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A study of the flexible space platform based deployable manipulator Marom, Itshak 1993

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A STUDY OF THE FLEXIBLE SPACE PLATFORMBASED DEPLOYABLE MANIPULATORITSHAK MAROMB.Sc., Technion - Israel Institute of Technology, 1974M.Sc., Technion - Israel Institute of Technology, 1977A 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 requ'red standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1993© Itshak Marom, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of my depart-ment or by his or her representatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without my written permission.(Signature)Department of Mechanical EngineeringThe University of British ColumbiaVancouver, B.C. CANADADate: JulOf 1993ABSTRACTA relatively general formulation, based on Lagrange's approach, for studyingthe nonlinear dynamics and control of a flexible, space based, Mobile DeployableManipulator (MDM) has been developed. The formulation has the following distinc-tive features:(i) it is applicable to a two-link deployable manipulator translating along a flex-ible space platform in any desired orbit;(ii) the revolute and the prismatic joints are flexible with finite gear ratio androtor inertia;(iii) the nonlinear, nonautonomous and coupled equations of motion are presentedin a compact form which provide insight into the intricate dynamical interac-tions and help achieve a highly efficient dynamic simulation.Validity of the formulation and the computer code were established through checkson conservation of the system energy, and comparison with test-cases.Results of the dynamical analysis clearly show that under critical combinationsof parameters, the system can become unstable.An inverse kinematic technique, accounting for the linear and angular displace-ments of the the flexible platform, has been developed to meet the desired positionand orientation requirement of the end effector with respect to the chosen referenceframe.A nonlinear control strategy, based on the Feedback Linearization Technique(FLT) has been developed to ensure accurate tracking of a desired trajectory in pres-ence of the system libration as well as the base translation and vibrations. The con-troller also ensures desired attitude motion in presence of the manipulator's transla-tional and slewing maneuvers. The closed loop simulation results show that a desireddegree of system stability and high tracking accuracy can be achieved.111TABLE OF CONTENTSABSTRACT ^LIST OF SYMBOLS ^  viiiLIST OF FIGURES  xiiiLIST OF TABLES ^  xixACKNOWLEDGEMENT  xx1. INTRODUCTION ^ 11.1 Preliminary Remarks 11.2 A Brief Survey of the Relevant Literature ^ 31.2.1^Robotics^  31.2.2^Flexible structures in space ^ 41.2.3^Control of flexible manipulators 61.3 Scope of the Proposed Study ^ 81.3.1^Introductory comments 81.3.2^Present investigation ^ 122. FORMULATION OF THE PROBLEM ^ 14^2.1^System Modelling ^  142.2^Modal Discretization  162.3^Kinematics ^  192.3.1 Reference frames ^  19iv2.3.2 Position vectors and transformations ^ 21^2.4^Kinetics ^  232.4.1 The kinetic energy ^  232.4.2 The potential energy  252.5^Equations of Motion ^  273. OPEN LOOP STUDY 313.1 Numerical Approach ^ 313.2 Formulation and Program Verification ^ 343.2.1^Total energy conservation 343.2.2^A comparison with a particular case ^ 413.2.3^Closed-form solution ^ 413.3 Open Loop System Dynamics 523.3.1^Numerical data used in the simulation ^ 523.3.2^MDM undamped response for the rigid platform case 533.3.3^MDM on a flexible platform ^ 623.3.4^Damped response for the flexible MDM ^ 723.3.5^Flexible system response to open loop torques ^ 773.4 Summary  894. DESIRED TASK DEFINITION ^  904.1^MDM Kinematic Equations  914.2^Inverse Kinematics for Point Tracking ^  954.3^Matching of the End Effector Position and Orientation^. ^ 1004.4^Task with respect to the orbit ^  1024.5^Parametric Study ^  1034.5.1 Trajectory tracking for the rigid system ^ 1034.5.2 Trajectory tracking for the flexible system ^ 1074.6^Summary ^  1175. CLOSED LOOP STUDY ^  1185.1^Preliminary Remarks  1185.2^Nonlinear Control ^  1185.2.1 Computed torque technique ^  1195.2.2 Joints stiffness control  1275.3^Controlled System Study ^  1285.3.1 Stationkeeping  1325.3.2 Trajectory tracking ^  136^5.3.3 Special cases   1415.4^Summary ^  1506. CLOSING REMARKS ^ 1526.1^Concluding Comments  152vi6.2^Original Contributions ^  1556.3^Recommendations for Future Study ^  156BIBLIOGRAPHY^  158APPENDICESI^MODE SHAPES ^  165II^THE SYSTEM INERTIA MATRIX^  167III MASS MATRIX [M] AND THE NONLINEAR VECTOR {N} ^ 169viiLIST OF SYMBOLSposition vector locating the (i + 1)t1 frame w.r.t. the ith frame-(10^shift in the system center of massdeployment generalized coordinatea5d^required deploymentEcij^transformation matrix from the jth frame to the (j — 1)th frameci^damping coefficient associated with the ith body generalized coordinatedmi^mass element in the ith bodycross-section dimension of the platformd4^cross-section dimension of the armDi^damping generalized force in the ith bodytotal energy of the systemEJ^flexural rigidity of the platformF0^system frameinertial coordinate frameFp^platform truss fixed reference frameFr^orbital coordinate frameith body fixed coordinate frameGM^earth gravitational constant[1-1]^homogeneous transformationangular momentum per unit mass w.r.t. the inertial framehi^location of the manipulator base along the platformviiih2^distance between joint 1 and the center line of the platform[I]^system instantaneous moment of inertia w.r.t. the system frame[-ii]^moment of inertia matrix of the ith body w.r.t. the system frameA. Aunit vectors along the X,Y,Z coordinates, respectivelyk3, k5^equivalent torsional stiffness of joints 1 and 2, respectivelyKm3,Km5^torque motor gains for joint 1 and joint 2, respectively[Kr], [Ku]^position and velocity feedback gain matrices, respectivelyirc^direction cosine vector for fc w.r.t. the system framesemi-major length of the platform14^length of arm 1total mass of the systemmi^mass of the ith body[NI]^system mass matrixnonlinear force vector; Eq. (2.37)number of bodies in the systemnom^number of flexible modesn3, n5^gear ratios of joints 1 and 2, respectivelyqd^required generalized coordinate vectorgeneralized coordinate vectorqi^generalized coordinate associated with the ith mode?2-^generalized force vectorrc^radius of the orbitRi^position vector of a mass element w.r.t. the inertial frameixposition vector of a mass element w.r.t. the body framer5^effective linear actuator radiustimegeneralized control torque exerted on the ith bodysystem kinetic energyTi3, Ti5^kinetic energies of joints 1 and 2, respectivelysystem potential energypotential energy stored in the ith bodyUg^gravitational potential energyUe^elastic potential energythe unit matirxith body fixed coordinate frameal^local slope of the deformed platformrotation angle of the ith body fixed coordinate frame/33^rotation of joint 1 w.r.t. the base04^slew of arm 1 w.r.t. the base134c1^required slew/35^rotation of joint 2 w.r.t. the armso^platform deflection at midpoint61^platform transverse deflection8(—)^denotes change in (—)orbit eccentricitytrue anomalywiil-Tdamping ratio in the i1 degree  of freedomposition vector of a mass element w.r.t. the ith body framevector containing the mode shapes of the platformsystem rotation about the orbit normal Zr; system librationrotation of the ith body fixed frame w.r.t. the (i — 1)th frameeigenvalues of the ith generalized coordinatesystem frame angular velocityorbital periodxiABBREVIATIONS C (—)^cos (—)c.m.^center of massd.o.f.^degree(s) of freedomdeg^degreesFLT^Feedback Linearization TechniqueHz^HertzI.C.^Initial ConditionJoulekg^kilogrammeterMDM^Mobile Deployable ManipulatorMSS^Mobile Servicing systemNm^Newton-meterr.h.s.^right hand sidesecondsSPDM^Sepecial Purpose Dexturous ManipulatorS (—)^sin (—)w.r.t.^with respect toDot ( • ) and prime ( / ) represent differentiations with respect to time and spatialcoordinate, respectively. Subscripts 0 and d indicate initial conditions and requiredvalues, respectively. Subscript i indicates the ith body or the ith mode. SuperscriptT represents transpose of a matrix. Overbar ( ) represents a vector.xiiLIST OF FIGURES1-1^The proposed Space Station Freedom configuration as of 1988. It hasundergone several revisions and is in the process of further change.^. . 21-2^A schematic diagram of the Space Station based Mobile ServicingSystem (MSS)^  91-3^A schematic diagram of the proposed Space Station based MobileDeployable Manipulator (MDM)^  102-1^A schematic diagram showing exploded view of the motor-gearassemblies driving the arms. ^  152-2^Reference frames and position vectors for the MDM system. Theframes are designated as Fi (i=r,0,1,...,6)^ 173-1^The MDM dynamics simulation flowchart  333-2^The MDM free joint response and energy exchange for a rigidplatform subjected to an initial pitch disturbance of 0=0.1 rad:(a) system response; (b) energy exchange ^  373-3^The MDM system free joint response and energy exchange for an initialslew angle P4=0.5 rad: (a) system response; (b) energy exchange.^. 393-4^^Response and energy exchange for the MDM, with free joints, whensubjected to a combined initial conditions of #4=1.0 rad, anda5=0.01 m: (a) system response; (b) energy exchange ^403-5^Simplified configuration of the MDM system ^  423-6^A comparison between the numerical and analytical solutions fordamped response of the simplified MDM system:(a) platform response; (b) link response^  51^3-7^MDM free joints undamped response when supported by a rigidplatform and subjected to various initial conditions: (a) '34 = 0.1 rad;(b) a5=0.5 m; (c) = 0.1 rad; (d) 04 = —1.57 rad. ^ 563-8^Undamped response of the MDM, with free joints and supported by arigid platform, when the base is located at h1=30 m from the platformc.m. The initial conditions are: (a.) 04 = 0.1 rad; (b) a5=0.5 m;(c) = 0.1 rad; (d) 04 = —1.57 rad.^ 583-9^Undamped free response of the system during translational maneuverfrom h1 = 0 at a constant velocity of 0.01 m/s. The deployable armis held fixed at a5 = 10 m. Initial conditions are: (a) /34 = 0.1 rad;(b) 0=0.1 rad, 04 = —0.1 rad; (c)Ik 0.1 rad; (d)^= —1.57 rad. . . 613-10 Effect of the platform flexibility on the free undamped systemresponse. The payload is initially located at a5 = 0 and disconnectedfrom the linear actuator gear (k5 = 0). The slewing arm isinitially oriented at /34 = 0.1 rad^  633-11 Effect of the platform excitation on the system response. Thedeployable arm is held fixed at a5=10 m. The initial conditions arefi4 = 1.5 rad and q=0.01 m^  653-12 Effect of a larger platform excitation of qi = 0.1 m on the free, undampedresponse of the system. The deployable arm is held fixed at a5 = 10 m,while the slewing arm is initially oriented at ,84 = 1.0 rad. ^ 673-13 Free undamped response of the system with the deployable d.o.f.frozen at a5=10 m. Initially 04 = 1.5 rad and qi=0.1 m^ 69xiv3-14 Free undamped system response with the deployable arm held fixed ata5=10 m. Initial orientation of the slewing arm is 04 = 1.5707 rad.The platform is excited in the first mode with q=0.1 m ^ 703-15 Effect of the initial orientation of the slewing arm on the system responsewith the deployable arm fixed at a5=10 m. Initially 04 = 0 and q=0.1 m. 713-16 System open loop damped response with the deployment d.o.f. frozenat a5 = 10 m. The platform is initially excited in the first-modewith qi = 0.1 m ^  733-17 Damped response of the system with the deployable arm fixed ata5 = 10 m, the MDM base held at hi = 60 m, and the slewing arminitially located at 04 = 1.0 rad. The platform is set vibrating withan initial disturbance of qi = 0.1 m ^  753-18 Effect of the MDM's translational maneuver, at —0.02 m/s on the flexibleplatform, from the initial location of hi = 60 m. The deployable armis fixed at a5 = 10 m. The initial conditions are: 04 = 0; q = 0.1 m. . 763-19 Response of the MDM system to a constant torque of 1 Nm appliedat joint 1. The deployable d.o.f. is frozen at as = 10 m, themanipulator is located at the midpoint of the flexible platform•(hi = 0). The platform flexibility is represented by the first mode.^. 793-20 Open loop response of the MDM with a constant torque T3 --= 1 Nmapplied at joint 1. The deployable d.o.f. is frozen at a5 = 10 m. Thebase is located at hi = 60 m and the platform flexibility is representedby the first mode. The initial arm position is 04 = 1.0 rad^ 803-21 System response to a constant torque T3 := 1 Nm applied at joint 1.XVThe deployable arm is held at a5 = 10 m. The manipulator is locatedat the midpoint of the platform and the platform flexibility isrepresented by the first two modes    823-22 System response to open loop torques T3 = 1 Nm, T5 = -1 Nm,assuming one flexible mode. The MDM base is located at h1 = 60 m.The initial conditions are: 134 = 1 rad; a5 = 10 m. ^  843-23 System response to open loop torques T3 = 1 Nm, T5 = -1 Nm,assuming two flexible modes. The MDM base is located at h1 = 60 m.The initial conditions are: 04 = 1 rad; a5 = 10 m. ^  853-24 System response to open loop torques T3 = 1 Nm, T5 = -1 Nm,assuming three flexible modes. The MDM base is located at h1 = 60 mwith the system initial conditions as: /34 = 1 rad; a5 = 10 m^ 863-25 System response to open loop torques 7'3 = 1 Nm, T5 = -1 Nm,assuming four flexible modes. The MDM base is located at h1 = 60 mand the initial conditions are: 04 = 1 rad; a5 = 10 m    873-26 System response to open loop torques 7'3 = 1 Nm, T5 = -1 Nm,assuming five flexible modes. The MDM base is located at h1 = 60 mand the initial conditions are: Al = 1 rad; a5 = 10 m ^ 88^4-1^Reference coordinate frames established for the inverse kinematicsstudy of the MDM system. ^  924-2^Kinematic relationships  994-3^Time histories for the required positions of the degrees offreedom with a constant rate of base motion^  1054-4^Time histories of the required degrees of freedom, for the rigidxvisystem, with an accelerating base and a specified trajectory. ^ 1064-5^^Time history of the required degrees of freedom for the flexiblesystem with an accelerating base and prescribed trajectory.Initial modal excitations are: qi = 1 m; q2 = 0 m ^  1084-6^Time history of the required degrees of freedom for the flexiblesystem with an accelerating base and prescribed trajectory.Initial modal excitations are: qi = q2 = 1 m^  1094-7^Required time histories of the generalized coordinates with theplatform excitation of qi = q2 = 0.1 m. The manipulator is located 50 mfrom the center (h1=50 m) ^  1114-8^The effect of increased platform excitation of qi = q2 = 0.5 m on therequired time histories of the generalized coordinates AI and a5to track a desired trajectory. The base is located at h1 = 50 m ^ 1124-9^Required slew arm position, velocity and acceleration for theflexible system, with initial excitation of qi = q2 = 0.5 m, to trackthe specified trajectory. The manipulator is positioned 50 mfrom the center of the platform ^  1144-10 Required deployable arm position, velocity and acceleration inabsence of the platform excitation (qi = q2 = 0). The base isinitially located at h1 =50 m ^  1154-11^Effect of the platform excitation of qi = q2 = 0.5 on the required position,velocity and acceleration time histories of the deployment generalizedcoordinate a5. The manipulator starts to translate from h1=50 m. . . 1165-1^Closed loop block diagram for the slew degree of freedom^ 125xvii5-2^Closed loop block diagram for the deployment degree of freedom.^. . 1265-3^MDM closed loop simulation flow chart^  1315-4^Closed loop response during stationkeeping for the rigid platformcase with an initial disturbance of 1,b = 0.02 rad. ^  1335-5^^Closed loop response for point-tracking (stationkeeping) of the payload,supported by the manipulator, located on the flexible platform.The manipulator is at the center of the platform and arm 2 isdeployed 10 m. The initial disturbance is 0.017 rad in pitch. ^ 1355-6^Closed loop response during the trajectory tracking, with the baseheld fixed and arm 2 initially deployed 10 m along the platform. . . . 1375-7^Closed loop response during the trajectory tracking for the MDMwith the arm perpendicular to the platform ^  1405-8^Closed loop response for trajectory tracking with initial position errors. 1425-9^Controlled response for trajectory tracking with the base motion at0.5 m/s and the initial condition of = 0.1 m imposed on the platform. 1445-10 FLT controlled response for the trajectory tracking with theplatform along the local horizontal and the base translation at 0.5 m/s.The initial condition is 41 = 0.1 m ^  1465-11 Controlled response during the trajectory tracking for the localhorizontal configuration of the platform. The initial condition is= 0.5 m for the platform. The initial position errors are: 2 m indeployment; and 0.3 rad in slew. ^  1485-12 Control effort time histories for the system described in Figure 5-11:(a) gear ratio n3 = 1; (b) gear ratio n3 = 10^  149xviiiLIST OF TABLES^3-1^Typical numerical values for the MDM system parameters. ^ 53^3-2^MDM configuration and typical initial conditions.   545-1^MDM closed loop velocity and position feedback gains. ^ 129xixACKNOWLEDGEMENTSI would like to thank Prof. V.J. Modi for his useful guidance throughout thisresearch project.Thanks are also due to colleagues Dr. Alfred Ng, Mr. Satyabrata Pradhan andMr. Anant Grewal who facilitated the progress of this work by sharing their technicalknow-how.Special appreciation is extended to my family for their understanding and support.The investigation reported here was supported by the Natural Sciences and En-gineering Research Council of Canada, Grant No. A-2181; and the Networks ofCenters of Excellence Program, Institute of Robotics and Intelligent Systems, GrantNo. IRIS/C-8, 5-55380.XX1. INTRODUCTION1.1 Preliminary RemarksThe proposed Space Station "Freedom", as shown in Figure 1-1, is intended tobe operational in the late nineties. Spanning over 110 meters in its permanentlymanned configuration, and carrying solar panels extending to 66 meters tip-to-tip,the Space Station is a highly flexible structure. The primary design requirements ofthe space station is to serve as a permanent operational base for scientific explorationsas well as processing and manufacture in the favourable microgravity and high vacuumenvironment. It will also be used as a platform for satellite launch and repair, as wellas assembly of space structures.Canada's contribution will be through the Mobile Servicing System (MSS), a com-plex robotic manipulator, which will be used to assemble, service and maintain theentire station. The MSS comprises the maintenance depot and the special PurposeDexterous Manipulator (SPDM). The SPDM, when attached to the station manipu-lator, will give the MSS an additional precision capability for tasks requiring a higherdegree of accuracy.The normal operation of the Space Station is designed to be by remote or teleop-eration of several robotic manipulators from a central control post, where the MobileServicing System (MSS) will operate as the main handling tool. Forming an inte-gral part of the space station, the MSS may have one or two arms supported by amobile base, which traverses the power boom through translational and rotationalmaneuvers.The Space Station based MSS is inherently a highly complex and variable inertiasystem. Structural and joints flexibilities further accentuate the problem. The inter-10eacc0cnUiacnFigure 1-1^The proposed Space Station Freedom configuration as of 1988. It hasundergone several revisions and is in the process of further change.2active dynamics involving the space station and the MSS is challenging as it involvesrelative slewing and translational motion of the flexible manipulator on a highly flexi-ble platform. Both the structural elasticity and joint compliance may adversely affectits performance in following a desired trajectory as well as assuring precise orientationof the payload.The study of an orbiting two-link deployable manipulator with elastic and dissi-pative joints, operating on a highly flexible space station and carrying an arbitraryshaped payload represents a challenge that has never been encountered before. Thisclass of problems is particular to space-based systems and seldom, if ever, encounteredby robots operating on the ground. They normally tend to be rigid and operate fromfixed, rigid supporting platforms.1.2 A Brief Survey of the Relevant LiteratureThe literature review focuses on the topics that are considered relevant to thestudy in hand as follows:- robotics in general, with emphasis on deployable manipulators;- dynamics of flexible structures in space;- control of space station based flexible manipulators.1.2.1 RoboticsA ground based robot is a general purpose, computer controlled, manipulatorconsisting of several rigid links connected in series by revolute or prismatic joints.Each link is connected, at the most, to two others; and a closed chain is usuallyavoided. The joints normally have only one degree of freedom. A revolute jointpermits rotation about an axis, while a prismatic joint allows sliding along an axis3with no rotation. The manipulator is attached to a fixed, rigid base; while its end isfree and equipped with a grip to manipulate objects, or to perform assembly tasks.In general, motion of the joints leads to link motion resulting in to spatial scanningby the grip.Unlike the ground based manipulators which are thoroughly described in thetextbooks by Paul [1] and Fu [2], space manipulators tend to be flexible and oftenoperate from a mobile base as discussed by Chan [3]. Recently, in order to meet thedemands of higher speed and efficiency, the new generation of fast robots have beendeveloped with lightweight, low inertia arms which are usually flexible [4]. In general,flexibility arises from two sources: elasticity of the structure and compliance of themotor and the transmission unit at the joint. The structural and joint flexibilities ofthe manipulator can cause end-effector oscillations, thus limiting its ability to performa given task with a desired degree of accuracy.Dynamic analysis of a flexible robot is complicated by coupling between the non-linear rigid body motion and elastic displacements of the deformed structure [5].Derivation of the coupled, nonlinear governing equations of motion is enormouslytime consuming, and efforts have been made to simplify the process. Book [6] devel-oped a recursive Lagrangian approach to generate full nonlinear dynamics of multilinkflexible manipulators using homogeneous transformation matrices. Centinkunt et al.