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

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PLANAR DYNAMICS AND CONTROL OF TETHERED SATELLITE SYSTEMS by  SATYABRATA PRADHAN  B.Sc.(Engg., Honou’rs), Sambalpur University, India, 1987 M.E. (Distinction), Indian Institute of Science, Bangalore, India, 1989 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December 1994  © Satyabrata Pradhan, 1994  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  M€cRc.fliC.*I  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  9  5  Ereev  ABSTRACT A mathematical model is developed for studying the inpiane dynamics and control of tethered two-body systems in a Keplerian orbit. The formulation accounts for: • elastic deformation of the tether in both the longitudinal and inplane trans verse directions; • inplane libration of the flexible tether as well as the rigid platform; • time dependent variation of the tether attachment point at the platform end; • deployment and retrieval of the point mass subsatellite; • generalized force contributions due to various control actuators (e.g. momen tum gyros, thrusters and passive dampers); • structural damping of the tether; • shift in the center of mass of the system due to the tether deployment and retrieval.  The governing nonlinear, nonautonomous and coupled equations of motion are ob tained using the Lagrange procedure. They are integrated numerically to assess the system response as affected by the design parameters and operational disturbances. Attitude dynamics of the system is regulated by two different types of actu ators, thruster and tether attachment point offset, which have advantages at longer and shorter tether lengths, respectively. The attitude controller is designed using the Feedback Linearization Technique (FLT). It has advantages over other control methods, such as gain scheduling and adaptive control, for the class of time varying systems under consideration. It is shown that an FLT controller based on the rigid system model, can successfully regulate attitude dynamics of the original flexible  11  system. A hybrid scheme, using the thruster control at longer tether lengths and the offset control for a shorter tether, is quite attractive, particularly during re trieval, as its practical implementation for attitude control is significantly improved. Introduction of passive dampers makes the hybrid scheme effective even for vibration control during the retrieval. For the stationkeeping phase, the offset control strategy is also used to regu late both the longitudinal as well as inpiane transverse vibrations of the tether. The LQG/LTR based vibration controller using the offset strategy is implemented in conjunction with the FLT type attitude regulator utilizing thrusters as before. This hybrid controller for simultaneous regulation of attitude and vibration dynamics is found to be quite promising. The performance of the vibration controller is further improved by introduction of passive dampers. The LQG based vibration controller is found to be robust against the unmodelled dynamics of the flexible system. Finally, effectiveness of the FLT and LQG based offset controllers is assessed through a simple ground based experiment. The controllers successfully regulated attitude dynamics of the tethered system during stationkeeping, deployment and retrieval phases.  111  TABLE OF CONTENTS ABSTRACT  .  TABLE OF CONTENTS  iv  LIST OF SYMBOLS  ix  .  LIST OF FIGURES  xv  LIST OF TABLES ACKNOWLEDGEMENT  xxii  INTRODUCTION 1.1  Preliminary Remarks  1.2  Review of the Relevant Literature  1.3 2.  .  1  6  1.2.1  Dynamical modelling  1.2.2  Control of the TSS  10  1.2.3  Control algorithms  13  Scope of the Present Investigation  16  FORMULATION OF THE PROBLEM  20  2.1  Preliminary Remarks  20  2.2  Kinematics of the System  21  2.2.1  Domains and reference coordinates  21  2.2.2  Position vectors  23  2.2.3  Deformation vectors  26  iv  2.3  3.  Librational generalized coordinates  • 28  2.2.5  Angular velocity and direction cosine  • 30  Kinetics  • 32  2.3.1  Kinetic energy  • 32  2.3.2  Gravitational potential energy  34  2.3.3  Elastic potential energy  36  2.3.4  Dissipation energy  .  38  2.4  Equations of Motion  39.  2.5  Generalized Forces  40  2.6  Summary  44  COMPUTER IMPLEMENTATION 3.2  Preliminary Remarks  3.2  Numerical Implementation  3.3  Formulation Verification  3.4 4.  2.2.5  45 45 45  .  50  3.3.1  Energy conservation  52  3.3.2  Eigenvalue comparison  54  Summary  DYNAMIC SIMULATION  57  .  .  4.2  Preliminary Remarks  4.2  Deployment and Retrieval Schemes  4.3  Simulation Results and Discussion v  •  .  .  .  •  .  .  .  •  .  .  .  •  .  .  •  58 58 58 60  Number of modes for discretization  4.3.2  Tether length  4.3.3  Offset of the tether attachment point  4.3.4  Tether mass and elasticity  74  4.3.5  Subsatellite mass  78  4.3.6  Deployment and retrieval  83  4.3.7  Shift in the center of mass  87  .  64 71  .  Concluding Remarks  89  ATTITUDE CONTROL  90  5.1  Preliminary Remarks  90  5.2  Thruster Control Using Dynamic Inversion  4.4 5.  61  4.3.1  5.3  93  .  5.2.1  Controller design algorithm  93  5.2.2  Control with the knowledge of complete dynamics  97  5.2.3  Inverse control using simplified models  5.2.4  Comments on the controller design models  102 110  •  .  111  Offset Control using the Feedback Linearization Technique 5.3.1  Mathematical background  •  112  5.3.2  Design of the controller  •  114  5.3.3  Results and discussion  •  119  •  128  •  130  5.4  Attitude Control using Hybrid Strategy  5.5  Gain Scheduling Control of the Attitude Dynamics  vi  .  5.6 6.  .  VIBRATION CONTROL OF THE TETHER  .  132 137  6.2  Preliminary Remarks  137  6.2  Mathematical Background  137  6.2.1  Model uncertainty and robustness conditions  139  6.2.2  LQG\LTR design procedure  142  6.3  6.4 7.  Concluding Remarks  Controller Design and Implementation  145  6.3.1  Linear model of the flexible subsystem  145  6.3.2  Design of the controller  146  6.3.3  Results and discussion  150  Concluding Remarks  EXPERIMENTAL VERIFICATION 7.1  Preliminary Remarks  7.2  Laboratory Setup  7.3  7.2.1  Sensor  7.2.3  Actuator  7.2.3  Controller  Controller Design and Implementation 7.3.1  FLT design  7.3.2  LQG design  170  7.3.3  Controller implementation  171  vii  7.4  7.5 8.  Results and Discussion  173  7.4.1  FLT control  175  7.4.2  LQG control  183  Concluding Remarks  189  CLOSING COMMENTS  191  8.1  Concluding Remarks  191  8.2  Recommendations for Future Work  194  BIBLIOGRAPHY  196  APPENDICES I.  MATRICES USED IN THE FORMULATION  204  II.  ELEMENTS OF MATRICES ‘M’ AND ‘F’  217  III.  NONLINEAR AND LINEARIZED EQUATIONS OF MOTION FOR THE RIGID SUBSYSTEM  IV.  226  CONTROLLER DESIGN USING GRAPH THEORETIC • APPROACH  V.  VI.  233  LINEARIZED EQUATIONS OF MOTION AND CONTROLLER FOR THE FLEXIBLE SUBSYSTEM  242  LABORATORY TEST SETUP  249  vii’  LIST OF SYMBOLS  Jp  offset of the tether attachment point; distance between the origins of coordinate systems F and Ft y—component of J,, D + D  J,  2 d  z—component of  Js  distance between the tether attachment point and the subsatel  +D  lite centre of mass at the subsatellite end flexible deformation vector of the tether at a distance of y along the axis Y unit vector along X, n Jn  unit vector along Y, n  =  =  p, t, s p, t, s  unit vector along the local vertical unit vector along Z, n {l}  =  p, t, s  direction cosines of the local vertical (Yo-axis) with respect to the  body fixed frame  1 m  L -I- ptL 8 m /2 2  m  +pL 8 m  ma  ptL+mo+ms  mc  1—maiM  0 m  mass of the offset mechanism  mp  mass of the platform  3 m  mass of the subsatellite  rat  mass of the deployed tether, ptL(t)  mat  +ptL 2 msL / 3 3  ix  q  vector of the generalized coordinates or parameter for the LQG/LTR design positioi of an elemental mass on the th body with respect to the th body fixed frame  t  time  ut(yt, L, t)  deformation of the tether along the Xt—direction  vt(yt, L, t)  deformation of the tether along the Yt—direction  wt(yj, L, t)  deformation of the tether along the Zt—direction distance measured along the Yt—axis  A  cross—sectional area of the tether or the system matrix in the linear state equation th  mode generalized coordinate for the out-of-plane transverse  tether vibration {B}  {Be}  equillibrium value of {B}  B  th  mode generalized coordinate for the longitudinal tether vi  bration {C}  {c}.Nt  Cdl  damping coefficient of the passive damper along the tether  Cdt  damping coefficient of the passive damper perpendicular to the tether  C  mode generalized coordinate for inpiane transverse tether vibration  {D}  offset vector required by the controller  {D}  specified offset vector specified offset along the local vertical, y—component of {D} x  specified offset along the local horizontal, z—component of {D}  offset required by the controller along the local vertical, y—component of {D} offset required by the controller along the local horizontal, z—component of {D} equivalent Young’s modulus of the tether material in the pres 1 ence of structural damping, E + iE  real component of E*; Young’s modulus of the tether material  E  in the absence of structural damping imaginary part of E* F(q, , t)  vector of nonlinear terms in the equations of motion due to centrifugal, gravitational, and coriolis force contributions damping force along the undeformed tether due to passive  Fdl  damper damping force perpendicular to the undeformed tether due to  Fdt  passive damper F, i  =  p, t, s  th body fixed frame  0 F  orbital frame  1 F  inertial frame  F  platform body—fixed frame  Ft  tether body—fixed frame  3 F  subsatellite body—fixed frame  {F} {F}  {iYt L)}  xi  G  universal gravitational constant  [IJ, i ‘Pz’  =  p, t, s  Ipy, ‘PZ  inertia dyadic of the th body with respect to the frame F moments of inertia of the platform about the Xp, Y and Zp axes, respectively  ‘Pyz  product of inertia of the platform about Yp, Z axes  L  unstretched length of the tether  L  deployment/retrieval velocity  M  total mass of the system, mp + m 0 + ptL -I- m 8  Me  mass of Earth  M(q,t)  mass matrix  M  torque applied by the control moment gyros about the Xp axis  1 N  number of admissible functions used to represent longitudinal vibration of the tether number of admissible functions used to represent inpiane trans  Nt  verse vibration of the tether Nj  Nt+Nj  Qq  vector of generalized forces corresponding to generalized coor dinates q orbit radius  Rdmj,  i  =  p, t, s  position vector of an elemental mass in the th body with re spect to the inertial frame  .j,  i  =  p, t, s  i body fixed frame (F) 2 distance between the origin of the 1  and the orbital frame RSM, {RSM }  shift in the centre of mass of the system; also distance between the origins of F and F 0  xii  RSM  shift in the centre of mass along the local vertical  RSMZ  shift in the centre of mass along the local horizontal  T  kinetic energy of the th body  T  kinetic energy of the entire system  [Tj]  2 transformation matrix from the frame Fj to the frame F  control thrust along the undeformed tether applied at the subsatellite end control thrust perpendicular to the undeformed tether applied  Tat  at the subsatellite end gravitational potential energy of the th body  UG  gravitational potential energy of the entire system U  strain energy of the system  U  potential energy of the system, U + UG  14  volume of the unstretched tether, AL(t)  j,  ,i 13, 7  =  p, t, s  0 2 and F Euler angles between the frames F  6  true anamoly  6  orbital angular velocity  6  orbital angular acceleartion  {w}, i  =  p, t, s  angular velocity of the th body fixed frame with respect to the inertial frame  pt  mass per unit length of the tether  E  strain in an elemental tether mass  IL  3 GMe/2Rc stress in an elemental tether mass without energy dissipation  04  stress in the tether causing energy dissipation  xlii  E E / 1  77  L)  th  shape function for the tether transverse vibration  ‘Ji(y, L)  th  shape function for the tether longitudinal vibration  i(Yt,  Dot above a character refers to differentiation with respect to time Subscripts ‘p’, ‘t’,. and ‘s’ refer to platform, tether and subsatellite, respectively  xiv  LIST OF FIGURES 1-1  Schematic diagram of the Space Shuttle based Tethered Satellite System (TSS)  2  1-2  Some applications of the tethered satellite systems  5  1-3  Dumbbell satellite and the associated forces  7  1-4  A schematic diagram showing the overview of the plan of study.  2-1  Domains and coordinate systems used in the formulation  2-2  Diagrams showing relative orientations of two different frames:  .  .  22  (a) sequence of Eulerian rotations; (b) rotation about the 0 -X a.xis. 2-3  19  29  A schematic diagram of the Tethered Satellite System (TSS) showing different input variables  41  3-1  Flowchart showing the structure of the main program  47  3-2  Structure of the subroutine FCN to evaluate the first order differential equations required by the integration subroutine  3-3  49  Flowchart for the subprogram CONTROL to compute actuation inputs  3-4  51  Figure showing variations of kinetic, potential (gravitational plus elastic) and total energies of the system during the stationkeeping phase: (a)L  =  20 km,  =  =  0; (b) L  =  5 km,  =  1 m,  d=0 4-1  53  Dynamic response of the system with the higher modes included in the discretization of the system  4-2  System response for a shorter tether with higher modes included in the flexibility modelling  4-3  63  65  Dynamical response of the tethered satellite system in the reference  xv  67  configuration with the offset set to zero 4-4  Response results showing the effect of decreasing the tether length 68  on the system dynamics 4-5  Plots showing the effect of increasing the tether length on the system 69  dynamics 4-6  System response during stationkeeping showing the effect of 70  flexibility on the tether attitude motion 4-7  Simulation results during offset motion along the local vertical 72  with the offset along the local horizontal fixed at zero 4-8  System response showing the effect of offset motion along the local 73  horizontal 4-9  Dynamical response of the system with fixed offsets along both 75  the local horizontal and local vertical directions 4-10  Plots showing the effect of decreasing the tether mass per unit 76  length on the system response 4-11  System response with an increase in the mass per unit length of the tether  77  4-12  Diagram showing the effect of decreasing the elastic stiffness  79  4-13  Response of the system with an increase in the elastic stiffness.  4-14  Response results showing the effect of decreasing the subsatellite mass. 81  4-15  System response with an increase in the subsatellite mass  4-16  Response of the system during deployment of the subsatellite  .  using an exponential-constant-exponential velocity profile 4-17  .  80  82  84  System response during exponential retrieval of the subsatellite. ) 1 Note instability of the system in rigid (aj) as well as flexible (B 86  degrees of freedom  xvi  4-18  Plots showing shift in the center of mass during three different phases of the system operation:(a) deployment; (b) stationkeeping; and (c) retrieval. Total deployed length is 20 km  88  5-1  Controlled response of the system during the stationkeeping phase.  5-2  System response during controlled deployment with an exponential-  .  constant-exponential velocity profile 5-3  100  Controlled response during the exponential retrieval without a passive damper  5-4  98  101  System response during the controlled retrieval with passive dampers. Note, the flexible modes are controlled quite effectively keeping the tether tension positive  5-5  103  System response during the stationkeeping using the rigid nonlinear model for the controller design  5-6  105  System behaviour during the stationkeeping with the outer PT control loop and FLT controller using the rigid nonlinear model.  5-7  .  107  Deployment dynamics with the controller based on the rigid nonlinear model  5-8  .  108  Controlled response during exponential retrieval of the subsatellite. The controller is designed using the rigid nonlinear model and applied in presence of passive dampers  5-9  Schematic diagram of the closed-loop tether dynamics with the FLT based controller  5-10  109  119  Controlled response in presence of the modelling error introduced by neglecting the shift in the center of mass terms during the controller design  5-11  121  System response with the shiftin the center of mass included  xvii  in the controller design model 5-12  122  Schematic diagram of the closed-loop system with the outer PT control loop for the tether dynamics  123  5-13  System response in presence of the outer PT control ioop  124  5-14  Controlled response of the system during deployment. The offset control strategy in conjuction with the Feedback Linearization Technique (FLT) is used  5-15  125  System response during controlled retrieval with an exponential velocity profile  127  5-16  Controlled response during deployment using a hybrid strategy.  129  5-17  System response during retrieval with a hybrid control scheme.  131  5-18  Controlled stationkeeping dynamics using the graph theoretic approach  5-19  133  Gain scheduling control of the retrieval maneuver using the thruster augmented strategy  5-20  134  Offset control of the retrieval dynamics using the gain scheduling approach  135  6-1  Standard feedback configuration  139  6-2  Diagram showing different unstructured uncertainties: (a) additive; (b) output multiplicative; and (c) input multiplicative  141  6-3  Closed-loop system with the LQG feedback controller  143  6-4  Flow chart for the controller design  151  6-5  Comparison of gain and phase of the return ratio at the plant output with those of the KBF loop tranfer function  152  6-6  Robustness property of the vibration controller  153  6-7  Three-level controller structure to regulate rigid as well as  xviii  transverse and longitudinal flexible motions of the tether 6-8  154  Response of the system using a three level controller in absence of a passive damper  155  6-9  Controlled response of the system in presence of a passive damper.  156  7-1  A device to measure the tether swing through rotation of the potentiometer shaft  161  7-2  A schematic diagram of the experimental test—facility  163  7-3  Photograph of the test—rig constructed to validate offset control strategy: (a) aluminum frame; (b) inplane motor; (c) carriage for out-of-plane motion; (d) wooden stand; (e) linear bearings; 164  (f) tethered payload 7-4  Carriage and sensor mechanism: (a) potentiometer on mounting bracket; (b) movable aluminum semi—ring mechanism with slots for tether; (c) tether; (d) inpiane traverse with linear 165  bearings; (e) payload 7-5  Digital hardware used in the experiment: (a) translator module, deployment and retrieval; (b) translator module, offset motions; (c) power supply; (d) function generator  7-6  166  Photograph showing the recently constructed larger test—facility which can accomodate tethers upto 5 m in length. The present smaller set—up on which the experiments reported here were carried out can be seen in the foreground to the left  7-7  Flow chart showing the real time implementation of the attitude 174  controller 7-8  167  Plots showing the comparison between numerical and experimental response results during the stationkeeping phase: (a) uncontrolled and controlled system; (b) subsatellite and carriage positions xix  176  7-9  A comparative study between numerical and experimental results for the system during stationkeeping at 1 m: (a) uncontrolled and FLT controlled system; (b) subsatellite and carriage positions.  7-10  .  179  A comparative study between numerical and experimental results for the system during stationkeeping at 2 m: (a) uncontrolled and FLT controlled system; (b) subsatellite and carriage positions.  7-11  181  Uncontrolled and controlled experimental results for retrieval of the subsatellite: (a) retrieval time of 15s; (b) faster retrieval in 5s.  7-12  .  .  .  .  184  A comparative study with the LQG controller during stationkeeping at 1 m: (a) uncontrolled and controlled performance; (b) subsatellite and the tether attachment point positions  7-13  Experimentally observed system response during retrieval 188  with the LQG controller 7-14  186  Subsatellite and carriage positions during control of the spherical pendulum using the LQG regulator  xx  190  LIST OF TABLES 3.1  Inpiane dimensionless natural frequencies  (L  EA 3.2  =  =  3 i0 kg, m  =  .EA  500 kg, Pt  5.76 kg/km,  2.8 x 1O 5 N)  55  20 km, mp  =  2.8 x  =  cc, m 3  =  576 kg, p  =  5.76 kg/km,  N)  55  Comparison of natural frequencies of a tethered two-body system with different end masses (L EA  5.1  =  Inpiane dimensionless natural frequencies (w/6) of a 2-body system  (L  3.3  100 km, mp  =  (w/Ô) of a 2-body system  =  =  20 km, p  =  5.76 kg/km,  2.8 x iO N)  56  Comparison of the time required (s) by the controllers using different design models  6.1  110  Comparison of the open-loop and closed-loop eigenvalues of the system  149  xxi  ACKNOWLEDGEMENT I am thankful to Prof. V. J. Modi and Prof. A. K. Misra (McGill University, Canada) for their guidance throughout the research project. A word of appreciation must be extended to my colleagues, Dr. C. A. Ng, Dr. A. K. Grewal, Dr. S. Hokamoto and Dr. F. Karray, for useful discussions and suggestions as well as to Dr. I. Marom, Mr. S. R. Munshi, Mrs. M. Seto and Mr. Y. Chen for their advice in planning of the experimental test—program. Assistance of Mr. John Richards in construction of the electronics hardware used during the experimental study is gratefully acknowledged. The research project was supported by the Natural Sciences and Engineer ing Research Council’s grant (A—2181) held by Prof. V. J. Modi.  2CXii  1.  1.1  INTRODUCTION  Preliminary Remarks  Over the past decades, a number of proposals have been made for the space exploration using Tethered Satellite Systems (TSS). The concept involves two or more satellites connected by a tether upto 100km in length (Fig.1-1). In a typical mission, the Space Shuttle carries the tether connected subsatellite payload. After a desired orbit is achieved, the subsatellite is deployed from the orbiter to the required altitude. Intended scientific measurements are carried out at a fixed tether length. Subsequently the subsatellite is retrieved back to the Shuttle (Fig.1-1). The innovative idea of the TSS is attributed to the Russian scientist Tsi olkovsky, who explored the effect of gravity force on an individual climbing up a tower extending upto geosynchronous altitude and beyond [1]. This analysis uncov ered the principle that a string orbiting in a central force field always remains in tension due to the gravity gradient, and led to the development of the TSS. So far as the actual application is concerned, interest in the system was initially associated with the retrieval of a stranded astronaut by throwing a buoy on a tether from a rescue vehicle and reeling in the tether. Starly and Adihock [2] have shown that the rotational motions of the tether grow continuously as it was reeled in. A proposal was made to use tether for stationkeeping between two orbiting space vehicles [3], however, the idea was abandoned due to the difficulties involved in determining and controlling the required tether tension. On the other hand, Gemini XI and XII flights have successfully demonstrated useful applications of a tethered system [4]. The former used a rotating configuration aimed at artificial gravity generation while 1  ORBITER  BOOM TRAJECTORY  TETHER  SUBSATELLITE  Figure 1-1  Schematic diagram of the Space Shuttle based Tethered Satellite System (TSS).  2  the latter had a gravity-gradient stabilized configuration. After one and half decade, a joint U.S.A.