Ck4};1 0{1rk}=ptLI f yt{D2F}dyt + L{DF,(L)} — {F(L)} J1 0{iIk}=ptL2f{{F#} + + L{F(L)} J0[Ilk3] =ptL— J’{{DFçj,}{DF}T +{F1,}{D2F}T}dyt—{F,(L)}{DF,,h(L)}Tf{{DF,ç}{DFq}T + {1\t}{D2Fc5}T}dyt02130[Hk4] ptL f{{vF}{F}T + {F}{DF}T}dyf{{F}{DF#}T + {DF}{F4}T}dy+{F(L)}{F(L)}0o 0 0[1}(aà) 0 —sin(at—cp) cos(atap)o—cos(at—ap) sin(cxtap)where:8{DF.} Ô{DF4{D2F4,}=‘‘ +ôytô{DF1,} ô{DF}{D2F1}= c9yt+ ÔL eRN1.Derivatives of Matrices used in the Potential Energy[Pi]cxp = 3M[Vpr,jap;[P2]ap = — 3ma[Vpo]ap;P— 2pt{J’{Fp(yt)}dyt}{Wtp}’p— 6Pt cos(ct){f{F,,,(yt)}dyt}{iV0’[3lap-2{fL{F}d}{U}T + 6pt214-I- 2ms[Api]T[Ttp]ap — 6ms[ApijT{Wto}{Wpo}’;{P4}ap = {O};[P5]cxp= [0];{P6}= (2m8 + ptL)L{Wtp}ap — 3(2m8 + ptL)Lcos(ct){1’Vpo}a;= [0];[P2]at = [O];— 2pt{f{F,t,(vt)}dyt}{Wtp}’ + 6pt sin(at){f{F(yt)}dyt}{17po}Tlt— {f{F(yt)}dyt}{utp} + 6Pt cos(at){f{F(yt)}dyt}{Wpa}T+ 2m8[A1]T [Ttp]at — 6m8[A1]T{TVp7to}at {l:vpa}T;{P4}at = — 6m8Lcos(a)[Ap1]T{Ti7jo}a + 6msL sin(czt)[Ap1]T{Wto};[P5jat = - 3m[Ap1]T[ o}at{Wo}T+ [Ap1];{P6} = (2m8 + ptL)L{Wtp}at + 3(2m8 + ptL)Lsin(at){T’’po},where:()ai?’ =p’o 0 0[Ttp]crp = 0 sin(at—czp) — cos(at—cip)0 cos(at—ap) sin(at—ap)[Ttp1c =215o 0 0[Vpo]czp = 0 — sin(2cp) — cos(2ap)o — cos(2a) sin(2c)1 0{Wtp}ap = sin(a— ap) ;I. — cos(c — cp) J{Wtp}at = {Wtp}ap;1 0{ Utp}ap = cos(at — cp)I. sin( — cxp){Utp}at = {Utp}ap;1° ‘I{T’Vpo}crpz —sin(ap) ;—cos(ap) J(o=— sin(a)—cos(at)216APPENDIX II: ELEMENTS OF MATRICES ‘M’ AND‘F’The matrices M(q, t) and F(q, v, t) appearing in Eq. (2.37) can be partitioned asfollowsMr(11)+MSMr(1,1) Mr(l,2)+MSMr(2,1)M(q,t) = Mr(2,1)+MSMr(2,1) Mr(2,2)+MSMr(2,2)Mfr(:, 1) + MSMf (:, 1) Mfr(:, 2) + MSMf (:, 2).Mrf(1:)+MSMf(1:)MTf(2,:) + MSMf(2:) E R(2t1)x(2ti);Mf+MSMfFr(1) + FSMr(1)F(q,,t) = E R(2tl);Ff + FSMfwhereNtz= Nt+Nz;M(1, 1)= ‘P2 + ma{dp}T[UkIT[Uk]{dp};Mr(1,2) = {dp}T[Uk]T[Ttp]T{L2{BkO}+msL{Dk+ [[Bk1]+ms[Uk][Api]j{X}};{Mrf(1, :)} = {dp}T[Uk]T[Ttp] [K2];Mr(2,1) Mr(1,2);217Mr(2, 2) = It + m8L2 + ms{X}T[Ap1IT[Uk]T[Uk][Apl]{X}+ 2msL{Dk}[Uk][Ap]{X};{Mrf(2, :)} = {X}T {[Hk4IT + ms[Apl]T[Uk]T[Apl]]+ {Hk2}T + msL{Dk2}[Ap1];{Mfr(:, 1)} = {Mrf(1, :)}T;{Mfr(:,2)} = {Mrf(2,:)}T;Mf = 2[K4];Fr(1) = (Mr(1 1) + Mr(12))8 + mQ{dp}T[U]T{Jp}+ {dp}T[uk]T[Ttp]T{L{AkO} + ms{Dk1}+ [Ak1]{X}}LH- ma{4}9’[Uk]{4} + 2mawpz{dp}T[Uj$,]T[Uk]{dp}+ {dp}T[Uk]T[Ttp]T{L2{AkQ} + 2LLwt{Bk0+ msLwt{Dk2}+ [[Ak2] + [K3]] {J} + [L[Akl] + wt[Ekl]] {x}}+ {dpT[uk]T[tp]T + {a}T[Uk]T[TtIT}{ {K} + [K2]{5} + [K3]{X}}- {{}T + wp{dp}T[uk]T}[rtp]P{{Kl} + [K2]{i} + [K3]{x}}+ ({dp}T[P2lap{dp} + {X}T[P3]ap{dp}+{P6}’{d} + 6{lp}’p[Ip]{lp});Fr(2) = (M(1 1) + Mr(1, 2))+ ltwt + 2msLLwt + {L2{BkO} + msL{Dk2}}[tp]{4}+ {L2{BkO} + msL{Dk2}+ [[Bk1]+ ms[Uk][Api]] {x}} [Tt]{J}218+ ({x}T[Hk3]{x} + {{Hkl}T + ms{Dkl}T[Uk][Apl]}{X}) L+ {2LL.Bk0 + msL{Die2}} [T]{4}+ {dp}T[Uk]T[Ttp]T{2LL{BkQ} + m$L{Dk2}}wpz+ {.1apT[uk]T[Ttp]T + {dp}T[Uk]T[ipJT}{L{Bj0}+ msL{Dk2}}wpz+ {4}T[i][[Bk1]+ ms[Uk][Api]j {X}+ {4}T[Tt]T [[Bk] + ms[Uk][Api]] {;t} + {ã}T[Tt]T[Ek]{x}+ [{ã }T[Uk]T[Tt]T + {dp}T[Uk]T[ip1T][[Bk1] ms[Uk][Api]j {X}p+ {dp}T[Uk]T[Ttp]T[Bkl]{X}wpz+ {dp}T[Uk]T[Ttp]T [[Bk1]+ ms[Uk][Api]] {}wp+ {i(}T[[Hk4]T + ms[Apl]T[Uk]T[Apl]] {} + {5}T[iIk4]{X}+ {k}T [4mswt[Apl]T [Uk]T[Uk] [A1 + L[Hk3j+ L[Hk3ITj {X}+ {ik2 + msL{Dk2}T[Apl]}{iC} + L{X}T [ilk3]{X}+ {L{Hkl}T + msL{Dk1}T[Uk] [A1+ 2msLwt {Dk2}T[Uk] [A1+ {L{ilkl}T + 2msLwtz{Dk}T[Uk][Api]}{X}- {{}T + wpZ{dp}T[UkjT}[Ttp]{{Kl} + [K2]{} + [K3]{X}}+ ({X}T[P3]at{dp} + {X}T{P4}cxt + {X}T[P5]czt{X}219+ {P6}cxt{dp} — 3mgL2(Vto)ct -I- 6{lt}t[It]{lt});{Ff} = (Mfr(:1)+Mfr(:2))+ [K2]T[Tt]{i} + {{Ck5}+ ms[Ap1]T{Dk}++ [[K2]T[] + [Ak2]T[Ttp}] {{a} +wP[Uk]{dP}}+[KTtp][Uk]{p}wp + [[Ok2]+ [Ks]] {}+ [L[ak3]+ [ilk4]] {X}+ {i{Ok5}+ + msLwtz[Apl}T{Dk2}-[K3]T[TJ{{4} +wP[Uk]{dP}} - [K5]T{}- [[K6]+ [K6]T]{X}- {K8}T- a{x} ({W}T[I]{W})c9Us OMe 1+ ã{X} + 2R j[P3]{dp} + {P4} + [[] + [P5JT]{X}— â{X} (tr[Itl) -- a{X} ({lt}T[It1{l}) }+ () f 8{} }dt 1° 1 fo){ 8{ó} }dYt j;TMSMr(1,1) =_MfôM} {CM};1.. &pTMSMr(1,2) _M1t3M} {CMczt};T{M(1, :)} = — MI ÔRSM } [CMx];1 pTMsM(2, 1) = — MI ÔRSM } {CMcxp};1 atTMSMr(2,2) = — MIf9RSM } {CMcxt};1. at220{ :)} = — Mf ÔRSM }T[cMx];l &t{MSMf(:1)} = {MsMf(1,:)}T;{MSMf(:2)} = {MsMf(2,:)}T;[MsMff] = _M[’9RSM1 [CMxJ;FSMr(l) = (MSMr(11)+MSMr(12)){ d (8RSM f-M{1sM}- ptL (L{Dki} + [A1]{X}) }+ dt p Jj 1.I ÔRSM (LL + + L[A1]{X} +—pt.I p , I— MI ÔRSM T{cM}I p J- J aR }T{_M{SM} - PtL (L{Dki} + [A1]{X}) }IGMe ( ÔRSMT2[p ]{R } + 2[P]{d}+2R3 I c9api I+ [P3]T{X} + {P6}})GMeI+ 2R{RSM}T{[Pl]ap{RSM} + 2[P]cxp{dp}+ [P3]{X} +{P6}op});FSMr(2)= (MSMr(21)+MSMr(22))TI d (ORSM-M{1SM} - PtL(L{Dki} + [A1i{X})}at jJ II ÔRSM (LL + + L[A1]{X} +—ptI at )— MI ÔRSMTI at J- I aRSM}T{M{ } - pL (L{Dki} + [A1]{X}) }I at221GM /IORSM+ 2R oat}T{+ [P3]T{X} + {P6}})GMeI+ 2R +{P6}at});{FSMf} = {MSMf(:1)+MSMf(:2)}rd fOR\)] {_M{1SM} - PtL(L{Dki} + [A1]{X})}roRSM1- pt [ . j + + L[A1]{X} +T— MFORSM] {CMF} + ptL[Apl]T{i?sM}raRSM1T- [ {_M{SM} - PtL(L{Dki} + {A1]{X}) }GMe I roRSM+ 2R U ]T{2[Pl]{RSM}+2[P{dP}+ [P3]T{X} + {P6}})GMeI2R [P31{RsM});where:I 8u •‘{ous’a{Bfl1 a{X} ) = 8U/8{C} )‘I ou8 r L(8F, 1OFIT 1t — EAI I —-j dutj{B}8{B} Lb I 8YtL/r8FT \2(OF+ (EA/2) f (j-__) {C}) i_a—)dut;O \(oU3} L(8F,T (OFTEAI !—.--$ {B}j-_? {C}}dtO{C}.‘ I ôt j8FT \3(OF+(EA/2)-— {C}) j-_-dt;222t8F1T Idjaj jT dãF T({)}T)(11 {O}+{_{_}} {C});i ayt J( a •‘ 1aFto{E}J ia1’( o •‘ /(ÔFT \1âF’jo{E} z(i.aj {C})t_j;{RSM} = — {ma{dp} + (msL +ptL2/2)[Ttp]T{Dk1}+ [Tt]T [ms[Api] + [Ak2]] {X}};{1SM} = - {ma[ip]{dp} + ma[Ttp]{4} + (ms + ptL)L{Dk1}+ [Ak2]{X} + [ms[Api] + [Ak2]]+ {wt} x {[Ttp]{RSM}};ma{CMap} ={CMcxt} = — + [Uk][Ttp]{RSM};[CMx] = — [ms[Api] + [Ak2]];{CMF} = — -{ma(àt — a)2[T2]{d} + 2ma[i’tp]{4,}223+ ma[Ttp]{äp} + ((ms + pL)L + pL2){D}+ [L[Aki] + L[Ak1]]{X} + 2[Ak]{}}+ [Uk] [T] {RSM } + wt [Uk] [i] {RSM }+ w [Uk][Ttp]{isM};IÔRSM’I ma1 a = [T1j{dp};dI8RSM1 ma______maa. )=[Tj{ãp}+[T] dp};ma mat àtàp){Ti]ap{dp} [TtpJap{dp}+ wtz[Uic][Tp]ap{RSM} + wtz[Uk][Ttp]{R5M}cz ;[t3RSM1 maaa =— j[Ti}{dp} + [Uk][Ttp]{RSM};d 1ÔRSM1 ma maôt=—[Tl]{dp}—[Tl]{dp}+[Uk]{sM};[ÔRSM1 ma mai ac ) = — p)[Ti]at{dp} — y[Ttp]at{dp}+ wt [Uk] [Ttp]at {RSM } + Wtz [Uk] [Tt] {RSM }at;raRSM1 1[ = — [ms[Api] + [Ak21j;d [aisM1dtl 9j jM224IÔRSM1__________ÔRSM] 1L ax j Wta [Uk] [Tt] [ ax j —IÔRSM1 11. aa j = — yj{(msL +PtL2/2)[Ttp]{Dk1}+ [T]’ {ms[Api] + [Ilk2]]{x}};(8RSM’l 1=— j{(msL +ptL2/2)[Ttp]{Dk1}+ [Ttp]at [ms[Apil + [Ilk2]]{x}};raRsM1 1L ax j = — [Tt]T [ms[Api] + [Ak2]];ro 0 0 1[T1] = lo —sin(at—ap) cos(at—cxp) I[o—cos(ot—ap) —sin(at—ap)j[0 0 0 1[T2] = 10 COS(tap) —sin(at—ap) I;[0 sin(t—cp) _cos(at_ap)][t1] = (at — a) [T2};[Ti]ap = — [T2];[Ti]at _[T2].The matrices used in the above equations were defined in Appendix I.225APPENDIX III: NONLINEAR AND LINEARIZED EQUATIONS OF MOTION FOR THE RIGIDSUBSYSTEMNonlinear EquationThe nonlinear equations of motion for the rigid degrees of freedom can be expressedas[Mu M12 f&,jfFj_ M j III1LM21 M22 1&tJ 12JLTatJ’where:M11 =mamc(d + d2) + ‘;M12 =mlmc (d1 cos(c) + sin());M21 =M12;M22 =pL3/3+ m8L2 —F1 =(M11 + M12)S + mamcdp’ydpz + 2mamcdpzdpzwpz+ { (mrn + pL)L + pL2 — mlwmc} (d sin(a) — cos(a))+ { (msmc + ptL +m2c)Lwt } (d cos(a) + d sin(c))+ (_msL cos(a) + m1wj sin(cz))ma4z/M+ 