[7] have proposed the use of symbolic manipulation programs to overcome algebraiccomplexities, and thus facilitate the formulation.1.2.2 Flexible structures in spaceWith the advent of large space structures, flexibility has become increasingly im-portant, and accurate modelling of the elastic behavior is fundamental to the dynam-4ics and control studies, as discussed by Nurre [8]. The problem is further accentuatedby the demanding performance requirements associated with these systems.A comprehensive literature review on the dynamics of flexible satellites was givenby Modi [9]. In treating flexibility, the continuous system is described in terms ofdiscrete and distributed coordinates. The resultant governing equations of motionare transformed into a set of ordinary differential equations, using assumed modemethods with time dependent generalized coordinates [10-12].Dynamic simulation codes have been developed for multibody systems consistingof rigid and flexible components. However, these programs have certain limitationsas pointed out by Singh et al. [13]. Kane et al. [14] have investigated dynamicsof a moving elastic cantilever beam, and observed conventional methods to predicterroneous divergence under extreme conditions.Of particular interest is the class of spacecraft with deployment of flexible ap-pendages from the central body. This class of systems involve transient inertia dy-namics, similar to that experienced by the space station during the MSS maneuvers.Reddy et al. [15], and Bainum and Kumar [16] have provided considerable insight intothe behavior of complex large space system, by modelling basic structural elementssuch as flexible beams and plates in orbit.Systems with deployable appendages have been analyzed, with varying levelsof simplifying assumptions, by Lang and Honeycutt [17], Cloutier [18], Hughes [19],Sellpan, Bainum [20] and others. These investigators have treated flexible members aspoint masses or rigid bodies. A general formulation for studying librational dynamicsof a spacecraft, with a rigid central body, deploying an arbitrary number of flexibleappendages was presented by Modi and Ibrahim [21]. The formulation also accountedfor a shift in the centre of mass.5Slewing appendages impose transient inertias in the study of system dynamics.Hablani [22] derived equations of motion for a chain of hinge-connected bodies inthe gravitational field. Conway [23] included dissipative elastic joints in the systemwhich was subjected to arbitrary external forces. Both the studies considered onlyrigid bodies and are based on the Newton-Euler formulation procedure.The Canadarm on the space shuttle represents a flexible manipulator systemcapable of relatively fast and large slewing maneuvers. The associated literature [24-30] would be of some value in understanding the dynamics of a large scale manipulator,although it is smaller than the MSS and fixed w.r.t. the space shuttle, which isassumed to be rigid unlike the Space Station.Meirovitch and Quinn [31] derived the equations of motion for a maneuveringflexible spacecraft using a perturbation approach. Longman et al. [32] addressed theproblem of slew induced reaction moment on the librational response, and modifiedthe joint angle commmands, through the kinematic equations, to position the end-effector of a rigid remote manipulator at its desired target.1.2.3 Control of flexible manipulatorsWith the high precision positioning critical to the success of certain missions, theproblem of maneuvering a flexible spacecraft while suppressing the induced vibrationsand the attitude librations is becoming increasingly important. Hale et al. [33]have discussed simultaneous optimization of structural and control parameters formaneuvering flexible spacecraft.Nonlinear feedback control was explored by Carrington [34], while Yuan [35] hasstudied the robust beam-pointing and attitude control of a flexible spacecraft. Theresearch related to the control of flexible orbiting structures intensified with two ex-6periments proposed by NASA: Spacecraft COntrol Laboratory Experiment (SCOLE)and COntrol of Flexible Spacecraft (COFS). Recently these have been replaced byControl Structure Interaction (CSI) studies, however, the scope of the project remainsessentially the same.Control of flexible robots, considering the elasticity at the joints, has been inves-tigated by many researchers [36-41]. Garcia [36] presents a formulation for modellinga single-link flexible beam. It undergoes slew motion with an actively controlledpinned end while the other end of the beam is kept free. Position and velocity sensorsprovided input for proportional feedback control about the slew axis. The motor char-acteristics, gear ratio and the position feedback determined the equivalent torsionalspring constant (servo-stiffness). The investigation concluded that, for moderate orlow values of the ratio of the servo stiffness to beam flexibility, it is necessary toconsider effects of the servo-drives on the dynamics of the flexible beam.In studying control of flexible space station based manipulators, Modi et al. [42]considered a mobile, two link manipulator with flexible joints. The inverse controltechnique was used to achieve high tracking accuracy of the end effector in the pres-ence of maneuvers induced as well as other external and internal disturbances. Twodifferent schemes based on the Quasi-Closed Loop Control (QCLC) and Quasi-OpenLoop Control (QOLC) were developed. Though quite effective, authors suggestedmore attention be given towards the robustness issue.This brief literature survey provides a synopsis of the research activities in thefield of robotics in space, with particular application to the proposed MSS. The mainconclusion from the survey is that more accurate mathematical models and simulationprocedures are needed to successfully perform prescribed robotic tasks in space.71.3 Scope of the Proposed Study1.3.1 Introductory commentsAll the present and proposed space based manipulators are of revolute type,usually consisting of two slewing arms on a rotating and translating base, with anend-effector attached to the tip of the second link, similar to the MSS, as shown inFigure 1-2.To carry out a typical task, such as following a desired trajectory, is a challengeeven for a rigid, fixed, ground-based manipulator. In its utmost generality, a space-based MSS presents a rather formidable problem in mechanics. Dynamics and controlof a flexible Space Station, supporting a flexible mobile manipulator, represents aclass of problems never encountered before. Both the structural elasticity and jointcompliance may adversly affect the performance in following a desired trajectory aswell as the positioning and orientation of the payload.Because of the complex nature of the space based revolute manipulator, it hasbeen impossible to present a closed form solution, even for a simplified model, whichmay help gain insight into the intricate dynamics involved. To overcome some of theproblems encountered with revolute type manipulators, a two arm Mobile DeployableManipulator (MDM) system is proposed here for space application, as schematicallyshown in Figure 1-3. It appears to have several inherent advantages which may beclassified as: software related benefits; and hardware simplification.Software Related Advantages — The kinematics and the inverse kinematics are less complicated in the deployabletype manipulator.8PayloadArm 2OrbitJoint 2\Joint 1Mobile Base....-,......,.., ...,■ ■■■\A410.str ■tle PlatformFigure 1 -2^A schematic diagram of the Space Station based Mobile ServicingSystem (MSS).9PayloadOrbitArm 2PrismaticJoint 2Arm4400114-1°°Mob„e Base4•4119eVPlatformRevoluteJoint 1■■Figure 1 -3^A schematic diagram of the proposed Space Station based MobileDeployable Manipulator (MDM).10- A unique solution exists for each of the degrees of freedom which define the desiredposition and orientation of the payload. In revolute type manipulators, there isusually more than one possibility to attain a desired position, thus requiring adecision based optimization algorithm.- In the system dynamics, there is no inertia coupling between the links of thedeployable manipulator. This results in simpler equations of motion demandingrelatively less effort in their derivation and, of course, integration.These advantages lead to a significant decrease in the computational effort andtime. Now one can obtain real time output even with a relatively small computer,thus improving the possibility of an online controller.Hardware Advantages- In the deployable manipulator, the actuators (including the prismatic joint) arelocated as one unit at the manipulator's base, thus limiting the inertia of theslewing arm only to that needed for construction and stiffness. In the revolutetype manipulator, the torque motor for the second link is located at the tip of thefirst link thus increasing the inertia. As a result a larger torque motor is requiredat the base.- In the deployable manipulator there are no singular positions, as in the revolutetype when the links are 180° apart. To compensate for the singular or close tosingular position, a larger torque motor is needed.A possible disadvantage of the MDM is the limited work space. However, thiscan be compensated quite readily by a moving base.11SYSTEMDYNAMICS1.3.2^Present investigationThe research project aims at studying dynamics and control of a typical SpaceStation based MDM system as described in the following block diagram:DESIREDTASKINVERSEKINEMATICSThe emphasis throughout is on the development of a simulation tool, applicable toa large class of space based MDM, leading to appreciation of the system performancein the presence of structural flexibility and disturbances.In Chapter 2 a comprehensive general formulation for studying the dynamics ofthis class of systems is discussed. The discretization of elastic deflections using theassumed mode method is described and the governing equations of motion obtainedusing the Lagrangian procedure.The formulation is applicable to a flexible space based deployable manipulatorin any desired orbit. It allows for slewing and deployment of an arbitrary payload,as well as base translation along the flexible platform, accounting for the transientsystem inertia and shift in the center of mass. The nonlinear, nonautonomous andcoupled equations of motion are not amenable to any closed-form solution.Chapter 3 discusses development of the numerical code for integration of theequations of motion. The emphasis is on a time efficient simulation. Validity of theformulation and the computer code is first established by checking conservation of the12system's total energy, and through comparison with results for particular configura-tions. A closed-form solution for a simplified case, using the variation of parametersmethod, not only provides better appreciation of the system behaviour but also helpsin checking the numerical code. The chapter ends with a study of coupling effectsbetween the flexible platform and the manipulator maneuvers.In Chapter 4, the inverse kinematic approach is developed. The objective isto express the required MDM degrees of freedom in terms of the desired positionand orientation of the end effector w.r.t. the chosen reference frame. To meet therequirements of both the desired position and orientation, a coordinated moving basestrategy, which takes into account the local deflection and slope of the platform, issuggested.The ultimate objective is to achieve minimum tracking error in the presence ofspace station maneuvers and vibrations. Conversely, to maintain position and atti-tude pointing within an acceptable limit, in presence of the manipulator's slewing anddeployment maneuvers. With this in mind, Chapter 5 proposes a nonlinear controlstrategy based on the Feedback Linearization Technique (FLT), where the unknowndesired values are obtained by real time numerical differentiation. To eliminate theneed for real time numerical differentiation, a control approach based on variablestiffness at the joints is suggested.The concluding chapter summarizes more important results and presents recom-mendations for future studies.132. FORMULATION OF THE PROBLEMThis chapter presents the development of a relatively general formulation partic-ularly suitable for studying dynamics and control of flexible space based deployabletype manipulators.2.1 System ModellingA space platform based two link deployable manipulator is considered in thisstudy. The slewing upper arm (first link), and the deployable lower arm (second link)carrying the payload, are activated through a system of motors and gears. Note, boththe arms go through the same slewing maneuver, however the deployment is confinedto arm 2 assembly. The manipulator is permitted to translate along the platform,orbiting around the earth.The space platform supporting the manipulator is taken to be in an arbitrary orbitaround the earth. The platform is considered to be a flexible beam type structure,and the momentum wheel(s) aboard provide the necessary control torque Po about itsprincipal axis. The system libration, platform vibration, as well as the manipulator'smaneuvers are confined to the orbital plane.The slewing arm is attached via a flexible and dissipative gear system, with ration3, to the joint 1 torque-motor-rotor 13, as schematicaly shown in Figure 2-1. Itundergoes slewing maneuvers 164 w.r.t. the manipulator base. The revolute jointflexibility is modelled as a torsional spring with an equivalent stiffness k3.The second arm assembly is deployed by an amount a5 w.r.t. arm 1. It is con-nected via a flexible and dissipative linear gear assembly, with ratio n5 and equivalentradius r5, to a torque motor attached to the slewing arm, as shown in Figure 2-1.The prismatic joint's flexibility is modelled by a torsional spring with an equivalent14Arm 2payload)^Arm 1Torque Motor StatorMobile BaseFlexible Platform#41.,77/7RotorGears135n5^Linear ActuatorFigure 2-1^A schematic diagram showing exploded view of the motor-gear as-semblies driving the arms.15stiffness of k5. The ratio of the joint flexibility to that of the arm is considered high,i.e. the arms are taken to be rigid in this study. The end effector, with the arbitrarilyshaped payload, forms an integral parts of arm 2, and is also treated as rigid.2.2 Modal DiscretizationThe flexible platform is considered as a free-free Euler-Bernoulli beam with length2/1 and square cross-section of dimension c/1. The platform's flexural deformation Siis discretized using the admissible functions .1.(xj.) satisfying geometric and naturalboundary conditions, in conjunction with generalized time dependent coordinates41(t) associated with 61. This procedure approximates elastic deformations by aseries of spatial functions,63. (x, t) = OT(xi)i(t), (2.1)where Si(x, t) is the transverse deformation w.r.t. the platform body fixed referenceframe F1 located at the c.m. of the undeformed platform, as shown in Figure 2-2.Here x1 is the coordinate along the longitudinal axis of the platform.The mode shapes are obtained by applying the separation of variables method tosolve the beam partial differential equationa281 + 04EJ^_ 0&2mp(2.2)where:EJ^flexural rigidity of the platform;Mp^mass per unit length of the platform.Considering the mass and inertia of the MDM to be small compared to that ofthe platform [12], and taking into account only the lower natural frequencies (lowacceleration levels), it is possible to neglect the effect of the manipulator's inertia on16YsX5 \X3F5 \I\^;^,'ili x2Y2^. ØF2,3,4\ 40/.."'I 1Idmi 1(4Fr , Fo• ,,°4041441/4,arcE and bodiesI -- Inertial frameLegend for framer -- Orbital frame0-- System1 -- Platform2— Base3 — Joint 14-- Arm 15— Joint 2^I. 6 -- Arm 2 assemblyXIFigure 2-2^Reference frames and position vectors for the MDM system. Theframes are designated as F (i=r,0,1,...,6).Orbit17the boundary conditions.Introduction of the appropriate boundary conditions, i.e. zero moment and shearat the ends of the platform:^(; ^E Jo, si(x , = OX 1=±11;^E J^= 01.1,±11;^(2.3)a x2 ax,  yield a trancendental equation for the spatial frequency parameter 'p',^cos(2pi li)cosh(2pi 11) = 1,^ (2.4)with the modes given by [46](x ) = coshpi(x + ii) cosPi(x^11) — cri[sinhpi(x + ) sinpi(x + )] • (2.5)Here:24 "IPwi = E Jcosh(2pili) — cos(2pili) • =sinh(2pi/i) —where:2/1^length of the platform;platform natural frequency associated with the ith mode.The various modes and the modal integrals are given in Appendix I.The orientation of the MDM base w.r.t the platform, at any location h1 from thecenter of the platform, is determined by the local slope,asi(hl, t)tanai =axi^•(2.6)182.3 KinematicsKnowledge of the system kinematics is the first necessary requirement to formu-late the system equations of motion.2.3.1 Reference framesThe reference frames are carefully chosen in order to establish the state of thesystem at any given instant. This is necessary to undertake the kinetic study.Consider the space platform and the MDM model as shown in Figure 2-2. Theinstantaneous center of mass of the system, C.M., negotiates an arbitrary orbit aboutthe center of force coinciding with the earth's center, where the inertial referenceframe Fr is located. As the system has finite dimension, i.e. it has mass as wellas inertia, it is free to undergo librational (pitch) motion about its c.m. This canbe defined as the rotation of the system frame F0 w.r.t. the orbital frame F.Here X0, Yo are aligned with the principal moments of inertia directions for the entiresystem in its reference equilibrium configuration.For the orbital frame, Xr coincides with the local vertical, Yr is aligned with thelocal horizontal, and Zr is along the orbit normal in accordance with the right handrule. At any instant, the position of Fr is determined by the orbital elements e,rcand O. Here e is the eccentricity of the orbit; rc, the distance from the center of forceto F0 (C.M.); and 61, the true anomaly of the orbit. As the dimensions of the systemare negligible compared to the orbital altitude, the effect of librational, slewing andtranslational motions on the orbit is expected to be negligible [47,48], therefore theorbit can be represented by the classical keplerian relation,h2rc =^GM(1 ecos9) (2.7)19where:the angular momentum per unit mass of the system about F1, dej;orbit eccentricity;true anomaly;rc^radius of the orbit;GM earth's gravitational constant.Attached to each member of the system (platform, base, joint 1, arm 1, joint2, arm 2 assembly) is a body coordinate system helpful in defining relative motionbetween the memebers. Thus F1 is the platform body fixed reference frame located atthe c.m. of the platform and taken to be parallel to the frame Fo. Here X0 coincideswith X1, the longitudinal axis of the platform.For the undeformed platform with the MDM, in absence of deployment (a5-=0),located at the geometric center of the platform (h1=0), the system C.M. coincideswith the platform c.m. In general there is a shift between this two centers of massdue to the MDM maneuvers.The position and orientation of the base frame F2 w.r.t. the platform frame F1is determined by the translation h1, the platform transverse deflection 61(h1, t), andthe local slope al, as shown in Figure 2-2.At joint 1, the rotor body fixed frame F3 rotates by an angle 03 about Z2. Theorientation of the slewing arm body fixed reference frame F4 w.r.t. the base is fig,where X4 is pointed in the direction of the prismatic joint 2. Joint 2 rotor fixedreference frame F5 is rotated through angle /35 w.r.t. X4, about Z4. The end effectorbody fixed reference frame F6 is parallel to F4 and translates through as along X4.The centers of mass of the payload, end effector and arm 2 assembly coincide withthe origin of F6.202.3.2 Position vectors and transformationsThe system and orbital frames were defined with their origins at the system C.M.while the local body fixed reference frames were attached to the members constitutingthe MDM system.The position vector Ri to the mass element dmi in the ith body, w.r.t. the inertialframe, can be expressed as (Figure 2-2)where:fc position vector of the system frame F0 in the inertial reference FT;fi position vector to a mass element in the ith body w.r.t. the system frame Fo.Expanding fri in terms of the body fixed reference frame,= ãj +: H[c Jet + H[c ]pi^i = 1, ...n,^(2.9)j.1where:[c -] rotation transformation matrices from the jth frame to the3^ - 1)th frame;a, j position vector locating the (j 1)th body fixed frame w.r.t. the jth frame;position vector for the mass element dmi w.r.t. the ith body frame Fi;number of bodies in the system.The transformation matrix for the[cd =planar case has the form^[cos 2,b i^— sin Vii^0sin 1&j^cos i,bi,^00^0^1(2.10)where 1P-i is the rotation angle of the body fixed frame Fi w.r.t. the F(i_1)) frame.21When dealing with a flexible body, angle çb can be expressed as the sum of rigid andflexible body rotations,= ai-i +^(2. 11)where:rotation due to bending of the flexible body (i — 1);rigid body rotation.The position vector which locates a mass element in the flexible beam is obtainedby superposing j (the undeformed position) and Si (the transverse deformation),neglecting the fortshortening effect,= +^(2.12)where:= 0.27:(x)(t),^ (2.13)and the -Mx) vector contains the modes of the ith body (Appendix I).The term ao in Eq. (2.9) represents shift in the system c.m., caused by the flexi-bility of the platform coupled with the MDM's translational and slewing maneuvers,1 x-■ x-■ao =^22 2_, mi)(11ci)ai + E(11 ci)^Pidmirn j=i+i^j=1^j=1where:rni^mass of the ith body;rn^total mass of the system.(2.14)Note, imi pidmi = 0 when the body fixed reference frame is located at the c.m. ofthe body.222.4 KineticsApplication of the Lagrangian procedure to formulate the governing equations ofmotion requires evaluation of the total kinetic and potential energies of the system.2.4.1 The kinetic energyThe kinetic energy accounts for the contributions from five major components ofthe system: the platform; the slewing arm 1; the deployable arm 2, end effector andpayload assembly; and the two torque motors at the joints:T^kidmi^Ti52 jm.i^zi = 1,4,6.^(2.15)Here Ti3 and Ti5 are the kinetic energies associated with the motors at joints 1 andjoint 2, respectively; and ki is the inertial velocity of the mass element dm, obtainedby differentiating Eqs. (2.7) and (2.9) with respect to time,(2.16)The orbital velocity of the system, fc, is given by7.•fc = c-r^rc3r, (2.17)and the angular velocity^of the system frame F0 due to the orbital velocity 9 andthe librational rate as= (1.p +O)ko.^ (2.18)Substituting for Ili from Eq. (2.16), the expression for the kinetic energy can bewritten in the form23T =12- I (f:c .72•c)dm + -12- I (C2 x fi) • (S2 x fi)drn -12=^(fi • f:i)dm+ f fic •^x fothn +^(fic • "fi)dm^• (f2 x fi)dm Ti3 Ti5.^(2.19)rnThis is the general expression of the kinetic energy. By looking at each term sep-arately, physical appreciation can be gained and some simplifications may be in-troduced. Note, definition of the c.m. requires im fidm. = 0. Hence the terms:fmf.', • (ft x fi)dm and fm(7c • i:i)dm vanish. The kinetic energy contributions canreadily be identified with the orbital (To) and librational (TL) motions; rotationswith respect to the body frames (Ts); angular momentum with librational coupling(TH); and angular velocities with respect to the body fixed reference frames (Ti3, T35).Thus the kinetic energy expression can be written in the form^T = To + TL Ts +^Ti3 Tj5.^(2.20)The kinetic energy contributions from the various sources can be expressed as follows:1 4 ,To = -2mrc -rc;^ (2.21)-?I = 1 -T-12 [./]52;^ (2.22)23^Ts = E 1 1 (-ziT • i:i)dmi;^ (2.23)i=1^rni3TH =^E (fi X f:i)drrii;^ (2.24)i=1 rni1^•^;^T3-3 = -1.3(0 u^+ 21T.35 = -/5(14+^+ (34 + ij5)2.^(2.26) 2where:(2.25)24[I]^matrix of instantaneous total moment of inertia about the system frame F0(Appendix II);13^joint 1 motor rotor inertia;joint 2 linear actuator rotor inertia;a^angular velocity of the MDM base due to platform deflection;/3^angular velocity of joint 1 torque-motor-rotor w.r.t. the base F2;45^angular velocity of joint 2 linear-actuator-rotor w.r.t. the slewing arm.«1 can be expressed as the rate of change of the local slope in the flexible platform,d o -Tal = -clif-a-x-j7[01 (h1)C01} (2.27)The system inertia diadic [I] w.r.t. the system frame can be obtained from therelation[I] = E f PT • fi)[U] — • iT)Jdmi,i=1where [U] is the unit matrix. Details of the inertia matrix are presented in Appendix2.4.2 The potential energyThe potential energy contribution arises from two sources: position in the inversegravitational field; and the strain energy.The librational response is dominated by the gravitational potential energy Ug,while the vibrations are primarily dependent on the elastic or strain energy, stored inthe flexible platform and the joints,U = Ug +Ue.^ (2.29)(2.28)25The strain energy contributions come from three members,Ue = + U3 + U5 ,where:the strain energy stored in the deflected platform;U3, U5^the strain energies stored in the twisted joints 1 and 2, respectively.Gravitational Potential EnergyThe system potential energy in the gravitational field is given bydUg = —GM-dm.^ (2.30)Substitute R from Eq. (2.8) into Eq. (2.30), expanding according to the Binomialpower series, and ignoring terms of order 1/4 and higher, the expression for thepotential energy can be rewritten as [49]GMm GM fU^trace[I] — 341:c[I]irclg — —^ --{trace[Iwhere irc is the direction cosine vector of fc w.r.t. the system frame,= [coszk — sinik 0].rc(2.31)(2.32)The first term in Eq. (2.31) represents the potential energy due to attraction be-tween two bodies represented as point masses. In the present study this term willbe eliminated, as no generalized coordinate is associated with it. The second termcorresponds to the contribution due to 3-D character of the body, i.e. the inertiaeffect. The third term is the contribution due to the librational motion (microgravityterm).26Elastic Potential EnergyThe platform contribution to the strain energy comes mainly from bending de-formation and can be expressed as= (EJ) 111 [dd2 (1)T2] 2 dxi -d(t).^(2.33)The strain energy due to flexibility of the joints (primarily contributed by torsionaldeformations of the transmission shafts) can be expressed as:1 L fa^\2U3 = —2 A.:3^03,— —) ,n31„a5^ 2.\U5 = —x5k— – /85---)2 r5 n5where:k3, k5 equivalent torsional stiffnesses of joints 1 and 2, respectively;n3, n5 gear ratios at joints 1 and 2, respectively;r5^effective linear actuator radius;/33^joint 1 torque motor rotation w.r.t. the MDM base;slew angle of arm 1 w.r.t. the MDM base;35^prismatic joint 2 torque motor rotation w.r.t. the slewing arms;a5^translation of the deployable arm w.r.t. the frame F4.