—Japan space project called the TPE (Tethered Payload Experiment) used a sounding rocket based tether to conduct a series of tests aimed at near earth environmental studies [5]. The technical and scientific data obtained in the TPE supported the electrodynamic tether mission TSS-i (Tethered Satellite System-i) launched in August 1992 [6]. The TSS mission is a collaborative project between NASA and the Italian space agency (ASI). Because of the mechanical failure, the mission objectives of the TSS-i were not met, but the information gathered during the short deployment upto 259 m, as compared to the desired length of 20 km, demonstrated the fundamental concepts of orbital tether flights in many ways. Another series of sounding rocket experiments called the OEDIPUS (Observation of Electrified Distributions in the Jonosheric Plasma  —  a  Unique Strategy) are being carried out by the Canadian Space Agency (CSA) [7]. The objective is to make passive observation of the auroral ionoshere and gain better insight into pane and sheath—wave propagation in plasmas using a radio transmitter. The first mission, called OEDIPUS-A, was successfully launched in January 1989 which deployed a tether upto 958 m. The next mission, OEDIPUS-C, planned for January 1995, will include a payload for tether dynamics measurements. The deployment of 20 km long tether was successfully completed in SEDS (Small Expendable Deployment System) missions [8]. SEDS-I and SEDS-Il missions flew in March 1993 and March 1994, respectively. Each deployed a 26 kg instru mented probe to a distance of 20 km from the second stage of a DELTA II rocket while in orbit around the Earth. These tether missions have successfully demon strated the feasibility of tether deployment. On the other hand, retrieval of a tether still remains to be realized in a real flight. With the advent of the Space Shuttle and 3  the proposed Space Station, several applications of the tethered subsatellite system has been suggested which can be summarised as: • sophisticated scientific experiments aimed at gravity gradient, magnetic, iono spheric, aero-thermodynamics and radio astronomy experiments (e.g. OEDIPUSA and C); • use of tethered system as a flying wind tunnel [9]; • deployment of payloads to new orbits [10] and retrieval of satellites for ser vicing; • provision of a desired controlled microgravity environment for scientific ex periments and space manufacturing [11]; • generation of electricity (electrodynamic tether, TSS-1); • power and cargo transfer between two orbiting bodies; • collection of atmospheric dust by a small probe tethered to a larger orbiting spacecraft thus avoiding landing [12]; • expansion of the geostationary orbit resource by having tethered chain satel lites [13, 14]; • large antena reflector  (  1 km aperture) using tethers for gravity gradient  stabilization and shape control [15]; • orbiting optical astronomical interferometer consisting of three telescopes at the corners and one at the center of a tethered triangle [16]; • use of multiple tethers for attitude stabilization of satellites in elliptic orbit [17]; and many others [10, 18]. Some of these applications are illustrated in Fig.(1-2). The fundamental principle governing the tether dynamic can be illustrated 4  Low Altitude Science Experiment  Cargo Transfer  Tether Cargo  Manipulator 100 km  -  Earth  Retrieval of Satellite for Servicing  Tether  Figure 1-2  Some applications of the tethered satellite systems.  5  by a simple two-body tethered system as shown in Fig.1-3. Consider a ‘dumbbell’ satellite in a circular orbit around the Earth. The top mass experiences a larger centrifugal force than the gravitational force, being in an orbit higher than that of the centre of mass. The reverse occurs at the lower mass. The resultant moment causes The system to oscillate like a pendulum with the tether maintained taut under tension. Now, as the length of the tether decreases during retrieval, the conserva tion of angular momentum dictates the swinging oscillation of the tether to amplify. For a space platform supported tethered satellite system, the swing motion could increase to the point that the tether wraps itself around the platform. Furthermore, reduced gravity induced tension at smaller lengths together with large amplitude os cillations may render the tether slack. The presence of any offset between the tether attachment point and the centre of mass of the end-bodies (platform and subsatel lite) imposes an additional moment on the bodies involved. Flexibility of the tether, along with the above mentioned factors makes the system dynamics rather chal lenging. Obviously, a fundamental understanding of complex interactions between librational dynamics, flexibility, deployment and retrieval maneuvers, as well as de velopment of appropriate control strategies is essential for successful completion of the proposed tether missions.  1.2  Review of the Relevant Literature  The possible applications of the TSS being numerous and diverse, consider able amount of literature has developed, particularly during the past 30 years, which was reviewed quite effectively by Misra and Modi [19, 20]. The growing importance  of this new technology is reflected in a special issue published in The Journal of the Astronautical Sciences [21]. Objective here is to briefly touch upon more important 6  F, F 1  CENTRIFUGAL FORCES 2  Fg F, 2 1 g  GRAVITATIONAL FORCES  FCl  FCl >F  C.M.  FC 2 F> FC 2 F  ‘ ‘ ‘  Figure 1-3  I I I ‘I ‘I  Dumbbell satellite and the associated forces.  7  contributions directly relevant to the present study.  1.2.1  Dynamical modelling  The dynamics of a TSS consists of three major phases of operation depending on whether the unstretched length of the tether is constant (stationkeeping phase) or varies with time (if increasing, deployment; when decreasing, retrieval). The dynam ics during stationkeeping is simpler than the other two phases and is quite similar to that of other cable connected bodies, e.g. the space station-cable-counterweight sys tem [22]. It has been investigated extensively by various researchers who concluded that the gravity gradient can excite longitudinal as well as transverse vibrations of the tether [22, 23]. However, a small amount of damping (1% of the critical damping) is quite effective in stabilizing the system [24]. Perhaps the first systematic effort at understanding dynamics of the TSS was made by Rupp [25]. In his pioneering study, the librational motion in the orbital plane was analyzed and the growth of pitch oscillations during retrieval phase noted. The system was further investigated by Baker et al. [26] taking into account the three dimensional character of the dynamics as well as the aerodynamic drag in the rotating atmosphere. During either deployment or retrieval, the out-of-plane rotational motion cannot be neglected if the orbit is in a nonequatorial plane. It is excited by the rotating atmosphere induced aerodynamic drag. Even when the orbit is in the equatorial plane, the initial out-of-plane disturbances may couple the inpiane and out-of-plane dynamics. Fortunately, in general, such disturbances are found to be small. Furthermore, their effects can be minimized through the design of an appropriate active control system. Thus the coupling between the inpiane and out-of-plane motions can be neglected, for systems in the equatorial orbit, at least 8  in the preliminary design stage. The other important parameters, which make the system dynamics compli cated, are mass and flexibility of the tether. The mass of the tether is expected to be of the same order of magnitude as that of the subsatellite, particularly when the tether is long, and hence it can not be neglected. Because the tether diameter is ex pected to be small (around few millimeters) and the length quite long (upto 100km), the system shows considerable flexibility (the longitudinal stretch for a 100km long tether can be around hundred meters). The axial vibration has been represented in an approximate fashion by a single longitudinal displacement similar to that of a spring mass system by several authors [26, 27, 28]. However, as the mass is dis tributed along the tether, a more accurate representation in terms of combinationof axial modes, similar to that of an elastic bar, would be more appropriate [29, 30]. Furthermore, the tether can have transverse displacements causing it acquire curvature. This happens mainly due to two reasons. The aerodynamic drag not only causes the subsatellite to lag behind the platform but also forces the tether to assume a curved equilibrium position. Secondly during deployment or retrieval, the Coriolis force acts in the transverse direction. Since the tether has distributed mass and elasticity, transverse vibrations are also excited. They can influence system dy namics substantially, particularly during retrieval [31]. The transverse oscillations of the tether have been studied using two distinct approaches. Quadrelli and Lorenzini [32] discretized the system using the lumped-mass approach. On the other hand, Kohler et al. [33], Modi and Misrà [27], and Xu, Misra and Modi [30] adopted the continuum model with admissible functions. To reduce the computational effort involved in simulation, a combination of lumped-mass and continuum model has been proposed called the semi-bead model [34]. 9  When the masses of the two end bodies are comparable, the system centre of mass does not remain fixed with respect to any of the body fixed frames. The shift in the centre of mass may become an important parameter in such situation and has to be accounted for. The major conclusions based on the available literature may be summarized as follows: • stationkeeping phase is normally stable; • deployment can be an unstable operation if velocity exceeds the critical rate; • retrieval is inherently unstable • particularly during the deployment and retrieval phases, the transverse vibra tions can build up due to the Coriolis force induced excitation even though there may not be any initial disturbance. The other aspects of practical importance are the damping in the tether [35], aerodynamics drag [36, 37], and orbital lifetime analysis [38]. The dynamics of tethered systems in different configurations, which has less relevance to the present study, have been addressed by Misra and Diamond [39], and DeCou [40, 16]. In real systems, there is always a chance of accident. The dynamics of the tether connected to the orbiter after a catastrophic failure has been studied by Bergamaschi [41].  1.2.2  Control of the TSS Control of the TSS, particularly during deployment and retrieval, is a chal  lenging problem. Successful control of the unstable retrieval dynamics is of great concern since it is directly related to the success of the mission. Over the years,  several control strategies have been proposed, which can be categorized into three types: (a) tension control and the law based on rotational rate of the tether reel (length  10  rate control or torque control); (b) thruster augmented active control; (c) offset control. (a) Tension and length rate control Among the control laws mentioned, the tension control law was developed first. Rupp [25] formulated this strategy to control the inplane rotation during deployment and stationkeeping. The retrieval problem was also touched upon briefly. Here the tension level is modulated as a function of the instantaneous length, length rate, and desired lengths of the tether. Several investigators have subsequently modified Rupp’s tension control procedure, however the inherent approach remains the same [26, 42—44]. For example, Bainum and Kumar [44] developed an optimal control law, based on an application of the linear regulator theory, which modulates the tether tension to achieve acceptable level of the tether swing. As opposed to the tension control law, in the length rate scheme the ‘nomi nal’ unstretched length or its time derivative are modulated to achieve the desired system response. The law corresponds to modulating the rotation of the reel of the deployment mechanism. This can also be implemented by monitoring the torque applied to the reel mechanism. In principle, tension control, length rate control and torque control affect the system dynamics by changing the tension in the tether. The differences are in the mathematical modelling of the system used for the controller design and its implementation. This type of control law was originally proposed by Kohier et al. [33], and used by Misra and Modi [45] and others to regulate the planar longitudinal and transverse vibrations.  11  (b) Thruster augmented active control  During retrieval, as length of the tether reduces to a small value, the equi librium tension in the tether due to gravity gradient approaches zero; and during a dynamical situation, the tether may become slack. Thus a tension control law or its modification, such as a length rate law, becomes ineffective. To alleviate this diffi culty, Banerjee and Kane [46], and Xu et al. [47] used a set of thrusters to control the retrieval dynamics. In this active control scheme, the thrusters are placed at the subsatellite end to help reduce the motion and speed up the retrieval process. The thrusters provide control forces in both transverse and longitudinal directions.  (c) Offset control strategy It has been shown that, when the tether length is small the thruster aug mented control is quite effective; however firing of thrusters in the vicinity of the shuttle or space platform is considered undesirable due to plume impingement, safety and other considerations. To overcome this difficulty the “offset control strategy” was proposed by Modi, Lakshmanan and Misra [48]. Here, the point of attachment of the tether at the platform is moved to control the tether swing. The coupling of the offset acceleration and tether swing makes this control strategy a success. In their study, Modi et al. considered a 3-dimensional rigid model without any shift in the centre of mass. The results obtained from the numerical simulation were verified by a ground based experimental set-up. From the comparison between numerical and experimental results, they concluded the control strategy to be quite promising for shorter tethers. Because of relative advantages and disadvantages of different schemes, a desirable solution would be to use a hybrid strategy, where the offset control is implemented at shorter lengths while the thruster, tension or length rate  12  control scheme is used for longer tethers [49].  1.2.3  Control algorithms  An important factor which needs careful attention is the methodology used to obtain the control laws. In the pioneering work by Rupp [25], the feedback gains were selected to achieve an appropriate stiffness and damping in the closedloop system. This approach is feasible only when the system dynamics is relatively simple, which is not the case in practice. Since then, linear and non-linear control laws have been developed by many investigators (for example [26]), where the control gains are obtained by trial and error. Among the wide variety of control methodologies, the Linear Quadratic Reg ulator (LQR) has received considerable attention [44, 48, 49]. On the other hand, a feedback control law using the second method of Liapunov was used by Vadali and Kim [50, 51]. It was concluded that the tension control law is sufficient dur ing fast deployment, however, thruster augmentation in the out-of-plane direction is required during the terminal phase of the retrieval [51]. The existence of limit cycles was avoided by the out-of-plane thrusting in conjunction with the tension control [50]. Monshi et al. [52] used the reel rate control law with nonlinear roll rate feedback, based on energy dissipation method. It was found to have better per formance than that for the law developed using the Liapunov method. The study showed that the retrieval constant c (c  =  i’/l;  ,,  nondimensionalized retrieval rate;  1, instantaneous tether length), in case of an exponential retrieval, does not have a significant effect on limit cycle amplitude. As the retrieval rate increases the peak value of the pitch oscillation also increase. So the maximum retrieval rate is limited by a maximum allowable pitch angle. The limit cycle amplitude found in [52] is 13  quite close to that obtained by Vadali and Kim [50]. In the study by Liaw and Abed [53], tension control laws were established using the Hopf bifurcation theorem, which guarantees stability of the system during the stationkeeping mode. They. also suggested a constant inplane angle control method which results in stable deployment but unstable retrieval. In this method, the instantaneous tether length is treated as an external control input, and the length rate is obtained as a function of some desired inpiane equilibrium angle. The length rate expression required to control the system during deployment does not stabilize the retrieval dynamics. Fleurisson et al, [54] designed an observer (Kalman filter) based controller to follow a predetermined retrieval length history. The length history was obtained to minimize a combination of final pitch angle, pitch rate at docking and total retrieval time. The control strategy used feedforward trajectories motivated by optimal control arguments. Optimal control procedure was employed to determine an invariant final approach path, or randezvous corridor, which the satellite must follow during the terminal phase to dock with the specified final conditions. The study focused only on the last 2 km of the retrieval. In the study by Fujii et al. [55], the dynamics during deployment and retrieval is controlled by regulating the mission function, a positive definite quadratic function of the mission states. The desired value of the system states are referred to as the ‘mission states’. For the closed-loop system, the non-dimensional time derivative of the mission function is set to be negative definite, guaranteeing stability of the system. Thus, the approach is quite similar to the Lyapuno’v’s second method. Onoda and Watanabe [56] as well as Fujii et aL [57] studied control of teth 14  ered systems in the presence of atmospheric drag. Attitude control of the tethered systems after blocking of the attachment point has been explored by Grassi, et al. [58]. The offset control, as discussed before, is for regulating the attitude motion of the tether. However, this strategy can also be used for precise control of the subsatellite attitude, where the control moment can be generated by a combined effect of tether tension and the attachment point motion [59]. Numerous variations of the control algorithms discussed here are also reported in the literature. Almost all the investigations reported so far deal with the control of attitude motion of the tether or the endbodies. Little attention has been directed towards the control of structural vibrations of the tether. Xu et al. [30, 47) addressed the problem of control of tether oscillations using length rate and thruster control. On the other hand, the study by Thornburg and Powell [60] considers the control of only transverse vibration in conjunction with the offset strategy. As seen in this literature review, the models developed so far do not include the attachment point offset for a flexible tethered system capable of deployment and retrieval. The study in reference [60] incorporates the motion of the attachment point, of a flexible tether, along a line parallel to the end body’s floor (i.e. motion along one bodyfixed axis), however, it does not include the most critical operational phases, deployment and retrieval. The tether is modelled as an arbitrary number of point masses connected by elastic members. Use of the lumped mass model for a continuum system may be appropriate for a preliminary study, however, any detailed analysis of the system needs the more accurate continuum model. With the offset motion restricted to be parallel to the subsatellite floor, the strategy is limited only to the control of transverse oscillations. Furthermore, the analysis does not account for the shift in the center of mass, an important parameter for long tethers during 15  deployment and retrieval maneuvers.  1.3  Scope of the Present Investigation  With this as background, the thesis aims at studying dynamics and control of two-body tethered systems, using a relatively general model, which is the funda mental requirement for their successful realization. Distinctive features of the model may be summarized as follows: (i) A flexible tether, with arbitrary mass distribution and finite dimensional rigid end bodies, is taken to be in a general Keplerian orbit. The tether is treated as an elastic continuum during the system discretization. (ii) Offset of the tether attachment point from the end body’s center of mass and its time dependent variations are included in the formulation. This permits development of several offset control strategies. Furthermore, the formulation is amenable to thruster augmented active control and its hybrid synthesis with the offset control procedure. (iii) The formulation accounts for time dependent variation of the tether length, thus permitting analysis of the tether performance during all the three op erational phases of importance  —  deployment, stationkeeping and retrieval.  The deployment/retrieval time histories are considered arbitrary. (iv) Kinetic and potential energy expressions are obtained for the general three dimensional (i.e. in the orbital plane and out-of the orbital plane) motion of the system. However, the governing equations of motion and hence the study based on them are purposely confined to the inplane dynamics. This helped focus on more important aspects of the system performance which are governed by a large number of variables. Furthermore, some appreciation of 16  the coupling between the inpiane and out-of-plane dynamics is already avail able through early studies (by the U.B.C. group and others) using simpler models. (v) The formulation is based on the Lagrangian procedure which can account for both holonomic and nonholonomic constraints. (vi) As can be expected the governing equations of motion are highly nonlin ear, nonautonomous and coupled. Their decoupled, linearized forms are also obtained to facilitate the controller design. (vii) Numerical integration codes for nonlinear and linear systems are so struc tured as to help isolate effects of design parameters on the system perfor mance. To begin with, kinematics and kinetics of the system, leading to a set of highly Coupled, Nonlinear and Nonautonomous (CNN) equations of motion, are discussed in Chapter 2.  This is followed by a brief account of the numerical integration  methodology as applied to the nonlinear as well as decoupled linearized system (Chapter 3). Validation of the formulation by energy check and comparison of frequencies with reported results in the literature are also included here. Chapter 4 focuses on the parametric dynamical study using the complete CNN set of equations. Objective is to assess effect of the system design parameters on its performance, and establish critical conditions leading to unacceptable response or instability. This sets the stage for an effective controller design. As pointed out before, the governing equations of motion are coupled, non linear and nonautonomous. To assist in the controller design, rigid and flexible parts of the system are often (but not always) decoupled due to their widely separated  17  frequencies. However, it must be emphasized that the coupling between the rigid degrees of freedom, and among the flexible generalized coordinates, was retained. Two different control methodologies, linear eigenvalue assignment and the nonlinear Feedback Linearization Technique (FLT), in conjunction with the actuators in the form of offsets, thrusters and their combinations were used. Attitude controller was developed first (Chapter 5) followed by a composite attitude-vibration regulator de sign (Chapter 6). In all the cases, effectiveness of the controller was assessed through its application to the original nonlinear, nonautonomous and coupled system. Finally, in Chapter 7, a ground based facility for studying dynamics of a teth ered system as well as its control using the offset strategy is described. Experimental results are compared with numerically obtained simulation data. The thesis ends with a summary of results and recommendations for future work. Figure 1-4 presents an overview of the research project.  18  I  I  DLN 2) and Attitude (FLT  PARAMETRIC DYNAMICS USING THE CNN  *  *  *  I  GROUND BASED EXPERIMENT  A schematic diagram showing the overview of the plan of study.  Validation of the offset Control (FLT, LQG)  Offset control; DNN (Platform) and DLN (Tether) er control, CNN, DNN, DLN controI;NN, DNN, DLN *  1 CNN (Coupled Nonlinear Nonautonomous) 2 DLN (Decoupled Linear Nonautonomous) 3 DNN (Decouplled Nonlinear Nonautonomous)  Figure 1-4  I  Attitude ( FLT or Pole Placement)  4,  CONTROLLER DESIGN  4,  PROBLEM FORMULATION Kinematics Kinetics Equations of Motion (CNN  PLANAR DYNAMICS AND CONTROL OF TETHERED SATELLITE SYSTEMS  tlled Dynamics (CNN)  I  I  I  2.  