3uma(D— D) sin(2p) + 6/.LmaDtDt2cos(2ap)+ 2m1 (D sin(a) — Dt2 cos(c))+ 6iimi cos(t) (Dt sin(ap) + Dt cos(czp))+31pz — Ipy) sin(2ap)—cos(2cp)226+3itmp(R%—R%)sin(2ap)+ 6IImPRSMYRSMZ cos(2ap) + ii + up2;F2 =(M21 + M22)8 + mlmc cos()Jz+ (pL2L + 2m3LL— 2mimL/M)wt+ m1mc (d sin(cz) — d2 cos(a))+ mi(1 + mc)4zwp sin() — maptLLwp cos(a) + sin(cx))/M—maptLL4z cos(a)/M— 2mi (D sin(a)—Dt2 cos(cz))+ 6mi sin(ot) (Dt cos(ap)— .Dt sin(ap))+34u(msL2+pL3/3)sin(2czt) — up—P1=() { —2mDt, sin(a) (i — 3 cos2(ap)) + 2maDt cos(a) (i — 3 sin2(cp))+ 3ma sin(2ap) (D cos(a)— Dt sin(cx)) + 3m1 sin(2czt)};mpm1 2P2= ( M ) {_2 (RSMY sin(a) — RSMZ cos(a)) +6RSM sin(a) cos (ap)— 6RSMZ cos(o) sin2(ap) + 3 sin(2ap) (RsM cos(a)— RSMZ sin(a)) };RSM = (—madpy — m1 cos(a)) /M;RSMZ = (_madpz — ml sin(a)) /M;1%, tRSM + d;Dt RsM + dpz;ma =ptL + m0 + m3;m1 =m3L+ ptL2/2;m2 =m3 + ptL;mc =1 — maiM;M=m+m0+ptL+m;227GMe2RLinear EquationThe nonlinear equations of motion for the attitude dynamics of the tethered systemare linearized about some arbitrary trajectory for the platform and tether pitch, andoffset motion along the local horizontal. The offset along the local vertical is keptfixed at The governing equations of motion can be represented as[ML11 ML12]Jl [CL11 GL12 f’i[ML21 ML22j & J GL CL22 j 1 &+ FKL1i KL12 I ‘ 1 + f MD1 jLKL21 KL22j cz f MD2 f 2(111.2)where:ML11 = ma(D + D) + ‘;ML12 m1 (D cos(ã) + D2 sin(a));ML21 =ML12;ML22 =m5L2+ ptL3/3;CL11 = 2maDpzDpz;GL12 = — 2m1Ô(D sin(ä) — D2 cos(a)) + 2mL(D cos(ã) + D2CL21 = 2m18 sin(ã) — D2 cos(ã)) + 2miDpz sin(a);CL22 = 2m3LL + ptL2L;KL11 = m1(D sin(ä) — D2 cos(ä)) — m2L cos(ä) + D2 sin(ä))228— ptL2 (D cos(ä) + D2 sin()) + miÔ2 (D cos(ã) + sin(ã))+ 2mL8 (D sin(ä) — cos(a)) + 6ma1u(D — D) cos(2äp)— 12matDpyDpz sin(2ãp) — 2mip (D cos(a) + sin(ã))+ 6miILcos(ät)(Dpcos(ãp) D2sin(aP))—6j(Ip— Ip)cos(2ap)+ 12,Ulpyz sin(2ap);KL12 — m1 sin(ã)—cos(a)) + m2L(D cos(ã) + D2 sin(a))+ ptL2 cos(ä) + D2 sin(ã)) — m1Ô2(Dm1cos(ä) + sin(ä))— 2mL9 sin(ä) — D2 cos(a)) + 2mi (D cos(a) -I- Dpz sin(ä))— 6m1sin(ät) (D sin(ãp) + cos(ap));KL21 = m1(D11 sin(ä) — cos()) + mliipz sin(ã)— m1Ô2 cos(ä) + sin(ä)) — 2mi8bpz cos(a);+ 2mt (Dpy cos(ä) + D2 sin())— 6m1s1n(t) sin(äp) + cos(p));KL22 = — mi(Dpy sin(ä) — cos()) — miiipz sin(ã)+ miÔ2 (D cos(a) + sin(ä)) + 2miODpz cos(a)— 2m1i(D cos(ã) + sin(ã))+ 6mi cos(ät) cos(äp) — sin(ãp))+ 6(msL2+ ptL3/3) cos(2at);MD1=maDpy;MD2 = ml cos(a);229GD1—2maSDpz;GD2 = 2m18sin(a);KD1 = 2ma6Dpz + m16sin(ã) — m2Lcos(ä) + 2ma8Dpz— pL2 cos(ã)-I- mÔ cos(a) + 2mLÔ sin(ã) — 6maDpz sin(2p)+ 6maitDpy cos(2p) — 2m1cos(ä) + 6m1,ucos(ãt) cos(äp);KD2 = m16sin(ä) + m1(2 — 2) cos(ä) — 6m1psin(äj) sin(äp);FL1 = (M11 + M12)8+ ma(D + D2)ö + Ipö + mi (D cos(a) + sin(a))+ maDbpz + m2L sin(ã) — D2 cos(ä))+ 2maSDpzDpz+ (p.L2 — mlÔ2)(Dpij sin(ä) cos(ã))+ 2mL6 (D cos(ä) + D2 sin(a)) + 3ma(D — D2)sin(2äp)+ 6mafLDpyDpz cos(2ãp) + 2m1 sin(ä) — cos(ä))+ 6mlI4cos(ãt)(Dpsin(äp) + cos(p))—3P(Ipy — Ip)sin(2a)— 6,.iIp cos(2ãp);FL2 (Mi1 + M12)8+ m1(D cos(a) + sin(ä)) + (msL2 + ptL3/3)+ mliipz cos(ä) + (2m3LL + ptL2L)8+ mÔ2 (Di sin(ä) — cos(a)) + 2miÔi3pz sin(ã)— 2m1 sin(ã) — cos(a))+ 6m11usin(ãt) cos(äp) — sin(äp))+ 3(mL2+ pL3/3) sin(2ät);230a =a—a3,.Here, ã,, ä define the reference trajectories for platform and tether pitch angles,respectively; cz3, and a are the differences between the actual and reference valuesfor the platform and tether pitch angles, respectively; D3, and D3, are the referencevalues for the offsets along the local vertical and local horizontal, respectively; andD2 is the offset required by the controller