(2.34)(2.35)2.5 Equations of MotionThe earlier multibody derivations were based on the Newton-Euler approach,which involves physically visualizable quantities represented by vectors. Usually lesstime and effort are required to arrive at a compact and explicit form of the govern-ing equations of motion. This also makes the Newton-Euler method computationally27more attractive. However, the method is useful only when dealing with relatively sim-ple configurations. For complex flexible mechanical systems with shifting center ofmass, the Eulerian approach has limitations. On the other hand, the Lagrangian pro-cedure automatically satisfies holonomics constraints, and nonholonomic conditionsthrough the Lagrange multipliers. This is in sharp contrast to the Newton-Eulermethod which requires introduction and subsequent elimination of the constraintforces. Obviously, it would be almost impossible to achieve this in the present studydue to the complex character of the system. Furthermore, Silver [50] has shown thatwith a proper choice of generalized coordinates, the Lagrangian formulation can be,numerically, as efficient as the Newton-Euler approach. However, the Lagrangianprocedure involves considerable amount of algebra. This is especially true in the caseof multibody systems where the position vectors are presented as a product of matrixand vector, and first as well as second derivatives are required.The governing equations of motion can be obtained fromd,aT, aT au,dt^aq aq (2.36)where T and U are the kinetic and potential energies of the system, respectively;the generalized coordinates; and Q, the associated generalized forces.The generalized coordinates and the corresponding generalized forces for theMDM system are as follows:system libration angle w.r.t. the orbital frame;-41^generalized coordinate vector associated with platform deformation;03^joint 1 motor rotation w.r.t. the MDM base frame F2;slew angle of arm 1 w.r.t. the MDM base frame F2;■35^prismatic joint 2 motor rotation w.r.t. the slewing arms;28a5^translation of the deployable arm w.r.t. the frame F4;To^resultant control torque acting on the platform due to momentum wheels;7'3^electro-magnetic control torque at joint 1;T5^electro-magnetic control torque at joint 2 (the linear actuator torque motor);Do^system damping in the pitch d.o.f.;D1^platform damping in the vibrational d.o.f.;D3^damping at joint 1;D5^damping at joint 2;D4^slewing arm damping torque;D6^deployable arm damping force.The total number of the generalized coordinates is 5 + nom,, where nom is thenumber of the modes chosen to represent the deflection of the flexible platform.Note, the translation h1 of the mobile base is treated as a specified coordinate,governed by the desired payload orientation, as discussed in Chapter 4. However, ifrequired, it can easily be treated as a generalized coordinate resulting in one additionalequation.The resultant governing equations of motion can be rewritten in a compact formas[M(q,t)]i+ 1Tt(q,4,t) = Q,^ (2.37)where:[M(q,t)] non-singular symmetric mass matrix (Appendix III);generalized coordinates vector with elements: 0, ih, /33, /34, /35, a5;nonlinear force/torque vector (Appendix III);generalized force vector, evaluated using the virtual work principle.29The contributions to the generalized force vector are from the control torques andthe non-conservative damping forces/torques:control torque vector with elements 70, i1(x)710, 0, T3, 0, T5, 0;nonconservative generalized damping vector with elements D0,D1,D3,D4,D5,D6.Here, -Sii(x) is the value of the first derivative w.r.t. xl of the mode shapes at thelocations of the momentum wheels.The equations of motion are explicitly derived in a compact form in order to iden-tify the inertia terms associated with the mass matrix and the nonlinear gyroscopic,Coriolis, centripetal and gravity related contributions. This helps in identifying in-compatible terms in the formulation, if any. Of course, as can be expected, thegoverning equations of motion are highly nonlinear, nonautonomous, and coupled.An analytical closed-form solution is not possible unless substantial simplificationsare introduced. Therfore, one is forced to resort to a numerical approach. However,search for an analytical solution is also attempted using the variation of parametermethod for a particular case [43].The validity of the governing equations of motion will have to be establishedthrough checks with similar mathematical formulations documented in the literature.Obviously, the same model has not been reported in the literature, and hence cannot be compared. However, some special cases may be available for comparison. Theconservation of system energy and momentum are also important avenues to follow.As pointed out before, the equations of motion are solved numerically, afterchecking the integration code against a benchmark case to ensure its validity.303. OPEN LOOP STUDYThe objective of this chapter is to study the dynamic response of the system. Tothat end, parametric studies with an increasing order of system complexity have beencarried out. This provides better physical understanding of the system behaviour,helps establish relative significance of the various system parameters, and suggestscritical combinations of parametric values which may lead to an unacceptable re-sponse.As mentioned before, the nonlinear, nonautonomous and coupled equations ofmotion are not amenable to any closed-form solution and hence must be solved nu-merically. A FORTRAN computer code has been written in a modular form to helpisolate effects of the system variables.3.1 Numerical ApproachIn order to solve the equations of motion numerically, they have to be rearangedinto a set of first order ordinary differential equations. Consider Eq. (2.37), a set ofN = 5 + norn second order differential equations,= F(q,  t).^ (3.1)Here(q, ,t) = [111(q,t)]-1{Q(t) — Ts1(q, 4, OERewriting Eq. (3.1) in the state space form gives 2N first order differential equationsto be numerically integrated:(3.2)31The subroutine IMSL (International Mathematical and Statistical Library): DGEARwas chosen for the integration procedure. The advantages of this subroutine include:automatic adjustment of the iteration step-size; two built-in integration approachesavailable to the user: the implicit ADAMS method and the backward differentiationprocedure, also refered to as GEAR's stiff method.The prime consideration is the stiffness of the problem, i.e. situations involvingvastly different time constants. For the space station based MDM, the orbital periodis about 100 minutes, while the structural vibration period can be less than 5 s.For the reasons of efficiency and speed, the ADAMS method is used with non-stiffsystems, whereas the GEAR method handles stiff configurations. For each step, theDGEAR subroutine checks for the possibility that the step-size is too large to passthe error test based on the specified tolerance. The integration step-size specified bythe user (10-8-10-12 in this study) is employed only as a starting value, and it isadjusted automatically by the subroutine.In order to understand the program architecture, a brief background to the nu-merical integration methodology may be useful. Let gn, be the solution of Eq. (3.2)at the nth integration interval (T), such that t = nT. Use of the multivalue methodto obtain gn may lead to better accuracy but the execution speed is reduced. Themulti-value method consists of three stages: prediction, error-test and correction asshown in the open-loop simulation flowchart of Figure 3-1.Given n - 1 7 Ign-k the prediction of gn, denoted by gn,o, is obtained by a linearinterpolation. A negative result of the error-test prompts the corrector to determine arefined value gn,i. The error-test and the correction procedure are repeated m timesuntil .Vn,771 satisfies the tolerance -Vn=gn,m. The error TOlerance Level (TOL) in therange of 10-8 to 10-12 used here is governed by the system under study.32CORRECTIONCALL FCN, 'n,m-NUMERICALINTEGRATION1INPUTr SYSTEMCONFIG.IMSL:DGEARPARAMETERSCALL IMSL:DGEARSTATE SPACEOUTPUTTIMESTEPGm.YESSTOP• FCNFigure 3-1^The MDM dynamic simulation flowchart.33Consider the flowchart showing computational steps for the MDM system dynam-ics simulation (Figure 3-1). The INPUT DATA block supplies the initial conditions,the MDM system properties (dimensions, mass, inertia, stiffness, damping, gear ra-tio), orbital elements and the open-loop input torques to the MAIN block where thesystem configuration is determined and the modal integrals of the flexible platformcalculated. The numerical integration IMSL:DGEAR routine calls the FCN subrou-tine to assess dynamics of the system (governing equations of motion). The actualstate of the system is obtained from the OUTPUT block.3.2 Formulation and Program VerificationIn modelling a complex multibody, flexible system such as the space based MDM,an obvious question concerning validity of the formulation and the computer codearises. Assessment of their accuracy presents a challenging task. In absence of aformulation for the same system, obtained independently by an other investigatorand reported in the open literature, three possible avenues are available:(i) continuouse monitoring of the system total energy to assure its conservation;(ii) comparison with the reported particular cases obtainable from the present"general" formulation;(iii) comparison between the numerical simulation and analytical closed-form so-lution results for a simplified configuration.3.2.1 Total energy conservationThe total energy for a conservative system is constant,E T U = constant.^ (3.3)Substituting T from Eqs. (2.21-2.26), and U from Eqs. (2.31-2.35) into Eq. (3.3),34the general expression for the system total energy is obtained as3^ 31 •T •^1 -^1E = -2 mfc • fc + -2nwl/^-2^(ri ri)dmi r2 • Ef -(-ri X )dmi^i=1 mi1^;^•^ 1^•^•+-2 1-3(IP + + + i33)2 + 2 NO + 0 + + 44 + 45)2rc+(EJ) foli (12F 2^_2^12^ dxiqi(t) ilc3C84 73.) +^Tii) •/33 2^1^a5^N2^(3.4)dxiTo facilitate the computational procedure, the total energy was monitored for thecase of a system with a rigid platform in a circular orbit, neglecting the shift in theC.M. The potential and the kinetic energy terms can be presented for the simplifiedcase as follows:GMm GM U =^{trace[I] - 317:c[ifircl + -1 k3(84 - P-3-)2 + -13 /C5( a5- — P-1)2; (3'5)rc^2',"^ 2^n3^z.^r5^n51 GM1 - - 1^•T = -2 m—rc + 2ftVg2 + -2 /30,b + e + al + /3.3)2 + -1 /5(0. + 0. + al + 44 + 45)222TM4 r7'.1^- •^h2^r;. 1,^r 1-;-;^,+—tai al -Fai Lc2.1,4 P4 iff -71.V.61 1- V-2.1(45 -r Lc2J(.65)T(.61-r t-2(1,2 c2a2).i -2-46z2ri4^i4 7, 14^14^r(-Entrn4Lt-,4 —3 p4,.1-2 cr4 - .1-2 sr-,4,^a5c04)(44a5c04 + a5s134)-a544(iii - 44a5s/34 a5c34)] 44/6z}.^(3.6)The system total energy was monitored for the gravity gradient stabilized orien-tation where the axis of minimum inertia Izz is aligened with the local vertical (inthe equilibrium position), while the axis of maximum inertia is aligned with the orbitnormal. The orbital energy Eo can be obtained from Eqs. (3.5), (3.6) as follows:GMm^-GMm=^ U0 =^2rc rc(3.7)GMm GM ftrace{n - 341:c[Ifirc}35For a system of 120, 000 kg mass in a circular orbit at 400 km altitude, the orbitalenergy (E0) is around —3.55 .1012 J with the potential energy Uo = —7.1 • 1012 J, andthe kinetic energy contribution To 3.55 1012 J. The orbital energy due to finitedimensions of the system (moments of inertia) represented by a 120 m long platformis around —180 J. The changes in the potential, kinetic and the total energies duringdynamical response to initial disturbances were monitored. Both rigid as well as freejoint systems were simulated for several initial conditions.The independent variable (t) can be expressed in terms of the true anomaly(0). This transformation is useful in studying spacecraft dynamics. The simulationwas performed with respect to the orbital unit of time (orbit). The transformationbetween the two variables is readily accomplished using the following relations:d^dcit — ded2^a 2( d2^2csinedt2^c182 1 + ecos4)-(3.8a)(3.8b)Eq. (3.8) is derived from Eq. (2.7). For a circular orbit, t = [orbit] • T,where: 0 = orbit • 2r;Figure 3-2 shows the MDM free joint response and the energy variations for aninitial pitch angle of 1,b0=0.1 rad. The initial slew (04) and deployment (a5) were setto zero. The mass of the deployable arm/payload assembly was taken to be 500 kg,with the mass of the 10 m long slew-arm set at 400 kg. The effective radius of thelinear gear was r5=0.1 m and the gear ratios at the two joints were n3 n5 = 1.As can expected, a periodic response was obtained for the librational motion, 0, asshown in Figure 3-2(a). The maximum energy exchange of 2.8 J between the potentialand the kinetic energy is the dominant feature of this response and the system's total366U(a) System response0.0^ Orbit^1.0(b) Energy exchange0.0^ Orbit^1 .0Figure 3-2^The MDM free joint response and energy exchange for a rigid plat-form subjected to an initial pitch disturbance of t,b=0.1 rad: (a)system response; (b) energy exchange.0.4rad; m0.0-0.43.00Joule0.00-3.0037energy is conserved, i.e. SE = 0 (Figure 3-2b). Note, the coupling effect leads tothe slew response (04), as can be seen after one orbit. The linear actuator rotor(15) induces shift of the deployable arm (a5) from the unstable equilibrium position(near the system c.m.), and causes the payload to deviate from the platform with anincreasing acceleration, resulting in further rotation of the motor (p5), as shown inFigure 3-2(a). The Coriolis effect induced moment, due to the deploying arm coupledwith the librational and the orbital motion, affects the slew response 04.Figure 3-3 shows the MDM open loop response and energy variation for aninitial slew arm position at 04=0.5 rad. The system parameters are as in the previouscase, and the initial conditions for the deployment and the platform pitch are zero.As before, a periodic response of the arm with a peak energy exchange of 0.008 Jbetween the potential and the kinetic energy was obtained for about 0.6 orbit, asshown in Figure 3-3(a). As the departure of the payload (a5) from the platformbecomes significant after 0.6 orbit, a continious increase in the system energy isobserved (Figure 3-3b). This is directly related to the shift in the c.m., and itsrate, which were purposely neglected (although the formulation account for it), toemphasize their importance. The increase in the energy of the system is due to thecontribution of the rate of the center of mass shift to the kinetic energy.Figure 3-4 shows the MDM response and energy exchange for an initial armposition at 04 = 1.0 rad and the payload displaced by 0.01 m (i.e. a 5 =0.01 m).In this case the payload departs from the platform with an increasing acceleration,while affecting the periodic motion of the slew d.o.f. due to the Coriolis reaction, ascan be seen in Figure 3-4(a). The maximum energy exchange of 0.024 J between thepotential and the kinetic energy is periodic, and is dominated by the slew motion upto around 0.8 orbit (Figure 3-4b). However, the secular increase in the energy follows38(a) System responserad;3.00.0-3.00.0 Orbit 1.0(b) Energy exchangeFigure 3-3 The MDM system free joint response and energy exchange for aninitial slew angle 04=0.5 rad: (a) system response; (b) energy ex-change.39(a) System response5.0rad; m0. 0IV-5.0 ^0.00 Orbit^ 1.0(b) Energy exchangeFigure 3-4 Response and energy exchange for the MDM, with free joints, whensubjected to a combined initial conditions of 04.1.0 rad, and a5=0.01m: (a) system response; (b) energy exchange.40due to a large and fast shift in the c.m., caused by the payload moving away fromthe platform, the effect neglected in the simulation as pointed out before.In the actual practice, this would not lead to any significant inconsistancy even ifthe c.m. shift is neglected as the system is, generally, actively controlled. The closed-loop response will maintain displacements, accelerations, etc. within the permissiblelimits and restablish the operational configuration in a short specified time. Of course,as emphasized before, the formulation developed here accounts for a. shift in the c.m.3.2.2 A comparison with a particular caseA simplified system as shown in Figure 3-5 was considered for comparison witha relevant case studied by Pradhan [51]. The platform is assumed to be rigid withthe MDM located 30 m from its center (C.M.).Only one rigid joint for the slew d.o.f. (04) is considered here. A 500 kg pointmass payload is attached at the tip of a 1000 m massless arm. The deploymentd.o.f. a5 is assumed to be frozen at that value. The configuration is similar to arigid tethered satellite model (also a degenerated case) investigated by Pradhan. Thesimulation response results for several different initial conditions were found to beidentical for all the generalized coordinates. This provided further confidence in theformulation and the associated computer code.3.2.3 Closed-form solutionA closed-form solution of the problem if available, can be useful to independentlycheck the validity of the formulation and the numerical simulation. Therefore, anattempt was made to arrive at an analytical solution of a rather simplified model ofthe system used in the previous Section 3.2.2.Consider the MDM system with a rigid platform in a circular orbit around the41PayloadArm 1^1334341Joint 1Orbit,....................■,........Xr■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ \ ■PlatformXIFigure 3-5^Simplified configuration of the MDM system.42earth. It supports a rigid arm carrying a point mass payload as shown in Figure3-5. The platform and the arm are free to librate and rotate through angles and/33, respectively in the orbital plane. The model accounts for the joint's torsionalflexibility by assuming that the generalized coordinate #3, associated with the joint'srotor, is fixed in any specified position. Thus the system has two degrees of freedomand i3.4. For this particular case the governing equations of motion were obtainedindependently, and can be presented as follows:d2[m1( ^+ (m4 + m6)14 + 2(7 + m 6 )h 1 l4cos (133 + fig) +^+ m6)/4i17)+^+ m6)h1/4cos(03 + 04)+ (7231 +m6)/(3i].4(711221 ms)h1/4sinG33 +134)44(2/i; + 2a + 04)+ 3(—m4 + ms)hiLisin(2/,b + fl3 + /34)02 + -3 (71-4 -Fm6Asin2(0 +#3 + ,84)e22^ 2 33 /2^d2i[(m4 + m6)11? + m1(-I - 1-1-)jsin200 -c04•;3^6KT +mad& K 7-i-4 .1 + m6 )hi /4cos(03^+^+ „AN;+ (-7-4 + m6)hiloin(33 +134)(11) +^+^+moisin2(0 +,83 + /34)0+ (7-124- +mohi4[sin(20 +0.3 +oho+ cosOinOk + 133 + 1340+ k3■34 = —C444.^ (3.9b)Here: a = IGM • Cn and C4 are the viscous damping coefficients in V) and ill d.o.f •,r3respectively.To make these highly nonlinear and coupled set of equations of motion amenableto known analytical procedures, it is necessary to introduce, rather judiciously, some• simplifications while retaining the significant dynamics of the system. To that end,transcedental terms were expanded in series, and third and higher degree terms ne-glected. In the present case, the nominal operating point considered is 03 = 0, i.e.(3.9a)43the arm is initially aligned with the platform. With this approximation the equationsof motion reduce to:^+ Ai& + 1304 + 41i) = A1(0444) ii31/);^(3.10a)^4.4 + A21.4 + B21,6 + 454 = 1i2(13414) + /1444;^(3.10b)where:B + CA1= E ;B + CA2= a'= 39.2B+C.EB2 = `3uon.2B+ CC ;2^E — 2Dn1 = 361 E2 + C+ k3n2 = 3614^C20B^—29B^—Co^—C4= E ; = c^ 113 = E 12A= m1.-1 + hj.(m4 + m6);3C = li("--i31 +m6);B = hi/4(n22± + m6);4D =16 'E = A + 2B + C + D.The set of Eq. (3.10) though simplified, are still coupled and nonlinear. The variationof parameter method as proposed by Butenin [45] is applied to obtain a closed-formsolution. Casting into a form similar to that discussed by Butenin:1:6 + A1134 + B104 + n?1,1) = f(/3, 44,V));^(3.11a)4.4 + A2"1 + B20 + n304 = /19034, 44, li;);^(3.11b)where ji is a small dimensionless parameter indicating the degree of nonlinearity, andf, g are nonlinear functions. By continuity, solution of the nonlinear system maybe expected to have approximately the same form as that of the linear homogeneoussystem obtained for /2 = 0:= asin(wit (1)1) bsin(w2t 4)2);^(3.12a)44i34 aiasin(wit^1) a2bsin(w2t 4'2);^(3.12b)where c4,1 and co2 are the natural frequencies of the system obtained from the charac-teristic equation,(1 — A1A2)w4 — (n? 77,3 — A1.B2 — A2B1)w2 + nn3 — B1B2 = 0.^(3.13)Here a, b, 4)1, 4)2 are integration constants obtained from the initial conditions andal, a2 are:^A2w2 — B2^A2w3 — B2al = ,.,2^2;^a2^,„2 "2 •^(3.14)— —2The solution of the nonlinear Eqs. (3.11a) and (3.11b) has the same form as Eq.(3.12) where a, b, 4)1, 4)2 are considered to be slowly varying parameters. Imposingthe constraint conditions such that the derivatives of and 04 are the same as whenthe parameters are treated as constants:awicospit + 4,i) + bw2cos(w2t + 4.2);^(3.15a)04^aiawicos(wit + >1) a2boi2cos(cv2t + 41)2).^(3.15b)The remaining terms are set equal to zero and yield the first two of the four simulta-neous equations for a, b, 431,iisin(co + 4)0+ bsin(cv2t 4)2) + ai)icos(wit (DO+ b(i2cos(cv2t 4.2) = 0; (3.16a)aiasin(wit 4)].) a2bsin(w2t .1)2) +^icos(wit^0+a2b(i2cos(w2t 4132) = 0. (3.16b)The remaining two equations are found by substituting 1.,6 and A into Eq. (3.11):awl (1 + ai Ai )cos(wi t 4.1) bw2(1 a2 Ai)cos(w2t 4.2) — cola( 1(1 + (Ilk)sin(wit (Pi) — w2b4)2(1 a2Ai)sin(w2t 4)2) = 14034,04,^(3.16c)45al24.24 — B24)2 = ^ f + (A1co3 — B gjsinnw2ba(c4 — c.o?)[^a24'1 =^[ A2u)? — B2^f + (AA — B i)g]sinC;wiao- (w3 — co?)w2o-(4 — co?)^a2[1124 B2F2+ (A1co3 — )G2] ;aw cr (w3ILbw20-(w3^[A24 — B2  F3 + (Aiw? — )G3];—w?)^al^ [A2w3 — B2  F4 + (A1w3 — B1)G4]— a2awi(ai Ai)cos(wit 4)0+ ixv2(a2 A2)cos(w2t 4)2) — w1a4)1(a1 + A2)sin(wit 4)1) — w2142(a2 A2)sin(cv2t 4)2) = 44,14)- (3.16d)The above four equations can be solved simultaneously to obtain the following fourrelations for a, b , 41and42:_ ^ ,A2u.)? — B2  f + (kw? B i)g]cosC;w 1c r (w3 wi.)[^al24 — B2-= ^ r Aa2^f + (A1co3 — Bagicosn;Here: o- = 1 —111A2;^C = colt + 1; and 77=^w2tEqs. (3.17) represent a transformation of the system (3.11) to a set of newvariables. It can be seen that for sufficiently small p the derivatives a, b, i , (i2 arealso small, consequently a, b, (1)1 and 4)2 are slowly changing functions of time. Forsmall variations compared to the oscillations in the resulting dynamic system, theapproximate equations for a, b, (1)1, 4.2 can be obtained by averaging the r.h.s. ofequations (3.17) over the periods -?Lr and 22-‘1 thus giving:w2A2ce? — B2= ^+ (^2Aiwi — )Gi ;wicr(c4 — co?) [w2a-(w3 — ?)(3.17a)(3.17b)(3.17c)(3.17d)(3.18a)(3.18b)(3.18c)(3.18d)46F3^27r /27r1= 4r2 Jo fsin(d(d77;where:=^127127 frosCd077;^47i2 0^02r /27rF2 —^fcomidC6/77;^472 ./0^027r /27r10 gcosCc1071;G2 = 1 /27r /27rgcosnd(c177;47r2 Jo^0G3 = 1 /27r /27rgsznCdCchi;47r2 Jo^01^27r^ 27rF4 = fsznndCdn;42 jo j 1^27r /27r^G4 = — gsiimd7.^*Cd^(3.19)4r2^0 -^  Equations (3.18a) and (3.18b) are independent of 4)1 and (h. Therefore, one cansolve them quite readily for a and b. Substiuting the results in equations (3.18c),(3.18d), and integrating, yields 4)1 and 4)2•Substituting for f and g, the averaging integrals (3.19) can be evaluated as:A3awi=^2 2ii3bw2 /2F2 =2114a2b1-02 pG2 =2^F3 = 0;^G3 = 0;F4 = 0; G4 = 0.^ (3.20)Introducing the expressions from Eq. (3.20) into Eq. (3.18) results in:1^r (A24 — B2) a =  ^ti3 awl + (kw? — Bi)Raiacui];^(3.21a)2wicr(w3 — 0.1?) l al...b = —1^(A2w2 — B2) ,i[ ^2^2 ^bo _L (A. 2^30 \^24.020A — w?)^a2^1-3 2 ' k 1W2 — '-'1)114a2bw2];^(3.21b)4)1= 0;^ (3.21c)ei2 = 0. (3.21d)The amplitude and phase angles can now be evaluated from equations47(3.21a), (3.21b):a = aoeHlt;b = b0e— H2tHere ao and bp are functions of the initial condition and1 =^[113 (A24 — B2) +^— BIN;2,7(4 —(3.22a)(3.221))(3 .23)= 2^ \ [113 (A24 — B2) + /44a2(A14 B1)]-^(3.24)2a-(w31 —w1) a2The amplitude parameters a and b exhibit exponential decay governed by the dampingat the joints and that of the platform. (Di and 4)2 are constants determined by theinitial conditions.Substituting a(t), b(t) from eqations (22) into Eqs. (12) yields the final form ofthe analytic solution:= aoeHltsin(wit (1)1) boe—H2tsin(w2t + (1.2);^(3.25a),64^ceictoeHltsin(wit 4)0+ a2b0e—H2tsin(w2t (1.2).^(3.25b)The amplitudes and the phase angles can be evaluated from thefollowing initial conditions:a0sin4.1 bosinch;^(3.26a)04,0 = a1aosin4q a2bosin4)2;^ 0.26011'0 =^aowicosstsi — b0H2sin(1,2 b0w2cos(132;^(3.27a)/34,0 = al (aoHisincbi aoce1cos4q) — a2(b0H2sinc2 — b0w2cos4)2).^(3.27b)The nonlinear analytical solution in Eq. (3.25) has the same form as that for thelinear homogeneous case given in Eq. (3.12), except for the time varying amplitudes48a(t),b(t). Note, for this particular case, where the third and higher degrees of non-linearities were neglected, the nonlinear system oscillates at the same frequencies asthe linear homogeneous case.Results and DiscussionThe response results obtained using the approximate closed form solution werecompared with those given by numerical integration of the original nonlinear Eq.(3.9). The following initial conditions were used: ifit,0 = 00, ikt=0 = 0, fi4,(t=0)44,0; 44,(t=0) = 0. Substituting the initial conditions into equations (3.26) and(3.27), the amplitudes and the phase angles are obtained as:= sin-1Wi VH? + w? 4)2 = sin-1( ^);VI/3 + 4#4,0 — a200 -a0 —— a2)'b0= 04,0 - al00 (3.28)sin4)2(a2 — al)•The numerical values used in the simulation are as follows:System Parameters: m1=120,000 kg; m4=900 kg; m6=5000 kg; /1=60 m; 14=17 m;c/1=5 m; C4=100 Nms/rad; Co = 1 x 103 Nms/rad; k3=10 Nm/rad.Orbital Parameters: e = 0,^T = 6 x 103sSpecified Coordinates Values: (33 = 0,^hi = 30 mInitial conditions: = 0.1 rad,^=^0, 44,0 = 0.2 rad, 44,0 = 049The response results are compared in Figure 3-6. As can be seen from the plat-form and the link responses, the numerical simulation results are quite close to theanalytical solution of the equations of motion. It is important to point out that asmall phase error observed in the link response does not accumulate over more thanone cycle of the platform pitch.With these three independent approaches suggesting the right-trend, thus sub-stantiating validity of the governing equations of motion and the numerical integrationcode, it was decided to undertake a parametric response study of the uncontrolledsystem.50_Numerical'Analytical .Numerical.:Analytical .2.0Orbits1(a) Platform response0.0^Orbits^2.0(b) Link response0.200.1w, rad0.0-0.11A comparison between the numerical and analytical solutions fordamped response of the simplified MDM system: (a) platform re-sponse; (b) link, rad0.00-0.200.0Figure 3-61513.3 Open Loop System DynamicsAn extensive study of the MDM open loop dynamic response was undertaken.For better appreciation of the system performance, it was analyzed in an increasingorder of complexity: it starts with a parametric study of the MDM, with free joints(free motion), supported by a rigid platform; this is followed by an assessment ofthe effects of damping and platform flexbility; and finally a parametric study of theflexible system with open loop torques.The investigation provided better physical undrstanding of the system behavior.It helped establish relative significance of the various system parameters. Further-more, it gave critical combinations of system parameters and disturbances leading toan unacceptable response, thus setting the stage for a control analysis.The amount of information obtained through a planned variation of the systemparameters and initial conditions is rather extensive. For conciseness, only typicalresults useful in establishing trends are presented here.3.3.1 Numerical data used in the simulationIn order to focus attention on the complex interactions between the system flex-ibility (joints, platform); base translation; slewing and deployment maneuvers; libra-tional motion of the entire system; etc. the study is purposely confined to the orbitalplane with the platform in the gravity gradient orientation. Circular orbit of 400 kmaltitude (orbital period=5550 s) is considered for the parametric study. The earthgravitation constant is taken as GM=3.986005 x1014 m3/82.The shift in the system c.m. significantly complicates the problem and increasesthe computational time, especially for flexible systems. Fortunately, it has relativelylittle effect on the system dynamics, even in the presence of flexibility and slew maneu-52vers, as shown by Chan [3]. Hence the shift in c.m. was neglected in the parametricstudy. However, it should be emphasized that the formulation accounts for the shiftin c.m. and its effect can be assessed quite readily if required.The system parameters as specified in NASA's space station Freedom referenceconfiguration manual [52] were used in the simulation. Mass of the torque motors andthe mobile base being relatively small is neglected. The equivalent linear actuatorradius (7.5) is chosen to be 0.1m in this study. Typical numerical values for the systemparameters are presented in Table 3-1, in Meter, Kg, Sec. (M.K.S.) units.Table 3-1^Typical numerical values for the MDM system parameters.parameter system platform joint 1 arm 1 joint 2 arm .