2.1  FORMULATION OF THE PROBLEM  Preliminary Remarks Considerable thought was directed in arriving at a model of the complex  tethered satellite system that would represent a logical step forward in understanding its dynamics and control. The model selected for study consists of a rigid platform connected by a flexible cable (tether) of finite mass density to a rigid subsatellite. The entire system is in an arbitrary Keplerian orbit and free to undergo librational as well as both longitudinal and transverse vibrations in the plane of the orbit. Furthermore, the tether can be deployed, retrieved or maintained at a constant length (stationkeeping). Deviation of the tether attachment from the platform center of mass is treated as a function of time permitting development of the offset control strategy. As in a practical situation, the platform based momentum wheels provide control torques for its attitude control. In addition, one may attempt to regulate the tether dynamics through an orthogonal set of thrusters at the subsatellite. Synthesis of the offset and thruster based strategies present interesting possibilities for a hybrid control design. As pointed out before, in a typical mission under consideration, the tether may be deployed from a length of few meters to around 100 km. This results in a large change in the system inertia as well as a shift in the system center of mass with respect to a body fixed reference frame. Thus, a realistic model must account for the effect of shift in the center of mass on the system dynamics. The structural damping of the flexible tether is also an important parameter and hence included in the present formulation. 20  This chapter can be divided into three major sections: kinematics; kinetics; and equations of motion. System configuration and position of an elemental system mass in the inertial space are established first. This is followed by evaluation of the kinetic and potential energies of the system. Finally the governing equations of motion are derived using the Lagrangian procedure. The generalized forces due to active and passive control inputs are obtained using the principle of virtual work.  2.2 2.2.1  Kinematics of the System Domains and reference coordinates Four distinct domains can be established for the system described in Figure  2-1. Domain ‘p’ represents the rigid space platform. The frame F  Z) is  attached to the centre of mass of the platform with its axes along any arbitrary directions. Vectors described relative to this rotating frame are distinguished by the subscript ‘p’. Mass of the platform is considered constant (rh  =  0).  The tether involves two major domains. The domain ‘o’ consists of the offset mechanism and undeployed tether that is wrapped around a reel. In general, mass of the offset mechanism is expected to be quite small compared to the mass of the platform (Space Shuttle, Space Station). So, without any loss of generality, the offset mechanism is treated as a point mass and its location is expressed with respect to F by the vector d. The mass of the offset mechanism is represented with subscript  ‘o’. As the tether is deployed or retrieved the mass of the spooi, around which the tether is wound, changes. The deployed portion of the tether is flexible and belongs to the domain ‘t’. The frame Ft (Xi, Y, Zt) has its origin at the point of attachment of the tether at the platform end. The Yt-axis is along the undeformed length of the tether, i.e. the direction of tether in the absence of transverse deformation. The 21  xs  Ys  SUBSATELIJTE  d  Zs dmt  yt i  Y  TETHER  Yo  rt  Zt  OFFSET MECHANISM (MANIPULATOR)  Rdmp  xi 1 Z Figure 2-1  Domains and coordinate systems used in the formulation.  22  Xt-axis is so selected that, in absence of the out-of-plane motion, it is parallel to the orbit normal. The Zt-axis completes the right-handed system. The mass of the deployed portion of the tether varies with time according to the deployment or retrieval rate. The total tether mass, composed of the deployed (domain ‘t’) and undeployed (domain ‘o’) portions, is constant (i.e. thj  =  —r’ao).  Domain ‘s’ consists of the rigid subsatellite attached to the end of the tether. The frame F ,Y 3 5 (X ,Z 3 ) is used to describe its orientation in space. The origin 5 of this frame is at the centre of mass of the subsatellite. Vectors measured relative to this rotating frame carry subscript ‘s’. Mass of the rigid subsatellite is constant (r  =  0). Two more reference frames are required to completely define the kinematics  of a mass element in the system: the inertial frame F 1 located at the centre of Earth; and the orbital frame F 0 (X ,Y 0 ,Z 0 ). The origin of the orbital frame is located at 0 the instantaneous centre of mass of the system and follows a Keplerian orbit. The orbital frame is so oriented that the Y -axis is along the local vertical (the line joining 0 the centre of Earth and instantaneous centre of mass of the system) and points away -axis is along the local horizontal (the line perpendicular to Y 0 from Earth; the Z 0 axis and in the plane of the orbit) and points towards the direction of motion of the system; the X -axis is along the orbit normal and completes the right-handed triad. 0  2.2.2  Position vectors With the reference frames selected, the position vector of an elemental system  mass can be defined easily. As pointed out before, the instantaneous centre of mass of the system follows a Keplerian orbit. This is based on the assumption that the orbital motion is not affected by the librational (attitude) and vibrational motions 23  of the system [61—63]. The position vector to the mass element with respect to the inertial frame can be obtained by a set of vectors added in sequence. First of all, the position vector is defined with respect to the frame attached to the domain in which the mass element is located (fj, i  =  p, t, s). For bodies like the platform  and subsatellite, f and f consist of only rigid components. But for the flexible tether, the position vector has contribution from two sources: rigid (ytt) and elastic deformation (ft(yt)), i.e. =  yt3t + ft(yt),  where 3t is the unit vector along Yt-axis, and y is the distance to the mass element from the origin of F. The position of the origin of the body fixed frames, Fj, i 0 by the vector R, i p, t, s, is defined with respect to the orbital frame F  =  =  p, t, .s.  Finally, the origin of the orbital frame is defined with respect to the inertial frame by Pc (i.e. radius vector of the Keplerian orbit). With these notations, position of any elemental mass in the th domain with respect to inertial frame can be expressed as Rdmj The vector R, i  =  =  +r. 1 Rc+R  (2.1)  p, t, s, can be expressed in terms of other vectors defined in the  body fixed frames as: (2.2)  RPRSM; =  RSM + d;  (2.3)  R$—RsM+dp+L,t+ft(L)—ds;  where:  4  distance between origins of frames F and Ft expressed in F; 24  (2.4)  L undeformed length of the tether;  unit vector along the fl-axis;  t  ft(L) flexible deformation vector of the tether at a distance L from origin of Ft;  J., offset of the tether attachment point at the subsatellite end and expressed in . 3 the frame F  In Eqs.(2.3 and 2.4) all the terms on the right hand side are not independent. 0 and F, represents The vector SM, the distance between origins of the frames F  the shift in the centre of mass and can be expressed in terms of other system vari ables. By taking the first moment of the masses about the instantaneous centre of mass and equating it to zero, it can be shown that  RSM  =  _{(mo + mt + ms)4, + ms(L3t + Jt(L)  —  )+ 8 J  J  tdmt},  where: ; 8 0 + mt + m M total mass of the system, mp + m  mp mass of the platform; 0 mass of the offset mechanism; m mt mass of the tether, ptL;  p mass of the tether per unit length; 3 mass of the subsatellite. m  It may be pointed out that the vectors defined above and in Eqs.(2.1-2.4) are not with reference to the same coordinate system. So, appropriate transformation matrices are used during vector operations.  25  2.2.3  Deformation vectors  Deformation at a point on the tether depends on its position and varies with time. At a given instant, it can be expressed as three orthogonal components ut, v and wt along Xt, Yt and Zt directions, respectively. The assumed mode method is used to discretize the deformations. The admissible functions are linearly inde pendent and satisfy geometric boundary conditions [64]. In a system with constant length, the admissible functions may represent the mode shapes. But, in the present case, where the tether length changes over time, the concept of mode shape does not apply. However, an admissible function can be chosen to satisfy the geometric boundary conditions. Since the diameter of the tether is very small (a few mm), the admissible functions depend only on Y. So the tether deformations can be expressed as: ut(yt,L,t)  =  mt,(t); 1  (2.5)  vt(yt, L, t)  =  n(yt, L)Bn(t);  (2.6)  n(yt, L)Cn(t);  (2.7)  Wt(yt, L, t) =  where: 4n(yt, L), ‘I’(yt, L) admissible functions for tether transverse and longitu  dinal deformations, respectively; A(t), B(t), C(t) generalized coordinates for out-of-plane transverse, lon  gitudinal and inpiane transverse deformations, respec tively. Theoretically, a complete set of admissible functions should include infinite 26  terms. Here the completeness implies that the energy of the discretized system is the same as the energy of the continuous system [64]. For most engineering systems, finite number of functions are sufficient to represent the dynamics. In the present study, the first N 1 and Nt functions from the complete set are considered for representing the longitudinal and transverse deformations, respectively. Here N 1 and Nt are arbitrary numbers. For the admissible functions, the kinematic boundary conditions dictate that the transverse deformations at the supported ends be zero, i.e. u(0,L,t)  =  wt(O,L,t)  ut(L,L,t)  =  and the longitudinal deformation at the boundary y vt(O,L,t) At the subsatellite end (y  =  wt(L,L,t) =  =  0;  (2.8)  0 is zero, i.e.  0.  (2.9)  L), a dynamic boundary condition relating  the stretch and static tension in the tether can be obtained [65]. Having defined the flexible deformations, elemental mass of the tether at any arbitrary unstretched distance y from the origin of Ft can be defined by the position vector =  u(yt, L, t)j + (Yt + v(yt, L, t))3t + w(yt, L, t)k.  (2.10)  The admissible functions for transverse oscillations of the tether correspond to those of a flexible string [65], (y,L)= %ñsin(Tht),  n  1,2,”,Nt.  (2.11)  For the longitudinal deformation, the admissible functions can be chosen as the eigenfunction of a elastic tether supporting a point mass [29], =  sin(/3nyt/L), 27  n  =  (2.12)  where 13 is governed by the equation /3ntan(i3n)=, 3 m  n=1,2,•,Nj.  (2.13)  Alternatively, it can be taken as  =  ()  2n— 1 ,  n  =  1,2,•• ,N , 1  (2.14)  which represents the vibration of a string with an end mass [36]. In the present thesis the latter form of the admissible function (Eq.2.14) is used for the numerical simulation.  2.2.4  Librational generalized coordinates The generalized coordinates defining librational motion (rigid body motion)  are identified here. The orientation of each frame, F (i  p, t, s), is obtained relative  to the orbital frame through three modified Eulerian rotations starting from the orbital frame. The sequence consist of: • rotation of 0 (X Y F , ) by angle a about the X 0 ,Z 0 -axis resulting in (X 0 ,Y 1 ,1 1 Z ) ; • rotation of (X ,Y 1 ,Z 1 ) by angle ,t3 about the 1 1 Y axis resulting in 2 ,2 2 (X Y , Z ) ; • rotation of (X ,Y 2 , 22) by angle 2  about the 2 Z axis resulting in (Xi, Y, Z).  These sequences of Eulerian rotations are indicated in Figure 2-2. Figure 2-2(b) shows a rotation about X 0 axis in Yo, Z —plane. In case of the tether, two 0 rotations at and ‘it are sufficient to describe its orientation in space (i.e. / t 3  =  0).  Therefore the generalized coordinates for the librational degrees of freedom are: ap; p 3 /  ‘in; at;  ‘it  (assuming the subsatellite and offset mechanism to be point masses).  28  zi  Zo  Z,Z2/\ /\i  / 7  \/L1 0 , x 1 x 2 X  0  (a)  Yo (b)  Figure 2-2  Diagrams showing relative orientations of two different frames: (a) -axis. 0 sequence of Eulerian rotations; (b) rotation about the X  29  2.2.5  Angular velocity and direction cosine Since the gcneralized coordinates are defined with respect to the intermediate  frames of the Eulerian rotations, their derivatives are not the same as the inertial angular velocities expressed in frame F. Before relating derivatives of the Euler angles with the angular velocities, the following discussion on the vector transfor mation between the reference frames is appropriate. 2 can The matrix which transforms a vector from the frame Fj to the frame F be described by the equation  {v}  =  [Tj]{vj},  (2.15)  where: } 2 {v  vector expressed in frame F;  {v,}  vector expressed in frame F;  [T] transformation matrix relating (Xi, Y, Z) and (Xi, }, Zj). The transformation matrix [T] has the following properties [66]:  [T}  =  [Tjk][Tk(k_1)}  [T][Tj]  =  [I] ;  [T]  =  [T]’  .  [T}[Tij], V integer k;  (2.16) (2.17)  =  (2.18)  . T [T]  Using Eq.(2.16), the angular velocity vector of the frame F can be obtained as  {w}  =  1wzi 1 5 (wzi) Li..),j  =  [T 21 ] 2 [T ] 10 ][T  1+at I  30  0 0  10  10.1 + [T ] 21 ][T 12  =  (oJ  I ( + at) cos(,6) cos(yj) +  j  —(Ô + àt)cos() sin(’yj) +  I where:  èt  ] 2 + [T  (6+àt)sin(i3)+’y  0  sin(y) cos(yi)  .1 J  (2.19)  6 is the orbital rotation rate; indices 1 and 2 in the transformation matrices refer to the intermediate frames in the Eulerian rotation sequence; and 2 T 2  sin(7j) cos(7j) 0  cos(yj) —sin(7j) 0  =  1 = 2 T  cos(f3j) 0 6) 1 sin(  0 T 1  1 0 0  =  0 1 0  0 cos() —sin(a)  0 0 1  —sin(/3) 0 ) 3 cos(/ 0 sin(a)  cos(a)  It is necessary to obtain the direction cosines {l} of the frame F relative to the local vertical as they are needed in evaluation of the gravitational potential energy later: {l}  .1 l (l J  1  I =  =  {i}  } 0 {i {j} {j°} .  } 0 ({k}.{j  10 =  =  where {i},  } 0 {j  {jJ, {k:}  ] 0 ] [Ti 21 ] [T 2 [T  I (  1 (0  sin(a) sin(i3j) cos(yj) + cos(a) sin(7j) sin(a) sin(/3) sin(7j) + cos(a) cos(’yj) —sin(a)cos(6)  4  ,  —  (2.20)  J  are unit vectors along the X, Y, Z axes, respectively; and  axis. -Y is the unit vector along the 0  31  2.3 2.3.1  Kinetics Kinetic energy The inertial velocity of any elemental mass in the th domain can be obtained  by differentiating Eq.(2.1). In the following development, a bar over a character to represent a vector quantity is dropped and replaced by brackets. The matrices are enclosed by square brackets. With these notations, the velocity of an elemental mass in the  domain becomes  1 {dm}F  =  1 + {ic}F  +  {} + {w}  x {r}.  (2.21)  Here, the terms with subscript F 1 refer to velocities with respect to the inertial frame while those without any subscript are with reference to the local frames. The kinetic energy of the elemental mass (Figure 2-1) can be expressed as Tdmj  =  •{Rdm}F  {Rdm.}Fdmj.  (2.22)  The energy of the th body (Ti) can be obtained by integrating the above equation over the mass of the body. The kinetic energy of the entire system (T) is the sum of the energies of the individual domains, T=T i=p,t,s =  i=p,t,s  J  m  {kdmj} {dm}dimi,  (2.23)  where mj is the mass of the domain i. Substituting Eq.(2.21) in Eq.(2.23) and after some algebraic manipulations, the total kinetic energy of the system can be written as T  =  -  {kc}F + M{RsM}F 32  {SM}F  (m + mt + 4 + 0 ms){ } F + +  —  } t j{  +  } 1 {it(L)}F  {t}dmt +  ({{Lt}}  J{. } t  .  } + 1 {it(L)}F  >  {w}T[I}{w}  {{wt} x {rt}}dmt  {{Lft}})  _ms({{Lt}}  + {SM}F 1  —  _{ML)}F})  {mo +mt 1 }F + 4 +ms){  +ms{{Ljt}}  1 _ms{{ds}F  {t}  -  +{wt} x {rt} }dmt  {it(L)}F}}  {t}+{wt}x{rt} }dmt + ms{{Ljt}}  -  ms{{s}F + {Jt(L)}FJ}}.  (2.24)  For conciseness, the kinetic energy is expressed as products of vectors and matrices. Note, the kinetic energy expression accounts for 3-dimensional rotation of the plat form, tether and subsatellite, as well as flexibility of the tether. For the particular case where the system dynamics is confined to the orbital plane, the kinetic energy expression reduces to  7’  =  M{ic}{Rc}F + {i?sM}{M{i?sM} + {i?m}} {w}T[I]{w} + (ms +  +  + +  i=p,t mp{cip}T[U]{dp}  {  2 pL).L  + rnsL2w + ma{4}T{4}  + mawz{dp}T[UkjT[Uk]{dp}  } + wpz{dp}T[UkjT[TtPJT{Kl} 1 }T[Tt ]T{K  + 2 [ {ã} [ T ] {..k} Tj K+T [ wpz{dp} [ [ K2]{X} Ttp} UkJ  33  + {}T [T]T [K ] {X} + w {dp}’ [Uk]T [Tt]T [K 3 ] {X} 3 +] 4 [ T {iC} {(} K+] 5 [ T {(} {X} K [ T {X} ] {X} K + {K }{i(} + {K 7 }{X}, 8 -i- 6  (2.25)  where:  ma  =  0 + pjL + m m ; 3  {1m}  =  ma{{4} +  {wp}  x {d}} +  +ms{{{Lt}} {X} =  { },  =  J  {{t}  + {wt} x {rt}}dmt  + {Jt(L)F};  { },  and are expressed with respect to Ft;  {d}  =  velocity of the offset point in the frame Fr,;  {B}  =  ; {C} 1 {B}  =  . 1 {C}t  Here, N 1 and Nt are the number of generalized coordinates for longitudinal and inpiane transverse vibrations, respectively. 1 {K } , 2 [K ] ,  ,  } are defined in 8 {K  Appendix-I. In the above energy expression (Eq.2.25), the first term represents the orbital kinetic energy; the second accounts for the shift in the centre of mass; while the rest of the terms arise from librational and vibrational motions of the system.  2.3.2  Gravitational potential energy As in the case of the kinetic energy, the total gravitational potential energy  is the sum of the energies of the individual domains. The gravitational potential energy for an elemental mass dm can be written as  dUG.  GMe =  IRdmjI 34  dm.  (2.26)  Using the binomial expansion with truncation of the series after the second degree terms, the potential energy for the i’’ domain can be written as Uc  —  J  +  =  ({R} {RJ —3 ({jo} {R})  2)  dm,  where: G universal gravitational constant; Me mass of Earth; .Rc magnitude of the orbital radius; m mass of the th domain;  {j°}  -axis (local vertical); 0 unit vector along the Y  and {R} as defined in Eqs.(2.3, 2.4). The gravitational potential energy for the entire system can now be written as —  U  GMeM + GMe  =  (M({RSM}. {RSM}  + ma(2{RSM} {d} + {d}. {d} -  —  —  3({jo} {RSM})2)  } {RM})({jo} 0 6({j  {dp})  } {d})2) 0 3({j  + ms({Ljt} {Lj} + 2{Lj} {ft(L)} + {ft(L)} {ft(L)}) + 2ms{{RSM} + {d}} {{Ljt} + {ft(L)}} -  3m ({io} {Ljt})  } 0 ({i  {Ljt} + 0 2{j {RSM} }  } {ft(L)}) 0 + 2{j } {d} + 2{j 0 -  3m ({i }. {ft(L)}) 0  + 2{{RSM} + {d}  -  (0  } {d}) 0 } {RSM} + 2{j 0 {ft(L)} + 2{j .  3({jo} {d} +  (tr[zj_3{z}T[I]{l})).  -  {jo}  .  {RSM}){jo}} {J{rt}dmt} (2.27)  i=p,t,s 35  Here the first term represents potential energy of the system treated as a point mass. The rest of the terms are due to the system’s finite dimension. The potential energy expression when simplified and expressed in matrix notation for the planar case has the form  —  UG  GMeM  =  ]{d} 2 ({RSM}T[Plj{RSM} + {d}T[P  +  {X}T[P ] {RsM + d} 2{RsM}T[P ] {dp} + 3 +2 {X}T[P j {X} + {p}T{R {X}T{P + 5 } +4 (1 2 + msL  —  to) 3T  (tr[i1  —  + d}  —  (2.28)  i=p,t  where [Ii] inertia dyadic of the domain i with respect to F (Appendix-I) -axis (local vertical) with respect to the frame F; 0 {l} direction cosine of the Y and [F ], [F 1 ],... , {P}, and Vto are defined in Appendix-I. In Eq.(2.28), terms 2 containing {RSM } account for the potential energy due to shift in the centre of mass.  2.3.3  Elastic potential energy In the linear elastic theory of strings, it is generally assumed that the ini  tial tension in the string is large enough and the transverse displacements cause negligible change in this tension. In a tethered orbiting system, the tension may be reasonably large for long tethers (length of the order of kms). But at shorter lengths, the tension is very low due to the weak gravity gradient force. In the ex treme case when the length approaches zero, the tension tends to zero. Therefore the effect of transverse vibration on the tether tension, and hence on the elastic 36  oscillations can not be neglected. The strain energy of the tether is based on the theory of vibrating string [67]. In order to account for interactions between the longitudinal and transverse modes, transverse displacement terms upto the second order are retained in the strain expression övt  1 1 / ãut 2 2 (—) yt 9 c  ow  2\  1 +—) Oyt yt 9 f / is the total strain in an elemental tether mass of volume dVt. Total strain E=—+—(  where  energy of the tether can be written as  usz_J  cdVt,  EAJ°  (  + 1((ôut)2 + (thvt)2))dY  (2.29)  where: u stress in an elemental mass of volume dVi, EE;  A area of cross-section of the tether; L(t) unstretched tether length; T4 volume of the unstretched tether, AL(t); E young’s modulus of the tether material. Substituting from Eqs.(2.6, 2.7) in Eq.(2.29), strain energy of the planar system can be obtained as JL(t) (d{F(Yt)}T{B})  5 U  JL(t) (d{F(t)} T +  JL(t) dyt +  ) 37  4 (d{Fb(vt)}T{ d yt c})  (d{Fc 5 1 (Yt)}T{C})  dyt.  (2.30)  Dissipation energy  2.3.4  The energy dissipation during tether deformation can be accounted for through structural damping. Because of the complex mechanism of energy dissipation, the stress—strain relation does not correspond to the elastic case. The system exhibits the hysteresis phenomena during vibration. The area enclosed by the hysteresis curve indicates the energy dissipation. In engineering applications, it is commonly accounted for by considering the structural damping coefficient determined experi mentally [68]. The structural damping can be expressed as an equivalent complex Young’s modulus, =  (2.31)  , 1 E + iE  where the real part E is the Young’s modulus without structural damping; the 1 contributes to the structural damping; and i imaginary part E  =  /1i. Now the  total stress can be written as at  E’c  =  € iE ) (E + 1  =  E(l + ii)c,  where  If  is harmonic with frequency  (2.32)  <<1.  =  ,  i  =  (2.33)  —  wo  and the total stress becomes =  where a  =  E[e+  (_)E] =  Ee is the stress in absence of the structural damping; and ad  is the stress causing energy dissipation. 38  (2.34)  u+ad,  =  The dissipation of energy can be expressed in terms of the Rayleigh’s dissi pation function [69], Wd= =  2.4  /  OdfdVt  2 L 71 EAi  (2.35)  Equations of Motion Using the Lagrangian procedure, the governing equations of motion can be  obtained from d  ÔT —  ÔT + c9U  ãWd  +  Qq,  (2.36)  where: q vector of generalized coordinates;  Qq  vector of generalized forces corresponding to generalized coordinates q;  U =UG+US.  The governing equations of motion account for: (i) inpiane rotation of platform and tether negotiating any arbitrary trajectory; (ii) longitudinal and inpiane transverse vibrations of the tether; (iii) inpiane (two dimensional) offset of the tether attachment point from the space platform’s centre of mass; (iv) effect of the controlled variation of the offset attachment point; (v) influence of deployment and retrieval on the system dynamics; (vi) effect of thrusters located at the subsatellite end and momentum gyros on the platform; 39  (vii) shift in the centre of mass due to rigid body librations and elastic deforma tions of the tether. They would permit parametric response analysis of the system as well as aid in devel opment of control strategies using thrusters, offsets and their hybrid combinations. In a compact form the equations of motion can be expressed as [M(q,t)]{ij} + {F(q,,t)}  =  (2.37)  {Qq},  where: [M(q, t)} mass matrix (Appendix-IT);  { F(q, t, t)}  nonlinear, nonautonomous terms accounting for gravitational, Coriolis and centrifugal forces (Appendix II);  {q} vector of generalized coordinates,  {ap,  at,  {B}T,  {C}T}T  ap platform pitch angle; at tether pitch angle; {B} vector of longitudinal elastic generalized coordinates;  { C} { Qq }  2.5  vector of transverse elastic generalized coordinates; nonconservative generalized force vector.  Generalized Forces The generalized force vector (Qq) accounts for the effects of nonconservative  external forces and moments due to active and passive control, and environmental disturbances. As mentioned before, the platform based momentum gyros provide control torque (M,) about the platform axis Xp (Figure 2-3). As shown in the 40  TL SU BSATELLITE  Tl:xt  TETHER  Ta  d’  ORBIT  ra Yi  0  xi  1 Z Figure 2-3  A schematic diagram of the Tethered Satellite System (TSS) show ing different input variables.  41  figure, the tether swing can be controlled by manipulating an orthogonal set of thrusters (Tat, TL) at the subsatellite, or the motion of the tether attachment point d) at the platform end. As discussed in Chapter 4, the control influence  matrix for the offset strategy can be obtained from the coefficient of the offset acceleration in the equations of motion. However, for thruster control and platform pitch equation, it has to be obtained as the generalized force vector. The generalized force vector, virtual work.  , F 1 , 2 Let F  ,  can be evaluated using the principle of  Q,  Fm are m external forces acting on the system  , ‘2, 1 at locations with position vectors r  ,  fm, respectively. The system has  qn). The position vectors are, in general,  n generalized coordinates (qi, q,  functions of qj’s. The virtual work, SW, by all the external forces can be expressed. as m  m  i=1  i=1  j=1  ( :=  p.  3 q  n  sqj  =  -  , q 8 3 Qq  (2.38)  where (2.39) is the generalized force corresponding to the generalized coordinate qj• The external forces and moments acting on the system are: M  control moment acting on the platform about the X, axis;  Tat thrust applied perpendicular to the tether line at the subsatellite end; TL control thrust along the undeformed tether applied at the subsatellite; Fdt damping force perpendicular to the undeformed tether;  42  Fdj damping force along the tether line. The transverse and longitudinal dampers are located at distances of dt and dl from the origin of Ft. Using Eq.(2.39), the generalized forces for the tether angle and elastic degrees of freedom can be obtained as:  { Qn} { Qc}  + {F(L)}T{B})  + Fdt (Ydt + {F(Ydt)}T{B}); (2.40)  =  Tat (L  =  TL{F(L)} + Fdl{F(Ydl)};  (2.41)  =  Fd{Fl(Yd)}.  (2.42)  Since the tether can not transmit any moment or transverse force, the com putation of Qap needs special attention. To that end, rather than considering the external forces acting on the tether for virtual work computation, the equivalent tension ( ”a) at the platform end of the tether is employed. The tension ta depends 1 On  Tat, TL, Fdt and Fdl, and attitude angles of the tether and platform. Tat and  ’a Fdt are perpendicular to the undeformed tether and hence their contribution to t can be neglected. This results in  The generalized force for ap can now be obtained from  where a is the position vector of the tether attachment point on the platform. After appropriate transformation of vectors and expanding the dot product it can be shown that  Qap  =  M + (TL + Fdl) (Dt sin(a) 43  —  .Dt cos(a)),  (2.43)  where: a  2.6  ap;  =  at  =  RSM + dp;  =  RSMZ +  —  4•  Summary A general set of kinetic and potential energy expressions are obtained for  a platform based tethered satellite system undergoing three-dimensional dynamics. These expressions are used to obtain, through the Lagrangian procedure, governing equations of motion for the system dynamics confined to the plane of the orbit. The highly nonlinear, nonautonomous and coupled equations of motion are extremely lengthy even in the matrix notation. They account for a shift in the center of mass, time dependent variation of the tether attachment point at the platform, as well as deployment and retrieval of the tether. The structural damping is modelled through Rayleigh’s dissipation function. The generalized force vector representing effects of external forces is evaluated using the principle of virtual work. The relatively general formulation can implement the offset and thruster control strategies to regulate both the rigid and flexible dynamics of the tether with the platform attitude motion controlled by momentum gyros.  44  3. 3.1  COMPUTER IMPLEMENTATION  Preliminary Remarks The governing equations of motion developed in the last chapter were inte  grated numerically to study the dynamics and control of the tethered systems. The computer code is quite lengthy  ( 6200  lines), mainly due to the highly time vary  ing nature of the system dynamics and moving tether attachment point. Moreover, the frequencies of attitude and elastic degrees of freedom are widely separated with a closely packed spectrum for the flexible system. The stiff characteristic of the system, which may present problems during numerical solution, demanded special attention. It was desirable to explore several different approaches to arrive at an efficient controller design. To that end, it was necessary to evolve a flexible program structure that would permit varied simulations with a few parameter changes in the input file. Of course, the program should be easy to debug and efficient to run. This chapter begins with discussion on the numerical algorithm and program structure developed. Next, validity of the equations of motion and the computer code are assessed by two different methods: conservation of total energy; and com parison of frequency spectra for a particular case of the linear system as reported in the literature. The chapter ends with a summary of salient features of the numerical code. 3.2  Numerical Implementation  Integration algorithm  A computer program is written to numerically solve the governing equations of motion of the tethered satellite system. The second order ordinary differential 45  equations representing the system dynamics are rearranged as first order equations to this end. The system dynamics as represented in Eq.(2.37) can be expressed as =  [M(q,t)]_1{Qq  —  (3.1)  F(q,i,t)}.  Defining  the above relation can be rearranged as  {  [M(q,t)]1{Qq  -  F(q,,t)}  }‘  (3.2)  which is a set of 2N first order equations. Here N is the dimension of the generalized coordinate vector {q}. The IMSL:DGEAR subroutine is used to integrate the above equation. The main advantage of this method is the automatic adjustment of the iteration step—size for stiff systems with error check in each iteration cycle [70]. The subroutine uses Gear’s predictor-corrector algorithm [71]. Program structure The flowchart showing the structure of the program to simulate- both un controlled and controlled dynamics of the system is presented in Figure 3-1. The program starts with initialization of the system parameters and generalized coordi nates. Special program parameters are introduced to identify the type of controller used in the simulation. The initialization is achieved by reading the input file. In each integration time-step, the generalized coordinates, length and offset variables are written into the output files. The main program calls the integration subrou tine DGEAR which in turn calls the subprogram FCN. The system dynamics as expressed in Eq.(3.2) is computed by this subroutine. In case of controlled simula tion, the actuator inputs are obtained from the subroutine CONTROL. Details of 46  INPUT  I I  Mass and Elastic Parameters  Orbital Elements Control  L  Initial Disturbances -  Deployment and Retrieval Parameters IMSL: DGEAR Parameters  - -  MAIN PROGRAM  Subroutine DGEAR (Numerical Integration)  Figure 3-1  Flowchart showing the structure of the main program.  47  the subroutines are given below. Subroutine FCN This subroutine computes the mass matrix and the nonlinear terms to for mulate the vector  ‘,  Eq.(3.2), required by the integration subprogram (DGEAR).  As shown in the flowchart (Figure 3-2), this subroutine calls a number of sub programs (such as KINETIC, POTENTIAL, etc.) which perform certain specific computation (e.g. computation of modal integrals used in Kinetic and potential energy expressions). This feature makes the program easy to debug. The velocity and acceleration of the tether deployment/retrieval maneuver are calculated by the subprogram LENGTH. Based on the specified parameters, the subprogram OFF SET computes the velocity and acceleration profiles for the specified offset motion These two subroutines define the system maneuvers. Next, the modal integrals and other matrices used in the formulation (Appendix—I) are evaluated using the subpro grams KINETIC and POTENTIAL. As explained in Chapter 2, some intermediate matrices are defined for concise presentation of the governing equations of motion. The matrices appearing in the potential and kinetic energy expressions are evalu ated through the subprograms SP and SK, respectively. The matrices used in the potential energy equation are ], 1 [F  ], 2 [F  ], 3 [F  }, 6 ], [F 4 [P ], {P 5  and those used in kinetic energy expression are ], {K 6 }, {K 7 }. 8 ], [K 2 ], [K 3 ], [K 4 ], [K 5 }, [K 1 {K  The contribution of the terms containing the inertia matrix (Appendix I) of the tether is computed by the subroutine INERTIA. The contribution of the strain 48  + Subroutine olNTROJ No Calculate  ‘  H  “F  Subroutine IMSL: DGEAR  Figure 3-2  Structure of the subroutine FCN to evaluate the first order differ ential equations required by the integration subroutine.  49  energy and structural damping are estimated in subprograms STRAIN and DAMP ING, respectively. The above subprograms completely define all the terms required to evaluate the governing equations of the system. Depending on the specified control param eters, the system inputs are established and the vector  computed. This value is  used by the integration subroutine DGEAR for the numerical solution.  Subroutine CONTROL The system inputs required to regulate the dynamics are computed by the subroutine CONTROL (Figure 3-3). This subroutine selects the appropriate con troller design subprogram for either offset or thruster strategy. The subprogram for offset control obtains the system model and then designs the controller. In the thruster based regulator, three different models can be used for the controller design. Once the model is selected, the procedure is the same as in the offset strategy. The control inputs are computed and returned to the subroutine FCN.  3.3  Formulation Verification Once the computer program is developed to integrate the equations of mo  tion, the next logical step is to validate the formulation as well as the numerical code. Two different approaches are used to this end. In the first place, total energy is computed for conservative configurations of the system. In the second approach, natural frequencies of the linear system are compared with those reported in the literature. As can be expected, numerical results for the model selected for study are not available. One is forced to be content with validation through comparison of a few simplified cases. 50  Offset control  Figure 3-3  Flowchart for the subprogram CONTROL to compute actuation inputs.  51  3.3.1  Energy conservation  In the study of the attitude dynamics of spacecraft, the effect of attitude dynamics on the orbital motion is very small [61, 62, 63J. Therefore, it is a normal practice to consider the satellite to be moving in a Keplerian orbit. This assumption does not affect the results of an attitude dynamic study. However, it imposes an ad ditional constraint and makes the system nonconservative. So, the attitude and the orbital motions are decoupled and the governing equations representing uncoupled attitude dynamics, which is a conservative system, is used for energy calculation. The inertia and elastic parameters considered in the analysis are the same as those used for the dynamic simulation (Chapter 4). With zero structural damping, the variations in kinetic, potential and total energies from the reference values are plotted in Figure 3-4. The energies at the beginning of the simulation are considered as the reference values. Figure 3-4(a) shows results for the stationkeeping at a tether length of 20 km and zero offset where as Figure 3-4(b) is for L  =  5 km and an offset of 1 m along the the local  horizontal. In both the simulations, initial disturbances of 0.5° is given to the tether attitude (at) and the platform pitch angle  (ap).  Initial conditions of 12 m and 0.8 m  ) of the tether for L 1 are given to the longitudinal oscillation (B  =  20 km and 5 km,  respectively. For the tether transverse vibration, the disturbance levels are taken as 1 m and 0.01 m for L  20 and 5 km, respectively. As expected for a conservative  system, change in the total energy remains zero and there is a continuous exchange between the potential and kinetic energies in both the simulations. Similar results were obtained for other lengths and offset positions of the tether. In all the cases conservative nature of the system was found to be preserved. It should be recognized that during deployment/retrieval and movement of the tether attachment point, 52  50.0  25.0  0.0  -25.0  -50.0  Orbit  0.3  2.0  0.0  -2.0  0.0  Figure 3-4  Orbit  0.3  Figure showing variations of kinetic, potential (gravitational plus elastic) and total energies of the system during the stationkeeping ,, = 1 = 0; (b) L = 5 1cm, d = 1 m, phase: (a)L = 20 km, d =  0. 53  there is energy input to the system. So the system is no longer conservative and the energy check can not be applied.  3.3.2  Eigenvalue comparison  In this case, linear models were obtained for decoupled rigid and flexible sub systems by neglecting the second and higher order terms and making appropriate substitution for the trigonometric terms. These linearized equations of motion for the rigid and flexible subsystems are given in Appendices III and IV, respectively. The frequencies of the system are calculated and compared with the results reported by Keshmiri and Misra [34], and Pasca and Pignataro [72]. In ref. [34], the number of elastic modes of a tether in each direction is limited to two. The higher frequencies• of the system are obtained by using what authors call the semi-bead model, where the tether is divided into a number of smaller segments and negligible masses are placed at the connection points. Ref. [72] analyzes a system consisting of an elas tic continuum tether with mass, and orbiter and subsatellite represented as point masses. The linearized equations of motion are solved by means of a perturbation technique.  -  The present analysis, which models the tether as a flexible string using the  assumed mode method, can include any arbitrary number of modes in both the longitudinal and transverse directions. The eigenvalues are computed for the same mass and geometric parameters as in the references. The governing equations are linearized about the static equilibrium position which is zero for the rigid and trans verse flexible modes. The equilibrium value for the first longitudinal mode is 100 m and 3.6 m for tether lengths (L) of 100 km and 20 km, respectively. The normalized frequencies are compared in Tables 3.1 and 3.2. The mass and elastic parameters 54  considered for the frequency computation are indicated in the tables. Table 3.1  Mode 1 2 3 4 5 6 7 8 9 10 11 12  Inplane dimensionless natural frequencies (w/Ô) of a 2-body system 3 500 kg, pt = 5.76 kg/km, (L 100 km, mp = i0 kg, m N). EA 2.8 x Present Study 1.732 5.958 11.917 17.876 23.835 29.794 35.753 41.712 47.670 53.627 54.266 59.589  Ref.[34]  Ref.[72j  Type  1.731 6.388 12.059 17.879 23.742 29.627 35.528 41.445 47.383 53.344 54.541 59.337  1.794 6.905 12.457 18.269 24.142  Lib. Tran. Tran. Tran. Tran. Tran. Tran. Tran. Tran. Tran. Long. Tran.  —  —  —  —  53.807 54.559 59.758  , 3 In the tables, w is the frequency of the system; 8, the orbital frequency; m the subsatellite mass; pt, the mass per unit length of the tether; E, the Young’s modulus and A, the cross sectional area of the tether. The first column represents the mode number and the last column is the mode type, i.e. librational (Lib.), transverse (Tran.) or longitudinal (Long.). Table 3.2  Mode 1 2 3 4 5  Inpiane dimensionless natural frequencies (w/6) of a 2-body system 3 = 576 kg, p = 5.76 kg/km, EA = (L = 20 km, mp = oo, m N). 2.8 x Present Study 1.732 12.640 25.281 37.922 50.563  Ref.[34]  Ref.[72]  Type  1.732 12.777 25.245 37.795 50.369  1.733 12.780 25.241 37.771 50.319  Lib. Tran. Tran. Tran. Tran.  55  The minor differences between the present results and those of ref. [34] are of the same order in both the cases. The discrepancies may be attributed to the different formulation methods used. The larger differences for L a parameter 7  =  =  100 km is due to  pL/ms defined in ref.[72], which plays an important role. y has a  smaller value in Table 3.2 than in the previous case. As pointed out by the authors, their results are more accurate for smaller -y, hence the correlation is better in Table 3.2. Frequencies of the systems with different platform and subsatellite masses are shown in Table 3.3. The discrepancy between the present results and those of Ref. [34] follow the same trend as before. Since  has a reasonable value, the results  are in better agreement with those of Ref. [72]. Results for three cases are reported here with different end masses. One case is close to the TSS configuration where the platform mass is iü kg and the subsatellite mass is 500 kg. Other two cases are for equal end masses. As shown in Table 3.3, the frequencies are the highest with the largest end masses (mp (mp  =  3 m  =  =  ms  =  iü kg) and the lowest for the smallest end masses  500 kg).  Note, the results of the linearized system match quite well with the reported data although the three studies use different methods to determine frequencies of the system. Furthermore, the results are in close agreement for different tether lengths as well as different masses of the endbodies. This with the conservation of total energ test provides considerable confidence in the validity of the governing equations of motion and the numerical code developed for their integration.  56  Table 3.3  Mode mp 1 2 3 4 5 mp 1 2 3 4 5 6 mp 1 2 3 4 5  3.4  =  =  =  Comparison of natural frequencies of a tethered two-body system with different end masses (L = 20 km, Pt = 5.76 kg/km, EA 2.8 x iü N). Present Ref.[34J Study 3 = 500 kg kg,m 1.732 1.732 11.917 11.956 23.835 23.578 35.753 35.286 47.671 47.019 kg ms = 1.732 1.715 10.028 14.432 113.453 113.430 226.908 227.023 340.364 340.934 453.813 455.167 ms = 500 kg 1.732 1.732 8.294 8.486 16.588 16.507 24.883 24.637 33.178 32.799  Ref.[72]  Type  1.738 11.993 23.644 35.368 47.112  Lib. Tran. Tran. Tran. Tran.  1.728 113,385 226.736 340.095 453.455  Lib. Tran. Tran. Tran. Tran. Tran.  1.742 8.341 16.191 24.155 32.142  Lib. Tran. Tran. Tran. Tran.  —  Summary The governing equations of motion for the tethered satellite system are in  tegrated numerically using the IMSL:DGEAR subroutine. The computer program is developed in a structured manner to reduce debugging and running time. The numerical code is validated by two methods: the total energy check for conservative systems; and comparison of frequencies with those reported in the literature. The excellent correlation provide confidence in the simulation model and the numerical code developed for its dynamical response and control studies. 57  4.  4.1  DYNAMIC SIMULATION  Preliminary Remarks  Understanding of the system dynamics is fundamental to the design and development of any engineering system. Furthermore, design of an appropriate controller requires appreciation of the system performance under a wide variety of operating conditions to guard against the possible instability. To that end, the governing equations of motion of the tethered satellite system are numerically inte grated and the system’s dynamical response studied for different parameter values and operating conditions. A major concern in the operation of tethered systems is the dynamical be haviour during deployment and retrieval phases. In modelling of a flexible system, the number of modes used for discretization is important. The model developed here can include an arbitrary number of modes for both the transverse and longi tudinal vibration of the tether. An acceptable number of modes for the analysis of the system is arrived at through comparison of simulation results including higher modes. This chapter focuses on results of a parametric study carried out by system atic variations of the tether length (L), offset of the tether attachment point (d and  Young’s modulus of the tether (E), density of the tether material (Pt) and  . (m ) mass of the payload 3  4.2  Deployment and Retrieval Schemes  Before proceeding with the parametric analysis of the system dynamics, some remarks on the deployment and retrieval time histories used in the study 58  would be appropriate. The deployment is carried out with an exponential-constant1 and L exponential velocity profile. Let L 2 are the lengths where the velocity (L)  profile changes from exponential to constant and from constant to exponential char acter, respectively. The deployment velocity profile is characterized by the following relations: LcdL, =  CdL1,  L), —Cd(Ll+L — 2  LaL<Li;  (4.1)  1 L  L <L ; 2  (4.2)  2 L  LLf;  (4.3)  0 and Lf are the initial and final tether lengths, and Cd is the proportionality where L  constant. The above equations can be integrated to obtain the length expression: L  =  Loecd(t_t0),  =  1 + CdL1(t L  =  1 +L L 2  —  , t ) 1  —  Lie_Cd(t_t2),  0 L  L <L ; 1  (4.4)  1 L  L <L ; 2  (4.5)  2 L  L  (4.6)  Lf,  , respec 2 1 and L and t are the time instants when the tether length is L  where  tively, starting from the beginning of deployment. Given the initial time t, final , the proportionality constant can be obtained as 2 1 and L , Lf, L 0 time tf, L Cd  (tftO){ln(t) +  ) (L i 2  Lf)} Ll+L 2 ( 1 _  ()  Similarly, expressions for the retrieval profile are  and  L  =  crL;  (4.8)  L  =  ed7(t_t0), 0 L  (4.9)  0 are the tether length and time, respectively, at the beginning of 0 and t where L 59  the retrieval phase. The proportionality constant, c,, now has the form cr =  ( )  in (t).  (4.10)  At times, particularly with the offset control in a hybrid scheme, only the exponential velocity profile is used. The corresponding length expressions for such deployment scheme are similar to the retrieval expressions with appropriate initial and final parameters.  4.3  Simulation Results and Discussion The parametric study was rather comprehensive, however, for conciseness,  only a few typical results are reported here to help establish the trends in the dynamic behaviour. The inertia and elastic parameters considered in the analysis are: I,  =  Inertia matrix of the platform, 8,646,050 —8,135 328,108  —8,135 1,091,430 27,116  328,108 27,116 8,286,760  ; 2 kg-rn  rnp  =  mass of the platform, 90, 000 kg;  0 rn  =  mass of the offset mechanism, 10 kg;  Pt  =  mass of the tether per unit length, 4.9 x i0 kg/rn;  rns  =  mass of the subsatellite, 500 kg;  EA= 61,645 N. The system negotiates a circular trajectory with a period of 90.3 minutes. In the simulation, the  ). The platform 0 X-axis is oriented parallel to the orbit normal (X 60  pitch,  ,,  is the angle between the Yp-axis and the local vertical (i.e. Y -axis). 0  Similarly, X.t is parallel to X 0 and the tether pitch is the angle between Y and the local vertical. The longitudinal and lateral elastic deformations of the tether are measured with respect to the frame Ft. The structural damping considered in the simulation corresponds to a damp ing ratio  () of 0.5% based on the first natural frequency of the longitudinal oscilla  tion of the tether. Computation of dissipative terms due to the structural damping 0 is considered and w 0 (Section 2.3.4). Here w  requires the values of the parameters as the frequency of the  st 1  longitudinal mode (B ) which can be computed from the 1  equivalent stiffness and mass of the decoupled system, wo  /--  =  4!  —,  tm V &  where:  m  =  (1, 1) + Wtl k 3 (1, 1) + Ck 3 Ck (l 1) 4  =  (1, 1). 4 2K  The structural damping parameter —  —  (1, 1) 6 2K  can be obtained from the damping ratio  ,  2Ewmb /  LI ‘ ’i \ 1  Jo  The matrices used in the above expressions are defined in Appendix I. 4.3.1  Number of modes for discretization In the modelling of a flexible system using the assumed mode method, the 61  system is complete in the sense that the energy of the mathematical model converges to the true energy when the number of modes approaches infinite [64]. However, in reality, one can employ only a finite number of modes. Fortunately, for most physical systems, it has been observed that only a few modes can represent the system dynamics with considerable degree of accuracy. The model with the first three transverse and the first two longitudinal modes are considered to be reasonably accurate. The disturbance induced relative amplitudes of different modes were used as a basis for selecting the acceptable number of modes. Simulations are carried out for two tether lengths. Figure 4-1 shows the response of the system with a tether length of 20 km and initial conditions as shown in the diagram. As expected, for the stationkeeping case, the rigid degrees of freedom (i.e. ap and at) exhibit pure oscillatory motion. The first and second longitudinal modes (B 1 and B ) show high frequency decaying oscillation due to the structural 2 damping, which has a stronger effect on the second mode 2 (B ) . Modulation of the 2 response is due to its coupling with the first longitudinal mode. Note, an initial B disturbance of 10 m is given to the first transverse mode 1 (C ) . With zero specified offset (D  =  =  0) of the tether attachment point, the tether dynamics is  not coupled with the platform motion. Therefore C 1 has a pure oscillatory motion with zero mean. As the structural damping has only the second order effect on the transverse tether vibration, the responses in the higher modes, C 2 and C , are 3 also non-decaying. As apparent from the at response, there is very small coupling between the attitude and flexible motion of the tether even for, a length of 20 km. The modulation of the second and third transverse modes (C 2 and C ) is due to the 3 coupling between the flexible degrees of freedom. The important aspect of this simulation is the relative magnitudes repre  62  (O) (0) 1 B  =  2°; a(0) = 20 12 m; B (0) = 0 2  C.