along the local horizontal direction.Equations for the Offset ControlThe dynamic model for the offset control of the tether attitude is based on the linearequation which is decoupled from the platform motion. However, since the platformdynamics is strongly coupled with the tether attitude in the presence of nonzerooffset, the platform pitch controller is based on the complete nonlinear equation.These can be presented asmap(cxp, x, t)&3, + p, x, th, t) = M; (111.3)= aa +a2D + a3c + €L4D2 + b+ cD2, (111.4)where:map(ap,x,t) =M11 —M1221/;f,(ap,àp,x,th,t) =F1 —M12F2/;a1 = — _?_{mi (_D3, sin(ä) + D3, cos(&))+ m1(2 — 2)(D cos(ä) + D3, sin(ä))+ 6i.tmi cos(ãj) (D cos(äp) — D3,2 sin(dp))-I-6fLmatcos(2at)};231a = — sin(ä) + mi(2.t — Ô2) cos(ä)mct— 6m1 sin(ãt) sin(ap)};a3 =—mat1 1 _a4 =— ____t2m1Ssmn(a);matb =—_--_{mi(Dpi, cos(ã) + sin(ä)) + maj + thatÔ+ mi(2 — 2t) (D sin(ä) — cos(ã))+ 6,um1 sin(at) (D cos(äp) — D2 sin(p))+ 3umat sin(2at)};— m cos(ã)matmat =ptL3/3+ m3L2;nat =ptL2LH- 2m8LL.The terms used to define the platform pitch equation, Eq.(III.3), were given in thesub—section titled “Nonlinear Equation”, Eq.(III.1).232APPENDIX IV: CONTROLLER DESIGN USING GRAPHTHEORETIC APPROACHSome details of the mathematical background and the controller design procedure which assigns specified eigenvalues to the linear time invariant tethered systemare described here.Mathematical BackgroundDefinitions *A directed graph (also called digraph) 13 = (V, E) consists of two finitesets• V, the vertex set, a nonempty set of elements called the vertices of 13; and• E, the directed edge set, a (possibly empty) set of elements called thedirected edges of 13;such that each e in E is assigned an ordered pair of vertices (u, v). If e is a directededge (also called edge) in the digraph 13, with associated ordered pair of vertices(u,v), then e is said to join u to v. u is called the origin or the initial vertex of e,and v is referred to as the terminus or the terminal vertex of e. If a number (oran edge weight) is assigned to each edge of a digraph, then the graph is called theweighted digraph. In G = (V, E), if the element e of E is assigned an unorderedpair (u,v), then G is called a graph.* Clark, J., and Holton, D. A., A First Look at Graph Theory, WorldScientific Publishers, Singapore, 1991, Chapters 1, 7.233A directed walk in the digraph D is a finite sequenceW = vjelv ..whose terms are alternately vertices and edges such that for i = 1,2, , k, theedge e has the origin v_1 and terminus v. If u and v are the starting and endingvertices, respectively, of the walk W then the u — v walk (i.e. W) is called closed oropen depending on whether u = v or u v, respectively. If W does not contain anedge then it is called a trivial walk, else it is nontrivial.A nontrivial closed walk in a digraph D is called a cycle if its origin and internal vertices are distinct. In detail, the closed walk C =v0e1 vk_lekvk(vk = vo)is a cycle if:(i) C has at least one edge; and(ii) v0, v1, v2, , v_ are k distinct vertices.Since C contains distinct vertices, it also has distinct edges. The integer k, thenumber of edges in the cycle, is called the length of C. A cycle C, of length Ic, isdenoted by Ck. A set of vertex disjoint cycles is referred to as a cycle family. Thewidth of the cycle family is the total number of vertices in it.A graph C is called connected if every two of its vertices are connected. Atree is a connected graph whose number of edges is one less than the number ofvertices. A spanning tree (or complete tree) of a connected