(d.o.f.) '0 (71 03 04 fis asdimension[m]d1=311=6014=7mass[kg]m* ml '=120000m3 = 0 M4 =500M5 = 0 M6 = 103[kgm2]inertia 13 = 10 /*4 15 = 10 /6x = 10016y = 10016., = 100stiffness[kgm2/s2rad][kgm3/.92]EJ=0.55. 109/c3 =1000les =1000dampingratio6 =10-56 =0.016 =0.005E4 = 0 6 =0.005E6 =0.005dampingcoefficient[kgm2/s/rad]Co = 5.6 C1 = 22 C3 = 10 C4 = 0 C5 = 10 C6 = 10[kg/s]gear ratio n3 = 1 n5 = 1* calculated during the simulation.3.3.2 MDM undamped response for the rigid platform caseThe purpose of studying this relatively simple configuration is to have someinsight into the intricate dynamics and gain better physical understanding of thesystem behavior. To this end the system was subjected to a variety of initial conditions53representing a wide spectrum of disturbances.There are several equilibrium positions for the MDM system, however the stableequilibrium configuration corresponds to the platform in the gravity gradient orienta-tion with ,84 = h1 = h2 = a5 = 0, i.e. the arm is aligned with the platform, and thec.m. of arm 2 assembly is assumed to coincide with the center of mass of the system.The unstable equilibrium position is obtained when the platform axis of the minimummoment of inertia coincides with the local horizontal, and the slew angle 04 =In all the simulations, the stable equilibrium position is considered as reference. Thegeometry and the common initial conditions are summarized in Table 3-2.Table 3-2^MDM configuration and typical initial conditionsi.c. of system platform base joint 1 arm 1 joint 2 arm 2position[rad), [m]16 = 0.1 41 = 0.1 li, = 60 /33 = n344 44 = 0.1 As =asnarsas = 10velocity[rad/s), [m/s)=0 il =0 ill = 0.1 43 = n344 44 = 0 i3s =asns/rscis = 0acceleration[m/s2) iti = 0torque[IC gm2 132]1j=00 T3 = 0 11 = 0Figure 3-7 shows undamped response of the system when the manipulator base(F2) is located at the center of the platform (hi =0). The arms are completely free,i.e. unconstrained, to swing and translate. Figure 3-7(a) shows the MDM responseover one orbit when the initial slew of 04=0.1 rad. is imparted, i.e. the arm 1 isrotated through 0.1 rad and released. The arm starts to oscillate with the initialperiod of 0.6 orbit. The arm, being connected through the shaft to the rotor at joint541, causes it to follow the arm's motion as shown by the 03 response. As can be seen,the linear actuator inertia 15 causes the rotor at joint 2 to rotate through an angle 05w.r.t. the slewing arm and, as a consequence, the deployable arm with the payloadassembly, originaly located in the unstable equilibrium position (on the orbit), startsto deviate (a5) with increasing acceleration. The rotor (15) is finally constrained bythe motion of the deployable arm and rotates according to the effective radius andgear ratio (7.5, n5). As the velocity of the payload w.r.t. the slewing link increases,the Coriolis reaction moment starts to dominate the link oscilations which is apparentafter 0.8 orbit. The platform pitch response ti) is not excited as the base is located atthe c.m. of the platform.Figure 3-7(b) shows the MDM response for an initial deployement of 0.5 m, i.e.the payload, originally located at the system c.m., is displaced through 0.5 m. Ascan be expected, the payload starts to move outwards (away from the c.m.) with anincreasing acceleration. The rotor 15 of the linear actuator follows the deploymentaccording to the linear transmission r5 = 0.1 m as shown by the /35 response. TheCoriolis reaction moment exerted by the deploying arm 2 causes the arm 1 (initiallylocated in the stable equilibrium position with /34 = 0) to rotate through 134 = —1rad where the gravity gradient moment balances the Coriolis torque. The rotor atjoint 1, being connected to the arm, is forced to rotate (133). As before, the platformpitch ik is not exited.Figure 3-7(c) shows the MDM response over one orbit for an initial pitch distur-bance of 0=0.1 rad. As can be expected the platform starts to oscillate about thelocal vertical (the equilibrium position) with a period of approximatly 0.6 orbit. Upto 0.2 orbit, arm 1 (/34) and the torque motor rotor (/33) follow the platform with thesame oscillation frequency. As arm 1 starts to oscillate with the platform, the inertia55Localverticala5i. c.i.c. (b)(a)rad0.300.00lif-0.3013 1-34---1.01.0rad4. OrbitOrbitP4=0.1 rad a5 = 0.5 ms\s\(33 ,134'N(c) i.c. w=0.1rad1.0Orbitrad (d)^1.c. P4=-1.57 rad133,14rad0.300.00-0.300.0 Orbit1.000.00-1.00-2.0(6.0^1.0%./(^•a 5 \ N:R5\Free motionRigid platformUndamped Responseh1 = 0.0Figure 3-7^MDM free joints undamped response when supported by a rigid plat-form and subjected to various initial conditions: (a) 94 = 0.1 rad;(b) a5=0.5 m; (c) b = 0.1 rad; (d) f = —1.57 rad.5615 causes the linear actuator to rotate w.r.t. the slewing arm and, as a consequence,shifts the deployable arm with the payload away from the unstable equilibrium po-sition. The payload deployment (a5) begins and gradually gains acceleration whichis reflected in the rotation of the linear actuator rotor 05. As the velocity of thedeployable arm increases, the Coriolis reaction starts to dominate the motion of theslewing arm and the torque motor rotor attached to it.Figure 3-7(d) shows the MDM response for an initial slew angle 13.4 = —1.57 rad.In this case both the slewing and the deployable arms are initially in the unstableequilibrium position; i.e. aligned with the local horizontal. Because of the arms' andpayload's inertias and the small gravitational moment, the angular acceleration ofthe slewing arm is relatively small in the begining (up to around 0.5 orbit). As theslew angle increases the payload starts to accelerate.From the dynamic response of the MDM described in Figure 3-7, it can be con-cluded that, even for a relatively simple system represented by a rigid platform,undamped and in absence of external torques applied at the joints, the behaviour ofthe system is rather complex and sometimes unpredictable. Hence a thorough dy-namic simulation study is necessary for a reliable design of the manipulator and itscontrol system.Figure 3-8 shows the undamped dynamical response of the system, in absenceof external torques, with a variety of initial conditions when the manipulator base islocated at h1=30 m from the center of the platform.Figure 3-8(a) presents the system behaviour, over one orbit, when the arm slewsthrough 0.1 rad in 0.05 orbit in a sinusoidal fashion. Note, the arm starts to oscillateand the torque motor rotor follows the arm (03). The deployable arm and payloadoriginally located approximatly 30m from the equilibrium position, deviates from57(a) i.c. 134=0.1 rad (b) i.c.100 0 Orbit 10Orbit\s,a5 = 0.5 m-1.00.0rad0.50.0-0.5rad0.50.0-0.5Fi 5a5(c) i.c. tv=0.1rad00^ 10Orbitrad (d)In1.0i • c • 04=-1.57 rad0.0-1.0'a5Orbit 10radr't11.0a 5 Free motionRigid platformUndamped responsehi = 30.0 m411-Figure 3-8 Undamped response of the MDM, with free joints and supportedby a rigid platform, when the base is located at h1=--30 m from theplatform c.m. The initial conditions are: (a) 84 = 0.1 rad; (b) a5=0.5m; (c) = 0.1 rad; (d) 134 = —1.57 rad.58this location with a relatively high velocity right from the beginning. The Coriolisreaction moment on the slewing arm dominates its motion. As can be seen, thegravity moment balances the Coriolis reaction moment on the arm at ,84 = —0.9 rad.The reaction on the base, due to the centripetal and Coriolis accelerations, inducemoment on the platform and a relatively small pitch motion (0) ensues.Figure 3-8(h) considers a case similar to that studied in Figure 3-8(a) with aninitial condition of a5=0.5 m. It is apparent that )544 = —0.9 rad is the equilibriumposition governed by the moments exerted at joint 1 due to the gravity field and theCoriolis reaction. The latter arises due to the rate of the deployment (a5) coupledwith the rotating system (0). The reactions on the base, located 30m from the centerof mass of the platform, excites the pitch motion tk as before.Figure 3-8(c) shows the system response for an initial disturbance of 0.1 rad inpitch. As can be expected, the effect of the reaction forces at the base on the pitchmotion is relatively small, due to the large platform inertia. As in the case of Figure3-8(b), the slew response (04) of arm 1 is affected by the deployed payload. Thegravity induced moment balances the Coriolis reaction moment as the arm reaches—0.9 rad from the local vertical. It may be pointed out that 04 is measured w.r.t.the platform.Figure 3-8(d) shows the system behaviour when arm 1 is originally located at[34 —1.57 rad. As in the previous cases, the slew motion is strongly affected by thedeployable d.o.f.From the dynamical response study in Figure 3-8 it can be concluded that thecoupling among all the system degrees of freedom is significant and cannot be ne-glected.Figure 3-9 shows the undamped free system response when the MDM base,59initially located at the c.m. of the platform, continuously translates with a constantvelocity of 0.01 m/s along the platform and the deployment d.o.f. is frozen at a5=10m.Figure 3-9(a) shows the system response for a slew initial condition of 04.0.1rad. As can be observed, the oscillation amplitude of the arm is amplified 8 timescompared to the initial disturbance of 0.1 rad, and the equilibrium is shifted fromzero to -0.4 rad. As the MDM base moves away from the c.m. of the platform,the frequency of oscillations increases and the equilibrium position is shifted. Themagnified view of the platform pitch response also shows a shift in the equilibriumposition from zero to -0.0015 rad after one orbit. The response is governed mainly bythe Coriolis reaction due to the base translation it1 coupled with the rotation of thesystem,Figure 3-9(b) shows the effect of simultaneous slew (fig = —0.1 rad.) and pitch(0 =0.1 rad) disturbances. As in the previous case, the Coriolis reaction excites theslew motion with a shift in the equilibrium position and an increase in the oscillationfrequency as the base moves away from the c.m. of the platform. It may be re-emphasized that the slew d.o.f. 04 is measured from the platform.Figures 3-9(c) and 3-9(d) show the system response for two other combinationsof initial disturbances. The responses are similar to those observed in Figures 3-9(a)and 3-9(b).The main conclusion from the system response described in Figure 3-9 is thatthe velocity of the base dominates the dynamic response of the slew d.o.f. and theplatform pitch irrespective of the initial condition of the arm. The significant featuresof the response are: a shift in the equilibrium position; and increase in the amplitudeand frequency of the arm's oscillations.60Localverticalas(b)rad^i.c w).1rac1.134=-0.1rad10rad  (a) i.c. 34=O.1 rad0.50.0 ^-0.51.000000150000etr—.0015^o3344Orbit 101.50.01304Orbit-1.50.0 10rad (d) i. c. 134 = —1.57 rad-ro.0015.0000.0015Free motionRigid platformUndamped Responsehi 0.0 for t =0hi =0.01 m/sas= 10.0mrad (c) i•c. w=0.1rad0.00 0Figure 3-9Orbit 10Undamped free response of the system during translational maneuverfrom h1 = 0 at a constant velocity of 0.01 . The deployable arm isheld fixed at as = 10 m. Initial conditions are: (a) /34 = 0.1 rad; (b)0=0.1 rad, #4 = —0.1 rad; (c) = 0.1 rad; (d) = —1.57 rad.613.3.3 MDM on a flexible platformObjective here is to assess the effect of platform flexibility on the dynamicalresponse of the MDM. The platform flexibility is represented, in this case, by onlythe first mode, which is symmetric [46]. As in the previous section, a parametricstudy has been carried out for the undamped system, subjected to various initialconditions, in an increasing order of complexity. The system parameters were shownin Table 3-1, and typical initial conditions used in Table 3-2.It is of interest to point out that a wide spectrum of initial conditions chosen forthe study is not aimed at mere generation of response data. Rather it provides infor-mation as to the possible critical system behaviour under diverse circumstances whichis considered essential to its design and operation under a variety of contingencies.Past experience suggests that slewing and deployment failures are not uncommon,and freeing of a locked member often demands a variety of maneuvers as shown byGEOTAIL, ANIK, GALILEO and other spacecraft.Figure 3-10 presents the free system undamped response when the flexibility ofthe platform is represented by only the first mode. The payload was initially locatedat the c.m. of the platform in the unstable equilibrium position, a5=0, and the initialslew angle was set at ill = 0.1 rad.Figure 3-10(a) shows the slew response over one orbit. As can be expected, aperiodic motion of around 0.7 orbit (3000 s) and 0.1 rad amplitude is observed forthe slewing arm.Figure 3-10(b) shows the flexible platform response, excited by the slewing armand coupling with the payload dynamics. The platform behaviour is characterized bythe deflection Si at the MDM base. In this particular case, h1 = 0 (i.e. the deflectiontakes place at the c.m. of the platform). Due to a relatively small mass of the arm and62Localverticala5.411-rad (a) i .c 04=0.1 rad,a5=0^m (b)i. C 4 . 1 r a d 5=0 .0 mOrbit 10Orbitrad (d)i. c .P4=0.1 rad, a500 in0.000(c) i...r3431.1 rad, a5=0.0 m10Free motionUndamped responseFlexible platfomn-fing modea5 = ao, k5 = 0.0Figure 3-10 Effect of the platform flexibility on the free undamped system re-sponse. The payload is initially located at a5 = 0 and disconnectedfrom the linear actuator gear (k5 = 0). The slewing arm is initiallyoriented at #4 = 0.1 rad.63the weak initial condition, the maximum displacement during the transient responseis only 10-5 m. The enlarged view of the 81 response is characterized by the naturalfrequency of 0.18 Hz, associated with the platform first mode.Figure 3-10(c) presents time history of the deployment degree of freedom (as)associated with arm 2, initially located in the unstable equilibrium position at thec.m. of the platform. The platform vibrations, excited by the slewing arm, force thedeployable arm-payload assembly to move and gain acceleration due to the gravita-tional force. The direction of the deployment towards the center of force (the earth)in this case, is determined by the platform displacement 81 and the slew motion, asthe payload motion along the slewing arm is unconstrained (Ics = 0). Of course,the deploying arm rotates with the slewing arm. From the enlarged view in Figure3-10(c) it is apparent that the deployment is coupled with the platform vibrations at0.18 Hz.Figure 3-10(d) suggests that the angular pitch librational motion is not excitedby the slewing maneuver of the arm, at least within the accuracy of the numericalsimulation and duration.From the response results in Figure 3-10 it can be concluded that the flexible andrigid d.o.f. are coupled. Flexibility of the platform changes the dynamic response ofthe system quite significantly and hence must be accounted for.Figure 3-11 presents free undamped response of the system, over one orbit, withthe platform flexibility represented by the first mode. The deployable arm is held fixedat as=10 m, and the MDM base is located at h1=0, i.e. the manipulator is locatedat the mid-point of the platform. The initial slew angle is 1.5 rad (86 deg.) and theplatform initial condition is qi = 0.01, i.e. the platform is initially deformed in the1st mode with a midpoint deflection of 0.012 m.64Localverticala5(b)i. c 434=1.5rad, 1=0.01 mNVVVVVvvv\Orbit8110 0,1 ■111 1 .^1111 111 10. 0^ —0.02 ^10 0.0(al0.03Orbit(c) i c 434=1.5 rad, 1=0 . 01Ma50.0^ 10Orbitrad (d)i c • 4=1.5 rad, q1=0 . 01 M0 .0—1.10-40.0^Orbit^10rri15.00.0-15.Free motionUndamped responseFlexible platform -first modea5= io.o m= 1.5 rad, q1=0 . 01 Mrad (a)i.c.Figure 3-11 Effect of the platform excitation on the system response. The de-ployable arm is held fixed at a5=10 m. The initial conditions are=-- 1.5 rad and q=0.01 m.65Figure 3-11(a) shows the slew response, i.e. angular motion of the arm in pitch.The relatively low frequency of the slew motion is due to the platform vibrations thatinduce a pseudo-gravitational field in the local horizontal direction. The oscillationperiod is about 1.8 orbit compared to 0.7 orbit in the undisturbed motion. As can beseen from the enlarged view of the /34 response, the rotational motion is modulatedat a frequency of 0.18 Hz due to the platform vibrations. The amplitude of themodulations is a function of the base displacement of 0.012 m, in this particular case,where the MDM base is located at the center of the platform and only the first flexuralmode is taken into account for the platform lateral vibrations. It also depends on thedistance between the common c.m. of the slew arm, the deployable arm assemblyand joint 1, and the slew arm position 04.Figure 3-11(b) shows the displacment and the slope of the flexible platform atthe base. As can be expected, with only the first mode, the slope at the center of theplatform where the base is located is zero.Figure 3-11(d) presents the platform pitch response. Note, although joint 1 islocated at the center of mass of the platform and the slew arm is free to respond with-out any constraint, the pitch d.o.f. is excited. The gravitational and the centrifugalforces coupled with the platform displacment induce torque about the c.m. of theplatform causing it to oscillate with a relatively large amplitude. The results suggestthat presence of even small platform vibrations, through the coupling dynamics, cancause the system to become unstable.Figure 3-12 shows response of the system (same as in Figure 3-11) but under anincreased disturbance corresponding to the flexible degree of freedom of qi = 0.1m.The arms continue to slew through 1 rad as before. Note, a rather unexpectedresponse of the slew arm (fin).66Localverticala 5rad (a) i.c.(34= 1.0 rad,2.01.0q1=0 .1mIII L4-0.20.0(b)i • C .134=1.0rad, q1=0 .10.03Orbit-0.100 Orbit 10Orbit(c) i.c.134=1.0rad, q1=0 .1Ma5rad (d)i • C.134=1.0 rad, q1=0.1m0.0in15.00.0-15.-5.10- 410^00Free motionUndamped responseFlexible platform -first modea5.10.0 mFigure 3-12 Effect of a larger platform excitation of qi = 0.1 m on the free,undamped response of the system. The deployable arm is held fixedat a5 = 10 m, while the slewing arm is initially oriented at /34 = 1.0rad.67The equilibrium position shifts from zero to 1.57 rad (90 deg.) and the period ofthe oscillations is 0.12 orbit compared to 0.65 orbit in the rigid platform case (Figure3-12a). The relatively large platform amplitude at the MDM base, creates a force fieldperpendicular to the earth gravitational field causing the arms to oscillate about thelocal horizontal. The frequency of oscillations is a function of the vibration amplitudeof the MDM base. It is apparent from the expanded view that the arm response 04 ismodulated, at the higher frequency of 0.18 Hz, due to the coupling with the platformvibrations as shown in Figure 3-12(b).Figure 3-12(d) presents the pitch response excited by the oscillating arm and itscoupling with the platform vibrations. The basic pitch response, with a period of 0.6orbit, is modulated due to the arm oscillations at a period of 0.12 orbit.Corresponding results for the slew arm, initially at 1.5 rad (04 = 1.5 rad insteadof 1 rad in Figure 3-12), are presented in Figure 3-13.Note, the equilibrium position for the slew arm is shifted to 90 deg as shown inFigure 3-13(a). The amplitude of the pitch response and its high frequency modula-tions are functions of the slew arm oscillation amplitude about the local horizontal.The pitch modulation virtually disappears and the librational amplitude decreases asthe initial slew angle is set close to the new equilibrium position (Figure 3-14).Figure 3-15 presents the dynamic response of the system, similar to that de-scribed in Figure 3-14, with the slewing arm initially set at 04 = 0. As can be seen inFigure 3-15(a), the arm swings between the two unstable equilibrium positions 134=0and 27r (local vertical). The arm response is coupled with the platform vibrationsas shown in the expanded view of the 134 time history. The platform pitch response(Figure 3-15d) is strongly affected by the slewing arm dynamics, and a significantlibrational amplitude of 0.001 rad is observed. Note, no torque is applied to the68 Free motionUndamped responseFlexible platform -first modea5.10.0 m(a)i.c. (34= 1.5rad, q1=0 .1m (b)i • c 334=1.5rad, q1=0 .1m0.1si0.0alOrbit 0 . 0 3(c) i c434=1.5rad, q1=0 .1m rad (d)i.c.134=1.5 rad, q1=0 .1m15.0-15.0 .00 .0^Orbita5-1.10-410^000.0Orbit^10Figure 3 -13 Free undamped response of the system with the deployable d.o.f.frozen at a5=10 m. Initially 04 = 1.5 rad and qi=0.1 m.69Localverticala51.5685P3'1341.5675133. 04\ivvvv\i\ j%/'\./v‘0 62 0.631 . 0OrbitIl0.03Orbit(a ) c .04.1.5707rad , q1=0 . 1 m (b) • e 4=1.5707rad, q1=0 . 1 m1 . 5 50.01.575radaWW\MAN0.0 Orbit 10Orbit(c)^c .134=1.5707rad, q1=0 . 1 ma5rad (d) c .134=1.5707 rad, q1=0 . 1 m0.00.0-15. --5-1.1010^00In15.0Free motionUndamped responseFlexible platform -first modea5= 10.0 mFigure 3-14 Free undamped system response with the deployable arm held fixedat a5.10 m. Initial orientation of the slewing arm is 04 = 1.5707rad. The platform is excited in the first mode with qi=0.1 m.70Localverticalas.411-rad2.0 A0.0 ItIIIal-2.000 0.031.0Orbit Orbit0 .0 Orbit 10Orbit• 04=0.0 rad, q1=0 .1mrad (d)i C^4:).0 rad, q1=0 .1m0. 0-15.0 .0(c)-1.10- 310^00In15.0Free motionUndamped responseFlexible platform -first modea5 = 10.0 m(a) i • c • 14=0.0 rad, q1=0 .1m^(b)i • c • 134= 0.0rad, q1=0 .1mFigure 3-15 Effect of the initial orientation of the slewing arm on the systemresponse with the deployable arm fixed at a5=10 m. Initially i34 = 0and qi=0.1 m.71motors and pitch response is the result of the platform vibrations.From the parametric study for the system configurations and the initial condi-tions as desribed in Figures 3-11 to 3-15 it is apparent that even a relativly smalldisturbance applied to the platform can adversly affect the dynamics. This can ex-cite significant slew and pitch responses which affect the performance of the MDMand the space station. Development of a presice and reliable dynamical simulationprocedure is necessary to predict unexpected system behaviour.3.3.4 Damped response for the flexible MDMThe parametric study carried out so far was for the undamped system with freejoints. As in real life there are no ideal systems, so in the next phase of the para-metric study viscous damping is introduced in all the d.o.f. The damping coefficientconsidered is realistic and accounts for the frequency spectrum associated with theindividual generalized coordinate. The numerical values for the damping coefficientare presented in Table 3-1. In general, damped systems have an inherent tendencyto be more stable.Figure 3-16 presents damped response of the system having free joints and theflexible platform represented by the first free-mode. The deployable arm is held fixedat a5 = 10 m while the slewing arm is initially oriented at 04 = 0. The flexibleplatform is initially disturbed, in the first mode, so that q=0.1 m.It is apparent from Figure 3-16(a) that introduction of the damping changes theslewing arm response substantially. Now the arm rotates in the positive direction(counterclockwise) about joint 1, as against the oscillatory motion in the undampedcase. The damping torque of the rotor 13, together with the inertia force field due toplatform vibrations, force the arm to the monotonic response. The arm's rotational72Localverticalas4111- -------rad20.00. 110.0 0 .0-0.10.000 -0.20.0 1 . 0Orbit 10 Orbit00 Orbit 10Orbit. 134=0.0 rad, q1=0.1ma 5(d)i.C.^1344.0 rad, q1=0.1m- 15.(c) rad0.0-1.10-310^0015.00.0Free motionDamped responseFlexible platform -first modeas= 10.0 m(a) i • c • 134=0.0 rad, q1=0.1m^(b)i•c. 134=0.0rad, q1=0.1mFigure 3-16 System open loop damped response, with the deployable d.o.f. frozenat a5 = 10 m. The platform is initially excited in the first-mode withqi = 0.1 m.73rate decreased as the vibration amplitude decayed due to the structural damping ofthe platform truss (Figure 3-16b). The pitch amplitude also showed a slight decreaseand the period of oscilation, as expected, approached /(orbital period) as shown inFigure 3-16(d).To accentuate the response, the MDM base was placed at the tip of the platform,i.e. h1=60 m. The response results for this case are presented in Figure 3-17.When the platform vibrations are relatively large (0.1 - 0.2 m) the slew-armequilibrium position is shifted from 04 = 0 to 1.5707 rad (90 deg). As the vibrationamplitude decays the slew-arm starts to oscillate about /34 = 0 with the amplitudedecreasing due to the damping of the platform truss and the torque motor (Figure.3-17a). The MDM base, being at the tip of the platform, experiences local rotationas indicated in Figure 3-18(c).The reaction on the MDM base due to the arm oscillations, coupled with theplatform vibrations, exert a relatively large moment about the center mass of theplatform, especially when the MDM is located at the tip of the platform. The momentinduce pitch oscillations of 0.01 rad ( 0.5 deg) as shown in Figure 3-17(d). The pitchand arm responses are modulated at a higher frequency of 0.18 Hz due to the platformvibrations as shown by the expanded view of and 04 time histories.Based on the results, it can be concluded that introdution of damping affectsthe system dynamics directly through changes in the moments at the joints, andindirectly by reducing the magnitude of the response.The MDM base can have a specified translation rate (h1) along the platform. Aresonable base velocity is 0.02 m/s. In that case, the MDM base translates along theentire length of the platform in approximatly one orbit. As apparent from Figure3-18(b), the base motion excites the platform vibrations at the beginning of the74a5 LocalverticalOrbit 1 . 