(0)=lOm  (0) 2 C  (0) 3 C  =  0  Stationkeeping: L D=0; D=O  2.0  =  20 km  2.0  cz  0.0 -2.0 0.0  =  2.0  0.0 -2.0 0.0  2.0  18.0  1 B 12.0 0.0  2 B  0.0 -0.6 0 10.0 0.0 -10.0  0.15  2 C  0.00 -0.15 0.05  3 C  0.00 -0.05  o.5 Figure 4-1  Orbit  2.0  Dynamic response of the system with the higher modes included in the discretization of the system.  63  senting the energy content of the higher modes. A comparison for the longitudinal 2 is an order of magnitude lower than that of modes shows that the amplitude of B . Moreover, the vibration of the second longitudinal mode decays in about 0.02 1 B  orbit as compared to one orbit for the first mode. As mentioned before, the sustained 1 response. So the energy content 2 is due to its coupling with the B oscillation of B  of the second longitudinal mode would be very small compared to that of the first one. Similar conclusion can also be made for the transverse modes. The amplitude of the first transverse mode (Ci) is higher by at least one order of magnitude from 3 response. Thus the , and two orders of magnitude from that of the C 2 that of C energy content in the higher modes of the tether vibration is minimal. Similar conclusion can be arrived at from the simulation results for a tether length of 10 km (Figure 4-2). Unlike the previous case, now the modulation of the third transverse mode is quite prominent, although the amplitude is rather small. The beat phenomenon is due to the weak coupling between the attitude and flexible motions through a shift in the center of mass. Note, in this case the differences between the fundamental and the higher modes are more than two orders of magnitude. From these two sample cases, it can be concluded that the energy content in the higher modes is relatively small. Therefore, in the subsequent dynamics and attitude control studies, only the first longitudinal and transverse modes are considered. 4.3.2  Tether length To facilitate comparison of results, a reference case is established for a tether  length of 5 km and zero offset along both the local horizontal and local vertical direc 64  (0) =0 3 (o) = C 2 c  x(O) = 2°; cx.(0) =2° (0) = 3m; B 1 B (0) = 0 2  Stationkeeping: L =0; =0  (0) =2 m 1 C  1 B  =  10 km  IWWWWVVvV  3  .5  Cl j 0.02 C2  3 C  0.01 0.00 -0.01 0.0  Figure 4-2  Orbit  4.0  System response for a shorter tether with higher modes included in the flexibility modelling.  65  tions. In the reference case (Figure 4-3), the rigid body rotations of the tether (ct) and the platform  (ap)  are essentially unaffected by the tether flexibility. For zero  offset it is expected that the platform dynamics is not coupled with the tether mo tion and hence free from its dynamical behaviour. The tether rotation is negligibly affected by its flexibility dynamics. As apparent from the figure, the longitudinal  ) consists of two frequencies. The lower frequency is due to 1 flexible response (B the coupling between the rigid body motion of the tether (ct) and the flexible dy namics, whereas the higher frequency corresponds to the flexibility itself. The high frequency component of B 1 decays due to the structural damping in the tether. The structural damping has a second order effect on the transverse tether vibration. So the C 1 response has an essentially constant amplitude. For zero offset, as expected, the transverse vibration (C ) is not coupled with other degrees of freedom. 1 The response of the system with a smaller tether length (L shown in Figure 4-4.  =  50 m) is  As in the reference case, the longitudinal mode contains  two frequencies. But the higher frequency of oscillation increases from 0.025 Hz (corresponding to the reference case) to 0.249 Hz when the length decreases to 50  m. For L  =  50 km (Figure 4-5), the higher frequency decreases to 0.0073 Hz. There  ) from 0.0045 Hz for 1 was also a change in the frequency of transverse vibration (C the reference case to 0.0448 Hz for L  =  50m and to 0.0014 Hz for L  =  50 km. The  stiffness of the tether, which is inversely proportional to its length, increases as the length decreases and hence results in a higher frequency of oscillation for a shorter tether. Figure 4-6 shows the effect of the flexible tether on the rigid body dynamics for a length of 500 m.  Simulation is carried out with zero initial condition for  the tether attitude motion.  at  response has a high frequency component due to it  66  cx(O) = 2°; cx(0) =2° (0)=0.8m 1 B  Stationkeeping, L  C ( 1 0)= 0.01 m  D =0; D=0  =  5 km  2.0 0.0 -2.0 0.0  2.0  2.0 0.0 -2.0  0.0  2.0  0.9  OO2  1 B 0.8 0.0  2.0  -0.00  Orbit  0.0  Figure 4-3  1.0  Dynamical response of the tethered satellite system in the reference configuration with the offset set to zero.  67  x(O)  x(0) =2° (0)= 0.81 xlO 1 B m 4  Stationkeeping, L =50 m  (0)=1.OxlO 1 C m 4  D=0; D=0  =  2°;  2.0 0cpo  0.0 -2.0  1.0  0.0 2.0 0.0 -2.0  1.0  0.0 X  B.,  .ff 4 I IVTJ  09  0.00  0.01  0.8 1.0  0.0  1.0 0.0 -1.0  Orbit  0.0  Figure 4-4  0.1  Response results showing the effect of decreasing the tether length on the system dynamics.  68  x(O)  =  2°;  tx(0) =2° (0)=75.Om 1 B  Stationkeeping, L = 50 km  (0) 1 C  =  20.0 rn  D=0m; D=0m  2.0  cxp  0.0 -2.0  2.0  0.0 2.0 0.0 -2.0  2.0  0.0  1 B  80.0  Cl  Orbit  0.0  Figure 4-5  2.0  Plots showing the effect of increasing the tether length on the sys tem dyanmcis.  69  a(b) =0; a(0) =0 (0)=8.133x10 1 B m 3 (0)=1.OxlO 1 C m 2  Station keeping, L = 500 m D=0; D=0  0.5  cxp° -0.0 0.0  1.0  1.0 x 0  I— -1.0 0.0  1.0  8.19 x10 m 3  1 B  8.16 8.13  0.1  0.0  1 C 0.5  Figure 4-6  System response during stationkeeping showing the effect of flexi bility on the tether attitude motion.  70  ). But the amplitude of oscillation is of 1 coupling with transverse tether oscillation (C the order of i0 degree, which can be considered negligible in practical applications.  Offset of the tether attachment point  4.3.3  ‘Effect of the tether attachment point offset was found to be quite significant. It couples the platform dynamics with the tether degrees of freedom. The offset along different directions, i.e. local horizontal and local vertical, was found to have  different coupling effects. Thus motion of the tether attachment point, which is required during the offset control, adds extra complexities to the system dynamics. Figure 4-7 shows the system response during the offset motion from the center of  2 is held fixed mass of the platform by 2 m in 120 s along the local vertical while D at zero. As shown in the figure,  is varied according to a sinusoidal acceleration  ) are unaffected by 1 profile. The tether pitch (c) and longitudinal elastic mode (B the offset motion, while the platform pitch response  (np)  shows amplitude modula  ) reaches around 0.08 m during 1 tions. The amplitude of the transverse vibration (C the offset motion and remains at that value subsequently. Note, here the response reaches a higher value than the initial disturbance of 0.01 m. Response results were also obtained for an offset motion from zero to 2 m in 120 s along the local horizontal (Figure 4-8). In this case there is strong coupling between the platform and tether dynamics. Particularly, the platform oscillates at much lower frequency and about a mean orientation at  _900  as compared to  1 response is modulated due to the local horizontal position in Figure 4-7. The C the coupling with the platform dynamics. The other degrees of freedom behave as before.  71  =  2°;  c(O) =20  Stationkeeping, L=5 km D: Oto2minl20s  B.(O)=0.8m C.(O)= 0.01 m  D=0  3.0  ap  °  0.0 -3.0 5.0  0.0  0.0  5.0  1 B 1.0  0.0  1 C Orbit  0.0  0.5  2.0  -0.0  0.0  Figure 4-7  Time, s  200.  0.0  Time,s  200.  Simulation result.s during offset motion along the local vertical with the offset along the local horizontal fixed at zero.  72  cz(O)  =  2°;  (0) 1 C  =  0.01 m  a(0) =2° (0)=0.8m 1 B  Stationkeeping, L=5 km D=0 D: 0 to 2 m in 120 s  0.0  cxp -180.0 0.0  B1  0.0  5.0  0.80  1.0  0.0 0.02 0.01  Cl -0.00 -0.01  bm.. Orbit  0.0  5.0  2.0  0 D°• pz -0.0 0.0  Figure 4-8  Time, S  200.  0.0  Time, s  200.  System response showing the effect of offset motion along the local horizontal.  73  When a fixed offset is given along both the local horizontal (D) and the ) directions (Figure 4-9), there is a strong modulation of the 2 local vertical (D crp response like that shown in Figure 4-7 but the frequency is not modulated as  ) has a 1 indicated in Figure 4-8. As shown in Figure 4-9, the transverse vibration (C regular amplitude modulations due to the coupling with the platform pitch motion, however, The tether attitude and the longitudinal flexible modes are unaffected by the offsets. 4.3.4  Tether mass and elasticity  The flexible motion of the tether is significantly dependent on the mass and elastic properties of the material. The two flexible characteristics which are most affected by the change in mass and elastic stiffness are: static deformation of the tether; and the frequency of both the transverse and the longitudinal vibrations. To understand these effects, studies were undertaken with different mass densities and elastic stiffnesses for the stationkeeping phase at a tether length of 5 km. A decrease in the linear mass density (Pt) from 4.9 x i0 kg/rn (corresponding to the reference case) to 1.0 x i0 kg/rn does not significantly affect the longitudinal response (Figure 4-10). However, the frequency of the transverse elastic mode is increased from 0.0045 Hz, corresponding to the reference case (Figure 4-3), to 0.01 Hz. The platform and tether attitude responses are not affected by the change in the tether mass. There is an increase in the static elongation of the tether from around 0.81 rn to 0.88 m when the linear mass density is increased to 25 x 10 kg/rn from the reference value of 4.9 x i0 kg/rn. As shown in Figure 4-11, the most significant ), which 1 change is observed in the frequency of the transverse vibration mode (C 74  x(O)  =  2°;  c(0) =2°  Stationkeeping, L =5 km  B ( 1 0)=0.8m (0)= 0.01 m 1 C  D=2m D pz =2m ,  0.0  -50.0 0.0  5.0  2.0 cx°  0.0 -2.0 0.0  B1  5.0  0.80 0.0  1.0  0.01  -0.00 -0.01  Orbit  0.0  Figure 4-9  5.0  Dynamical response of the system with fixed offsets along both the local horizontal and local vertical directions.  75  cx(O)  2°;  cx(O) =2° (O)=O.Bm 1 B  Stationkeeping, L =5 m  (O)= 0.01 m 1 C  D=O; D=O; p=1g/m  =  2.0  cxp°  0.0 -2.0 2.0  0.0 2.0 0.0 -2.0  2.0  0.0  0.85  1 .B 0.80 0.5  0.0  0.01 Cl -0.00  -0.01  Orbit  0.0  Figure 4-10  1.0  Plots showing the effect of decreasing the tether mass per unit length on the system response.  76  x(O) = 2°; c(0) =2° (O)=O.Bm 1 B (0) = 0.01 m 1 C  Stationkeeping, L = 5 m D=0; D=0; p=25g/m  2.0 0.0 -2.0  2.0  0.0  2.0 0.0 -2.0 2.0  0.0  1.0  1 B  0.9 0.8  0.5  0.0  Cl  -0.00  Orbit  0.0  Figure 4-11  1.0  System response with an increase in the mass per unit length of the tether.  77  is decreased to 0.0021 Hz from the reference value of 0.0045 Hz. The rigid body responses are essentially unchanged. The response of the system with a decrease in the the elastic stiffness (EA) to 40,000 N from the reference value of 61,645 N, is shown in Figure 4-12. As expected, with the reduced stiffness the static elongation of the tether increased to 1.27 m compared to 0.81 m for the reference case (Figure 4-3). The frequency of the longitudinal vibration (B ) is also reduced to 0.02 Hz from the reference value 1 of 0.0249 Hz. The change in stiffness has only a small influence on the transverse oscillation (C ) of the tether. For a system with an increase in the elastic stiffness 1 of the tether (Figure 4-13), the trends are as expected. Now, the frequency of B 1 is increased to 0.0282 Hz from the reference value of 0.0249 Hz. As before, the transverse vibrations exhibit insignificant change from the reference response.  4.3.5  Subsatellite. mass  The mass of the subsatellite (ms) is an important parameter affecting, par ticularly, flexible dynamics of the tethered satellite systems. The tether tension is mainly governed by the subsatellite mass and length profile during deployment/retrieval which, in turn, affects the elastic response of the system. The system behaviour with a decrease in the end mass to 50 kg and its increase to 5,000 kg from the reference value of 500 kg, is shown in Figures 4-14 and 4-15, respectively. For m 3  =  50  kg (Figure 4-14), frequency of the longitudinal oscillation increased to 0.0733 Hz  from the reference value of 0.0249 Hz (Figure 4-3). This can be explained by mod elling the longitudinal oscillation by a spring—mass system which has a frequency inversely proportional to square root of the end mass. However, the frequency of the transverse vibration, which is proportional to square root of the tether tension 78  a(O)  =  2°;  c(0) =2°  Stationkeeping, L =5 m  (0)=1.2m 1 B  D=O; D=0; EA=4OkN  (0)= 0.01 m 1 C 2.0 cx° 0.0 -2.0 0.0  2.0  2.0 0.0 -2.0 2.0  0.0  1.4  1 B  1.3 1.2  0.0  0.5  0.01 Cl  0.00 -0.01 1.0  0.0  Figure 4-12  Diagram showing the effect of decreasing the elastic stiffness.  79  a(O)  =  0) B ( 1  =  2°;  x(0) =2° 0.6 m  Stationkeeping, L =5 m =0;  C.(0)= 0.01 m  =0; EA =80 kN  2.0 0.0  -2.0 2.0  0.0  2.0 0.0 -2.0 2.0  0.0 0.7  1 B 0.6 0.5  0.0 0.01  -0.00 -0.01 0.0  Figure 4-13  Orbit  1.0  Response of the system with an increase in the elastic stiffness.  80  (O) (o) 1 B  =  2°;  x(0) =2°  Stationi<eeping, L =5 m  0.09 C.(0)= 0.01 rn =  =0;  =0; m =50 kg  2.0  cxp  0.0 -2.0  2.0  0.0 2.0  cx°  0.0 -2.0  2.0  0.0  0.10  1 B 0.09 0.1  0.0  1 C Orbit  0.0  Figure 4-14  1.0  Response results showing the effect of decreasing the subsatellite mass.  81  x(O)  =  2°;  (o) 1 B  =  7.5  c(0) =2°  Stationkeeping, L =5 m D=0; D=0; m=5,000 kg  C.(0)= 0.01 m 2.0 0.0 -2.0  2.0  0.0 2.0 0.0 -2.0  0.0  2.0  0.0  1.0  1 B  0.01 1  -0.00 -0.01 0.5  0.0  Figure 4-15  System response with an increase in the subsatellite mass.  82  [73] (tension is higher for a larger subsatellite mass), has a lower value of 0.00 16  Hz as compared to 0.0045 Hz in the reference case. On the other hand, for an in crease in the subsatellite mass to 5,000 kg, the frequency of longitudinal oscillation reduced to 0.0081 Hz and that of the transverse vibration increased to 0.014 Hz. It is interesting to note that the frequencies of the transverse and longitudinal oscilla tions change in opposite directions (i.e. one decreases and the other increases) for a change in the subsatellite mass. As expected, the static elongation of the tether is more for higher subsatellite mass. 4.3.6  Deployment and retrieval From the dynamics point of view, deployment and retrieval are the critical  phases in a tether mission. As pointed out before, the retrieval dynamics is an inherently unstable operation where as the deployment can be unstable if the velocity exceeds a certain critical value. Figure 4-16 shows the dynamic response during deployment of the subsatellite from a tether length of 200 m to 20 km in 3 orbits. An exponential- constant-exponential velocity profile is employed for the deployment. The deployment parameters are obtained from Eqs.(4.4-4.7). The velocity profile switches from the exponential to constant at a tether length of 2.5 km and the second switch, from constant to exponential, occurs at 18 km (L 1  2.5 km and L 2  =  18  km). As shown in the figure, even with a zero initial condition, the tether pitch (at) grows to —30° during the initial phase of the tether deployment. Subsequently, it slowly decreases and oscillates between ±2° after the deployment is over. This growth in the amplitude of at is mainly due to the Coriolis force caused by the interaction between the orbital and deployment velocities. For zero offset, there is no coupling between the platform and the tether dynamics. Therefore, ap oscillates between 0 and  Q•50•  The mean value of 0.25° is caused by the nonzero product 83  a(O) =0; cx(0) =0 (0) = 0.1304 x 102 m 1 B  Deployment: 200 m to 20 km in 3 orbits D=O; D=O  (O)=O 1 C  0.0  0.5 a° P  0.0  -30.0 20.0 1.5  L  L  0.0  0.0 0.0  5.0  0.0  5.0  1 B  0.0  4.0  5.0  1 C  0.0 -5.0  Orbit  0.0  Figure 4-16  4.0  Response of the system during deployment of the subsatellite using an exponential-constant-exponential velocity profile.  84  of inertia about the Y, Zp axes  (‘P,z  0). As expected, the mean value of the  longitudinal oscillations, which is the static elongation of the tether, increases with 1 plot, the deformation of the tether is the length. As shown in the inset of the B  greater than zero implying that the tether tension is positive. Since tension in the tether is small at the beginning of the deployment, it is critical to use appropriate acceleration profile for deployment because higher acceleration may cause slackness (i.e. negative tension) in the tether. If the first switch in the velocity profile occurred 1 at 800 m (L  =  1 became negative at the beginning of 800 m) instead at 2.5 km, B  the deployment and the tether was slack (plot not shown). Because of the Coriolis 1 increases to —7 m even without any initial disturbance. In the terminal force, C 1 decreases because of the decrease in the velocity exponential deployment phase, C  (L).  1 oscillates with a constant amplitude about the After the deployment is over C  zero mean value. The system response for retrieval from L  =  20 km to 200 m in 4 orbits with  an exponential velocity profile is shown in Figure 4-17. The Coriolis force during the retrieval made the tether pitch  () unstable.  Since the offset in the simulation  is taken to be zero, the platform attitude angle is not affected by the tether motion. ) decreased with the length. As the length 1 As expected, the tether elongation (B ) increased as expected. The 1 1 and C decreased, frequencies of the elastic modes (B ), which grew to ±50 m. 1 instability in the system excited the transverse mode (C The magnitude of the retrieval velocity, and hence the Coriolis force, decreases with the tether length. This, along with the increase in the stiffness, led to the decrease 1 towards the end of the retrieval. However, at the beginning in the amplitude of C of the retrieval the instability due to the Coriolis force is stronger resulting in an increase in the transverse amplitude. The instability also excited the longitudinal , i.e. the slack tether. 1 oscillations resulting in negative B 85  a(O)  =  (O) 1 B  =  0;  c(O) =0  Retrieval: 20 km to 200 m in 4 orbits  13.0 m (O)=O 1 C  D=O;  D=O  250.0  0.5  a° P  0.0  0.0  0.00  20.0  L  L  -4.0  0.0 0.0  3.5  0.0  3.5  15.0  1 B 0.0 20  50.0  Ci  0.0  -50.0 0.0  Figure 4-17  Orbit  3.0  System response during exponential retrieval of the subsatellite. ) 1 Note instability of the system in rigid () as well as flexible (B degrees of freedom. 86  4.3.7  Shift in the center of mass  In a typical mission the length of the tether can vary from zero to 20 km. Experiments involving 100 km long tether are in the planning stage. Obviously, this would result in an extremely wide shift in the center of mass of the system. Figure 4-18 shows plots for the shift in the center of mass of the system during deployment, stationkeeping and retrieval for a maximum tether length of 20 km. As expected, at the beginning of deployment (L  200 m), the shift in the center of mass is small  (Figure 4-18 a). The shift along the local vertical (Rn) increases with the tether length and, at the end of the deployment, R becomes more than 120 m, which is 0.6 % of the tether length. The shift along the local horizontal (R) is mainly due to the tether libration. Though the tether pitch is large at the beginning of the deployment (Figure 4-16), because of a smaller length,  variations remain  essentially the same. As shown in Figure 4-16, there is a change in the mean value of  t  as the deployment terminates. This effect can be easily seen in a change in  the amplitude of R 2 at the end of deployment. Figure 4-18(b) shows the response during stationkeeping at L  =  20 km. The  coupling between the attitude motion of the tether and platform results in a beat response for both m, where as  and  variations are confined to one side of —121.26  oscillates about the zero mean. As expected,  decreases with  the tether length during retrieval (Figure 4-18 c). The retrieval is carried out from a tether length of 20 km to 200 m in 4 orbits with an exponential velocity profile. The tether pitch (c) being unstable, there is no specific pattern to the shift in the center of mass. Though at becomes very large during the retrieval,  does not increase  beyond —25 m. This is due to the fact that an increase in the tether attitude angle is associated with a decrease in the tether length. The combined effect of these two 87  (a) 0.0  15.0  R°.° -120.0 0.0  -15.0  Orbit  0.0  5.0  Orbit  5.0  (b) -120.9  10.0  -121.2  R°° pz  0.0  -10.0  Orbit  10.0  0.0  Orbit  10.0  Orbit  3.0  (c) 25.0  0.0  0 R° pz -25.0  -120.0  0.0  Figure 4-18  Orbit  3.0  0.0  Plots showing shift in the center of mass during three different phases of the system operation:(a) deployment; (b) stationkeeping; and (c) retrieval. Total deployed length is 20 km.  88  factors limits any increase in  4.4  beyond a certain value.  Concluding Remarks Results of a comprehensive parametric study suggest the following: (i) Higher modes carry insignificant amount of energy.  (ii) In absence of the tether attachment point offset, tether flexibility has little effect on the rigid body dynamics. (iii) Offset of the tether attachment point couples the platform dynamics with the tether degrees of freedom. (iv) Flexible dynamics of the tether is substantially affected by the mass and elastic properties of the tether material. In particular, they affect static deflection of the tether, and frequency of both transverse and longitudinal oscillations. (v) The tension of the tether is primarily governed by the subsatellite mass and length of the tether and, in turn, affects its natural frequencies. (vi) Deployment and retrieval represent critical phases in a tethered mission. De pending on the deployment/retrieval rate, the system is susceptible to insta bility and the tether may become slack. (vii) There can be significant shift in the system center of mass during deployment, stationkeeping and retrieval.  89  5.  51  ATTITUDE CONTROL  Preliminary Remarks The instability during retrieval, large amplitude swinging motion at the time  of deployment and undesirable performance in the stationkeeping phase would de mand some active control strategy for a successful tether mission. Depending on the mission requirements, the objective of the controller is to regulate the attitude and/or flexible motion of the system. The design and implementation of the attitude controller is addressed in this chapter. As mentioned in Chapter 1, in the past, control of tethered systems has been approached through three different strategies: tension or length rate control; thruster control; and offset strategy. The tension or length rate control scheme depends on the differential gravity field, and hence is ineffective at shorter tether lengths. Although the thruster based control procedure is unaffected by the length of the tether, it is not advisable to use thrusters in the vicinity of the space platform due to plume impingement and safety considerations. Therefore, the offset control strategy is the most effective choice for shorter tethers. The offset motion required for controlling the tether attitude dynamics increases with the length and hence the applicability of this strategy is limited by the space available to move the tether attachment point on the platform. The advantages associated with the three control strategies over different lengths lead to a hybrid scheme where the offset method can be used for shorter tethers and thruster or tension based approaches can be applied at longer lengths. Such a hybrid strategy was originally used by Modi et al.