graph C is a subgraphwhich is a tree that involves all the vertices of C. A digraph is said to have a rootr if r is a vertex and, for every other vertex v, there is a path which starts in r andends in v. A digraph D is called a rooted tree if D has a root from which there isa unique path to every other vertex.234Mapping of linear state-space model into diagraphs *Consider the Linear Time Invariant (LTI) equationx—Ax+Bu, (IV.1)where z e R is the state vector; u E R is the control input vector; A Eand B E RnXm. For the state feedback situation the control law can be expressedasu=Fx, (IV.2)where F e R>< is the controller matrix. The closed-loop system of Eqs. (IV.1)and (IV.2) can be mapped into the digraph (GS) defined by a vertex set and an edgeset as follows:• the vertex set consists of m input vertices denoted by Ui, U2, ..., urn, and nstate vertices denoted by 1, 2, •, n;• the edge set results from the following rules:— if the state variable occurs in the x-equation, i.e. 0, then thereexists an edge from the vertex j to the vertex i with ajj as its weight;— if the input variable Uk occurs in the z-equation, i.e. 0, then thereexists an edge from the input vertex Uk to the vertex i with the weight— finally, if the state variable x occurs in the ui-equation, i.e. f 0,then there exists an edge from the vertex i to the vertex Uk with theweight fj.* Reinschke, K. J., Multivariable Control: A Graph Theoretic Approach,Lecture Notes in Control and Information Sciences, Edited by: M. Thomaand A. Wyner, Berlin, Springer-Verlag, 1988, Chapters 1,2.235Here, ajj is the row and th column entry of the matrix A; and bk and fk arethe entries of B and F, respectively.For illustration, consider a system defined by the matricesA= [ai a32 a33 a34]; B = []; F = [fi f2 f f i•a41 a42 a43 a44The digraph C for this dynamic system can be represented as in Figure IV-1.Obviously, the diagraph C5 contains less information than Eqs. (IV.l) and(IV.2). Actually, G8 reflects the structure of a closed-loop system with state feedback. As far as the small scale systems are concerned, it seems unnecessary toinvestigate their structures separately. In case of the large scale systems, however,one should start with a structural investigation. Thus, particularly for higher ordersystems, the diagraph approach is extremely useful.A typical feature of large scale systems is their sparsity. This structuralproperty becomes evident in the digraph C5. The digraph reflects only the non-vanishing couplings of the system. So, instead of n2 state edges and nm controledges one has really to take into account only a small percentage of this number inmost sparse systems. Moreover, the diagraph C5 gives an immediate impression ofthe information flow within the closed ioop system. So for higher order systems, thedigraph approach to controller design is advantageous from computational point ofview.For lower order systems, which is the case in the present study, this designmethod gives a simple closed-form expression for the controller that can be readilyincluded in the dynamic simulation program or implemented in real system. Thisfeature allows for efficient simulation with a gain scheduling control for the highly236— — — — — ———I”b3731i\f4a33 afia1 a32 —-2Figure IV-1 Digraph representation of a linear, time-invariant system.time varying dynamics of the tethered systems.ControllabilityA class of systems characterized by the structure matrix pair [A, B] is saidto be structurally controllable (or s-controllable) if there exists at least oneadmissible