0Free motionDamped responseFlexible platform -first modea5= 10.0 mhi = 60.0 m(a) .0 . 4= 1.0 rad q1=0 .1 m(c)i .c. 134=1.0 rad, q1=0 .1m(b)i .0 134= 1.0rad, q1=0 . 1 m(d)i c^134=1.0 rad, q1=0 . 1 m. 01rad0 . 00-0.01. 0^Orbit-1.10- 21 0^0.0rad0.0Orbit^1.0Figure 3-17 Damped response of the system with the deployable arm fixed ata5 = 10 m, the MDM base held at h1 = 60 m, and the slewing arminitially located at #4 = 1.0 rad. The platform is set vibrating withan initial disturbance of qi = 0.1 m.75as Localverticalrad3.0133,134 0.0 -••••••-".-3.000 OrbitFree motionDamped responseFlexible platform -first modea5= io.o mhi = 60.0 mhi = -0.02 m/s(a) i • c •^=10 rad , q1=0.0^(b)i•c• 134=0.0rad, q1=0.0(d)i • c •^134.0 rad, q1=0. 0-1-10-21 0^0.0^Orbit^1 0Figure 3-18 Effect of the MDM's translational maneuver, at -0.027,1t- on the flex-ible platform, from the initial location of h1 = 60 m. The deployablearm is fixed at a5 = 10 m. The initial conditions are: 04 = 0;41 0.1 m.-8-2.1 0 ^. 0 Orbitrad0.00(c)i.^1344.0 rad, q1=0.0 radrad0 .076translational maneuver; but the oscillations decay as the base advances along theplatform. The movement of the reaction point along the vibrating platform, and thefact that the direction of the reaction on the flexible platform remains the same (asconcluded from the slew arm response shown in Figure 3-18a), cause the vibrationsto decay and the platform deflects according to the reaction moment.The Coriolis and the gravitation moments about joint 1 cause the slew arm torotate to the second equilibrium position when the resultant c.m. of the payload andthe slew arm cross the orbital trajectory. In other words, the base is located 50 mfrom the second tip of the platform as can be seen in Figure 3-18(a). The reaction atthe base excites the pitch motion as shown in Figure 3-18(d).The free vibration analysis results suggest that the system response can be un-predictable and the instability may result under certain combinations of configurationand initial disturbances. Lateral inertia force field due to the vibrating platform canbe used to orient the slewing arm in a predefined position.3.3.5 Flexible system response to open loop torquesThe next logical step is to introduce generalized forces and represent the platformflexibility more precisely through the necessary number of modes that would convergeto the right response.The MDM system is simulated with open loop torques applied through the motorslocated at the joints. The system parameters and configuration are according to thedata in Tables 3-1 and 3-2, respectively. To start with, a constant torque of T3 = 1Nm is applied at joint 1, when joint 2 is locked with arm 2 deployed to 10 m. Thesystem response is presented in Figure 3-19. In this reference case, the MDM islocated at the center of the platform (h1 = 0) and the flexibility is still represented77by the first mode. The initial conditions for all the d.o.f. are zero.The applied torque causes the rotor 13 and arm 1 to rotate through angle 03 and#4, respectively (Figure 3-19a). From the expanded view, it can be seen that therotation of 13 deviates by 0.001 rad w.r.t. the slew arm and oscillates at its naturalfrequency of 9.6 rad/s (1.5 Hz). The natural frequency is detemined by the inertiaof the rotor and the joint stiffness, taking into account the gear ratio n3 = 1 in thiscase. The relative rotation (,83 — /34) is the actual twist of the torsional shaft due tothe joint flexibility. The damped response of 03 is due to the dissipation at the joint.The rotation of the arm excites the flexible platform, with increasing frequencyand amplitude, according to the rate of the slew (Figure 3-19b). From the expandedview it is apparent that the response is modulated at a frequency of 0.18 Hz associatedwith the first mode. As the MDM base is located at the geometric center of theplatform, and only the first free-free mode is taken into account, the local slope al iszero (Figure 3-19c). This, of course, does not reflect the reality as the applied torquemust deform the platform asymmetrically.The rotation of the arm, together with the platform vibrations, excite the pitchd.o.f. by a small amount as the MDM is located at the c.m. of the platform (Figure3-19d). Note, the oscillations are about the new equilibrium position of —1.6 • 10-5rad.More general case is presented in Figure 3-20, where the MDM base is locatedat the tip of the 'platform (hi = 60 m), and the slewing arm's initial condition is(34 = 1.0 rad.The input torque to joint 1 balances the gravitational torque at Al = 0.4 rad,about which the slew arm is oscillating (Figure 3-20a). The deflection of the platformis affected at the begining by the torque motor oscillations, and later by the oscilla-7844-Open loopDamped responseFlexible platform -first modea5= 10.0 mT3. 1 NmFigure 3-19 Response of the MDM system to a constant torque of 1 Nm applied atjoint 1. The deployable d.o.f. is frozen at as = 10 m, the manipulatoris located at the midpoint of the flexible platform (h1 = 0). Theplatform flexibility is represented by the first mode.791 0Orbit(b)i.c. 4 =0.0 rad, q1=0 . 0-5-510 ^00ItI1 . 0Orbitm810.0o(d)i • c •^1300.0 rad, q1=0 .0rad0 . 0 0-_2.108^ -5.1 0- 300 10^0 0Orbit(a) i . c . 134.0 rad, q1=0 . 0(c)i .c . 13.4.0 rad, q1=0 .0alrad0.0as Localvertical0 00 . 99700Orbit13 4=1.0 rad, q1=0 . 0 p4.1.0 rad, q1=0 . 0(c)i.c.rad (a)i.c.1 . 54=1.0 rad, q1=0 0 (b)i.c. 4= 1.0 rad, q1=0 . 00 . 01.003—1.50.005 1 0radrad0 .00 . 0-7-2.101 00 . 0-5.10-21 . 0^00Orbit OrbitFree motionDamped responseFlexible platform -first modea5= 10.0 mhi = 60.0 mT3 = 1 NmFigure 3-20 Open loop response ofNm applied at joint 1.m. The base is locatedrepresented by the firstrad.the MDM with aThe deployable h1 = 60 m andmode. The initial80constant torque T3 = 1o.f. is frozen at a5 = 10the platform flexibility isarm position is 04 = 1.0tions of the arm as shown in Figure 3-20(b). The time history of the local slope ath1 = 60 m is presented in Figure 3-20(c). Both the platform deflection as well asrotation (libration) remain quite small.Next, to better model the system, the platform flexibilty was represented by thefirst two modes. For the present case, the natural frequency of the first (symmetric)mode is 1.13 rad/s (0.18 hz) and the second (antisymmetric) mode is 3.11 rad/s (0.5Hz). The corresponding simulation results are presented in Figure 3-21. Systemconfiguration and parameter values are the same as those for the case discussed inFigure 3-19.Comparing the response results in Figures 3-21(b), 3-21(c) with those in Figure3-19, clearly reveals the discrepancies. The local slope al is the result of the reactiontorque —7'3 acting on the platform at h1 = 0. The expanded view in Figure 3-21(c)clearly shows, that the slope is modulated at a higher frequency of 0.5 Hz, whichcorresponds to the second mode. Obviously, it can be concluded that one mode isnot enough to represent the flexible platform.To progress further towards a more realistic simulation of the system behaviour,open loop torques 7'3 = 1 Nm and T5 = —1 Nm were applied to joints 1 and 2,respectively. The MDM base was located at the tip of the platform, h1 = 60 m, andthe initial conditions of Al = 1.0 rad and a5 = 10 m for the slew and deployable arms,respectively, were applied. The platform flexibility was simulated with an increasingnumber of modes to assess the effect on the dynamical response. This set of resultsare presented in Figures 3-22 to 3-26.The response of joint 1, the slew-arm and the deployable-arm, as representedby the generalized coordinates /33, 04 and a5, respectively, is presented in Figures3-22(a) to 3-26(a). The deployable-arm is retrived as a result of T5, where the small81• I/ ,ttfI0 rad, q1=0 0(a) i• 13 4=0.Orad, q1=0 . 0-5  (b)i •5.108100.rad50.0 0.00.0050. % -3-o . o -51 d5^10^00 1 . 0Orbit Orbit1340.0 rad, q1=0.0 (d)i•C•^134.0 rad, q1=0 . 0rad'011104464•40.4Y-0.0al0 0^ 0.0151.01 . 0^0.0Orbit Orbit2-10rad0 . 0——2.1080.0—5.10-3Open loopDamped responseFlexible platform -two modesa5 = 1o.o mT3= 1 Nm Figure 3-21 System response to a constant torque 7'3 = 1 Nm applied at joint1. The deployable arm is held at a5 = 10 m. The manipulator islocated at the midpoint of the platform and the platform flexibilityis represented by the first two modes.82fluctuations are due to coupling with the slew arm oscillations. As can be seen, theequilibrium position is shifted as a result of a decrease in the gravity moment (payloadapproaches the joint). In general, the system response remains virtually unchangeddue to inclusion of the second mode (Figure 3-23). However, the platform deflectionSi and local slope al do show some sensitivity to the number of modes (Figures 3-23,3-24 and 3-25). Now the joints oscillations at 9.6 rad/s (1.5 Hz) exite the third andthe forth flexible modes with natural frequencies of 1 Hz and 1.6 Hz, respectively.Adding the fifth mode with a natural frequency of 15 rad/sec (2.4 Hz), the platformbecomes less sensitive to the oscillations of the joints, as apparent in Figure 3-26.The main conclusion based on the last parametric study may be stated as follows:For flexible systems with multiple degrees of freedom (flexible and rigid), the couplingeffects are considerable. For the cases demanding extreme accuracy, it is necessaryto take in account additional modes beyond the fundamental, to accurately modelthe platform deflection and slope. However, in most situations, representation of theflexibility by a single mode may be adequate. In the present study, the accuracy isdictated by the desired tracking performance of the MDM and the system attitude.For tracking of a specified trajectory with the end effector, an accuracy of 0.01 m isconsidered adequate. On the other hand, for the antenna pointing and stationkeep-ing the librational control with an accuracy of 0.001 rad (0.05 deg) is an acceptedstandard. Most of the elastic energy contribution comes from the first mode. In thesubsequent closed loop study, the first two modes representing the symmetric andantisymetric character of the system are used to model flexibility of the platform.83T5Localvertical1543.4Ir1.0Orbit0.0-5-1.1010^0.0_5 (b)i. C.1 *1 0 44=1.0 rad, a 5=10m81P4=1.0 rad, a 5=10m10Orbit-7-5.100.0_7(c)i. c. 04=1.0 rad, a5=10m5.10rad0.0(d)i • C •^13 4=1.0 rad, a5=10mOrbit-1.10-21 0^00rad0.0Open loopDamped responseFlexible platform -first modehi = 60.0 mT3= 1.0 NmT5 = -0.1 NmFigure 3 -22 System response to open loop torques 713 = 1 Nm, T5 = -1 Nm,assuming one flexible mode. The MDM base is located at h1 = 60m. The initial conditions are: i34 = 1 rad; a5 = 10 m.84a5 LocalverticalT5.4--1.101 . 0^0.0Orbit0 . 00 . 0 10Orbit(a) i .c.^4=1.0 rad a 5=10m -5 (b)i •^4 = 1.0 rad, a5=10m1-10a503 4340.01 0 . 0rad5 . 01 0Orbit-7-51 00 . 0_7(c)i. c. 04=10 rad, a5=10m5.10-1.10-21 . 0^0.0rad0.0(d)i • c •^(34=1.0 rad, a 5=10mOrbitrad0 . 0Open loopDamped responseFlexible platform-two modeshi = 60.0 mT3= 1.0 NmT5 = -0.1 NmFigure 3-23 System response to open loop torques T3 = 1 Nm, T5 = -1 Nm,assuming two flexible modes. The MDM base is located at h1 = 60m. The initial conditions are: 134 = 1 rad; as = 10 m.85134LocalverticalT51 . 0Orbit(a) i .0^134=1.0 rad a 5=10re (-5 b)i^4=1.0 rad , a 5=1 om1 -1 00.0—1•101 . 0^0.0(d)i • c •^134=1.0 rad, a 5=10m134=1.0 rad, Ei5=10m11000 . 01 . 0Open loopDamped responseFlexible platform -three modeshi = 60.0 mT3= 1.0 NmT5 = -0.1 NmFigure 3-24 System response to open loop torques T3 = 1 Nm, T5 = —1 Nm,.assuming three flexible modes. The MDM base is located at h1 = 60m with the system initial conditions as: 134 = 1 rad; a5 = 10 m.86Localverticala5 ,T531 -1 0-5  (b)i•c• R4=1.0rad, a5=10m0.0—1-10Orbit^1 0Open loopDamped responseFlexible platform -four modeshi = 60.0 mT3= 1.0 NmT5 = -0.1 Nm(a) i • c • 04=1.0 rad, a 5=1 OmFigure 3-25 System response to open loop torques T3 = 1 Nm, T5 = —1 Nm,assuming four flexible modes. The MDM base is located at h1 = 60m and the initial conditions are: 04 = 1 rad; a5 = 10 m.87a5 LocalverticalT5114—1-101.0^0.0Orbit0 . 00 . 0 1.0Orbit(a) i.C. 1i41.O rada5lOm —5 (b)i.c• 134=1.0rad, a5=10m1.101.0030.t970 0^ 0.005 03,13 40,010. 0rad5. 0133aAk(d) i • c •^134=1.0 rad, a5=10mOpen loopDamped responseFlexible platform -five modeshi = 60.0T3=1.O NmT5 = -0.1 Nm_7(c)i.c. 04=1.0 rad, a5=10m5.10rad0.0—7—5100.0Figure 3-26 System response to open loop torques T3 = 1 Nm, 7'5 = —1 Nm,assuming five flexible modes. The MDM base is located at h1 = 60m and the initial conditions are: (34 = 1 rad; a5 = 10 m.Orbit^1. 0883.4 SummaryThe chapter focused on the open loop dynamics of the MDM system using aversatile simulation tool. In general, derivation of the equations of motion and de-velopment of a computer code demand considerable amount of effort. In the presentstudy it was minimized by structuring the equations judiciously to make the pro-gramming process more efficient. Furthermore, the computer code is structured in amodular form, thus facilitating the debugging procedure and the parametric study.The validity of the formulation and the computer code have been establishedthrough: comparison with particular cases; checking the conservation of system en-ergy; and by comparison with the analytical closed-form solution of a particular case.An extensive parametric study, with an increasing order of complexity, providedinsight into the complex and, at times, unexpected behaviour of the system. It helpedassess the effects of platform and joints flexibilities, damping and generalized forceson the system dynamics.From the open loop study it can be concluded that, for the space based manip-ulators, control of the generalized coordinates is a necessity imposed by the complexnonlinear, nonautonomous and coupled dynamics, extreme flexibility, and demandingaccuracy. Compared with the conventional ground based robots, it is not possible torely even on the so called 'stable' equilibrium position.Description of the desired tasks and their realization through control are theissues addressed in the following chapters.894. DESIRED TASK DEFINITIONAs concluded in Chapter 3, a controlled MDM system is required in order to ac-complish a predefined task. In the case of the proposed space station the manipulatorwill perform tasks usually w.r.t. the platform or the platform based modules.The MDM as described in Chapter 2 operates in the joint-variables space. On theother hand, tasks to be performed, such as the space station assembly, operation, andmaintenance, are usually expressed in the space station fixed coordinate system. Insome cases, tasks are defined w.r.t. the orbital or the earth based coordinate system,e.g. aiming of an accurate instrumantation such a telescope and a communicationsantenna, transfer of payload from the space station to the shuttle, etc.In order to control the position and orientation of the MDM's end effector so thatit can perform the desired tasks, the inverse kinematics solution is more important.In other words, given the desired position and orientation of the end effector w.r.t.the selected reference coordinate system (station based, orbital or inertial frame) andthe MDM parameters, it is necessary to find the corresponding MDM d.o.f. b , /34and a5, so that the end effector can be positioned as desired. Thus, it is necessary tofind the required controlled variable Od, ,84d, a5d in order to close the loop.A number of methods are available to tackle an inverse kinematic problem. Theyinclude inverse transform, quaternion, iterative and geometric approaches, and oth-ers. Paul et al. [1] have presented an inverse transform technique using the 4 x 4homogeneous transformation matrices to solve the kinematic problem. This approachis adopted in the present study to define the required MDM d.o.f. for performing thedesired tasks.904.1 MDM Kinematic EquationsIn this section the homogeneous transformation, that represents the position andorientation of the end effector coordinate frame F6 with respect to the chosen stationbased coordinate frame, has been developed.The MDM is treated as a two degrees of freedom robot, on a translating androtating base, with revolute and prismatic flexible joints as shown in Figure 4-1.The position and orientation of the end effector frame F6 with respect to a ref-erence coordinate frame (i.e. the station or the orbital frame in the present case) isdescribed by the homogeneous transformation matrix [11],= [B] • [T2] • [E].^ (4.1)T2^E Hhere:[B]^homogeneouse transformation matrix relating the manipulator base frame F2to the reference frame;[T2] homogeneouse transformation matrix relating the manipulator end frame F6to the base frame F2;[E]^homogeneous transformation matrix relating the end effector to the manipu-lator end. This, in the present case, is a unit matrix as a separate end effectoris not considered.[T2] can be expressed as a product of the arms' transformation matrices,91F1^XIFigure 4- 1^Reference coordinate frames established for the inverse kinematicsstudy of the MDM system.92[T2] = [A1] • [A2],^ (4.2)where:[A1] homogeneous transformation matrix relating the coordinate frame F4 of theslewing arm to the base frame F2;[A2] homogeneous transformation matrix relating the deployable arm frame F6 tothe slewing arm frame F4.From Figure 4-1, which establishs the arm based coordinate frames, it is possibleto determine the homogeneous transformation for each arm:cos 04 0 sinN^0[A1]sinN001—cosi34^00^0 ' (4.3)0 0 0^110 100 00^0[A2]—0 0 1^a51 (4.4)0 0 0^1Substituting [A1] and [A2] into Eq. (4.2) gives the arms' homogeneous transformationmatrix [T2] ascosi54^0 sin/34— cos (34a5sini34[T2] =[sin0134 01 0—a5cos,840 • (4.5)0^0 0 1The task to be performed by the MDM can be defined w.r.t. the platform refer-ence frame Fp (Figure 4-1), the orbital frame Fr or to the inertial frame F1 (Figure2-2). The platform reference frame is fixed to the geometric center of the platformand displaced by an amount So (due to the platform deflection at h1=0) from F1,93which is fixed to the center of mass of the platform as shown in Figure 4-1,60 = g(0)-411,^ (4.6)where So is the platform's deflection at its center.The orbital refernce frame Fr is displaced through Eto and rotated by an angle 0w.r.t. F1. The inertial reference frame Fr is translated through fc and rotated by anamount 0 w.r.t. Fr as shown in Figures 2-2, and 4-1.As most of the tasks planned for the MDM are confined to the space station,they are defined in this study w.r.t. the frame F.The MDM base homogeneous transformation [Bp] takes into account the transla-tions (TRANS) and rotations (ROT) of the base frame F2 w.r.t. the chosen referenceframe Fp as follows,[Bp] = T RAN S[(h1 - h2sinai),(81 + h2cosa1 — So),01ROT[Zp,(721 + at)], (4.7)where:[Bp] MDM base homogeneous transformation to the platform reference frame Fp;h1^location of the base along the platform;h2^distance between joint 1 and the center line of the platform;61^platform deflection at h1.In the matrix representation, the base homogeneous transformation has the fol-lowing form,^—sincxi —cosai 0^h1 — h'2'sinaicosai —sinai 0 Si — So + h2cosa1[Bp] =^0^o^1^0^.^(4.8)0 o^0 194The homogeneous transformation matrix [Hr] , which specifies the location of theMDM end coordinate frame F6 w.r.t. the platform coordinate frame, is the chainproduct of the successive coordinate transformation matrices [Bp] and [T2], and canbe expressed as[Hp] = [Bp][T2].^ (4.9)Substituting transformation matrices from Eqs. (4.5) and (4.8) into Eq. (4.9) gives—y 0 cosy a5cos-y h1 — h2sina1[Hp] = [cos70001sin-y0a5sin-y — So^h2cosai0 (4.10)0 0 1where [Hp] is the homogeneous transformation between the MDM end and the plat-form based frame. Here: -y = al + 04.4.2 Inverse Kinematics for Point TrackingIn the previous section the relationships between the position and orientation ofthe coordinate frame that defines the MDM end and the actual system d.o.f. havebeen established. The relationships that define the position are also functions ofthe MDM base instanteneous location along the platform (h1) and the shift of therevolute joint w.r.t. the platform center line (h2).In this section the above mentioned relationships are used with the inverse trans-form technique to determine the requied MDM d.o.f. (required control variables) fora desired position of the payload.A given task for the MDM is defined by the instanteneous desired position andorientation of the end effector w.r.t. the chosen reference frame (Fp in this case).For time dependent variable position the desired task describes a trajectory in thereference frame.95A desired task, in the planar case, can be represented by a homogeneous trans-formation matrix [Hpd] with known elements as[Hpd] =nzflyU000000 1azay0pxpy0 (4.11)where px,py are the components of the desired position, with nz,ny, and ax,ay as thecomponents of the desired orientation. The required angular position #4d of the slewarm and the required linear position asd of the deployable arm, to impart specifiedposition and orientation to the payload (end effector), can be found using the inversetransform technique as follows.Rewriting Eq. (4.9) for the desired case as[Hpd] = [Bp] [T2d],^ (4.12)and premultiplying both sides with [Bp]-1 gives[B p]-1[H pd] = [Bp]-1 [Bp][T2d] .^ (4.13)After some mathematical manipulations, it is possible to present Eq. (4.13) in theformh(n)h(n)f3(n)[^00000h(a)12(a)13(a)0h(p)12(P)f3(p)1Yr= 1.- 2d1i , (4.14)[T2d] is the homogeneous transformation [T2] of Eq. (4.5) with the required values of04 and a5;^fi = -sinai(x) cosai(y) hisinai - (Si - 80)cosai - h2;^(4.15)^f2 = -cosai(x) - sinai(y) hicosai (Si - 60)sina1;^(4.16)96134d = tan(Pm — h )s in a + (Si — 60 — py)cosai h2 (61 — 60 — Py)sinai (h1 — Px)cosal I •13 = 1(z);^ (4.17)and x,y,z refer to components of the vectors given as arguments of fi,f2, and h.Equating the fourth column on both sides of Eq. (4.14) gives the followingrelations:*A (7)) = a5dsini34d;12(p) = —a5dcos134d.Dividing Eq. (4.18) by Eq. (4.19), 134d can be evaluate fromh(P) tan/34d12(p)•(4.18)(4.19)(4.20)Substituting fi(p) and 12(p) from Eqs. (4.15) and (4.16), respectively into Eq. (4.20)gives the required value for the slew,(4.21)Note, P4d is a function of the trajectory parameters px and py, the instanteneousbase location h1 along the platform, the location of the joint w.r.t. the platform h2,the instanteneous platform deflections 60, Si., and the local slope al. It should berecognized that 04d is a function of time even for the stationary end effector (pz, pyconstants) w.r.t. the reference frame.To determine the required value of the deployment a5, Eq. (4.13) is premultipliedby [A1]-1,[Airl[Bpri[Hpd]=Eitirind]=EA2di.^(4.22)Algebraic manipulations lead to the relationr3(n) 0 ii(a) MP)]f2(n)^12(a) 12(P)f(n) 0 f3(a) 13(P)^[A2c],0^0^0^1(4.23)97where [A2d} is the homogeneous transformation of Eq.(4.4) with the required valueof a5.Here:h = —3(a1+04)(x)-Fc(a1-F■34)(0-Fh1s(cei-04)—(51-80)c(a1+04)—h204; (4.24)f2 = 1(z);^ (4.25)h = c(cei. + 04)(x) + s(ai + 04)(y) — hic(ai + i34) — (Si — ao)s(al + 04)—h2s04.^ (4.26)Equating the third element of the fourth column on two sides of Eq. (4.23) gives thefollowing relation for a5d (with 04=04d as obtained from Eq. 4.21),a5d = (pz — hi)cos(ai + 040+ (PY — 61+ 60)sin(al + 134d) — h2sini34d. (4.27)As in the case of the P4d, a5d is also a function of time, even for the stationary endeffector (pz, py constants) w.r.t. the reference frame.Let the MDM degrees of freedom 134 and a5 be the required value as given inEqs. (4.21) and (4.27), respectively. Now the end effector will meet the desiredtrajectory position. For the control purpose, it is important to have required velocityand acceleration for the degrees of freedom. From Figure 4-2, the following kinematicrelationships for the desired trajectory can be obtained,utan/34d = (4.28)with u and v defined from Eq. (4.21) as:u = (h1 — px)sinai + (py + 60 — Si)cosai — h2;^(4.29)98_^Trajectory ____ _________(pz — hi)sinai + (61 — 50 — py)cosai + h21tanAid = [ , ,loi — 00 — py)sinai + (h1 — pz)cosal iu = (h1 — pa)sinai + (py + So — 51)cosai — h2v = (py + 60 — 61)sinai + (px — hi)cosala5d = licosP4d + iisinAidii5d= ikosi34d-1- iisinf34dihcosAid — Vsini34d44d = a5dfLcosi34d — ilsin,84d134d = a^5dFigure 4-2^Kinematic relationships.99v = (py 60 — 61)sina1 (px — hi)cosai.^(4.30)Now the required velocity and acceleration can be obtained from the correspondingcomponents of the trajectory as:a5d 1.)c08184d il3in134d;^(4.31)a5d =Ald —le4d —fizinP4d;iicosAld — imini34da5diico8P4d iisinfi4d a5d(4.32)(4.33)(4.34)where IL,^are obtain by differentiating u,v in equations (4.29) and (4.30) w.r.t.time.4.3 Matching of the End Effector Position and OrientationIn the previous section, the required MDM degrees of freedom 04d, a5d and theirfirst and second time derivatives were evaluated in order to negotiate the desiredend effector trajectory. For some tasks, orientation of the MDM end effector, w.r.t.the reference coordinate, may also be importent, in addition to its position. Forthe planar case, this would require an additional degree of freedom besides the slewand deployment. To meet this requirment, the proposed space station based MobileServicing System (MSS) is equipped with the Special Purpose Dexterous Manipulator(SPDM). In the present study, the obvious solution is to consider the translation h1along the platform as the additional degree of freedom. Of course, addition of ageneralized coordinate will further complicate the simulation. As no flexibility isinvolved in the base translation system, and the base's mass is negligible comparedto those of the platform, arm and the payload, it is taken to be specified. Thusthe motion of the base (h1) is specified in such a way that the MDM orientation1001 [ (az — hi)sinai + (Si — h — ay)cosai + h2 (4.38)154c1 = tan— (61 — 60 — ay)sinai (hi — ax)cosai J •requirment is met. Note, the system degrees of freedom are coupled with the basemotion hi) as apparent from the r.h.s. of Eq. (III.1). However, as mentionedbefore, the base motion is not affected by the system generalized coordinates.To evaluate the required base position hid, the third columns from two sides ofEq. (4.14) are equated giving:h(a) = sin/34d;^(4.35)Ma)^ (4.36)Dividing Eq. (4.35) by Eq. (4.36) gives 134d as a function of the desired end effectororientation134d = tan—^.f2(a)^l[ fl (a)]^ (4.37)Substiting for fi(a) and f2(a) from Eqs. (4.15) and (4.16), respectively into Eq.