[49j for a rigid tethered 90  system. In this thesis, the hybrid strategy is extended to a system with flexible tether. The thruster augmented active control is used here to regulate the attitude dynamics for longer tethers. The dominant feature, which governs the system characteristics, is the time varying length of the tether. If the time dependent variation of the system param eters is relatively small, the Linear Time Invariant (LTI) robust control techniques can be used to regulate the dynamics with the parameter variations treated as un certainties. But, in the present situation, the length of the tether varies from few meters to 20 km, and hence the robust LTI controller will not be effective. There are mainly four different approaches to control such highly time varying systems: • gain scheduling control; • adaptive/self-tuning control; • simultaneous control of multiple plants; • control based on the Feedback Linearization Technique (FLT). In the gain scheduling approach, a number of LTI controllers are designed for different operating conditions and are stored on-board. In the real time operation, the controller selects and implements the appropriate gains corresponding to an operating condition. The major difficulties with these controllers are the design and storage of the controller gains for a number of operating conditions. Any change in the mission objective may need redesign of the controllers for new operating points. In the adaptive control, a simple system model is estimated from the knowl edge of the system inputs and outputs. Based on the estimated model, the controller is designed and implemented in real time [74]. This approach has advantages for systems whose dynamics can not be modelled accurately.  91  Another alternative to these approaches is the simultaneous control of a number of LTI plants, over the entire range of parameter variation, by a single controller. Here, the time varying system is considered as a collection of controllable and observable LTI plants described by: th  =  (5.1)  Ax + Bu;  y=Cx,  (5.2)  i=l,2,...,np;  where: x, yj and u are the state, output and control input vectors, respectively; and A, B and C are the LTI matrices of appropriate dimensions. The index ‘i’ corresponds to the th plant and Tip is the total number of plants. Considering the output feedback, the objective is to design a single controller, u 1,2,  -.  ,  Tip,  =  —Ky,  i  =  so that all the plants are simultaneously controlled. The control can also  be achieved by state or dynamic output feedback. Several algorithms to accomplish this objective have been reported in the literature [75—81]. In the FLT, the nonlinear and time varying dynamic equations are trans formed into an LTI system by the nonlinear, time-varying feedback. Thus a single controller structure can regulate the highly time varying and nonlinear dynamics of the system.  References [82-88] discuss detail mathematical background and design  procedures for the feedback linearization control. The FLT based controller is quite effective for systems which can be modelled accurately. The controller performance is sensitive to the modelling errors [89]. In the present study, attitude dynamics of the system can be modelled quite accurately and hence the FLT is selected for the controller design. In a general problem with feedback linearization, the control algorithm consists of three steps: transformation of the state space; control variable transformation making the system linear in the 92  new coordinate system; and control of the linear system in the new state space. The dimension of the transformed linear controllable system depends on its properties and type of the linearization procedure applied, such as the input-state or the inputoutput feedback linearization [83]. The attitude controller for shorter and longer tethers are designed using the offset and the thruster strategies, respectively. The following section discusses the design algorithm and implementation of the thruster based controller. The design and implementation issues related to the offset strategy are addressed in the next section. Finally, a hybrid control scheme is presented followed by some concluding remarks.  5.2  Thruster Control Using Dynamic Inversion Selection of an acceptable design model is important in the development of  a controller using the dynamic inversion. From the implementation point of view, it is advantageous to use a simpler model for the controller design. However, an approximate model may affect the system stability [89] and performance. In the present study an acceptable dynamic model for the controller design is obtained via numerical simulation. Three different models nonlinear, and rigid linear  —  —  complete nonlinear flexible, rigid  are selected for the controller design. Simulation results  for each case are compared to arrive at an acceptable model. 5.2.1  Controller design algorithm  As pointed out before, the thruster control strategy is used to regulate the attitude motion of the longer tether, and momentum gyros control the platform pitch motion. The rigid body modes, i.e.  ap  93  and aj, are controlled actively through  feedback and the flexible degrees of freedom are regulated by passive dampers. The control of the flexible generalized coordinates becomes important particularly during retrieval when the elastic modes are unstable. The controller for the rigid degrees of freedom is designed using the inverse control procedure, which is a special case of the Feedback Linearization Technique (FLT). In case of the thruster control, the system is already in the canonical form. So only control transformation is necessary for linearization of the system. The system model for the controller design can be expressed in the form 1j + F 3 8 M 8  =  Q,  (5.3)  3 is the vector of nonlinear terms; and q 3 and where: M 3 is the mass matrix; F  are the vectors of generalized coordinates and forces, respectively. If the number of independent control inputs in  3 Q  is the same as the dimension of q , the above 3  system can be transformed into a linear time invariant form by the generalized force vector =  (5.4)  . 3 v+F 3 M  This transforms Eq.(5.3) into a linear form of =  (5.5)  v.  Here v is the new control input required to regulate the transformed decoupled linear system. A linear control theory can be used to design the control input. In the present study, v is chosen in such a way that the error (e  =  3 q  —  q3)  dynamics  has poles at desired locations in the s-plane. This is accomplished by  =  d + Kv(sd 1 ‘  —  s)  94  + Kp(q$  —  qs),  (5.6)  which results in the error equation (5.7)  ë+Kvé+Kpe=O.  Here K and K are the diagonal matrices of velocity and position feedback gains, respectively, and q, sd and sd represent the desired trajectory. The controller expression in Eq.(5.4) is referred to as the primary controller while that in Eq.(5.6) is called the secondary controller.  The next step in the controller design is to decide an acceptable structure for 8 and , F 8 M  Q  for the system model in Eq.(5.3). As mentioned before: nonlinear  flexible; nonlinear rigid; and linear rigid models are used for the controller design. The linear and nonlinear rigid models are in a form which can directly be used for the controller design. But the nonlinear flexible model has a different form. However, it can be transformed to have the required structure as follows. The system under consideration has the form  M(q, t)  + F(q, , t)  Qq,  (5.8)  where: q and Qq are vectors of the generalized coordinates and forces, respectively; M(q, t) is the nonlinear time varying mass matrix; and F(q, , t) is the vector con  taining nonlinear gravitational, Coriolis and centrifugal terms. Here the dimension of q is more than the number of independent control inputs in Qq. Let q, and denote the components of q; Fr and Ff the components of F(q, , t); and Qq,. and the rigid and flexible modes, respec Qqf the components of Qq corresponding to tively. Here  Qq  Qq,.  represents the generalized forces due to the control inputs, and  corresponds to the generalized forces due to the passive dampers in the system. 95  Partitioning the mass matrix appropriately, the above equation can be rewritten as [M,.,.  1 Here: q,  =  {crp,  ct}T;  qf  Mrf1f4ri> ff J 1q J =  , B 1 {B ,• 2  fFrfQqr l f J 1.. q •  ,  , C 1 B , C 1 ,• 2  •  ,  CNt}T; and Nt and  N represent the number of flexible modes in transverse and longitudinal directions,  respectively. If the controller is to be designed based on the rigid model of the system, Mrf and the flexibility terms from Mrr, F,. and Qq,. are neglected and the resulting rigid body (qr) equation can be used to design the controller using Eqs.(5.4) and (5.6). But if the effect of flexibility on the rigid modes is to be accounted for by the controller, the equations of motion for the rigid degrees of freedom (q,.) can be obtained as [84] 8 &+Fs—Qqr, M  (5.10)  where: 8 M  and F  =  M,.,.  —  —1 MrfMff Mf,.;  F,. + MrfMJ(Ff + Qqf)’  Eqs.(5.4) and (5.6) can be used to design the controller for this system. In the implementation of this control strategy, the submatrices of M and F are to be computed, which need the knowledge of the flexible system model and the flexible generalized coordinates. Using the assumed mode method to discretize the elastic deformations, the dynamic model can be obtained with a considerable degree of accuracy. The problems associated with the observation of the flexible modes and their resolution is a subject in itself. Here, it is assumed that the flexible generalized coordinates are available for the control purpose.  96  5.2.2  Control with the knowledge of complete dynamics The system model considered for the controller design includes all the non  linear time varying terms including the effect of tether flexibility (Eq. 5.10). The controller is designed using the dynamic inversion (Eqs. 5.4 and 5.6) and imple mented on the complete nonlinear flexible model of the system (Eq. 5.8). The numerical values for the system parameters considered in the simulation are the same as those used for the dynamic study in Chapter 4. Only the first longitudinal (Bi) and lateral (Ci) modes are considered in the simulation, however, the program developed can incorporate an arbitrary number of modes in both directions. Figure 5-1 shows controlled response of the system during the stationkeeping phase at a tether length of 20km. Since the dynamics of the rigid system is exactly cancelled by the controller, the platform pitch  (ap)  and tether swing (as) behave  as required. The desired performance specifications for the closed ioop system are characterized by the rise and settling times, which are 0.1’r and 0.4T, respectively, for both the platform and tether librations. Here r is the orbital period. The thrust (Tat) requirement is quite small for an initial disturbance of 2° in the tether pitch (at). The nonzero mean value of M is required to regulate  ap  about zero which is  not its equilibrium value, and to cancel the extra moment on the platform due to a nonzero offset. The control inputs (M and Tat) cancel the effect of flexible dynamics on the rigid degrees of freedom. This introduces the high frequency component in the time histories of M and Tat. The coupling between the rigid and flexible generalized coordinates excites the transverse vibration (C ) even in absence of any 1 initial disturbance. As expected, the longitudinal elastic mode decays due to the structural damping. The controlled response during deployment from L 97  =  200m to 20km in 3  ct(O)  =  2°;  ci(0) =2°  Stationkeeping, L =20 km  (0)=13m 1 B  D=0;  C(0)=O 2.0  2.0  0.0  0.0  M x 45.0  D=1m  .IIIIlIIIIIIjIIIIIIIIiIlI, lvlIIIIiIIIIIIIIIIlIuiIIIlI 1  40.0 0.0  1 B  2.0  14.0 12.0  1.0  0.0  10.0  1 C  0.0 -10.0 0.0  Figure 5-1.  Orbit  2.0  Controlled response of the system during the stationkeeping phase..  98  orbits is shown in Figure 5-2. An exponential-constant-exponential velocity profile  is used for the deployment. The first switch from exponential to constant velocity occurs at L  =  2.5 km and the second switch takes place at L  behaviour of the rigid modes, c, and  ,  =  18 km. The  is similar to the stationkeeping case.  It is interesting to note the similarity between the deployment velocity (L) and control thrust (Tat) profile governed by the Coriolis force. The moment M is very small due to zero offset of the tether attachment point from the platform center of mass (D  =  =  ) increases 1 0). As expected, elongation of the tether (B  -plot 1 with the length. With the present deployment profile, inset within the B shows the tether elongation at the beginning of deployment to be greater than zero. Sometimes, if the deployment acceleration is high, tether may become slack. The ) is governed by two effects: change in 1 response of the transverse flexible mode (C the tether tension due to the deployment acceleration profile; and the Coriolis force effects due to the deployment velocity.  The rigid body dynamics during the retrieval is controlled quite successfully (Figure 5-3). However, it is important to recognize the unstable dynamics of the ) becomes nega 1 flexible subsystem. The uncontrolled longitudinal elastic mode (B 1 response also becomes tive implying that the tether is slack. The amplitude of C very high (±80m) due to the Coriolis force. The retrieval is carried out with an exponential velocity profile (Chapter 4). As the velocity decreases the effect of the Coriolis force becomes small. Furthermore, frequencies of the flexible degrees of freedom increase with a decrease in the tether length. The decay in the amplitudes of B 1 and C 1 responses after certain time during retrieval is due to the combined effect of the tether length, changing Coriolis effect and structural damping. The  99  cx(O)  2°;  cs(O) =2° (O) = 0.1304 x 10.2 m 1 B C(O)=O =  Deployment: 200 m to 20 km in 3 orbits D=O ; D=O  2.0  2.0  ap°  xto  -0.0  -0.0  20.0  L 0.0  M  1 B 0.0 0.0 10.0  1 C  4.0  0.0 -10.0 0.0  Figure 5-2  Orbit  4.0  System response during controlled deployment with an exponentialconstant-exponential velocity profile.  100  cz(O)  =  2°;  (O) 1 B  =  13.0 m  O) =2°  Retrieval: 20 km to 200 m in 4 orbits D=O ; D=O  (O)=O 1 C 2.0  2.0  -0.0  -0.0 0.0  20.0  L  L  -4.0  0.0  M 5.0  5.0  13.0 Eii1 1 B 0.0 5.0  0.0 80.0  1 C  0.0 -80.0 0.0  Figure 5-3  Orbit  5.0  Controlled response during the exponential retrieval without a pas sive damper.  101  control thrust cancels effects of the flexible dynamics on the rigid modes resulting in the high frequency oscillations of Tat. In the present study, the unstable dynamics of the flexible degrees of freedom is controlled only by passive dampers. Passive dampers are provided for both the transverse and longitudinal oscillations. The transverse damper (damping coeffi cient  =  0.03 Ns/m) is placed at the middle point of the tether and the longitudi  nal damper (damping coefficient  =  1.5 Ns/m) is located at the subsatellite. The  magnitude of the damping coefficients considered in the simulation are within the practically achievable range [90]. The results for controlled retrieval with passive dampers are presented in Figure 5-4. The rigid body responses are the same as the previous case without damper. However, the flexible responses (B 1 and C ) 1 now settle to their equilibrium values rather quickly. Note, the magnitude of B 1 is always greater than zero implying that the tether has positive tension.  5.2.3  Inverse control using simplified models  The controller design in the last section is based on the complete nonlinear model of the system. The implementation of this controller needs knowledge of all the flexible modes, which are difficult to obtain, and requires considerable amount of the real time computational effort. These limitations may make the controller difficult to implement. Therefore, to obtain a readily implementable controller, two simplified models are considered here. The results of the uncontrolled dynamic simulation (Chapter 4) serves as the guideline in selecting the simplified model. As seen in Figure 4-6, the rigid body responses are not significantly affected by the tether flexibility. Since objective of the attitude controller is to regulate the rigid body modes, the nonlinear rigid body model is considered for the controller design. 102  2°; cx(0) =2° (0) = 13.0 m 1 B (0)=0 1 C  Retrieval: 20 km to 200 m in 4 orbits D=0; D=0  =  2.0  2.0  (:xp°  cx° -0.0  -0.0  20.0  0.0  L  L  -4.0  0.0 2.0  M  -0.0  0.0  Tat -6.0  -2.0 0.0  1 B  5.0  0.0  5.0  13.0  0.0 70.0  1 C 0.0 0.0 Figure 5-4  Orbit  5.0  System response during the controlled retrieval with passive dampers. Note, the flexible modes are controlled quite effectively keeping the tether tension positive. 103  The governing equations of motion can now be expressed as [Mn 21 [M  M 1 2 22 M  f&pj+JF 1 1 _f M j &tJ 1F2J1LTatJ’  (.511)  where the coefficients are defined in Appendix III. This system is in the canonical form and hence Eqs.(5.4 and 5.6) can be used directly to design the controller. The other simplified model considered for the design corresponds to the rigid linear system, which can be expressed as [ML 11 21 [ML  1) 1ii CL 1 Ia 12 fz’l+ & 21 [GL ] 22 CL 1 J at 11 KL [KL 1 1 I 1 + I FL 12 f + KL 21 KL j 1cJ 22 J 2 FL  1 12 ML  j 22 ML  —  —  M 1LTatJ’  (5 12 ‘  with the coefficients defined in Appendix III. This controller is also designed using Eqs.(5.4 and 5.6). The controllers, designed by dynamic inversion using simplified models, are implemented on the complete nonlinear flexible system. Figure 5-5 shows the system response during stationkeeping with the controller based on the nonlinear rigid model. As expected, ap and at settle to the desired value in about 0.5 orbit. Note, in the present case, the control inputs do not cancel the effect of flexibility on the rigid modes. Therefore the high frequency components of M and Tat have much lower amplitudes than those in Figure 5-1. As can be seen from the 1 and at, the tether pitch has a high inset of at-plot, due to the coupling between C  frequency component with very small amplitude. These oscillations slowly decay 1 through coupling due to active control and, in turn, decease the amplitude of C  effects. As shown in Figure 5-5, the controller based on the rigid nonlinear model results in a steady state error in the platform pitch (ap). An outer Proportional Integral (PT) controller loop is used to take care of this error. With this, the primary 104  a(O)  =  2°;  a(O) =2°  Stationkeeping, L =20 km  (O)=13m 1 B C. (0)  =  D=O;  0  2.0  2.0  0.0  0.0  D=1m  2.0  M  44.0  T°° at 42.0 -2.0 0.0  1 B  2.0  0.0  2.0  14.0 12.0 1.0  0.0  10.0  1 C  0.0 -10.0 0.0  Figure 5-5  Orbit  2.0  System response during stationkeeping using the rigid nonlinear model for the controller design.  105  controller of Eq.(5.4) becomes  =  v+5 8 M F+  I KpT(ap—pd)+KITj(ap_apd)dt 0 ( J  ,  (5.13)  where KPT and KIT are the proportional and integral gains, 2.0 and 0.05, respec tively. The response of the system with the P1-controller is presented in Figure 5-6. Note, the steady state error in the platform pitch response is reduced to zero. The other responses remain essentially unchanged. The response of the controlled system during deployment, with dynamic inversion, using rigid nonlinear model is shown in Figure 5-7.  The deployment  velocity profile is the same as in Figure 5-2. Comparing Figures 5-2 and 5-7, it can be concluded that the system dynamics is virtually identical except for two effects: amplitude of the steady state ocillations of the control thrust (Tat) is very small; and the C 1 response decays slowly with the controller based on the rigid nonlinear model. In Figure 5-7, the controller does not cancel the effect of flexibility on the rigid model. It leads to a smaller fluctuation of Tat as compared to that in Figure 5-2. The tether transverse oscillation (Ci) decays slowly due to its coupling with the a response which has a small diminishing amplitude. Similarly, response of the system with the controller based on the rigid linear model (Eq. 5.12) was found to be almost identical to that in Figure 5-7. Thus the controllers based on the rigid nonlinear and rigid linear models lead to essentially the same performance during deployment. The simplified linear and nonlinear models of the rigid system were also used to design the controller for regulating the attitude motion during retrieval. Response of the system with the controller based on the rigid nonlinear model and in presence of passive dampers is shown in Figure 5-8. The dampers used here are the same 106  a(O)  2°; x(O) =2° (0)= 13m 1 B  Stationkeeping, L =20 km  (O)=O 1 C  D =0;  =  2.0  D= 1 m  2.0  0.0  0.0 2.0  M  44.0  Tat 0.0 42.0 0.0  1 B  -2.0 0.0  2.0  2.0  14.0 12.0 0.0  1.0  1  1 C -10.0  0.0  Figure 5-6  Orbit  2.0  System behaviour during stationkeeping with the outer PT control loop and the FLT employing the rigid nonlinear model.  107  a(O)  =  (0) 1 B  =  2°;  x(O) =2° 0.1304 x 102 m  Deployment: 200 m to 20 km in 3 orbits D=O ; D=O  (0)=0 1 C 2.0  2.0  -0.0  -0.0  xpo  20.0 I  1.5  L  L  0.0  0.0 2.0  {N-m 1.5  M  0.0  T  -2.0  0.0  0.0  4.0  0.0  4.0  13.0  1 B  10.0 Cl  0.0 -10.0 0.0  Figure 5-7  Orbit  4.0  Deployment dynamics with the controller based on the rigid non linear model.  108  Retrieval: 20 km to 200 m in 4 orbits D=O; D=O  cx(O) = 2°; cx(0) =2° (0)=13m 1 B C.(0)=0 2.0  2.0  -0.0  -0.0  20.0  0.0  L  -  -4.0  0.0  2.0  0.0  M  Tat -2.0 0.0  -8.0 5.0  —  0.0  5.0  1 13.0 B 0.0 70.0  1 C 0.0  Orbit Figure 5-8  5.0  Controlled response during exponential retrieval of the subsatel lite. The controller is designed using the rigid nonlinear model and applied in presence of passive dampers. 109  as those for the simulation in Figure 5-4. A comparison of results in Figures 5-4 and 5-8 shows similar behaviour except for reduced Tat fluctuations in the present case. Similar trends were observed with the controller based on the rigid linear model. Identical observations can be made for the controller design aimed at the stationkeeping case. 5.2.4  Comments on the controller design models As seen in the previous section, the controllers based on the complete flexible  nonlinear, rigid nonlinear, and rigid linear models result in very similar system performance. Hence, the computational time may serve as an important criterion in selection of the model. To that end, the controlled dynamics was assessed over five orbits with a time—step of 1 s. This corresponds to the situation where the controller can be implemented as a sampled data system with a step—size of 1 s. The computational time with the controller design based on different models are compared in Table 5-1 for stationkeeping, deployment and retrieval phases.  Table 5.1  Comparison of the time (s) required by the controllers using different design models.  Stationkeeping Deployment Retrieval  Nonlinear Flexible Model 402.6 5195.1 5332.2  Nonlinear Rigid Model 67.6 68.1 68.4  Linear Rigid Model 70.4 71.6 71.7  Each computation loop involves calculation of the control inputs, and record ing of the system output as well as input values in a file. Calculations for the sta tionkeeping case correspond to L  =  20 km. Deployment is carried out from a tether 110  length of 200 m to 20 km in 3 orbits using an exponential-constant-exponential velocity profile with the switch over at tether lengths of 2.5 km and 18 km. During retrieval, the tether length decreases from 20 km to 200 m in 4 orbits with an expo nential velocity profile. The study was carried out on a SUN Sparc-2 workstation. As expected, in all the three modes of tether operation, the controller based on the nonlinear flexible model demands the maximum computational effort. The controller based on the rigid nonlinear model takes the minimum time. In the linear rigid model, the equations are linearized about an arbitrary reference trajectory. So the governing equations involve more algebraic and trigonometric operations than the rigid nonlinear case.  This results in a longer computational effort for  the controller based on the rigid linear model as compared to the riid nonlinear. case. As seen in the previous section, the system performance with the different controller is essentially similar. So the controller based on the rigid nonlinear system is preferred. Of course, it should be recognized that irrespective of the system model chosen for the controller design, its effectiveness is assessed through application to the nonlinear, nonautonomous and coupled flexible system.  5.3  Offset Control using the Feedback Linearization Technique  As mentioned before, because of practical limitations, it is advantageous to use the offset strategy for attitude control when the tether length is small. This section presents the design procedure and simulation results for the offset control of a class of tethered systems. The Feedback Linearization Technique (FLT) is used to design the controller for this highly time varying tether dynamics.  111  5.3.1  Mathematical background For two vector fields  f  and g on R”, the Lie bracket  [f, g]  is a vector field  defined by [83]  ôg  of  [f,g]f—g, where Of/Ox and  ag/ax  are the Jacobians. It is also denoted by ad}(g), and by  induction  ad(g) with ad°f(g)  =  g.  [f,  =  (g)], 1 ad ,..., xm} is said to be involutive 1 {x  A set of vector fields (x)  if there are scalar fields  ajjJ  such that  j, =  or in other words [xi, Xj] E point, assuming x1,  ,  s(X).  It is called completely integrable if for every  x are linearly independent, there exists an rn-dimensional  manifold M in R such that at each point of M the tangent space of M is spanned  byx,•,xm. Let h  R”  —‘  R be a scalar field. The gradient of.h, denoted by dh, is a  row vector field  fOh ’ 1 Ox  Oh ‘8  on R”. A set of scalar fields are linearly independent if their gradients are a linearly independent set of row vector fields. For a scalar field h and a vector field (fi,  ,  f)T,  ••  the dual product of dh and  f,  f  =  denoted by (dh, f),is a scalar field  defined by  Oh  Oh  The feedback equivalence for control systems is based on three operations 112  [86]: coordinate transformation in the state space; coordinate transformation in the control space; and feedback. For linear systems, with the operations defined in a linear fashion, it is well known that every single input system which is time invariant and controllable is feedback equivalent to the canonical form Z[  0  Z2 =  dt  Z_1  Z  Zn  0  +  0 1  (5.14)  V.  Any nonlinear system which is feedback equivalent to the above form is called linear  equivalent.  Theorem 1 (Su, 1982 [86)): A system  =  f(x, u) is a linear equivalent if and only  if (i) J(x,u)—_ f(z)+g(x)4’(z,u) where f(O)= 0, 4(0,O)=0 and 84/ôu  (ii) the vectors g, ad}(g),.  ,  0;  ad)(g) span R about the origin;  (iii) the set of vector fields {g, ad}(g),... , ad?)(g)} is involutive.  D  For systems satisfying the above theorem, the state transformation  T(x)  , 1 {T  .  ,  z  =  Tn}T required for linearization can be obtained from the following  conditions:  (dTi,adjc(g)) =0,  i  =  0,1,,n—2;  (5.15)  ,ad’ 1 (dT ( g)) 0; 0  2 T dt  +  =  (5.16)  .  Tn  (5.17)  Eqs. (5.15) and (5.16) can be used to get the function T . The complete transfor 1 113  mation, z  =  (x,u) is selected as 5 T(x), can be obtained from Eq.(5.17). If q v —P(x)  q(x,u) where:  (5.18)  n  P(x)  =  n  and j1  Eq.(5.17) acquires the same canonical form as that of Eq.(5.14). The expression for (x,u), Eq.(5.18), which transforms the system into the canonical form, is referred to as the primary controller. Now the task is to design a feedback controller (called the secondary controller) to generate the control input, •• z , v, so that the transformed states 1  ,  ’ 1 Z  • x , and hence the original states 1  ,  follow the desired trajectory. This can be achieved by using the linear time invariant control procedure. 5.3.2  Design of the controller The purpose of the attitude controller is to regulate the rigid body rotations  of the platform and tether. The dominant characteristic, which governs the system dynamics, is the time varying length of the tether during deployment and retrieval. As discussed in the previous section, the controller based on the rigid model of the system gives almost the same performance as that obtained using the flexible nonlinear model. Furthermore, as shown in Chapter 4, the tether response is not significantly coupled with the platform dynamics; however, for a nonzero offset, the platform attitude is strongly affected by the tether motion. Therefore the controller for the tether dynamics is designed based on the rigid model decoupled from the platform motion. On the other hand, the model for the platform controller includes 114  the effect of tether libration. As shown in the thruster control case (Section 5.2), performance of the system with the controllers based on linear and nonlinear rigid models is almost identical. Therefore, it is sufficient to consider only the linear model for design of the tether controller. The rigid body equation for the tether dynamics, decoupled from the platform motion, is linearized about the quasi-equilibrium trajectory (at) and the specified offset motion. The total offset position, {d}, is the sum of the specified value, {D}, and the controller coordinate {D}. The resulting dynamic equations for the controller design are: ma(ap, x, t)& + f(ap, p, x, , t) =  =  D + a3at + a 2 Dz + b + cDi, 4 +a  alat  (5.19) (5.20)  where a, -‘jiT  r x =jat,  Here D 2 is the z-component of {D} which is the offset along the local horizontal required by the controller. The coefficients of Eqs.(5.19) and (5.20) are defined in Appendix III. The equilibrium value for the platform pitch angle (p) is specified by the mission requirement, and the quasi-equilibrium angle for the tether pitch rotation (at)  is given by (Ô 1 m {miDpz + 2 =  —  2ji)D + 6 imi(Dp 1  mi(—Dpy + Da) 2 + mi(Ô  —  —  mcrtS  —  —  Da) + 8 that  2u)(Dpyäp + D ). 2 115  (5.21)  Here Dpv and  represent the ‘y’ and ‘z’ components, respectively, of the vector  {D} and other coefficients are defined in Appendix III. In Eq.(5.20), the accelera  tion of the tether attachment point along the local horizontal (Di) is used to control the tether swing. Since it is necessary to regulate the position and velocity of the tether attachment point, the dynamic model for the tether pitch is augmented by the identity (5.22)  ut.  =  The alternate method to control the offset motion is to include the physical be haviour of the offset mechanism into the system dynamics. It should be pointed out that the time constant of the offset mechanism is usually much smaller than that of the tether libration. Hence the use of Eq.(5.22) to regulate the offset motion does not affect the implementation of the control strategy. The governing equation for the platform dynamics (Eq. 5.19) is already in the canonical form. Therefore, only the control variable transformation is required to linearize the system. The structure of the primary controller which accomplishes the linearization is  Qcrp  =  m(cp, x, t)v  The secondary control input,  Vp,  + fap(p, &p, Z, , t).  which asymptotically drives the error (e  (5.23)  =  p—apd)  dynamics to zero, can be expressed as  Vp  where:  1 (Pd k  —  + kp (àpd 2  —  p)  + &pd,  (5.24)  and k 2 are coefficients of the desired closed loop polynomial for the  error dynamics; and cpd, &pd, cpd represent the desired trajectory. Now the error 116  equation becomes +kp 1 ê 2 ë+kp e  =  0.  The model used for the tether controller design, Eqs.(5.20) and (5.22), can be rep resented by =  f(x, t) + g(x, t)ut,  (5.25)  where:  I ‘  ‘  X3 X4  a+4 x a+b x 1+2 ax a+3 x 0  —  (U  and g(x,t)=  Jo  Ii In general, the coefficients in Eq.(5.25) are time varying. The transformation of this system to time invariant canonical form can be obtained using the algorithm of Hunt and Su {91j. However, in the present case, the coefficients of the tether dynamics are slowly time varying parameters and hence a quasi-static approach is sufficient to achieve the objective. This results in a quasi-static controller structure which transforms the system into the canonical form. The functions f(x) and g(x) satisfy the conditions in Theorem 1, and hence the system is feedback equivalent. The state transformation, which maps the system into the canonical form, is z  =  T(x)  =  [t]x,  where: =  12 t  =  =  a + a3(a4 + a3c); c a+1 —  c(a2 + aic) + a4(a4 + a3c); a4  —  a3c; 117  (5.26)  4+a c(a c); 3  =  and for i  =  2,3,4 til  =  2 tj  =  3 tj  =  t(_l) a ; )+3 1 t(_  . 3 + a4t(_l) The structure of the transformed system is similar to Eq.(5.14), where the primary controller has the form P(x)  —  Ut  (5.27)  ,  =  with: P(z)  =  3 )x 4 t a ; 42 + 4 3)z + (t 4 t 3 z + (t 43 41 + a 3 —i- at x 4 t a 1  44 t 43 + . Q(x) = Ct  The secondary controller for the system is designed to place the closed ioop eigen values at some desired locations. This leads to the following expression for the transformed control input  Vt  =  —  where kt and Zid  zld)  =  Vt,  (z 2 + kt  1,  ,  —  Z2d)  (z 3 + kt  —  Z3d)  (z 4 + kt  —  Z4d),  (5.28)  4, are the coefficients of the desired closed loop polyno  mial and the desired values for z, respectively.  Zid  can be computed from Eq.(5.26)  with the knowledge of the desired x. The structure of the tether pitch controller is shown in Figure 5-9.  118  I  1  FLT controller  Figure 5-9  5.3.3  Schematic diagram of the closed-loop tether dynamics with the FLT based controller.  Results and discussion  The controller designed for the platform and tether pitch angles are imple mented on the complete flexible and nonlinear model. As mentioned before, the offset control strategy has advantages at shorter tether lengths. In this section, con trolled response results are presented for operations with a maximum tether length of 200 m. The inertia and elastic parameters are the same as those used in the dynamic simulation of Chapter 4. One possible drawback of the inverse control pro cedure may be its lack of robustness against the model uncertainty [89]. To assess this aspect, intentional modelling error was introduced by neglecting the shift in the center of mass terms from the controller model, but retaining the corresponding terms in the simulation model. The desired system performance is characterized by the settling and rise times. The settling times for platform, tether and offset motion are 0.4 ‘r, 0.8r and 0.82T, respectively. The rise times are O.lr and 0.15r for the platform and offset  119  motions, respectively. The rise times for the tether libration are 0.2r, 0.16r and 0.18r during deployment, stationkeeping and retrieval, respectively. Here r is the orbital period. The maximum allowable offset is taken as ±15m. Figure 5-10 shows the controlled performance of the system for stationkeeping at L  =  200 m with the modelling error in the controller design. Though the rigid  modes are stabilized to some steady state values, the performance of the system is not satisfactory. The desired steady state values for  ap  and  are 0 and 1 m,  respectively, which are not achieved by the controller. Moreover, the at and d 2 ,, i.e. M, is also 1 response is quite oscillatory. The moment required to control a oscillatory because of the coupling between the platform and the tether dynamics caused by a nonzero offset. When the shift in the center of mass terms are included in the controller 1,2 responses is reduced substantially and model, the oscillatory nature of a and d 1, become zero (Figure 5-11). The steady state value of the steady state value of a 1, is still less than the required magnitude of 1 m. An outer Proportional-Integral d (P1) loop was introduced to reduce this steady state error. With this, the structure of the tether primary controller becomes = Ut  1,0 K + (d  Q(r  j 1 0 1, 1,2 K D )+ (d  —  —  1,2 D )dt,  (5.29)  10 are the proportional and integral gains, respectively. The sec where K 1,0 and K ondary control input  Vt IS  obtained as before using Eq.(5.28). In the present sim  1,0 ulation, the gains are: K  =  1.0 x 10—6 and K 10  =  —1.5 x i0. The schematic  diagram of the controllers with the integral loop is shown in Figure 5-12 and re sponse of the system is presented in Figure 5-13. Now, the steady state offset is 1.0 m. Note, the offset requirement is a little higher than the previous case, however 120  a(O)  2°;  cx(0) =2° (0)=0.1304x10 1 B m 2 C.(O)=0 =  Stationkeeping, L = 200 m  D=O  ; D=1.0m  2.0  xp° -0.0 0.0  2.0  2.0  M  0.0 -2.0 0.0  2.0  0.0  2.0  1 B  0.1  0.0 0.01  Cl -0.00 -0.01 0.0  Figure 5-10  Orbit  0.5  Controlled response in presence of the modelling error introduced by neglecting the shift in the center of mass terms during the con troller design. 121  c(O) =2° (0)=0.1304x10 1 B m 2  a(O)  =  2°;  Stationkeeping, L =  (0) =0 1 C  0  =  200 m  ; D = 1.0 m  2.0  xp° -0.0 0.0  2.0  0.0  2.0  M  2.25  1 B 1.25 0.0 0.01  1 C  0.1  -0.00  -0.01 0.0  —  Figure 5-11  Orbit  0.5  System response with the shift in the center of mass included in the controller design model.  122  D+ Plant  CL  Figure 5-12  p  Schematic diagram of the closed-loop system with the outer P1 control loop for the tether dynamics.  ) decays due 1 the response is substantially improved. The longitudinal oscillation (B to the structural damping in the tether. Although the transverse vibrational degree of freedom is not subjected to any initial disturbance, the offset motion during the  1 response resulting in a small amplitude oscillation. control excites the C Response results were also obtained for the controlled deployment of the subsatellite from a tether length (L) of 50 m to 200 m in one orbit (Figure 5-14). An exponential-constant-exponential velocity profile was used for the deployment. The first switching of the velocity from exponential to constant profile occurs at L m and the second switching is at L  =  80  180 m. The controller used in this case includes  the outer P1 loop for the tether dynamics with the gains as mentioned before. As shown in Figure 5-14, the platform pitch angle is controlled quite successfully. The 123  a(O)  2°;  a(O) =2° (0)= 0.1304 x 10.2 m 1 B  Stationkeeping, L  C,(0)=0  D=0  =  =  200 m  ; D=1.0m  2.0  xp° -0.0 0.0  2.0  2.0  2.0  M  0.0  0.0  -2.0 0.0  -2.0 0.0  2.0  2.0  1 B  0.1  0.0 0.01  1 -0.00 C -0.01 0.0  Orbit  L  Figure 5-13  System response in presence of the outer PT control ioop.  124  0.5  cx(O)  =  0) B ( 1  =  a(0) =00 8.14 x i0 m  2°;  Deployment: 50 m to 200 m in 1 orbit D=O ; D=O  C(O)=O 2.0  0.0  -0.0  -10.0  xp°  0.04 0.2  L  L  0.00 0.0  M  1 B  5.0 ‘Cl  x10  0.0 -5.0 0.0  Figure 5-14  Orbit  1.5  Controlled response of the system during deployment. The offset control strategy in conjunction with the Feedback Linearization Technique (FLT) is used. 125  tether pitch  (at)  is controlled about the quasi-equilibrium trajectory as defined by  Eq.(5.21). After deployment, at settles to zero. The total offset motion required is within ±5  m,  which is much less than the limiting value of ±15  m.  The control  moment M cancels the coupling between the tether and platform dynamics, and drives  ap  towards its desired value. The coupling between the tether and platform  . This leads to similarity between the moment (d ) dynamics is due to the offset 2 2 time histories. The nonzero steady state value of M is needed to (Mi) and d stabilize  a,  about zero, which is not its equilibrium state.  As expected, the mean value of the longitudinal oscillations increase with the tether length. Note, frequency of the transverse vibrations decrease as the tether length increases. Effect of the Coriolis force during deployment is stabilizing and keeps amplitude of the transverse oscillations quite small, however the mean value of C 1 is not zero. In the post—deployment phase, amplitude of the transverse oscillations increase with zero mean. Simulations were also carried out for several other deployment rates (0.7 and 1.5 orbit, plots not shown). As can be expected, the maximum offset requirement was found to be higher (±7 deployment in 0.7 orbit and lower (±3  m)  m)  for the faster  for the slower deployment rate.  The controlled response of the system during the exponential retrieval from a tether length of 200  m  to 50  m  in 1 orbit is shown in Figure 5-15. The generalized  ) grows to around 0.02 1 coordinate for the transverse vibration (C 0) C ( not excited initially, i.e. 1  =  m  even when it is  0. The initial disturbance in the first longitudinal  ) decays quite rapidly due to the structural damping, however a small 1 mode (B amplitude oscillation persists due to coupling with the transverse vibration (inset in 1 plot). In the present case the destabilizing Coriolis force is not enough to make B 1 < 0). The platform pitch (ap) response settles to zero the tether slack (i.e. B  126  a(O) (0) 1 B  2°;  cx(0) = 0° = 0.1304 x 102 m (O)=O 1 C =  Retrieval: 200 m to 50 m in 1 orbit D=O ; D =O 2 10.0  2.0  0.0 -0.0  L  ::  -0.00  L -0.05 15.0  4.0  M  d pzO.O  0.0  3.0  0.0  3.0  0.0 m 3 x10  1.500  1 B  1.505  1.5  0.1 Cl  -0.0 —-  -0.1 0.0 Figure 5-15  Orbit  1.5  System response during controlled retrieval with an exponential velocity profile.  127  within 0.4 orbit while the tether pitch (at) is stabilized about the quasi-equilibrium trajectory, which is zero after the retrieval. The moment (Mi) required to control ap is only 4 Nm. The offset position required to control the tether swing is between  +12 m and —5 m, which is within the limit of ±15 m used in the present study. As in the case of deployment, simulation results were also obtained for several retrieval times (results not shown). As anticipated, the maximum offset required to regulate the tether swing was larger for a faster retrieval.  5.4  Attitude Control using Hybrid Strategy  As discussed earlier, thruster and tension control schemes have disadvantages at shorter tether lengths which leaves the offset strategy as an efficient alternative. This section presents simulation results of a tethered system implementing a hybrid control scheme. The controller design procedures are the same as those described earlier. In the study, the offset strategy is used for tether lengths less than 200 m and the thruster control law is applied for longer (> 200 m) tethers. The thruster control strategy is based on the rigid nonlinear model of the system. Response of the system during controlled deployment from 50 m to 20 km using this hybrid strategy is shown in Figure 5-16. Deployment from 50 m to 200 m is carried out in 1 orbit with an exponential velocity profile while the rest of the  deployment (200 m to 20 km) is completed in 3 orbits with an exponential-constant exponential velocity profile. In the second deployment stage, the first switch (from exponential to constant velocity profile) takes place at a tether length of 2.5 km and the second switch occurs at 18 km. The tether pitch is controlled about its quasi-equilibrium value during the first orbit. In this period, the maximum offset required along the local horizontal is within ±3 m from the steady state value of 1 128  2°; cx(O) =2° = 0.814 x m C.(O)=0  (O) (0) 1 B  Deployment: 50 m to 20 km in 4 orbits  =  D=O; D=1m  2.0 0.0 -0.0  -8.0 20.0  1.5  LL 0.0  0.0 60.0  2.0  T  M 0.0  -0.0 20.0  3.0  1 B 0.0 0.0 10.0  Cl  0.0 0.0  6.0  6.0  0.0 -10.0 0.0  Figure 5-16  Orbit  6.0  Controlled response during deployment using a hybrid strategy.  129  m. The nonzero steady state value of the offset leads to coupling between the tether and platform dynamics. This coupling exhibits through the small fluctuations in the cp response. As observed earlier in the thruster control section (Sec. 5.2), the M and Tct profiles have similarity with the L and  L  responses, respectively.  The simulation results for retrieval from a tether length of 20 km to 50 m in 5 orbits is shown in Figure 5-17. An exponential velocity profile is used for the re trieval. The initial retrieval from 20 km to 200 m is carried out in 4 orbits. To limit the offset motion, the final stage of the retrieval from 200 m to 50 m is completed in 1 orbit. The instability of the flexible modes is controlled by passive dampers. As in the case of the thruster control, the damper for the longitudinal oscillation is located at the subsatellite and for the transverse vibration at the center of the tether. The damping coefficients for the longitudinal and transverse dampers are 1.5 and 0.03 Ns/m, respectively. The thruster control is used for retrieval upto 200 m, and the offset scheme for the rest of the retrieval as well as the subsequent sta tionkeeping. During the offset control, the tether pitch angle is regulated about the quasi-equilibrium trajectory (Eq.5.21). As in the case of deployment, M and Tat profiles are similar to the L and L trajectories, respectively. The offset excursions to control the unstable tether attitude during the retrieval range from —4 m to +13 m, a significantly large distance than that required during deployment. Of course, this is expected due to unstable character of the retrieval maneuver.  5.5  Gain Scheduling Control of the Attitude Dynamics  For a comparison between the FLT based regulator and a linear controller with gain scheduling, design of an attitude controller using the eigenvalue assignment algorithm was undertaken. This linear time invariant controllers were implemented 130  a(O)  =  2°; x(0) =2°  O) B ( 1  =  13 m  Retrieval:  20 km to 50 m in 5 orbits = 1 m =0;  O) =0 C ( 1  8.0  2.0  cx° 0.0  0.0  20.0  0.0  L  L  -4.0  0.0  50.0  N-mj  0.0  Tat  M 0.0  -6.0  12.0  1 B  1  0.0 0.0  6.0  0.0  6.0  70.0  Cl  0.0 0.0 Figure 5-17  Orbit  6.0  System response during retrieval with a hybrid control scheme.  131  on the time varying nonlinear plant in conjunction with the gain scheduling. The controllers were designed using the Graph Theoretic Approach. The detail mathe matical background and the controller expressions are given in Appendix IV. Only a few typical results are presented here for comparison. Figure 5-l8shows the stationkeeping dynamics in presence of the thruster augmented active control. Note, the attitude degrees of freedom are controlled quite successfully. The control effort (M and Tat) requirements are rather modest. However, the system performance during retrieval with a fixed specified offset of 1 m along the local horizontal (D =1 m) is 2 not satisfactory (Figure 5-19). Particularly, the platform pitch response has a large overshoot of more than —5° at the beginning of the retrieval. This is attributed to coupling, caused by a nonzero offset, between the tether and platform dynamics. The linear controller (with gain scheduling) takes some time to compensate for the coupling effect. Similar response was also observed during the retrieval maneuver of shorter tethers using the offset strategy (Figure 5-20). The performance of the system during a hybrid control is essentially the same. Comparison of these results with those presented in Figure 5-17, where the FLT based hybrid control is used to regulate the retrieval dynamics, clearly shows better performance of the FLT controller compared to the gain scheduling linear regulator.  5.6  Concluding Remarks A controller based on the FLT is found to be adequate in regulating the  highly time varying attitude dynamics of the TSS with a flexible tether. Two dif ferent strategies, thruster and offset control, are used for longer and shorter tethers, respectively. Three different system models (nonlinear flexible, nonlinear rigid, and linear rigid) are considered to arrive at an efficient FLT controller design. Results 132  cc(O)  a(O) =2° (0)=0.8m 1 B =  2°;  Stationkeeping, L =5 km  D=o  (0)=0.1 m 1 C  ,  2.0  2.0 xto  cx°  M  D pz =0  -0.0  -0.0  0.5  0.2  Tat -0.0 0.0  0.0 2.0  0.0  2.0  0.85  1 0.80 B 0.75 0.0  1.0  0.3 Cl  -0.0 -0.3  Orbit  0.0  Figure 5-18  2.0  Controlled stationkeeping dynamics using the graph theoretic ap proach.  133  cz(O) =0; a(0) =0 (0) = 13.0 m 1 B  Retrieval: 20 km to 200 m in 4 orbits D=0; D=1m  C.(0)=0  0.0 -5.0  20.0  0.0  L -4.0  0.0  40.0 -0.0  M 0.0 0.0  -3.0 6.0  0.0  6.0  13.0  1 B 0.0 60.0 30.0 0.0 30Qo Figure 5-19  Orbit  6.0  Gain scheduling control of the retrieval maneuver using the thruster augmented strategy.  134  x(O)  =  (0) 1 B  =  0;  a(0) =0 1.304 x m  Retrieval: 200 m to 50 m in 1 orbit D=0; D=0  C(0)=0  14.0  5.0 0.0  0.0  -5.0 -0.00  0.2  L -0.05  0.0  M  3.0  10.0  -0.0  d0.0  -3.0  —  0.0 m 3 x10  3.0  -10.0 0.0  3.0  1 B 2.0  0.0 3.0 Cl  m1 2 x10  0.0 -3.0 0.0  Figure 5-20  Orbit  1.5  Offset control of the retrieval dynamics using the gain scheduling approach.  135  suggests that the rigid body model (either nonlinear or linear) is sufficient to de velop an effective thruster based regulator with a PT control ioop. The FLT based controller with an PT loop is also shown to be effective during the offset control. Both the offset and thruster strategies are also implemented in a hybrid fashion. The hybrid control strategy using the thruster control at longer tether lengths and the offset control for shorter tethers appears quite promising. Linear time invariant regulators, designed using the graph theoretic approach and implemented through gain scheduling, though effective are not as efficient as the FLT based control system.  136  6.  6.1  VIBRATION CONTROL OF THE TETHER  Preliminary Remarks  Tether missions involving controlled gravity environment or precise position ing of the subsatellite require regulation of the tether’s vibratory motion.  