realization (A, B) e [A, B] that is controllable in the usual numericalsense.The n x (n -I- m) structure matrix pair [A, B] is s-controllable if and only ifits associated digraph G([Q]) contains a set of i ( m) disjoint cacti, each of themrooted in another input vertex and, together, touching all the state vertices. Here237i_ ([A] [B]’\ 1V3LQJ - [F] [0])’ ( .)and the ‘cactus’ associated with G([QJ) is the spanning input-connected graphconsisting of a simple path with an input vertex as the initial vertex, p ( 0)vertex disjoint cycles and p distinguished edges each of which connects exactlyone cycle with the path or with another cycle.Note, the digraph G([Q]) associated with the structure matrix [Q] is the same asthe diagraph GS corresponding to Eqs.(IV.1) and (IV.2) defined earlier.Characteristic polynomial of a square matrixLet the characteristic polynomial of the closed-loop system be representedby+Pn_ls+Pn. (IV.4)The coefficients p, 1 i n, can be determined by the cycle families of width iwithin the graph G( [Q]). Each cycle family of width i corresponds to one term inp. The numerical value of the term results from the weight of the correspondingcycle family. The value must be multiplied by a sign factor (_].)d if the cycle familyunder consideration consists of d vertex disjoint cycles. In particular, pi results fromall cycles of length 1, with the common sign factor as —1; and P2 arises from allcycles of length 2, each with a sign factor —1, as well as all disjoint pairs of cyclesof length 1, each pair with a sign factor +1, etc.Controller Design AlgorithmBased on the preliminaries discussed above, the state feedback controller238which assigns the closed-loop poles at the desired locations in the complex planecan be obtained by the following procedure.Step 1: Consider the digraph G([Qj), Eq.(IV.3), and choose a subgraph consisting of m disjoint cacti as indicated before. Enumerate these cactiarbitrarily from 1 to ñì.. There are fiz.! possible different enumerations.Step 2: Choose an mx n feedback structure matrix [P’] whose th— 1 nonvanishingelements correspond to feedback edges leading from the final vertex ofthe path of cactus jto the root of cactus j + 1 (j = —1).Choose a unit structure vector {gj such that the input vertex associatedwith the column structure vector [Bg], is the root of cactus 1.The matrix [F] and the vector [g] assure the s-controllability of the singleinput pair [A + BP, Bg]. Moreover, for almost all admissible (A, B) e[A,B], P E [P’j, g E [g], the pair (A + BF,Bg) is controllable in thenumerical sense.Step 3: Choose admissible (A,B,P’,g) E [A,B,P,g] such that the pair (A +BF, Bg) becomes controllable. Set up the system of equationsP —[pl]Tf (IV 5)where: p = {Pi, P2, , }T is the vector of characteristic polynomial coefficients defined by the desired pole locations of the closed loopsystem; p0 contains the characteristic polynomial coefficients determinedby the cycle families in G(A + BP); and[F1]= [P,)]’ w = 1,2,••• ,n; j = ,n.The element of [F1], i.e. Pw,j)’ is the (sign weighted) sum of weights239of all those cycle families of width w within the diagraph of Eq.(IV.3)that contain one feedback edge (with weight 1), from state vertex j tothe input of cactus 1.[F’] has rank n, if and only if (A + BP’, Bg) is controllable. Hence,Eq.(IV.5) can be solved for f for almost all (A, B) E [A, B].Step 4: The overall feedback matrixF=P+gfT (IV.6)provides the desired eigenvalue placement.For single input systems, the controller can directly be designed from Eq.