(4.37) leads toNote, the required slew 134d is a function of the desired orientation parameters ax, ayand the instanteneous base location hi along the platform.The required base position hid is now found by equating the two expressions for134d in (4.21) and (4.38),—Nay -I- pyaz — (ax — py)(61 — 60 h2cosai)hid=^ — ayNote„ hid is a function of the desired trajectory (pz, py) as well as the orientation(az, ay). Furthermore, the hid depends on the platform deflections 61, 60 and thelocal slope which are functions of 41. From the equations of motion (2.37) andAppendix III, it is clear that the platform flexibility generalized coordinate qi itself is(4.39)^.101a function of hi, hi, h1. This suggests that the hid must be calculated by an iterativeprocedure.To have a constant arm orientation while the end is tracking a desired trajectory,the base must translate at a velocity hid which is proportional to the trajectoryvelocity and the platform deflection rate. It can be determined by differentiating Eq.(4.39) w.r.t. time.4.4 Task with respect to the orbitIn this case, the homogeneous transformation from the MDM base frame F2 tothe orbital frame Fr takes into account the system libration and shift in the centerof mass Etch[Br] = TRANS[(aox + hi — h2sinai), ((Loy +^h2cosa1), 0]ROT[Zr, (Ik +;+ al)],(4.40)where:[Br]^MDM base homogeneous transformation to the orbital refernce frame Fr;system libration;a0x )610y components of the shift of the system c.m. w.r.t. the frame Fo.In the matrix notation, the base homogeneous transformation has the form—s(0 + al) —c(V) ai) 0 (aoz hi)cik — (Si + aoy)s0 — h2s(0 + 01)[Br]= c(tk +0 al)—s(IP0ai) 01(aoz h1)s0 + ((Loy + 804^h2c(tk^al)00 0 0 1(4.41)The homogeneous transformation matrix [Hr], which specifies the location of theMDM end effector frame F6 w.r.t. the orbital coordinate frame, is the chain productof the successive coordinate transformation matrices [Br], [T2] and can be expressed102as[Hr] = [BriiT2i.^ (4.42)Substituting from Eqs. (4.5) and (4.41) into Eq. (4.42) gives the transformation as[N.—8(0 + 7) 0 c(lk + 7) a5c(0 +7) + (h1 + aox)ciP — (6). + a0y)s0 — h2s(ib + cei)c(cb + 7){0 c(0 + 7) a58(0 + 7) + (hi + aox)slib + (6). + a0y)4 + h2c(0 + al)0 0 .^1 00 0 0 1(4.43)were 7 = ai +134, and (34d, a5d, hid are obtained using the inverse kinematic techniqueas developed in Sections 4.2 and Parametric StudyIn the previous section the MDM required degrees of freedom 04d, a5d and thebase location hid to meet a desired payload trajectory and orientation w.r.t. theplatform reference frame were defined.An investigation to study the effects of the system parameters on the requireddegrees of freedom Aid and a5d has been carried out in this section. The main objec-tive in the parametric study is to present the effects of the platform flexibility on therequired d.o.f. when following various trajectories with different initial conditions.To begin with, the system is simulated with the rigid platform. The required d.o.f.obtained for this case will serve as the reference for the simulations that take into ac-count the platform flexibility. Both the stationary point tracking as well as trajectorytracking are considered for the flexible system parameter study.4.5.1 Trajectory tracking for the rigid systemIn this subsection the required degrees of freedom for the rigid system tracking a103predefined trajectory, while the MDM base is moving, are presented. This case, wherethe desired trajectory is defined w.r.t. the platform, is similar to a rigid ground basedrobot moving on its track.The simulation results for a constant speed base motion, as well as the acceleratedbase, starting from the center of the platform, are presented in Figures 4-3 and 4-4,respectively. In both the cases, the payload is initially located on the platform 10 mfrom its midpoint (arms' total length 10 m). The duration of the tasks are 0.02 orbit(111 s).Figure 4-3(a) presents a desired trajectory, a straight path traversed at a constantvelocity in the Yp direction at Xp = 10 m. The translation of the MDM base at 0.01m/s is shown in Figure 4-3(b). The required slew (04d) and deployment (a5d) timehistories are also presented (Figures 4-3c and 4-3d, respectively). As can be seen, •close to the constant rates of 0.001 rad/s and —0.01 m/s are required for the slewand the deployment, respectively.Figure 4-4 shows a more complex case where the desired trajectory forms a curvedpath. It represents the payload accelerating at —0.005 m/s2 and 0.002 m/s2 in theXp and Yp directions, respectively where, at the starting point (10,0), the velocity is0.2 m/s and 0.03 m/s in the Xp and Yp directions, respectively. The payload mustnegotiate the desired trajectory, with the base accelerating at 0.001 m/s2, as shownin Figure 4-4(b).The time histories of the requirmed slew (134d) and deployment (a5d) for the abovementioned task are indicated in Figures 4-4(c) and 4-4(d), respectively. Note, in thiscase, a relatively slow variation of the d.o.f. is required, hence the control problemcan be solved relatively easily. It is important to remember that the MDM base ismoving on a rigid platform while the joints, where the control input is applied, are104DesiredtrajectoryALocalverticalas411-Required degreesof freedomRigid systemh1 =0.01 m/s0.5Y m0.0-0.5(a) Desired Trajectory Base Motion hl10.0^X p m 15.0(c) Required Slew 0 4d—0.40.00^Orbit^0 .02Figure 4-3^Time histories for the required positions of the degrees of freedomwith a constant rate of base motion.0 . 4rad. 0150 (d) Required Deployment a5d10.05.00.0 ^^0.00 Orbit^0.02105Localverticala5Desiredtrajectory(b)10.0 ^Base Motion hl5.00.0(d) Required Deployment a5d15.010.05.0 ^0.00 0.02Orbitrad1.02.00.0(c) Required Slew 134d0.00 Orbit 0.02Required degreesof freedomRigid system••h1 = 0.01 m/s2(a) Desired Trajectory5.00.015.0Y m10.00.0^10.0^X p m 20.0^0.00^Orbit^0.02Figure 4 -4^Time histories of the required degrees of freedom, for the rigid sys-tem, with an accelerating base and a specified trajectory.106flexible.4.5.2 Trajectory tracking for the flexible systemIn this subsection the effects of platform flexibility, MDM base location and initialconditions on the required position, velocity and acceleration of the d.o.f. are studied.Figures 4-5 and 4-6 present the required d.o.f. time histories for a desiredtrajectory and base motion as described in Figure 4-4. The system is simulated fordifferent initial conditions for the two generalized coordinates, associated with thefirst two modes, that represent the' platform deflection.In Figure 4-5, a relatively large initial disturbance of qi = 1 m, for xi = 0 (initialbase location), is applied to the first mode. The initial condition for the second(antisymmetric) mode is zero.As can be seen from Figure 4-5(c), the relatively large amplitude platform vibra-tions (Si = 1.2 m for x1 = 0), do not affect the required values in the initial stage,when the base is located close to the center of the platform. As the base translatesalong the platform, oscillations of the local slope ai affect the required slew after 0.02orbit. The required deployment remains virtually unaffected by the flexibility effectrepresented here with the first vibration mode (Figure 4-5d).Figure 4-6 provides similar information to assess the effect of the second modeexcitation. Now the slope initial condition of q2 = 1 m is added to the displacement atxi = 0. Note, the local slope oscillations affect the required slew , 04d. The nominalposition is modulated at a high frequency of 0.5 Hz, the natural frequency of thesecond mode, as shown in Figure 4-6(c).The required deployment time history is essentially unaffected in the beginning,however, high frequency modulations, growing in amplitude, do appear and become107Localverticala5DesiredtrajectoryRequired degreesof freedomFlexible platform -two modesqi = 1.0, q2=0.0hi = 0.01 nits 2(a) Desired Trajectory10.05.000^10.0 x p m 20.0(c) Required Slew I 4dBase Motion hl(d) Required Deployment a5d0.02 . 015.0Y m0.00^Orbit^0.0215.0rad1 . 010.00 . 00.00^ 0.02Orbit5.0 ^0.00 Orbit^0.02Figure 4-5 Time history of the required degrees of freedom for the flexible systemwith an accelerating base and prescribed trajectory. Initial modalexcitations are: qi = 1 m; q2 = 0.108LocalverticalasDesiredtrajectory(b)10.0 ^Base Motion hl5.00.00.00 Orbit 0.02Required degreesof freedomFlexible platform -two modes= to, q2=1.0 mii = 0.01 m/s2(a) Desired Trajectory15.0Y m10.05.000^10.0^X p m 20.0(c) Required Slew 134drad1.00 . 0• (d) Required Deployment a5dUt15. . 0Orbit^0.020. 00^Orbit^0.02^0.00Figure 4-6^Time history of the required degrees of freedom for the flexible systemwith an accelerating base and prescribed trajectory. Initial modalexcitations are: qi = 1 m; q2 = 1 m.109significant rather quickly (Figure 4-6d). The vibrations affect the required deploymentwhen the displacement at the center of the platform, where the reference frame Fp islocated, is different from the displacement at the base (51-60).Next, the MDM base was initially located 50m from the center of the platform.The trajectory parameters and the base motion were kept the same as before. Effect ofthe magnitude of the initial conditions on the required time histories of the generalizedcoordinates )34 and a5 is discussed in Figures 4-7 and 4-8.In the case presented in Figure 4-7, the initial conditions are qi q2 = 0.1 mwith the base located at x1 = 50 m.It is apparent from Figure 4-7(c) that even a relatively small disturbance at theplatform affects the required slew. In this case, the modulations are of about 0.05rad amplitude and at the frequency reflecting the combined effect of the first (0.2 Hz)and the second mode (0.5 Hz). The required deployment is also affected as shown inFigure 4-7(d). Note, for the present case, the modulation amplitude appears to berapidly increasing.With an increase in the initial disturbance to qi = q2 = 0.5 m, the required timehistories of the generalized coordinates are strongly affected as can be seen in Figure4-8. The required slew is modulated at a high frequency with a peak amplitude of0.2 rad. The deployment time history is also modulated, at a high frequency, andthe amplitude of 2 m reached in 0.02 orbit is still increasing. Obviously, an efficientcontroller would be necessary to obtain the desired performance.As explained in the next chapter, for the control purpose, the required positionas well as the velocity and acceleration of the d.o.f. will be needed to achieve thedesired performance.Figure 4-9 presents the required profile for the slew degree of freedom, 134. The110LocalverticalDesiredtrajectoryrad1 . 00 . 00.00 Orbit 0.020.02OrbitBase Motion hl(d) Required Deployment a5d50.0 ^0.00(a) Desired Trajectory15.0Y P m10.05.060.0 x p m 70.0(c) Required Slew 134d2.0(b)60.0 ^55.0Required degreesof freedomFlexible platform -two modesq1=0.1, q2=0.1rnii =0.01 nVs2Figure 4-7^Required time histories of the generalized coordinates with the plat-form excitation of qi = q2 = 0.1 m. The manipulator is located 50m from the center (h1=50 m).111a5DesiredtrajectoryLocalvertical.41rad1.02.00.0(c) Required Slew 134d0.02Orbit0.00^ 0.02Orbit15.01 0 .05.0 ^0.0015.0Y m10.05.00.0 ^50.0 60.0 x p m 70.0 0.02(d) Required Deployment a5dRequired degreesof freedomFlexible platform-two modes(11 = 0.5, q2= 0.5 111ii = 0.01 I11/S2(a) Desired Trajectory Base Motion hlFigure 4-8^The effect of increased platform excitation of qi q2 = 0.5 m on therequired time histories of the generalized coordinates f34 and a5 totrack a desired trajectory. The base is located at hi = 50 m.112system parameters, the desired trajectory, and the base position as well as motionare the same as in the previous case. The system is simulated with qi = q2 = 0.5m as the initial values of the platform generalized coordinates, associated with thefirst two modes. The required slew time history, /34d, is presented in Figure 4-9(b).The slew velocity 04d is also recorded (Figure 4-9c). It is of interest to note thatthe nominal velocity is close to zero. Amplitude of the high frequency modulationsis rather significant, 0.4 rad/s. The modulation frequency of 0.5 Hz is associatedwith the second mode. The required acceleration time history, 44d is presented inFigure 4-9(d). Note, that the acceleration remains positive with a peak modulationamplitude of 1.4 rad/s2 at a frequency of 1 Hz.The required position, velocity and acceleration for the deployment d.o.f. a5 arepresented in Figures 4-10 and 4-11. For zero initial conditions the required valuesare smooth as shown in Figure 4-10(b), 4-10(c) and 4-10(d). However, for the initialconditions of qi = q2 = 0.5 m, the required velocity time history is modulated at highfrequencies associated with the first and second flexible modes. The peak velocityis 3.5 m/s as shown in Figure 4-11(c), while the maximum required acceleration is—2m/s2 (Figure 4-11d).1130.0rad1.00'1400.020.00 Orbit0.02Orbit-0.20.02 ^0.00Figure 4-9 Required slew arm position, velocity and acceleration for the flexiblesystem, with initial excitation of qi = q2 = 0.5 m, to track thespecified trajectory. The manipulator is positioned 50 m from thecenter of the platform.114Required slew: positionrate, accelerationFlexible platform -two modesqi = 0.5, q2 = 0.5 mh1 = 0.01 m/s2(b)2.0 ^required position 134d(d) Required acceleration 134drad sec20.10.0-0.1—1.00. 00 Orbit0 .0(a) Desired Trajectory15.0Y m10.^60.0 X p m 70.0(c) Required rateradsecLocalverticalDesiredtrajectoryRequired dep. : positionrate, accelerationFlexible platform-two modesqi = 0.0, q2 = 0.0• •h1 = 0.01 m/s2 ■as^Localvertical11‘Desiredtrajectory0.050.0^60.0 Xp m 70.0(c) Required rate a5d(a) Desired Trajectory15.0Y m10.05.0b) required position a5d15.^Orbit^0.02(d) Required acceleration a5d0.020.00 Orbit 0.020.5insec0.0—0.5Figure 4 -10 Required deployable arm position, velocity and acceleration in ab-sence of the platform excitation (qi = q2 = 0). The base is initiallylocated at h1 =50 m.115a5 LocalverticalDesiredtrajectory5.0 -^60.0 Xp m 70.0^0.00 Orbit^0.0250.015.0Y m10. Required rate a5dsec0 . 0—4.00.00^Orbit^0.02(d) Required acceleration a5drprOrbit^0.02Required dep. : positionrate, accelerationFlexible platform -two modesqi = 0.5, q2 0.5 mh1 = 0.01 m/s2(a) Desired Trajectory^(b) required position a5dFigure 4-11 Effect of the platform excitation of qi = q2 = 0.5 m on the requiredposition, velocity and acceleration time histories of the deploymentgeneralized coordinate a5. The manipulator starts to translate fromh1=50 m.1164.6 SummaryInverse kinematic relationships to evaluate the required MDM controlled variableswhen tracking a desired trajectory, defined w.r.t. the platform or the orbit, have beendeveloped. It imparts the desired position and orientation to the payload by movingthe MDM base to the appropriate location.From the simulations results it is concluded that the platform vibrations affectthe required degrees of freedom via the displacement and rotation at the base. Toobtain required time histories of the slew and deployment generalized coordinates, atleast the first two modes must be taken into account.1175. CLOSED LOOP STUDY5.1 Preliminary RemarksA relatively extensive open loop study was presented in Chapter 3 and the systemdynamic simulation established. From the open loop study it can be concluded thatit is impossible to achieve even an approximate trajectory tracking, and therefore itis necessary to close the loop with a controller.From the control point of view, a given task is usually accomplished in two dis-tinct steps: First, the gross motion control in which the arms move from an initialposition/orientation to the desired target position/orientation along a planned tra-jectory. Next, the fine motion control in which the manipulator end-effector (SPDM)interacts with the object to complete the task.The purpose of the control in this study is to achieve the desired trajectorymotion of the manipulator end effector with respect to the reference coordinate frame.The trajectory tracking is achieved by controlling the required degrees of freedom asdetermined by the inverse transform technique in Chapter 4. The control actiontakes place in the presence of the platform pitch and transverse vibrations, as well asoscillations at the flexible joints.This chapter focuses on development of a nonlinear control strategy, which utilizesthe dynamic simulation discussed in Chapter 3, to efficiently control the MDM.5.2 Nonlinear ControlThis section is concerned with the design of a reliable control technique in order toensure the desired trajectory tracking performance of the MDM. The highly nonlinearMDM dynamics were accurately modelled in Chapter 2. A control strategy accounting118for the complete nonlinear dynamical model is sought.Control strategies based on linearized system models have been found to be in-adequate because of the stability and robustness problems. Furthermore, there aresituations of practical importance, where the contribution of the nonlinear dynamicscannot be neglected and must be modelled accurately. For example, a problem mayinvolve precise trajectory tracking of a system governed by highly nonlinear dynamicsand maneuvering histories as in the present study. To meet this challenging problem,an inverse control technique is suggested to achieve high tracking accuracy of theMDM, in presence of the platform libration, base translation, and system vibrations.5.2.1 Computed torque techniqueThe inverse control method, also referred as the Computed Torque Technique,which is a particular case of the Feedback Linearization Technique (FLT), was firstinvestigated by Beijezy [53], and applied to the rigid arm control by Fu [2]. In thisapproach, the nonlinear and coupling terms in the equations of motion are eliminatedby judiciously selecting the control input to be a function of these terms. The ad-vantages of this procedure are: accounts for the complete nonlinear dynamics of thesystem; simplicity of the control algorithm and compensator. Spong [54] later ex-tended the method to control a robot with elastic joints. Recently the technique wasextended by Modi and Karray [42] to deal with a flexible space based manipulator.For its application to a flexible orbiting manipulator, authors proposed two differentcontrol schemes: quasi-open loop; and quasi-closed loop.Basically the computed torque technique is a nonlinear control with feedforwardand feedback components. The feedforward components compensate for the interac-tion torques/forces at the joints, while the feedback component provides the necessarytorque to correct any deviations from the desired trajectory.119For example, consider the MDM system described by the governing nonlinearequations in the form[M](q, 04- +TT (q, , t) = (t), (5.1)where and Q are vectors of the generalized coordinates and generalized forces (con-trol forces), respectively. [M] is the symmetric positive definite mass matrix. It is afunction of 4 and t due to the nonlinear and time varying nature of the system iner-tia. N is the noninertial, nonlinear force vector associated with Coriolis, centrifugaland gravitational forces. Note, the nonlinear force vector N also accounts for thestructural and joints damping.The main objective here is to design a control technique to implement valuesof the generalized coordinates for tracking the desired trajectory. To this end, aproportional plus derivative control is used to regulate the various MDM systemactuators: momentum wheels at the platform, and torque motors at the joints. Thusthe control law has the formQ(t) = [114-1(q, t)fid + [Kv][4d — + p][ci — +1TT (q , 4, t), (5.2)where qd, qd, 4d are the desired (required) system degrees of freedom: position, veloc-ity, and acceleration vectors, respectively, as obtained in Chapter 4 from the inversekinematics of the desired trajectory. For the system pitch motion, the required andthe desired values are the same as the platform reference frame Fp is parallel to thesystem frame F0 (Figure 4.1). [Kr] and [K pj are the n x n matrices, of velocity andposition feedback gain, respectively, where n is the total number of the controllabledegrees of freedom. Substituting the control torque vector Q(t) from Eq. (5.2) intothe equations of motion (5.1), the following error dynamics equation is obtained,^[M](q, ORO + [K }(t) + [K]E(t)]^= 0, (5.3)120where e(t) = q(t) - -4(t) is the tracking position error. The mass matrix [M](q,t)is always nonsingular, and [Kp], [Kv] can be chosen appropriately to get negativereal roots for the characteristic equation (5.3), so that the position error vector e(t)approaches zero asymptotically. The optimal choice for [Kv] and [Kp] is:[Kr] = diag[wP, ...,w]; [Kr] = diag[2w0, ...,2wn]. (5.4)This leads to a globally decoupled system with each generalized coordinate respondingas a critically damped second order oscillator. wi is the closed loop natural frequencyof the ith degree of freedom. In order not to excite the platform oscillations and reso-nance at the joints, and to ensure system stability, the closed loop natural frequencywi should not exceed one-half of the structural natural frequency as suggested by Paul[1].In the present study, the total number of the generalized coodinates is non-I + 5,where nom, indicates the number of modes representing the platform flexibility. Herethe first two free-free modes were taken during the closed loop simulation study, hencethe degrees of freedom are seven.In the present control study, the main objective is to maintain the desired atti-tude of the platform while tracking a prescribed trajectory of the payload, and toensure stable response of the flexible joints (elastic torsional deformations within thepermissible limit). Consistent with the main objectives mentioned before, the flexi-ble generalized coordinates are considered nonobservable. The stable response of theplatform is assured because of the structural damping, and by designing the controlto have a closed loop natural frequency less than one-half of the lowest structuralfrequency.The desired or required values of the system degrees of freedom must be assigned121-P1 + T0(Tbii(0)FO + TOF3 + T3 + T4F4 — T4F6 + 7'6 + T6F6 — 7'6^_(5.6)05_(15for application of the FLT to control the MDM. In the present study, required valueswere assigned to the platform pitch (0d), and to the arms' degrees of freedom - /34dfor the slew, and a5d for the deployment. The required values for the arms wereobtained from the inverse tranform of the desired payload trajectory. No desiredvalues were assigned for the joints degrees of freedom: fig, and (35. A method todetermine the MDM joints' desired values, from the given desired values of the arms,has been developed. The system equations of motion (5.1) can be written as[M](q, 04 = F (q , , t) + T (t) = Q(t) — N (q , q, t),^(5.5)where F is the nonlinear force vector N without the coupling force/torque, and T isthe control, coupling torque vector. Eq. (5.5) can be presented in a more detailedform as follows,[m(i , ) 1where:[M(i, j)] the mass matrix with elements as shown in Appendix III;the generalized coordinate vector associated with the platform flexibility;To^the control input torque of the momentum wheels attached at the center ofthe platform;1(0)^the value of the first derivative w.r.t. xi of the mode shapes at the plat-form's midpoint;7'3, T6^the control input torques acting at the joints of the slew and deployablearm, respectively;T4, T6^coupling torques at the joints.122T4 and T6 can be expressed as follows:T4 = k3( 3P403 \n3(5.7)a5T6 = /c5/35.— (5.8)r5 n5Applying the FLT (Eq. 5.2) to the: pitch (0); slew (f34) and the deployment (a5)degrees of freedom, and introducing the required values ,84c1 and a5d, respectively,the following control laws can be obtained:Tod = M(2, 2)[1.4d^Kv0("C•bd —;1))^KpO(Od—^— FO -F [M(2, 1)i+M(2, 3)/33 + M(2,4)/34 + M(2,5)/35 + m(2, *5]; (5.9)T4d = —M(4, 4)[44d^Kv4(44d — 44) + Kp4C84r^04] + F4— [M(4, 1)41 + M(4, 2)7,T) + M(4, 5)45]; (5.10)T6d _^NI.=^Tr^ /6)Lu5d^-II v6 V45d^(-45) + -Exp6ka5d — as)] + F6r5—[M(6, 1)6. + M(6, 2)1.41. (5.11)Here:Kpo, Kv0Ks, Kv4K6, Kv6position and velocity feedback gains in the pitch (0) degree of freedom;position and velocity feedback gains in the slew (f34) degree of freedom;position and velocity feedback gains in the deployment (a5) degree offreedom.Tod is the required control input in pitch; and Lid T6d are the required couplingtorques in slew and deployable joint output, respectively, in order to asymptoticallydecrease the tracking error to zero.123Substituting for T4d and T6d into Eq. (5.7) and (5.8), respectively, the requiredvalues for the joint rotor positions 03d and 05d, can be written as:a 714d133dd = (P4 — in3;135d = ( a 5 T6d)n5.r5^1c5(5.12)(5.13)The required velocity and acceleration can be obtained by differentiating (33d and ,85dwith respect to time:d^d243d = .c7t.133d,^43d = dt2)33d;^(5.14)d135d=d245d clif135d.(5. 15)Applying the FLT to the MDM joints, with the required values obtained in equations(5.12) to (5.15), the control laws for the joint motors take the form:T3d = M(3, 3)[/33dKv3(43d — 43) + Kp3(03d — 133)] + C3/33 —T4dn3+M(3,1)4.1 + M(3,2)1,.6;^ (5.16)T6d = M(5, 5)[45d Kv5(45d — 45) + Kp5 (05d — /95)] + C5+M(5,1)41 + M(5,2)7.,6 + M(5,4)44.