(IV.5) withP = [0] and g = {0}.Design of the ControllerThe state feedback controller is designed to regulate the rigid degrees offreedom, i.e. p and . The governing linearized equations of motion for the rigidsystem are used to that end (Appendix III, Eq.(III.2)). The platform and tetherpitch controllers are designed based on the decoupled equations of motion. For theoffset control, to regulate the offset motion, the tether pitch equation is augmentedby the identity= Ut.The equations governing the tether dynamics can be written in the state space formas±=Ax+bu, (IV.7)240where:x = {at, at, }T;0 0 1 0 00 0 0 1 b= 0a31 a32 a33 a34a41 a42 a43 a44The entries in A and b matrices are obtained from Eq.(III.2), Appendix III. Thedigraph of this system with the state feedback controller is shown in Fig. IV-1. Thecontroller, u = fx, can now be designed using Eq.(IV.5) with p° and [F1] as givenbelow:( —a33 — a44o— J a33a44 — a3 — a42 — a34a432’ ——a32a43 + a42a33 + a3a44 —a3441. a3142 —a32410 —b3 b3a44 — b4a34 b3a42 — b4a32rpll — 0 —b4 —b3a43 + b4a33 —b3a41 -I- &ia3iI— —b3 b3a44 — b4a34 b3a42 — b4a32 0—b4 —b3a43 + b4a33 —b3a41 -I- b4a3j 0Similarly, the state model of the platform dynamics can be represented by the matrices0 1 loand b=caj a22The corresponding matrices for the controller design are= { 22 } and [p1] = [—2 I241APPENDIX V: LINEARIZED EQUATIONS OF MOTIONAND CONTROLLER FOR THE FLEXIBLE SUBSYSTEMLinearized Equations of MotionThe governing equations of motion (Appendix II) are decoupled separatingthe rigid and flexible subsystems. Neglecting the nonlinear terms from the decoupledequations, the linear model for the flexible system can be represented by the vectorequationMZ + GZ + K2Z + Mdydpy + Gddpy + Kdydpy+ Mdzdpz -I- Gdpz + Kdzdpz + 1z = QzTL, (IV.1)where:Z = {{B — Beq}T, CT}T;{Beq} = equillibrium value of {B};M2 = 2{K4};C2=+ [K5] — [K5]T+ () [[i [AC3d]]F{F(Ydl)} {O} 1 FCdz{F(Ydl)}T {O} 1+ [ {Q} {F(Ydt)}j I {O} Cdz{F(Ydt)}Tj‘K2 = rrig[Apl]T[Uk] [A1 .+ + ++ Ô[1k4]— [K6] — [K6]T + [P5] -f- [p5]T242(—2—4i)[I] + [AB] [0] ]+ L [0] (2 + 2i)[Ici] + [AC]l{F(Ydj)} {0} 1 CdjL{DF(Ydl)}T Cdl{F(Ydl)}T+1L {0} {F(Ydt)}j [ Cdt{F(Ydt)} cdL{DF(yd)}T]Md [K2IT[Ttp];Gd = [Ak2]T[Ttp] +[K2]T[Ttp][Uk] — [K3]T[Tt]r {0} {0} 110 Cdl 0 1+ I {0} IL {0} {F(Ydt)}j L0 0 Cdt]’Kd = ë[K2]T[Ttp][Uk] + [Ak2TTtp][Uk] — [K3]TTtp][Uk] + [P3]r {0} {0} 1—Cdl 0+ S J F(Y1)} {0}[ {0} {F(Ydt)}j { 0 Cdi]’iz = {Hk2}+msL[Apl]T{Dk2}}+ {{Ck5}+ ms[Apl]T{Dkl}}L- {K8}T + {Ok5} + + mSLÔ[Apl]T{Dk2}+ {P4}I (Ô2 + 4){’b2}}+ { 1 Lcdl ,F.Ø(Ydl)} {0}-1 {} {0} {F(Ydt)}J { YCjr ,L(ãF. (ÔFI1T[AB]=EAIJ —‘-0 t 6t J dY];I T IÔF 18FT 1[AC}=EA’J ij_} {Be})t?t dYtj;L o[AC8d](8FlT8ytfJ-j ditj.Here: Cdl and °dt are the damping coefficients of the longitudinal and transverse243dampers located at distances dl and dt, respectively; and w0 are as definedin Eqs.(2.32) and (2.33); {B} and {C} are the vector containing longitudinal andtransverse modes of the tether; and the coefficient matrices are defined in AppendixI.Numerical Values of the Coefficient MatricesThe numerical values for the coefficient matrices in Eq.(IV.1) are given below. They correspond to the stationkeeping case with a tether length of 20 km, andmass and elastic properties as given in Chap.4.532.7 519.6= 519.6 514.0 [t1 e[0] [Diag.(98.0)][0.405 0.4051 c T[0.405 0.729] E R122;[G] [0]—1.023 —0.4010.512 0.434—0.341 —0.3180.256 0.246—0.205 —0.199Gt =0.1 X 0.171 0.168—0.146 —0.1440.128 0.127—0.114 —0.1130.102 0.10213.08 3.081K [3.08 5.54] [0] E R12X 2;[0] [Diag.