^(5.17)Here Ki93, Kv3 are the position and velocity feedback gains for joint 1 (43) degree offreedom; and K7,5, Ks are the corresponding gains for joint 2 (,85) degree of freedom.T3d and T6d are the required control input torques at the slew and deployment joints'motors, respectively, in order to decrease the trajectory tracking error asymptoticallyto zero. The closed loop block diagram for the slew and deployable arm degrees offreedom are shown in Figures 5-1 and 5-2, respectively.T6dn5124F313454T3d(13,[3,13)4dDesiredtrajectoryInversekinematicsSlewcontrollerEq. (5.16)SlewjointdynamicsEq. (5.6)K3(04—(33M3)SlewarmdynamicsEq. (5.6)Figure 5-1^Closed loop block diagram for the slew degree of freedom.125F5a, T5d11 Deploymentjointdynamics+ Eq. (5.6)Desired^Inverse^•••trajectory kinematicsVDeploymentcontrollerEq. (5.17)R5, 135Ts a5^,5 — -r5 n5DeployablearmdynamicsEq. (5.6)a5,isFigure 5 -2^Closed loop block diagram for the deployment degree of freedom.126As mentioned before, the process of getting the required velocity and accelerationof the joints degrees of freedom involves differentiation with respect to time. Inreal systems the signals contain noise; hence it is necessary to include some filteringprocedure in the numerical differentiation. To avoid this problem a different approachis suggested as discussed in the following section.5.2.2 Joints stiffness controlA new control approach is proposed in order to avoid the numerical differentiationof the required joint degrees of freedom when applying the FLT. It is suggested thatthe joint stiffness be controlled according to the actual twist at the joints and therequired input torque to the arms. Thus, the stiffness of the joint, represented by theflexural rigidity of the torsional shaft, is not constant. For example, an arrangementthat varies the equivalent length of the shaft can be adopted. This can be achievedby shifting the point of application of the torque to the shaft by the joint motor.Alternatively, a controllable clutch (say, a magnetic clutch) may be introduced tochange the equivalent shaft cross-section moment of inertia. A multiple elementclutch can engage with different sections of the shaft thus changing the cross-sectionarea effective in transmitting the torque.The main objective in this new approach is to determine the required joint stiff-nesses k3d and k5d in order to obtain the required torques T4d and T6d given in Eqs.(5.10) and (5.11), respectively. For this case the arms' required position and velocityas obtained from the inverse kinematics are assumed to be the required values for thejoints degrees of freedom. Applying Proportional plus Derivative (PD) control to thejoints' degrees of freedom, the control law takes the form:(33^fi.3 )1 re-T3d = [-Elv v3(P4d – n-3 ) v la^-n-3 11-Lxm3;(5.18)127a5dT5d = • [Kvs(---a5d — 4-5 ) Kp5^— )1 Km5r5^n5^r5^n5(5.19)T3d and T5 d are the control input torques at the slew and deployable joint motors,respectively, in order to decrease the error at the joints to zero. Kra, K m,6 are theslew and deployable torque motor gains, respectively.Application of the control inputs T3d, T6d to the joints with the nominal constantstiffness K3, K5 may not necessarly meet the requirement of the input torques to thearms, T4d and T6d, as found by the FLT (Eqs. 5.10, 5.11). As a consequence trackingmay not be possible. To overcome this problem, the joint stiffnesses must be changed,as proposed.The relations governing the input torques to the arms and the stiffness of thejoints are given in Eqs. (5.7), (5.8). The required joint stiffnesses can be calculatedas follows:T4dk3d = ^ •(04 —k5d = T6d—a5^ (5.21)T5 n5which provide the required input torques to the slew and deployable arms, respec-tively.5.3 Controlled System StudyAn extensive open loop study was presented in Chapter 3. The simulation resultsclearly showed that a controller must be incorporated in the MDM system in orderto successfully perform any desired task. Now the control law developed in Section5.1 is to be implemented to achieve a closed loop response. The system parametersare the same as for the open loop case (Table 3-1). The only additional parametersare the position and the velocity gains Kp K, respectively.(5.20)128As mentioned in the previous section, with an appropriate choice of the feedbackgain Kt, and .4, each degree of freedom can be made to respond as a decoupledcritically damped second order oscillator. In this case, the gains are functions of thesystem closed loop natural frequencies as shown in Eq. (5.4). For stability reasons,the closed loop natural frequency for each degree of freedom cannot exceed one-halfthe value for the associated open loop case. The feedback gains can be obtained fromthe open loop simulation results presented in Chapter 3, and are given in Table 5-1.Table 5-1^MDM closed loop velocity and position feedback gains.d.o.f Lao, rad/s Lac, rad/s Kv = 2wc Kp = 4qi 1.13 — — -'72 3.11 — — —IP 0.002 0.001 0.002 1.10-603(n3 = 1)9.6 1.5 3 2.25#4 0.002-0.01 0.001 0.002 1 - 10-6fi5(n5 = 1)9.6 0.6 1.2 0.36a5(r5 = 0.1)2.8 0.6 1.2 0.36Here:wo^open loop natural frequency as found from the simulation results;Loc^the desired closed loop natural frequency;Kv^the velocity feedback gain;ifp^the position feedback gain.It is apparent from Table 5-1 that not all the closed loop natural freequencies,129coc, have been chosen according to the rule of one-half. For example, in the case ofjoint 1 (03), the dynamic coupling between the platform and oscilations at the jointwill exite the second mode with a frequency of 3.11 rad/s. Hence, the joint 1 closedloop natural frequency is chosen to be 1.5 rad/s.An extensive study of the MDM closed loop response was carried out. For betterappreciation the system was analyzed in an increasing order of complexity. The studystarts with the stationkeeping mode (fixed desired position) for a rigid or flexibleplatform. This is followed by the response investigation during trajectory trackingwith stationary or moving base. Note, the effects of initial position error are accountedfor. Finally, the attention is focused on the trajectory tracking for system withexternal disturbances.The main goal of this closed loop simulation study is to assess the controlledtracking performance of flexible space based manipulator, i.e. trajectory tracking inthe presence of system libration, base translation and rotation.The amount of information obtained through a planned variation of the systemparameters in the above mentioned case is rather extensive. For conciseness, only thetypical results useful in establishing trends are presented here.The MDM closed loop simulation flow chart is shown in Figure 5-3. The systemproperties and the initial conditions are supplied by the input data to the mainprogram. The input to the inverse kinematics block is the desired trajectory, theactual state of the platform (deflections and slope) and the MDM base location.The output of the inverse kinematic analysis gives the required values for the systemdegrees of freedom which serve as an input to the control block (actual state variables).Output from the control block is the control efforts (generalized torques) for variousdegrees of freedom. Input to the dynamics block are the generalized forces, system130SYSTEM PARAMETERSINPUTDATA0M ccc)z(1) 0VDESIREDCONTROL INPUTS MAINBLOCKTRAJECTORYREQUIREDSTATESACTUAL STATESINVERSEKINEMATICSSYSTEMDYNAMICSINTEGRATIONVCONTROLFigure 5-3^MDM closed loop simulation flow chart.131parameters and the actual state of the system. The output from the integration blockis the new state of the system.5.3.1 StationkeepingThe system closed loop response, when the desired payload position is fixed withrespect to the platform, is studied here. The tracking error and the control effortshave been evaluated, for a rigid or flexible platform, when the pitch initial conditionis 0.02 rad (1.1 deg from the local vertical). The desired payload position in this casewas Px = 10m, Py = 0 in the Xp and Yp coordinates, respectively (Figure 4-1). Inothers words, the desired position of the payload is on the platform 10 m from itscenter. The desired position of the platform is along the local vertical (0 = 0).The closed loop responses for both the rigid and flexible platforms are shown inFigure 5-4 and 5-5, respectively.Figure 5-4 presents simulation results over 0.02 orbit (110 s). The parameters ofimportance are the degrees of freedom, tracking error, control efforts and twist at thejoints.Figure 5-4(c) shows controlled response of the degrees of freedom //), (33, 134. Thepitch (0) decreases asymptotically to zero. As a consequence, joint 1 motor (33)reacts relatively fast to drive the slew arm (fl4), in order to decrease the error. Figure5.4(b) shows the tracking errors in platform pitch, slew and deployment. As canbe seen, the initial pitch error of 0.02 rad is asymptotically decreased to zero withthe time constant determined by the feedback gains. The deployment error remainsessentially zero. The maximum slew error is about 0.0005 rad and decreases to zeroasymptotically as can be seen from the expanded view of ,84. Figure 5-4(a) showsthe desired and the actual positions of the payload in the platform reference frame1320.0Nm-20000J30,000. 00 0.02J5(a) Payload trajectory^(b) D.O.F. error0.005 ^0.020-0.0059.0Y m0.00010.0 X p rn^0.000.°^0.00 Orbitrad0.010[340.00^0.020.02(c) D.O.F. response (d) Control torques0.015rad0.000134-0.015^0.00^Orbit(e) Joint twist0.02^0.00^Orbit^0.02(f) Control torques-10.000.00Nm0.014rad0.000TiJ5actualdesired0.00T3-0. 0 ^0.00T50.02Closed loop response - rigid platformpoint tracking - px=10m, p,=00.00^Orbit^0.02^0.00^Orbit^0.02Figure 5-4^Closed loop response during stationkeeping for the rigid platformcase with an initial disturbance of = 0.02 rad.133Xi,, Yp. Note, the resultant position error is between -0.001 m to 0.0035 m along theYp direction.The control effort and twist at the joints are shown in Figures 5-4(d), 5-4(e),5-4(f). The maximum control effort at the slew joint T3 was found to be around -17Nm (Figure 5-4d). The torque decreased to zero asymptotically with an overshoot of4 Nm. The associated twist at the slew joint J3 is 0.003 rad, as can be seen in Figure5-4(e). The maximum control effort at the deployment joint T5 was approximately-0.01 Nm, and the steady state value of around -0.008 Nm is required in order toovercome the gravitational effect (Figure 5-4d). The associated twist at the joint J5remains relatively small due to the low torque value and the equivalent gear ratio of 10(r5 0.1 m). The control effort provided by the momentum wheels (T1) is —2.9 • 104Nm as shown in Figure 5-4(f). The huge demand is due to the large moment of inertiaof the platform and the relatively high feedback gains.Figure 5-5 presents the same case as discussed in Figure 5-4 but now accountingfor the platform flexibility. The platform is modelled by the first two free-free modeswith the natural frequencies of 1.13 rad/s and 3.11 rad/s, respectively.At the outset it is apparent from Figure 5-5(c) that the pitch response is notaffected by the platform vibrations. The initial pitch disturbance of 0.02 rad decreasesto zero asymptotically. The response of the slew joint motor (f33) is the reaction to theerror that has been developed by the pitch motion. The associated impulse excitesthe platform vibrations (Figure 5-5f). In this case, the MDM base is located at thecenter of the platform, hence the displacement Si and the rotation al of the base arerelated to the first and second mode, respectively. The deflections are modulated atthe high frequency associated with the second mode. The rotational oscilations ofthe base affects the required slew as apparent from the responses [33 and 04 in Figure134(a) Payload trajectory (b) D.O.F. error.0003rad. 00000.00^Orbit 0.020.005Y m0.000—0.0059.995 10.000 Xp mevq104000000Vk T5T30.00e)J3OrbitJoint twist0 .02J50.015radClosed loop response Flexible platformpoint tracking - px=10m, py=0(c) D.O.F. response (d) Control torques0.00Nm-10.000.015rad0.000-0.0150.00^Orbit^0.02(f) Platform response0.000581^al0.0rad0 .0000.00 Orbit 0.02^0.00 Orbit^0.005Figure 5-5 Closed loop response for point-tracking (stationkeeping) of the pay-load, supported by the manipulator, located on the flexible platform.The manipulator is at the center of the platform, and arm 2 is de-ployed 10 m. The initial disturbance is 0.017 rad in pitch.1355-5(c). The residual tracking error, because of the base vibrations, are negligiblecompared to the oscillation amplitude (Figure 5-5b). The small steady state error inthe deployment degree of freedom is attributed to the gravitational force acting onthe payload, which is deployed 10 m from the orbit along the local vertical. As canbe expected, the control efforts at the slew joint and the twist (T3, J3) are affectedby the base rotational oscillations as shown in Figures 5-5(d) and 5-5(e), respectively.The main conclusions from the closed loop simulation results for the stationkeep-ing case are: a) it is possible to achieve point tracking (stationkeeping) for this class ofspace-based manipulators in the presence of platform maneuvers and base vibrations;b) the tracking error is essentially unaffected by the flexibility of the platform; c) theerror in the controlled variables approaches to zero asymptotically.5.3.2 Trajectory trackingThe next logical step would be to assess performance of the controller in trackingthe desired trajectory. This subsection presents the closed loop response when thespecified payload position describes a trajectory with respect to the platform. Thetracking error and the control effort have been evaluated for the fixed as well as themoving base. The prescribed position of the platform is along the local vertical. Thespecified initial velocity components of the payload are 0.2 m/s and 0.03 m/s in the Xpand Yp directions, respectively. The corresponding desired acceleration componentsare -0.005 m/s2 and 0.002 m/s2. The base is initially located at the center of theplatform, and the payload is deployed 10 m along the platform (Px = 10 m, Py = 0).The closed loop responses for stationary as well as the moving base are presented inFigure 5-6 and 5-7, respectively.Figure 5-6 shows the simulation results over 0.02 orbit (110 s). The parametersdiscussed are the degrees of freedom response, tracking error, control effort and the1360.0rad-0.10.00^Orbit^0.02(d) Control torques0.00.0^10.0 Xp m(c) D.O.F. responseY m10.00. 000.020.00 0.005Orbit OrbitOrbitJoint twist (f)0.0005radPlatform5-0.0 0.0responseClosed loop Trajectory trackingr=• 2, py=.03m/s ,p.=-.005, py=.002m/s2(a) Payload trajectory^(b) D.O.F. errorFigure 5-6^Closed loop response during the trajectory tracking, with the baseheld fixed and arm 2 initially deployed 10 m along the platform.137twist at the joints.Figure 5-6(a) depicts the desired trajectory with respect to the platform referenceframe F. The controlled response of the degrees of freedom is shown in Figure 5-6(c).The large displacement at joint 1 (03) is due to the initial desired values (velocityand acceleration) when the initial conditions of the controlled variables are all zero.Note, the slew of arm 1, P4, follows the joint motion. The slew response excitesthe pitch with a maximum attitude error of —5 • 10-5 rad. The tracking error wasrelatively significant in the beginning and reached a steady state value of -0.025 m forthe deployment degree of freedom (Figure 5-6b). The slew error increases as the armapproaches the orientation of 90 deg w.r.t. the platform. In that position, a relativelylarge desired maneuver causes the tracking error to increase to 0.002 rad, as seen inthe expanded view of /34 (Figure 5-6b). The payload follows the desired trajectory(Figure 5-6a) with the tracking accuracy according to the error in the individual arm(Figure 5-6b).The time histories of the control efforts are presented in Figure 5-6(d). In thebeginning, the control efforts are relatively large, as expected, in order to decrease theinitial error in velocity and acceleration. As a result, the flexible platform is excitedwith linear and angular oscillations Si and respectively, as shown in Figure 5-6(f).A small reduction in amplitude is attributed to the structural damping. The angularoscillations of the platform affect the control effort of the slew joint T3 and platformpitch T1 as can be seen in Figure 5-6(d). The control effort of the deployment degreeof freedom is not affected because of the gear ratio and the relatively small oscillationsof 5.10-4. The twist at the joints is large at the begining in order to meet the demand,of required output torques, as shown in Figure 5-6(e).Figure 5-7 presents the system response while performing a task in which the138payload is required to translate, along the platform at a constant speed of 0.5 m/s,over a distance of 10 m. Simultaniously, the MDM base is moving in the samedirection at the same velocity. In others words, the slew arm is perpendicular tothe platform while the base is translating along the platform, and the deployablearm keeps the payload at a constant distance from the undeformed centerline of theplatform.The control efforts for this case are shown in Figure 5-7(d). To overcome theinitial error as a result of the fact that the payload has no initial velocity, the controleffort T3 at the slew joint is relatively large. Once the payload achieves the desiredvelocity, the demand on the controller decreases to the level required to compensatefor the gravitational force. As the arms and payload deviate from the system c.m.,more control effort is needed to overcome the gravitational effect. The control effortT5 at the deployment joint is required to compensate for the Coriolis reaction due tothe translation of the payload.As a result of the input torques, the platform flexible modes are excited as shownin Figure 5-7(f). Here 61 represents the linear displacement at the base and ai thecorresponding rotation. Note, while moving along the platform, the base crossesthe nodal points: i.e. zero slope and displacement, at 0.008 orbit and 0.016 orbit,respectively. The effect of the platform oscillations can be clearly discerned fromthe response of the deployment and slew degrees of freedom as presented in Figures5-7(c) and 5-7(e), respectively. As can be expected, at the nodal points, effect of theplatform vibrations is relatively small, as apparent from the control inputs.The tracking errors for the individual degrees of freedom are shown in Figure5-7(b). For all practical purposes they are negligible. The steady state error for thedeployment arm is 10-5 m, and the corresponding errors in pitch and slew are also of139Xp m 50.02.5E-rad0.000.00^Orbit^0.0113 40.00Nm2.00T3-AwoNOWARMT5^•,1.570750.001.57085rad04VrIYAVA"-^401010000004^4i010106Orbit 0.02Closed loop Trajectory trackingBase motion h1=0.5m/s^Px=0.5m/s(a) Payload trajectory^(b) D.O.F. error(c) D.O.F. response^(d) Control torques0.00^Orbit^0.02(e) D.O.F. response0.00^Orbit^0.02radin0.0(f) Platform response0.00020.00^Orbit^0.02Figure 5 - 7^Closed loop response during the trajectory tracking for the MDMwith the arm perpendicular to the platform.140the same order of magnitude. The actual and the desired trajectories are compared inFigure 5-7(a). Note, the total tracking error is small even compared to the vibrationsamplitude of the platform.The simulation results clearly point out effectiveness of the nonlinear controlstrategy even when applied to such highly flexible space-based manipulator. It isimportant to recognize that the tracking performance is essentially unaffected by thevibration of the MDM base.5.3.3 Special casesIt was decided to assess effectivness of the nonlinear controller under several,particular demanding situations such as: the system with a relatively large initialerror for the payload position; the system subjected to an external disturbance; andan inherently unstable orientation of the system. The issue concerning the choice ofthe feedback gains also must be addressed. With this objective in mind, to beginwith, the system with initial position errors of /34 = 0.02 rad in slew and as = 1.0m in the-deployment degree of freedom is considered. The simulation results for thiscase are shown in Figure 5-8.Figure 5-8(b) presents time histories of the errors corresponding to the generalizedcoordinates associated with the platform pitch (0), slew (f34) and deployment (as).Note, the initial errors decrease significantly as a result of the control effort. Thecontrol torques and the dynamic reactions excite the flexible platform (Figure 5-8f). The manipulator base is initially located at the center of the platform, hence,the contribution to the displacement 61 is mainly from the first mode. The base ismoving with a constant velocity of 0.01 m/s and the vibration amplitudes decrease.The actual and the desired payload trajectories are shown in Figure 5-8(a). It141200.0 k^Nm(c) D.O.F. response20.0^15radOrbitjoint twist0.00.00(e)0.2•Closed loop Trajectory trackingInitial error: a5=1.0m , 04=0.02rad(a) Payload trajectory10.0 Xp M13 4rad0.00-0.03^0.000.03 (b) D.O.F. error0.02Orbit10. 0134 -200.0-^0.02^0.00 Orbit 0.02(f)0.004 Platform responseOrbit -0.004 ^0.02^0.00•Orbit••0.0 0•• • •al0.02Figure 5-8^Closed loop response for trajectory tracking with initial position er-rors.1420.0J5(d) Control torquesrad'-0.0rad0.000is apparent from the expanded view of the start of the tracking maneuver that theMDM has a relatively slow response. This is a consequence of the relatively lowfeedback gains selected to avoid excitation of the structure. After 0.001 orbit (5.5sec) the initial error has diminished significantly. From Figure 5-8(c), it can be seenthat the joint degrees of freedom #3 and 05 react relatively fast and strongly to theinitial error. In real systems, for such a case, the joint motors may get saturated andhence may not be able to supply the required torques. To overcome this problem agear ratio of more than 1:1 has to be considered. After the transient maneuver, thearms follow the joints up to the twist angle as shown in Figure 5-8(e).In the earlier closed loop simulations, the platform initial conditions were taken tobe zero. Yet the platform was excited due to coupling between the degrees of freedom.In the present tracking simulation, the platform is initially disturbed equally in allthe modes to give 41 = 0.1 m. The effect of the platform initial condition and thefeedback gains on the tracking performance are presented in Figure 5-9. Note, theconfiguration here is similar to that described in Figure 5-7. The slew arm is forcedby the motion of the base and the desired payload trajectory to align perpendicularto the platform.Figure 5-9(f) shows the platform deflection Si and slope al at the location of thebase. The initial displacement for the base located at the center of the platform is0.12 m (due to the modal disturbances). The platform dynamics is affected by theMDM and the maximum amplitude is 0.2 m. When the base is close to the nodalpoint, the platform response at the base reaches a minimum. Note, in the presentcase, the vibration of the base primarily affects the deployable arm. The control effortat joint 2 is shown in Figure 5-9(b) for the velocity and position feedback gains of= 1.2 and .rfp = 0.36, respectively. The peak control effort is around 150 Nm.143if81cxiClosed loop Trajectory trackingPlatform disturbance q1=0.1m(a) Payload trajectorydesiredNm-^200.000.00-200.0(b) Control torquesT5Y m10.210.09.8'actual0.0^Xp m 50.0(c) Payload trajectoryYP m ^Nm10.2^ 200.0actualdesiredXp m 50.00.00^Orbit^0.02(d) Control torques10.09.80 . 0 1011T5-200.00.00^Orbit^0.02(f) Platform response0.2rad-0.0- 0.2II0.00^Orbit 0.02Figure 5-9^Controlled response for trajectory tracking with base motion 0.5 -7and the initial conditions of 4). = 0.1 m imposed on the platform.144The resultant tracking performance is shown in Figure 5-9(a). Note, even withsuch a severe disturbance, the controller is successful in limiting the peak error toaround 0.1 m. The performance can be improved further by increasing the gains toKv = 6 and Kp = 9. As a result the closed loop bandwidth has increased. Thetracking performance for the high feedback gains is shown in Figure 5-9(c). Themaximum tracking error is now reduced to 0.03 m and the peak control effort isslightly increased (Figure 5-9d).From the simulation results it can be concluded that: external disturbances to theplatform can adversely affect the tracking performance; the feedback gains dominatethe closed loop behavior; and by proper selection of the gains, the controller canprovide an acceptable performance.The question concerning preferred orientation of the platform from the manip-ulator's performance consideration has not been addressed yet. The obvious choicewas the gravity gradient configuration where the system is inherently stable. In thenext closed loop simulation, the platform was aligned with the local horizontal, i.e.,an unstable orientation, and the control effort in pitch compared with that of thegravity gradient case. The results are presented in Figure 5-10. The configuration ofthe system is similar to that described in Figure 5-9, except for the platform whichis now oriented along the local horizontal.The tracking performance data in Figure 5-10(a), clearly show that the orienta-tion of the platform has virtually no effect on the trajectory tracking. The attitudeerror in pitch and the associated control effort are shown in Figures 5-10(b) and 5-10(d), respectively. The control effort is a function of the base and payload locationsas well as the platform vibrations. As can be seen, the attitude error is kept lowwhile the control effort is increasing according to the MDM base location along the145(a) Payload trajectoryY m10.02^actualrils10 . 005.10- (b) D.O.F. errorrad. 