{K}]244—42.982 0.0211—41.928 0.08450.0 0.19020.0 0.33820.0 0.5284= 0.0 Kt 0.76090.0 1.03570.0 1.35270.0 1.71200.0 2.11360.0—549.0 0.0—524.5 —88.2310.0 0.00.0 —29.4100.0 0.0Md= 0.0 Md = —17.6460.0 0.00.0 —12.6040.0 0.00.0 —9.80310.00.00.0 —1.27330.2046 —1.21650:0 0.00.0682 0.00.0 0.00dy= 0.0409 Gd 0.00.0 0.00.0292 0.00.0 0.00.0227 0.00.0245—0.2215e 2 1.0—O.2116e — 2 1.00.0 0.00.0• 0.00.0 0.0Kd= 0.0 =0.0 0.00.0 0.00.0 0.00.0 0.01.0 0.0 2’1.0 0.00.0 0.2221e — 30.0 0.4443e — 30.0 0.6664e — 3— 0.0 0.8886e — 3C0 0.0 0.lllle—20.0 0.1333e — 20.0 0.1555e — 20.0 0.1777e—20.0 0.1999e — 20.0 0.2221e — 2where:b1(L)...0 •.. 0C0= .. 0 (a1,o)•..(8N /8t)Yt=0 Yt0[Diag.(a)] =[D];with D= 0, Vij;= a, if a is a scalar;= a(i), if a is a vector.246Design Parameters and Controller MatricesThe numerical values for the weighting matrices used for the controller designand final controller matrices are given below.B1- Controller (LQG/LTR)q = 1.0e5;=R = 1.0e5;= 1.5e2;—1.811e — 2 1.0— 7.277e— 1 9.480Afb—4.460e— 2 00—7.135e — 1 —9.199( —1.8110e — 2I —1.0165e—3Bfb j —4.4601e — 2I—8.1649e — 4Cfb= {0.7126 9.19933.Oe — 2 0.0 0.0 0.0— 0.0 3.Oe —4 0.0 0.0— 0.0 0.0 1.Oe — 2 0.00.0 0.0 0.0 1.Oe — 40.01.053e+11.0—1.022e + 1—1.811e— 21.029—4.460e— 2—1.0011.0002 10.2202 };247C1- Controller (LQC)0.0 0.01.Oe—2 0.00.0 1.Oe—10.0 0.01.Oe+9 0.0 1.— 0.0 1.Oe+9j’r—1.2363e 3I—2.3062e — 4Afc I 0.0L 1.5779e—5r 5.5654I 3.3292e — 3Bfc= I 0.0L 0.01.0—1.6433e —30.01.4645e — 30.0 0.02.8471e— 5 7.3920e — 3—4.4732e — 3 1.0—4.1623e— 5 —8.2105e — 3Cf= {1.5779e—5 1.4646e—3 —3.1623e — 5 —8.2105e — 3 };1.Oe — 10.00.00.0R=1.Oe+8;1.Oe+9— 0.00.00.00.00.00.01.Oe — 30.00.00.01.Oe — 10.0 0.01.Oe+7 0.00.0 1.Oe+10.0 0.00.00.01.Oe — 5248APPENDIX VI: LABORATORY TEST SETUPThe experimental setup used three step-motors with corresponding translatormodules and two optical potentiometers. The motors consist of permanent magnetrotors while the stators contain a stack of teeth with several pairs of field windings.The windings can be switched on and off in sequence to produce electromagneticpole pairs that cause the rotors to move in increaments. The sequential switchingof the windings is accomplished by a translator module. The translator module haslogic circuitry to interpret a pulse train and “translate” it into the correspondingswitching sequence for stator field windings (on/off/reverse state for each phaseof the stator). The details of the motors and translators are well documented inreference [110]. Relevant mass, geometry, and other system parameters used in theanalysis are listed below:Mass of Carriage = 70.43 kgMass of inplane traverse = 2.23 kgMass of the pendulum = 0.1108 kgDeployment/Retrieval reel diameter = 2 x 10_2 mPulley diameter = 4.4 x 10—2 mMaximum tether length = 2.25 mMaximum offset motion in X — Y plane = (± 0.7 m) x (± 0.7 m)Number of pulses required to move the motors = 200 pulses/revA pair of Softpot optical shaft encoders (Si series, U.S. Digital Corp.) isused to measure the angular deviation of the tether from the vertical position. The249Softpot is available with ball bearings for motion control applications or torque-loaded to feel like a potentiometer for front panel manual interface. Characteristicfeatures of the Softpot are given below:• 2-channel quadrature, TTL (Transistor — Transistor Logic) squarewave output;• 3rd channel index option;• tracking capability — 0 to 10,000 RPM;• ball bearing option tracks to 10,000 RPM;•— 40 to + 100°C operating temperature;• single + 5v supply;• 100 to 1024 cycles/rev;• small size;• low cost.250

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