02.0rad1.0^^0.00 Orbit^0 .0161alOrbit^0.0211Closed loop Trajectory trackingLocal horizontal configuration11^•9.98^desired0.00^Xp m^50.0^0.00^Orbit^0 .02(c) D.O.F. response^(d) Control torques1500.00.0-1500.0.00^Orbit^0.02(f) Platform response0.2rad]in-0.0-0.20.00Figure 5 -10 FLT controlled response for the trajectory tracking with the platformalong the local horizontal and the base translation at 0.5 T.17' . Theinitial condition is = 0.1 m.146platform and the local vibration amplitude as shown in Figure 5-10(f). As expected,the control effort required to keep the platform in the desired orientation, when thebase is close to the tip of the platform, is less compared to the corresponding gravitygradient case with MDM arm normal to the platform.To assess effectivness of the controller in enabling the MDM-payload to followa desired trajectory, even when subjected to disturbances of extreme severity, theinitial conditions were increased to 41 = 0.5 with the position errors for the slewingand deploying arms set at 0.3 rad and 2 m, respectively. The tracking performanceresults for the MDM are presented in Figure 5-11. The desired trajectory propertieswere described earlier in Figure 5-6.Figure 5-11(b) shows the resultant linear and angular oscilations at the centerof the platform where the MDM base is located. The displacement amplitude 6).associated with the first mode is 0.6 m and the slope angle al associated with thesecond mode is 0.04 rad. The tracking performance results are presented in Figure5-11(a). As the initial position error is large, the transient response is relativelyshort and the tracking error decays to less than the vibration amplitude. From theexpanded view, it can be seen that the tracking error in the transient is 0.1m. Itdiminished to almost zero after 0.02 orbit as shown in the enlarged view.The control effort time-histories for this case are shown in Figure 5-12. Figure 5-12(a) presents the control efforts for a gear ratio of n3 = 1 at the slew joint. Note, thetorque (2'3) required to control the slew arm is enormous due to the large vibrationsof the platform. When the slew arm approaches the 90° position, the effect of thevibrations is gradually diminished. In fact, now the vibrations begin to affect thedeployable arm. The control effort for this case is relatively small due to the effectivegear ratio of 10. The torque required to maintain the desired attitude of the platform147Closed loop Trajectory trackingLocal horizontal configuration(a) Payload trajectoryYp10.00.00.0 10.0 Xp In1.00radm(b) Platform response‘^810.00 ilk, ' AMI VIAMAWWWWVW1 VW,Ital-1.00 ^0.00 Orbit^0.02Figure 5-11 Controlled response during the trajectory tracking for the local hori-zontal configuration of the platform. The initial condition is 41 = 0.5m for the platform. The initial position error are: 2 m in deployment;and 0.3 rad in slew.148Closed loop Trajectory trackingLocal horizontal configuration(d) Control torques20000.Nm0.0T5-20000^0.00 Orbit^0.02(d) Control torques-5000.0.00 OrbitTS:0.025000.0 T3Nm0.0Figure 5- 12^Control effort time histories for the system described in Figure 5-11:(a) gear ratio n3 = 1; (b) gear ratio n3 = 10.149is the result of the reaction moment from slew joint torque motor.To reduce the control effort at the slew joint, gear ratio of n3 = 10 was introduced(Figure 5-12b). The higher gear ratio reduces the torque demand to slew the armfrom around 15,000 Nm to around 5,000 Nm. The joint rotor oscillates (83) at higherfrequency and amplitude when the gear ratio is increased in order to supply the torquein the right direction.Thus the results suggest that the gear ratio is an important parameter in thedesign and closed loop simulation of flexible space based manipulators.5.4 SummaryThe control study showed that it is possible to achieve high tracking performancewith the end effector of a flexible space based manipulator, even under extreme con-ditions of platform maneuvers and base oscillations.The strategy based on the FLT has been found to be effective in controlling themanipulator arms as well as the attitude motion of the space station. To successfullyimplement the controller, the MDM base must be provided with sensors to measurethe local slope and the deflection of the flexible platform. The robustness issue,though not addressed here, is a valid one for the present study where the system isrepresented by a relatively accurate model.The most important parameter is the feedback gain. Performance of the manip-ulator, tracking a desired trajectory, is not satisfactory when the gains are selectedaccording to the common recommendation of one-half the lowest structural frequency.When the system is subjected to external disturbances, for example docking of thespace shuttle, higher gain values may be necessary.To implement the FLT, desired values must be assigned to each degree of freedom.150A procedure for real time evaluation of the joint parameters has been developed. Toavoid numerical differentiation required for this purpose, introduction of the jointstiffness as a control variable is suggested.1516. CLOSING REMARKS6.1 Concluding CommentsA relatively general formulation and associated simulation tool for studying thenonlinear dynamics and control of a space based flexible, mobile, deployable manip-ulator have been presented. The formulation employs Lagrange's approach. It leadsto a set of nonlinear, nonautonomous and coupled equations of motion, governing thesystem dynamics.In general, implementation of the equations into a comprehensive computer codeinvolves enormous amount of effort. This is minimized through the use of vectoroperations and judicious synthesis of mathematical expressions leading to a relativelycompact form of the equations of motion. The system is amenable to further simpli-fication by neglecting the effect of the shift in the center of mass. This is justifiablefor relatively small slewing maneuvers and slow deployment of the second arm.To close the loop, a nonlinear control strategy based on the Feedback Lineariza-tion Technique (FLT) is adopted and incorporated in the program to form a completesimulation tool for design and performance evaluation of a large class of MDM sys-tems.Validity of the formulation and the computer code is established through com-parison with particular cases and by checking the conservation of the total energy.The computer program is so structured, in a modular fashion, to permit detailedparametric analysis. The versatile simulation is applied to study three major aspects:open loop response; inverse kinematics; and closed loop performance.A manipulator with a synthesis of revolute and prismatic joints is a novel idea,never explored in depth before, for application to space based platforms. Thus the152formulation, associated computer code, parametric analysis and control strategy rep-resent original contributions of far-reaching significance. Of particular importance isthe application of the FLT for control, which accounts for the complete nonlinear dy-namics of a class of complex flexible, orbiting MDM's. This has never been attemptedbefore, and presents possibilities of exciting developments in the areas of robotics andadaptive structures.Based on the open loop and control analysis, following general conclusions can bemade:Open-Loop Systems (i) Even for simple configurations the response of the MDM system is complexand unpredictable.(ii) The MDM system is unstable even when the platform and the slew arm arein the gravity gradient orientation, especially when the MDM base is locatedaway from the mid-point of the platform.(iii) The flexibility of the platform affects the dynamic response of the systemsignificantly and hence cannot be neglected. Platform vibrations change thefrequency, amplitude and the equilibrium position of the slew arm.(iv) Even small vibrations of the platform can create significantly different envi-ronment for the MDM, and the resulting coupling dynamics can cause thesystem to become unstable.(v) Introduction of damping in the degrees of freedom adversely changes the sys-tem's dynamical response by exerting moments at the joints.(vi) To obtain an accurate response, it is necessary to take into account morethan one flexible modes for discretization of the platform deformations, This153removes constraints on slope and deflection at any station along the platform.(vii) The vibration of the platform creates a pseudo-gravity field in the directionof the deflections which, in some cases, can be used to advantage for locatingthe slew arm in a position perpendicular to the platform.(viii) For space based manipulators of the class discussed here, it is necessary tocontrol the degrees of freedom to achieve desired performance. It is unsafe todepend even on the so called stable equilibrium positions to locate the arm,as in the case of ground based robots.Inverse Kinematics(i) It is possible to achieve trajectory and orientation tracking for a class of twodegrees of freedom manipulators with a coordinated base motion.(ii) The parametric study suggests that the platform vibrations affect the requireddegrees of freedom for tracking via the displacements and the rotations at theMDM base.(iii) In order to take into account the effect of the base motion (displacement,slope) on the required values at each location along the platform, at least twoflexible modes are necessary.Closed-Loop Systems (i) The control strategy based on the FLT is found to be effective in controllingthe manipulator arms as well as the attitude motion of the platform.(ii) The flexibility of the platform has significant influence on the control effort,which is proportinal to the vibration amplitude.(iii)^It is possible to reduce the control effort by increasing the gear ratio, implying154a need for the controller with a higher bandwidth.(iv) To achieve better tracking accuracy in the precence of extreme external dis-turbances, higher gains than that those normally recomended in the standardtextbooks are required.6.2 Original ContributionsA complete simulation tool that can handle a flexible space based deployable typemanipulator has been developed and analyzed in this thesis. The distinctive featuresof this simulation are:• two degrees of freedom slewing-deployable type robot, with two more degrees offreedom for the flexible, dissipative joints;• torque motors with gears at the joints;• moving base traversing a vibrating platform free to librate in orbit;• arbitrarily shaped payload supported at the tip of the deployable arm;• inverse kinematics for flexible systems;• payload capable of tracking any predefined trajectory.The presentation of a novel concept, demonstration of its advantages, and thor-ough dynamical as well as nonlinear control analyses provide a rather selfcontainedcharacter to this innovative contribution. It leads to some interesting observations,analysis and discussions concerning:• the pseudo gravitational field induced by the platform vibrations;• the coordinated base motion in order to position and orient the payload;• the variable joint stiffness and its use as a control variable.1556.3 Recommendations for Future StudyThe study of the mobile flexible manipulator, free to slew and deploy, thoughreasonably comprehensive with reference to the stated objectives aimed at laying asound foundation, represents only the first step in the development of an innovativeidea. The possible exciting developments are indeed many. Some thoughts concerningits future evolution, which are likely to be satisfying, are indicated below:(i) Introduction of the out-of-plane degrees of freedom will expand the capabilityof the MDM making it more versatile. It will be able to deal with the orbitnormal configuration and assess gyroscopic effects on the MDM's performance.(ii) In the present study, flexible character of the manipulator is reflected by theflexibility of the joints. For the out-of-plane study, it would be useful toincorporate flexibility of the deployable arm, as well as the torsional degree offreedom of the platform and the arm.(iii) When dealing with relatively heavy payloads, the boundary conditions at theplatform change with respect to time and must be taken in account in orderto get accurate mode shapes.(iv) For the time varying systems such as the MDM, an adaptive control strategymay be incorporate in order to improve the tracking performance.(v) Extension of the FLT for nonlinear control with flexible degrees of freedomparticularly with reference to the robustness, is necessary.(vi) A systematic approach to select the controller gains and gear ratios is required.The tracking performance in the presence of system constraints (maximumpower, bandwidth) needs further attention.vii)^Application of the concept to the development of adaptive structures appears156quite exciting. It can lead to the development of an entirely new field witha wide range of applications. Dynamics and control of variable geometrystructures obtained through slewing-deploying links have never been studiedbefore.157BIBLIOGRAPHY[1] Paul, P.R.,Robot Manipulators: Mathematics, programing and Control, The MITPress, Cambridge, Massachusetts, 1983.[2] Fu, K.S., Gonzales, R.C., and Lee, C.S.G., Robotics: Control, Sensing, Visionand Intelligence, McGraw-Hill, Inc., New York, 1987.[3] Chan, J.K.W, Dynamics and Control of an orbiting Space Platform Based MobileFlexible Manipulator, M.Sc. 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August 1991, Paper No. AAS 91-399.[13] Singh, R.P., Vandervoort, R.J., and Likins, P.W., "Dynamics of Flexible Bodiesin Tree-Topology - A Computer Oriented Approach," Paper No. AIAA-84-1024,AIAA/ASME/ASCE 25th Structural Dynamics, and Materials Conference, PalmSpring, U.S.A., May 1984.[14] Kane, T.R., Ryan, R.R., and Banerjee, A.K., "Dynamics of a Cantilever BeamAttached to a Moving Base," Journal of Guidance, Control, and Dynamics, Vol.10, No. 2, March-April 1987, pp. 139-151.[15] Reddy, A.S.S.R., Bainum, P.M., and Krisna R., "Control of a Large FlexiblePlatform in Orbit," Proceedings of the AIAA/AAS Astrodynamics Conference,Danvers, Massachussetts, August 1980, Paper No. AIAA 81-122.[16] Bainum, P.M., and Kumar, V.K., "On the Dynamics of Large Orbiting FlexibleBeams and Platforms Oriented Along the Local Horizontal," Proceedings of the31st International Astronautical Congress, Tokyo, Japan, September 1980.159[17] Lang, W., and Honeycutt, G.H., "Simulation of Deployment Dynamics of spiningSpacecraft," NASA TN-D-4074, 1967.[18] Cloutier, G.J., "Dynamics of Deployment of Extendible Booms from SpinningSpace Vehicles," Journal of Spacecraft and Rockets, Vol. 5, May 1968, pp. 547-552.[19] Hughes, P.C., "Dynamics of a spin-Stabilized Satellite during Extension of RigidBooms," CASI Transactions, Vol. 5, 1972, pp. 11-19.[20] Sellappan, R., and Bainum, P.M., "Dynamics of Spin- Stabilized Spacecraft dur-ing Deployment of Telescopic Appendages," Journal of Spacecraft and Rockets,Vol. 13, October 1976, pp. 605-610.[21] Modi, V.J., and Ibrahim, A.M., "A General Formulation for Librational Dynam-ics of Spacecraft with Deploying Appendages," Journal of Guidance, Control,and Dynamics, Vol. 7. No. 5, September-October 1984, pp. 563-569.[22] Hablani, H.B., "Attitude Dynamics of a Rotating Chain of Rigid Bodies in aGravitational Field," Journal of Guidance, Control, and Dynamics, Vol. 8, No.4, July-August 1985, pp. 471-477.[23] Conway, B.A., and Widhalm, J.W., "Equations of Attitude Motion for a N-BodySatellite with Moving Joints," Journal of Guidance Control and Dynamics, Vol.8, No. 4, July-August 1985, pp. 537-539.[24] Millar, R.A., Graham, W.R., and Vigneron, F.R., "Simulation of the Motionof a Shuttle Attached Flexible Manipulator Arm,"Proceedings of the Tenth In-ternational Association for Mathematics and Computers in Simulation, WorldCongresss on System Simulation and Scientific Computation, Montreal, Canada,August 1982, Vol. 3, pp. 225-227.[25] Ravindran, R., Sachdev, S.S., and Aikenhead, B., "The Shuttle Remote Manip-160ulator System and its Flight Tests" Proceedings of the 14th International Sym-posium on Space Technology and Science, Tokyo, Japan 1984, pp. 1125 - 1132.[26] Meissinger, H.F., and Spector, V.A., "The role of Robotics in Space SystemsOperations", A Collection of technical papers of the AIAA Guidance, Navigationand Control Conference, Snowmass, Colorado, U.S.A. August 1985, pp.223-236.[27] Gardner, B.E., McLaughlin, J.S., Tucker, T.E., and Williamson, R.K., "AerospaceInitiative in Robotics Research," A Collection of Technical Papers of the AIAAGuidance, Navigation and Control Conference, Snowmass, Colorado, U.S.A. Au-gust 1985, pp. 237-245.[28] Bejczy, A.K., Kan, E.P., and Killion, R.R., "Integrated Multi-Sensory Contol ofSpace Robot Hand," A Collection of Technical Papers of the AIAA Guidance,Navigation and Control Conference, Snowmass, Colorado, U.S.A. August 1985,pp. 253-259.[29] Chase, C., Chase, W., Lohr, M., Lee, G.K.F., and Dwyer, T.A.W., III, "AnOperational 1/16th Size Model of the Space Shuttle Manipulator," A Collectionof Technical Papers of the AIAA Guidance, Navigation and Control Conference,Snowmass, Colorado, U.S.A. August 1985, pp. 269-277.[30] Shiraki, K., Marumo, H., Sugasawa, Y., and Nishida, S., "Preliminary Con-cept of RMS for Japanese Experiment Module of the Space Station," JointAAS/Japanese Rocket Society (JRS) Symposium, Honolulu, Hawaii, Dec. 1985,Paper No. 85-662.[31] Meirovitch, L., and Quinn, R.D., "Equations of Motion for Maneuvering Flexi-ble Spacecraft," Journal of Guidance, Control, and Dynamics, Vol. 10, No. 5,September-October 1987, pp.453-465.[32] Longman, R.W., Lindberg, R.E., and Zedd, M.F., "Satellite Mounted Robot161Manipulators - New Kinematics and Reaction Moment Compensation," The In-ternational Journal of Robotics Research, Vol. 6, No. 3, Fall 1987, pp. 87 - 103.[33] Hale, A.L, Lisowski, R.J., and Dahl, W.E., "Optimal Simultaneous Structuraland Control Design of Maneuvering Flexible Spacecraft," Journal of Guidance,Control, and Dynamics, Vol. 8, No. 1, January-February 1985, pp. 86-93.[34] Carrington, C.K., and Junkins, J.L., "Optimal Nonlinear Feedback Control forSpace attitude Maneuvers, "Journal of Guidance, Control, and Dynamics, Vol.9, No. 1, January-February 1986, pp. 99-107.[35] Yuan, J.S.C., and Stieber, M.E., "Robust Beam-Pointing and Attitude Controlof a Flexible Spacecraft," Journal of Guidance, Control, and Dynamics, Vol. 9,No. 2, March-April 1986, pp. 228-234.[36] Garcia, E., and Inman, D.J., "Modeling of the Slewing Control of a FlexibleStructure",Journa/ of Guidance, Control, and Dynamics, Vol. 14, No. 4, Sep.-Oct. 1991, pp. 736-742.[37] Spong, M.W., and Vidyasagar, M., "Robust Linear Compensator Design forNonlinear Robotic Control", Proceeding of IEEE Conference on Robotics andAutomation, St. Louis, Missouri, March 1985, pp. 954-959.[38] Widmann, G.R., and Ahmad, S., "Control of Industrial Robots with FlexibleJoints," Proceeding of the IEEE International Conference on Robotics and Au-tomation, Raleigh, USA, 1987, Vol. 3, pp. 1561-1566.[39] De Luca, A., "Dynamic Control of Robots with Joint Elasticity,"Proceeding ofthe IEEE International Conference on Robotics and Automation, Philadelphia,USA, 1988, Vol. 1,pp.152-158.[40] De Wit, C.C., and Lys, 0., "Robust Control and Parameter Estimation ofRobots with Flexible Joints," Proceeding of the IEEE International Conference162on Robotics and Automation , Philadelphia, USA, 1988, Vol. 1, pp. 324-329.[41] Ahmad, S., and Guo, H., "Dynamic Coordination of Dual-Arm Robotic Systemswith Joint Flexibility ," Proceeding of the IEEE International Conference onRobotics and Automation, Philadelphia, USA, 1988, Vol. 1, pp. 332-337.[42] Modi, V.J., Karray, F. and Chan, J.K., "On the Control of a Class of FlexibleManipulators Using Feedback Linearization Approach," 42nd Congress of theInternational Astronautical Federation, Montreal, Canada, October 1991, PaperNo. IAF-91-324; also Acta Astronautica, Vol. 29, No. 1, 1993, pp. 17-27.[43] Marom, I, "A Closed-Form Dynamical Analysis of an Orbiting Single Link Ma-nipulator," Technical Report.[44] Ng, A.C., Dynamics and Control of Orbiting Flexible System: a Formulation withApplications, Ph.D. Thesis, The University of British Columbia, April 1992.[45] Butenin, N.V., Elements of Nonlinear Oscillations, Baisdell Publishing Company,New York, U.S.A. pp. 102-137.[46] Blevins, R., Formulas for Natural Frequencies and Mode Shapes, Robert E. KriegerPublishing Co., Malabar, Florida, U.S.A., 1984, pp. 107-109.[47] Yu, E. Y., "Long-term Coupling Effects Between the Librational and OrbitalMotion," AIAA Journal, Vol. 2, No. 3, March 1964, pp. 553-555.[48] Misra, A. K., and Modi, V. J., "The Influence of Satellite Flexibility on OrbitalMotion," Celestial Mechanics, Vol. 17, 1978, pp. 145-165.[49] Brereton, R.C., A Stability Study of Gravity Oriented Satellites, Ph.D. Thesis,The University of B.C., November 1967.[50] Silver, W.M., "On the Equivalence of Lagrangian and Newton-Euler Dynamicsfor Manipulators," The International Journal of Robotics, Vol. 1, No. 2, 1982,163pp. 60-70.[51] Modi, V.J., Pradhan, S., and Misra, A.K., "On the Parametric Response andControl of a Flexible Tethered Two-Body System," in preparation for publicationin Acta Astronautica.[52] Space Station Engineering Data Book, NASA Space Station Program Office,Washington, D.C., NASA SSE-E-87-R1, November 1987.[53] Beijczy, A.K., Robot Arm Dynamics and Control, JPL TM 33-669, CaliforniaInstitute of Institute of Technology, Pasadena, California, 1974.[54] Spong, M.W., "Modelling and Control of Elastic Joint Robots" Journal of Dy-namic Systems, Measurement and Control, Vol. 109, December 1987, pp. 310-319.164APPENDIX I: MODE SHAPESThe modal functions for the platform, treated as a free-free beam, as given inBlevins [46] are-) 1 (x) = coshpi(xi + 11) + cospi(xi + 11) — cri[sinhpi(xi + 11) + sinPi(xi + /1)], (1.1)withcosh2pil1 — cos2pil1cri =  ^ (. 1.2)sinh2pil1 — sin2pili'where pi is the spatial frequency parameter of the ith mode. 2pi11 is the ith root ofthe characteristic equation associated with the ith mode,co.52pi/1cosh2pi/1 = 1.^ (1.3)This transendental equation has infinite number of roots, thus leading to an infinitenumber of mode shapes, listed below:Mode 2pi11 al1 4.7300407448627 0.982502214576242 7.8532046240958 1.00077731190733 10.995607838002 0.999966450125414 14.137165491257 1.000001449897705 17.278596573990 0.99999993734438i > 5 (2i-1-1)7r 1.002165In formulating the governing equations of motion, it is necessary to evaluate thedisplacement and slope of the modal functions at the location of the manipulatorbase. Furthermore, it is required to evaluate integrals, involving the modal functions,over the length of the platform. These integrals are given below:166[13]= [00000000/3[15]= [00000000/5,^ (11.4)APPENDIX II: THE SYSTEM INERTIA MATRIXThe instantaneous system inertia diadic about the system frame is given by[1] = Ef {( f)[U] — (fif)ldmi,i..1where [I] is the unit matrix.According to the system model as described in Chapter 2, the MDM base mo-ment of inertia /2 is small comared to the platform moment of inertia I. As nogeneralized coordinate is associated with the base motion, 12 has been neglected inthe present study. The system moment of inertia w.r.t. the system frame, is the sumof contributions coming from the bodies constituting the system,=^[-13] + V41 + [4] + [16].^(11.2)The platform mass moment of inertia matrix is= ml2^-T szb -aoy -r qi—a0xaOy C41.0—a0za0y — 14E41^0/2^d2^aL +-+-13 dr 61 0012^d22^2^_Iaox + aoy + 3 + ql 4311q1 + 6 _(11 .3) where aoz, aoy are the shifts of the system c.m. w.r.t. Fo.Neglecting the mass of the joints, the torque motor rotor inertias are given as:167The moment of inertia of arm 1 is[aoy (n' (hi )qi] [(Loy +^(h1)11 + 14804]a+18204 +—Kaoy 1-6T ( )qi)(aox + +(aox + hi)^+ is24]—Kaoy OT (hi )qi)(aox + h1 + 11-44)±(aox + hi) ifso4+ 1-6182,,b4](ao. + hi )(aft + h1 ± 1444)+tc21,1400[4] .7-- m40 0with14(3 ,3 ) = (aox+h i )(aox +hi+ 14c7,b4)+(aoy +g (hi)41)(aoy+ OT (h ].) -41+14s04)+Here, 04 = al + P4, and 14 is the length of arm 1.The moment of inertia of the arm 2 assembly is(aoy^-(iT (hi )1 + assi,b4 )2^— (aox + hi + a6c04)(aoy^j)T(12.1)q1 + a5stP4)0[16] = m6 —(ao. + h1 + a5c04) (a01 + h1 + 61644)2 0(a0„ +^a5504)0 0 16(3,3)+ /6y + /6z 0^0[1610^161 + 1.6y^I6z^00 0 .16z^I6y I6z16m (6204 — C204) + I6y(C21P4 — $204) + 161^—25114C04(i6y 1.6m)0]—281P4C1P4(hy I6x)^16z (C204 — 3204) + 1.6y (8204 — C204) + 16z 02^ 0^ 0^ 000 0+00 00 0 hz — /6zwith^16(3,3) = (aoz + h1 + a5c04)2 (aoy rhi(h1.)th a5s/P4)2, (11.6)where /6x, 16y, 16z are the principal moments of inertia of the arm 2 payload assemblyw.r.t. the frame F6.168APPENDIX III: MASS MATRIX [M] AND THENONLINEAR VECTOR {N}The MDM system equations of motion have the formThe mass matrix is symmetric and positive definite, therefore M(i, j) = M(j,i)where:M(1,1) =m14)11 + maVT (h1))2^(h1)aln-72MbC0401 (h1)0 + (01 (h1))2 [Mc + 13 + + hz];M(1,2) ="114)Tx Ma[O]hl 7%44[0 +^(h1)]+ mbs044T(h1) 10 + AT [mc +13 + 15 + 16z l ;/IT^\M(1, 3) =I^/3P1 );M(1,4) =rnbc4P + (Pi (al)[mc +15 + 16.z];M(1,5) =1501 (hi);M(1,6) =m6s040;d2 TM(2,2) =13 + i +'6z +^+^+ 4)119.11+ ma[h? + (g(h1)q1)2] + mc+ 2mb[hict,b4 + g(111)-4004];M(2,3) =13;M(2,4) =mb[hiclk4 + 4'41441+ mc +15 +16z;M(2,5) =15;M(2,6) =m6[h1s04 + 9541c1k4];M(3,3) =13;169M(3,4) =0;M(3,5) =0;M(3,6) =0;M(4,4) =mc + /5 + 16z;M(4,5) =15;M(4,6) =0;M(5,5) =15;M(5,6) =0;M(6,6) =ms.Here:ma = m4 + ms;ma = m44 _L m6a5;2mc = m44-1 + m64;= g(hl) iq•The components of the nonlinear vector N, on the r.h.s. of the equation of motion,are as follow:N( l ) =h i{mai(hi).904}+ ( ;.b + e )2{m143 1 141 + mag(hi Xig(hi)+ marg(h1)s11)4 + (g(h1)-4144 — his04)4T (hal— 2(11, + a){maiii0 msaS [044 + a5AT(h1)]mb[-0(144 —^+ /14 4T(hi)c;b4}Ig{Orribs04}170—2144a5m6 {004 + a54T(h1)}—E.14Tql — Cla)7413GM 2 2 s203 frnlEt'llqi(c^-3-)+rc2^s20rna[g(h1)q173hii(hi)(c20 — —3) + —Thljl(1)0 +[mc + 16y — I6^Tx101 (hi)s2(^04) 2^+ mb[g(h1)Eclbs(0+ 04) — 041 + g(h1)447(//1)[cOcelk + 04) — —24413 3-IT+ hi 951 (hi )[sOc(V) 1P4)3-s04111;3N(2) =-111{ma(Aql sO4mb}1gimb[h1s04 — 011041—24a5rn6{h1cz,b4 a5 0144}—2(0){7n167411q1^g(h1)41q5T(hOihima▪ m6a5[h1c1b4 a5 4(h1)qls'04]mb[iilcib4 — h1z44.904 R:(h)16 804▪ g(h1)(1-111/44)4]}3GM^4 _ 5203 Im1Ki^— 1.1141) 2rcsiTzq1c21k]+ ma [(hi ((h1)1)2 ---3221k^-.T(/11)411c2li'l1+ —2 [mc + 16y — 16x}82(0 + 04)] + mb[h1s(20 + 04)+ Of (hi)1c(20 + 4)]}+^C00;N(3) =k3(#4 f33) + 1-13 — C3/33;171N(4) =hi [mbs04]—2(i.k + a){m6a5a5 + mb[iiim,b4 + 0400411- + j)2{mb[his04 — g(hi)qiczP4]}—2m6a5hb4a5 —^— k3(04 — 03) — C4043GM^s2(1,1) + 04) + mb[hi [43 {[mc +^/6x]^2^4'0 + 04)rc2+ —3 .504] + iff(h1)4.1[c0ce0 + 04) 244 3N(5) =1c5(-a—r55 — 05) + P5 — C5)615;N(6) = — himsclim + 1gm6a5 2(14 + a )m6 [ill 4)4 — 954104 — "Cb4a5]+ + )2m3 [12404 + ii)T( hl ) -41304 +k5- (--P5) C6a5r5 r5G3^ 3114m {a5 [32(0 + 04) — —2] + [s0s(0 + 04) — —2 c04]ra 3^3+ (1)T(hi)i [48(& 4- 04) - 2-3-44].172


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