The  tether may oscillate in both the longitudinal and transverse directions. The design and implementation of the controller to suppress tether vibrations is addressed in this chapter. The results presented here correspond to the stationkeeping situation where most of the mission objectives are carried out. Only the first longitudinal and transverse modes are controlled actively here as their energy content is domi nant. The higher transverse modes constitute critically stable degrees of freedom. The tether vibrations can be controlled either by some active methods (if control lable and observable) or using passive dampers at appropriate locations. In the present study, a passive damper is used to control the higher transverse modes. The rigid degrees of freedom are regulated by thrusters and momentum gyros, and the transverse and longitudinal modes are governed by the offset strategy. This chapter begins with some mathematical background for the controller design. This is followed by the system linearization. Finally design of the controller is undertaken which includes the choice of system inputs and outputs as well as some typical results showing its performance.  6.2  Mathematical Background  The nominal design model of a plant in the linear state space form can be 137  expressed as: th 0 +B x+1’; =A u  (6.la)  +q; y=C x 0  (6.lb)  E Rm; y E R’;  ; 1 where: x E R  B 0 , , 0 0 C and F are constant ; and A 1 E R  are the state, control input  real matrices of appropriate dimensions. x, u and  and measured output vectors, respectively.  is the state noise vector and  i  is the  measurement noise vector. These are white noises uncorrelated in time (but may be correlated with each other) with covariances E[T]  =  E  0,  E[T]  and  0 > 0,  =  E[T]  =  0,  where 0 is the null matrix of appropriate dimension. In the transfer function nota tion the system can be represented by (6.2)  where:  (s)Bo; C 4 0  C(s)  =  d  =  and 4(s)  =  [sU  —  . 1 J 0 A  Here, C(s) is an m x r transfer function matrix and U is the unit matrix. The standard feedback configuration of the system is illustrated in Figure 6-1. It consists of an interconnected plant C(s) and a compensator F(s) forced by the command input r, measurement noise  and disturbance d.  138  d  Figure 6-1  6.2.1  Standard feedback configuration.  Model uncertainty and robustness conditions  In reality, the true model of the system, G’(s), is not the same as the nominal design model, G(s). Hence, no nominal model can be considered complete without some assessment of its error, normally referred to as model uncertainty. The repre sentation of these uncertainties varies primarily in terms of the amount of structure it contains. For example, the uncertainty caused by variation of certain parame ters in the governing equations of motion is a highly structured representation. It typically arises from the use of linear incremental models, e.g. error in the moment of inertia of a spacecraft, variation in the satellite mass due to firing of thrusters, changes in the aerodynamic coefficients of an aircraft with flight environment and configuration, etc. In these cases, the extent of variation and any known relationship between the parameters can be expressed by confining them to appropriately defined subsets in the parameter space. An example of the less structured representation of uncertainty is the direct statement for the transfer function matrix of the model 139  such as G’(jw)  =  G(jw) + G(jw)  (6.3a)  with ã[G(jw)]  <la(w),  Yw  0,  where: la() is a positive scalar function confining the matrix C’ to the neighborhood of C with magnitude la(w); and  a(.) represents the maximum singular value of a  matrix. The statement does not imply a mechanism or structure that gives rise to G(jw). The uncertainty may be caused by parameter changes as above, or by neglected dynamics, or by some other unspecified effects. This is also referred to as additive uncertainty. The alternative statement has the multiplicative forms: G’(jw)  =  [U + Lo(jw)]G(jw);  (6.3b)  G’(jw)  =  G(jw)[U + L(jw)],  (6.3c)  or  with (jw)} <lmo(w), 0 ö[L where lmo() and  lm()  U[L(jw)] < lm(w),  Vw  0,  are positive scalar functions; U is the unit matrix of appropri  ate dimensions with Lo(jw) and L(jw) representing output and input multiplicative uncertainties, respectively. The structure of the system with these uncertainties is shown in Figure 6-2. The objective of the feedback design problem is to find a compensator F(s) such that: , is stable; 1 (i) the nominal feedback system, GF[U + GF] , is stable for all possible C’; and 1 (ii) the perturbed system, G’F[U + G’F] 140  (a)  True Plant G’(s) r  True Plant G’(s)  (b)  True Plant G’(s)  (c)  Figure 6-2  Diagram showing different unstructured uncertainties: (a) additive; (b) output multiplicative; and (c) input multiplicative.  141  (iii) performance objectives are satisfied for all possible G’. The requirement (i) demands that the encirclement count of the map det[U+ GF(s)], valuated on the standard Nyquist D-counter, be equal to the (negative) number of unstable open loop modes of GF(s) [92]. As shown by Yuan and Stieber [93], for the additive uncertainty used in the present study, the closed loop system satisfies the stability robustness requirement (ii) if la(w) where [R(jw)]  =  1 < ã[R(jw)]’  (6.4)  F(j)[U + 1 G(jw)F(jw)] Simillar conditions for other uncer .  tainties can also be obatained [94]. 6.2.2  LQG\LTR design procedure Linear Quadratic Gaussian (LQG) procedure is a widely used approach for  feedback design [95, 96]. The LQG controller is an ordinary finite dimensional LTI compensator with the internal structure as shown in Figure 6-3. It consists of a Kalman-Bucy Filter (KBF) which provides an estimate of the state, &. The KBF gain, Kf, is given by [94]  Kf  Pf C 0T-1 0 ,  (6.5  where Pf is the solution of the algebraic Riccati equation PfA + AOPf and Pf  =  PJ  —  PfC’O C 1 OPf + F E FT =0,  (6.6)  0. In general there are several solutions to Eq.(6.6),.but only one  of them is positive-semidefinite. The state estimate, I, is multiplied by the full-state Linear Quadratic Regu lator (LQR) gain, K, to produce the control command which drives the plant and 142  Compensator F(s) I  Figure 6-3  Closed-loop system with the LQG feedback controller.  also feed—backs internally to the KBF. The LQR gain K is obtained to minimize the cost function J =  where: z R  =  =  (zTQz + uTRu)dt,  j  Mx is some linear combination of the states; and  Q  =  QT  0,  RT > 0 are weighting matrices. The solution to this problem is I’  .LkC  1 D —  1-lTD  c,  6.7  where P satisfies the algebraic Riccati equation 0 + PA  -  PCBOR  BQTPC + MTQM =0,  (6.8)  and P=P’>0.  Both the LQR and KBF loops have good robustness properties [97, 98]. Therefore, it may be expected that the LQG compensator would generally display acceptable robustness and performance. Unfortunately, it has been shown that the LQG designs can exhibit arbitrarily poor stability margins [99]. However, there are 143  procedures to design either LQR controller or KBF so that the full state-feedback properties are recovered at the output or input, respectively, of the plant [100]. For a square minimum phase plant, design for the Loop Transfer (function) Recovery (LTR) at the plant output consists of two steps: (i) Design a KBF by manipulating the covariance matrices ratio C (sU 0  —  and 0 until a return  Ao)’Kf (i.e. the KBF loop transfer function) is obtained  which would be satisfactory at the plant output (i.e. at the break point (i) in Figure 6-3). (ii) The ioop transfer function obtained by breaking the loop at point (ii) in. Figure 6-3 is OF(s), where F(s) is the compensator transfer function. It can be made to approach Co(sU  —  f pointwise in s by designing the LQR Ao) K 1  in accordance with a sensitivity recover’y procedure due to Kwakernaak and Sivan [101]. To achieve this, synthesize an optimal state-feedback regulator by setting  Q  =  - qU and R Q H 0  =  0 (or R  Q  =  Qo  and R  =  0 + pU), and R  increase q (reduce p) until the return ratio at the output of the compensated plant converges sufficiently close to Co(sU  —  Aa)’Kf over a large range of  operational frequencies. For non-square plants, the inputs and/or outputs can be redefined to make the system square and the LQG/LTR procedure can be applied to the modified plant (if it is observable and controllable). For non-minimum phase plants the recovery may be achievable at those frequencies at which the plant’s response is very close to that of a minimum-phase plant, i.e. at frequencies which are small compared to the distance from the origin to any of the right-half plane zeros [94]. A simple strategy to use with the non—minimum phase plant is to follow the usual LTR procedure. If the right half-plane zeros lie well out side the required bandwidth, then adequate  144  recovery of the ideal characteristics would be achieved at all significant frequencies.  6.3  Controller Design and Implementation The prime objective of the controller is to regulate longitudinal and trans  verse tether vibrations using the offset strategy with the attitude dynamics regulated by the thruster augmented active control. As explained earlier, only the first longitu dinal and transverse modes are considered for the controller design. The LQG/LTR based procedure is used which can account for the model uncertainty due to the neglected dynamics. 6.3.1  Linear model of the flexible subsystem The model is obtained by linearizing the decoupled equations of motion for  the flexible subsystem about the equilibrium positions. The equilibrium position for the transverse vibrations is zero and for the longitudinal oscillations corresponds to the static deflection value. The linearized equations can be written as 2 Z + G Z + K Z + Mdy M  + Gddpy + Kdydpy  2 + Md dtpz + Gdzdpz + Kdzdpz + P 2  where: Z  {{B  —  =  QZTL,  (6.9)  Beq}T, CT}T is the flexible generalized coordinate vector;  2 TL, the control thrust along the undeformed tether line; d, (D + D) and cL  , the offsets of the tether attachment point along the local vertical and D ) 2+2 (D local horizontal, respectively;  the specified offsets; and D and D , 2  and  the offsets required by the controller. The expressions and numerical values of the coefficient matrices (for a system with 10 transverse and 2 longitudinal modes) are defined in Appendix V. The numerical values are for the system without any passive damping. Depending on the choice and controllability, the variables TL, D, D, 145  D and D 2 can be used as control inputs, either separately or in combinations. The selection of outputs completes system characterization in the linear statespace form. It should be recognized that placement of sensors on the tether to measure vibrations is impractical due to small diameter (one to two mm) of the tether as well as its deployment and retrieval maneuvers. The objective of the controller is to suppress longitudinal as well as transverse vibrations using the offset control strategy. The output vector consists of: longitudinal deformation from the equilibrium value of the tether at yt deformation at y  =  =  L; slope of the tether due to transverse  0; and offset positions along the local vertical and horizontal;  i.e.  {  { 6.3.2  0,  •..,  i(L),  0,  ...,  %(L), 0,  , 0 (aia)  ••,  ...,  0}{Z}  0 / 1 (aN } aYt) {Z}  (6.10)  Design of the controller In the nondimesional form, the lowest elastic and maximum attitude frequen  cies (w/O) are 12.6 and 1.732, respectively. This separation of frequencies allows the controllers for the rigid and flexible subsystems to be designed based on the decou pled equations of motion. The rigid body controller was designed using the Feedback Linearization Technique (FLT) discussed in Section 5.2.3. Now, the controller for the flexible subsystem is designed by the LQG based approach. Here, the objective is to regulate both the longitudinal and transverse vibrations by the offset control 1t transverse (C strategy. As mentioned earlier, the s ) and 1 146  st 1  longitudinal (B ) 1  1 degrees of modes are controlled. The nondimensional frequencies of C 1 and B freedom are 12.6 and 65.5, respectively. Again, the frequency separation permits 1 models. The control input for 1 and B the design to be based on the decoupled C 1 model is the offset acceleration along the local vertical the B  (.b)  and that for  1 model is D (offset acceleration along the local horizontal). The outputs are C 1 subsystem, the output vector is selected from Eq.(6.10). For the B ) (L)Z ,b ’ I1 D, J 1 where Z  =  1 B  —  (6.11)  , and Beg 1 Beg 1 is the equilibrium (static deflection) value of the  1t longitudinal mode (B s ). For the C 1 1 equation, the outputs are  I  (ôi/ôt) 0 Yt  =  1 ‘1 c .  (6.12)  ) 1 model, which is a two output and a single input system, is Controller for the B designed using the LQG/LTR method. The design approach is due to Doyle and Stein [100]. The controller matrices are obtained using the subroutines available in the Robust Control Toolbox of MATLAB [102]. This procedure requires the system 1 dynamics is rendered square to be square. The rectangular linear model for the B by defining the output as Yb  =  (L)Z + D. 1 b  (6.13)  The system and control influence matrices are obtained consistent with the first mode assumption. With this modified output, the system is found to be controllable 1 dynamics is obtained in and observable. The controller design equations for the C a similar way. In this case, the LQG/LTR controller, obtained after redefining the output, does not give satisfactory performance. The degradation was attributed to the modified output and can be avoided by using the original input-output structure 147  in conjunction with the LQG design procedure. To regulate the offset motion, the design equations are augmented by the following identities:  and  jjz  =  u;  =  u,  for B 1 and C 1 dynamics, respectively. The controller design models can be repre sented by:  Xb  =  Abxb + Bbub;  (6.14a)  Yb  =  Cbxb,  (6.14b)  Ax + Bcuc;  (6.15a)  and  th  (6.15b)  Yc  where:  Z  1 z{(B  , 1 z={C  —  Beq ) 1 ,  , 1 O  , 1 E  D, ]3}T;  , .Oz}T; 2 D  with Yb and Yc given by Eqs.(6.13) and (6.12), respectively. Here, the subscripts ‘b’ and ‘c’ refer to B 1 and C 1 dynamics, respectively. The dynamic feedback controllers have the form:  Xfb  =  Ub =  Afb Zfb + Bfb Yb;  (6.16a)  Cfb Xfb,  (6.16b)  148  and  Xfc  =  U =  Af Xfc + Bfc ye;  (6.17a)  Cf Xfc,  (6.17b)  where Xfb and Xfc are the state vectors for the B 1 and C 1 controllers, respectively. The numerical values of the design model are obtained from the higher order model. The complete linear model, controller and weighting matrices are given in Appendix V. The data are for a tethered system during stationkeeping at L  =  20 km with  mass and elastic properties as given in Chapter 4. Next, the closed-loop eigenvalues for the linear flexible system were obtained to have some appreciation as to the controller’s effectiveness. The eigenvalues of the open-loop and closed-loop systems are shown in Table 6.1.  Table 6.1  Comparison of the open-loop and closed-loop eigenvalues of the sys tem.  Mode Trans. 1 Trans. 2 Trans. 3 Trans. 4 Trans. 5 Long. 1 Trans. 6 Trans. 7 Trans. 8 Trans. 9 Trans. 10 Long. 2  Open-loop  —  —  0.0 0.0 0.0 0.0 0.0 3.797e-4 0.0 0.0 0.0 0.0 0.0 2.267e-2  ± ± ± ± ± ± ± ± ± ± ± ±  1.468e-2 2.937e-2 4.405e-2 5.874e-2 7.342e-2 7.601e-2 8.811e-2 1.028e-1 1.174e-1 1.321e-1 1.468e-1 5.870e-1  Closed-loop —  —  —  9.880e-4 0.0 0.0 0.0 0.0 1.144e-2 0.0 0.0 0.0 0.0 0.0 2.183e-2  ± ± ± ± ± ± ± ± ± ± ± ±  1.463e-2 2.937e-2 4.406e-2 5.874e-2 7.343e-2 7.614e-2 8.811e-2 l.028e-1 1.174e-1 l.321e-1 1.468e-1 5.912e-1  Additional controller was designed for the model with a passive damper hay 149  ing a damping coefficient (Cdt) of 0.2 N.s/m and located on the tether at a distance of 9.8 km from the platform. The damper was introduced to reduce the transverse vibrations. The controller design procedure is given in Figure 6-4. As before, the controller for the B 1 -dynamics is designed by the LQG/LTR method and the C 1 controller is designed using the LQG algorithm. The attitude controller is obtained through the FLT procedure. The recovery of the return ratio at the plant output 1 dynamics is shown in Figure 6-5. The gain with the LQG/LTR controller for the B and phase of the return ratio approaches those of the KBF loop transfer function 1 dynamics with an increase in the design parameter q. In the present case, the B is a non-minimum phase system with zeros at +0.045 and —0.042. As mentioned before, the recovery in this class of plants is not guaranteed as apparent in Figure 6-5. 1 and C 1 controllers is checked Finally, the robustness property of both the B against the additive model uncertainty G(j). Figure 6-6 compares the bound for G(jw), i.e. la(w), and the inverse of the maximum singular value of R(jw) F(jw) (u + L( w)) 3  ,  where: L(jw)  =  ) is the ioop transfer function; G(jw)F( w 2  F(jw) is the controller transfer function (both the controllers); G(jw) is the openloop transfer function; and U is the unit matrix. As shown in the figure, the stability condition of Eq.(6.4) is satisfied. The structure of the closed-loop system, with controllers for rigid body dynamics as well as longitudinal and transverse vibrations, is shown in Figure 6-7.  6.3.3  Results and discussion The controllers were implemented on the complete nonlinear system with  two longitudinal and three transverse modes. Figure 6-8 shows the response of the 150  Complete (rigid and flexible) nonlinear model  Nonlinear rigidbody model  Decoupled linear flexible model 28 states 4 outputs 2 inputs Truncated flexible model for controller design 8 states 4 outputs 2 inputs Controller design (LQG and LQG/LTR) Improve damping  8 dimensional controller  4,  Is controller acceptaIe?  No  I  I 1 Controller B -  Controller  C .  Three level controller for rigid and flexible model  Figure 6-4  FLT based controller design  Flow chart for the controller design.  151  Attitude Controller  100.0  Gain 100.0  q -200.0  ..  1  180.0  ....  102  =  .q=lx  0  ..  10.1  10°  101  102  10°  01  02  3 io  degree!  \  q=1  Phas: 0  -  .0 10  10.2  10.1  1  Frequency, rad/s  Figure 6-5  Comparison of gain and phase of the return ratio at the plant output with those of the KBF loop transfer function.  152  100.0  .S..-...-—  __/\  0.0  -100.0  1  V  1  Frequency, rad/s Figure 6-6  Robustness property of the vibration controller.  system without a passive damper and using a three level control structure for simul taneous regulation of flexible and rigid modes. The total longitudinal deformation, v(L), is controlled about its equilibrium value of 13.896 m. As shown in Figure 6-8, it returns to the equilibrium value in less than 0.05 orbit. The initial specified offsets for this simulation was set at zero (D  =  =  0.0). The offset motion  requirement for control of the flexible modes is much lower than ± 15 m limit set for regulation of the rigid degrees of freedom. The total offset motion along the local vertical (d) is less than ± 2 m. The first transverse mode settles to zero in around one orbit. The offset motion along the local horizontal (d), required by controller, is around ± 0.6 m. As discussed earlier, the platform motion is -C the 1 strongly coupled with the tether dynamics through  Since the FLT controller  for the platform pitch (ap) dynamics cancels all the coupling effects, the shape of 153  Input  I  M(q,t)’ + f(q,,t)  I =  Output  Qq  LQG/LTR Controller for Long. Vibration  Desired Trajectory  Figure 6-7  I  LQG Controller for Trans. Vibration FLT Controller for [e}  Three-level controller structure to regulate rigid as well as trans verse and longitudinal flexible motions of the tether.  the moment (Me) time history required to regulate  p  is similar to that of  The  magnitude of M is well within the acceptable limit. The thrust (Tat) required to control the tether pitch (at) is also very low. Figure 6-9 shows the response of the controlled system with a passive damper. The rigid body and longitudinal vibration responses are similar to those observed in the previous case. However, there is a substantial improvement in the transverse 3 modes. There is a room for 1 and C vibration responses, particularly in the C  further improvement through optimum location of the damper. From these simulations, it can be concluded that the tether vibrations in both longitudinal and transverse directions can be controlled effectively by the offset strategy. It is important to note that the offsets required to control the flexible modes 154  (O) (O) 1 B  = 10; =  12  Ex(0)  o 2 c  = 10  m; B (O) =0 2  (0)=lOm 1 C  (C 3 O) =0 Stationkeeping: L = 20 km D=0; D=0 =  1.0  1.0  cz 0.0  0.0 2.0  0.0  13.0 Eii1  18.0  v(L)  Cl 0.0  12.0 0.1  0.0  2 C  2.0  0.0  0.5  -13.0 0.0  2.0  3 C  0.0 1.0  0.0 40.0 M 20.0 0.0 _on A 0.0  Tat 2.0  1.0  0.0  ::  2.0  3.0  0.0  Figure 6-8  Orbit  0.1  0.0  Orbit  2.0  Response of the system using a three level controller in absence of a passive damper.  155  cz(O)  = 10; (0) = 10  (0) = C 2 C (0) = 0 3  (0) = 12 m; 2 1 B B ( 0) = 0  Stationkeeping: L = 20 km  (0)=lOm 1 C  D=0;  1.0  1.0  0.0  0.0 1.0  0.0  0.0  18.0  v(L)  D=0  1.0  13.0  1 C  12.0  -13.0 0.0  0.1  0.0  0.0 1.0  0.5  2 C  0.5  3 C  0.0  -0.5 0.0  1.0  0.0  ° 8 M  Tat  0.0  0.0  0.6 0.0  -8.0 0.0  1.0  0.0  2.0  0.2  0.0  0.0  -2.0 0.0  Figure 6-9  1.0  Orbit  0.1  -0.2 0.0  1.0  Orbit  1.0  Controlled response of the system in presence of a passive damper.  156  are less than ±2 m, which is small compared to the limit (±15 m) set for regulating the attitude motion.  6.4  Concluding Remarks Issues involving the control of elastic motion of the tether using offset strategy  were addressed in this chapter. To begin with a linear model of the flexible subsystem was obtained. Next, three controller ioops were designed for simultaneous regulation of attitude as well as longitudinal and transverse flexible modes. The controllers for flexible modes were obtained using the LQG based approaches while the attitude controller employed the FLT algorithm. Offset accelerations along local horizontal and vertical were used as inputs for transverse and longitudinal modes, respectively, while the attitude motion was controlled through thrusters. Effectiveness of the controllers was assessed through its application to the complete nonlinear coupled system. From the simulation results it can be concluded that the procedure is quite effective in regulating, during stationkeeping, both the rigid and flexible degrees of freedom.  157  7.  7.1  EXPERIMENTAL VERIFICATION  Preliminary Remarks  Since the first practical application of a tethered system in 1966 during the Gemini flight, several missions have been carried out to demonstrate the concept. As mentioned in Chapter 1, they include: a joint U.S.—Japan sounding rocket based Tethered Payload Experiment [TPE, 5]; U.S.A.—Italy TSS—i (Tethered Satellite System—i) mission in August 1992 [6]; sounding rocket based OEDIPUS (Observa tion of Electrified Distributions in the lonosperic Plasma—a Unique Strategy) ex periment launched by the Canadian Space Agency in January 1989 [7]; and the most recent studies called SEDS—I and II [Small Expendable Deployment System, 8], launched in 1993 and 1994, respectively by NASA. Several ground based lab oratory experiments have also been carried out to verify the tether dynamics and effectiveness of different attitude control strategies. Dynamical aspects of a spin ning tethered system were explored by Jablonski et al. [103] and Tyc et al. [104] particularly with reference to the OEDIPUS—C experiment scheduled for lunch in December 1995. Gwaltney and Greene [105] implemented a tension scheme by converting the control algorithm to a length rate law, for both inpiane and out-of-plane dynamics, as well as accounting for the deployment and retrieval phases. The control scheme, with measured tension feedback, was investigated by Shoichi and Osamu [106]. A laboratory set—up, for studying the attitude control of the subsatellite using the offset strategy, has been developed by Kline -Schoder and Powell [59]. Effectiveness of the offset control strategy was verified by Modi et al. [48] using a ground based 158  experimental facility. An LQR type offset control algorithm was used to regulate tether librations during deployment, stationkeeping and retrieval. Using the modi fled facility to improve the sensor accuracy, the present study assesses effectiveness of the FLT and LQG control methodologies. To begin with, the experimental set—up is described followed by details of the controller design and implementation procedure. Only a sample of typical results is presented to attest effectiveness of the two controllers. The chapter ends with some concluding remarks.  7.2  Laboratory Setup  The objective of the present experiment is to have some appreciation as to the effectiveness of the FLT and LQG based offset control schemes for regulating the attitude motion of the tether. The experimental setup consists of a spherical mass, representing the subsatellite, which can be deployed or retrieved from a carriage. The carriage, depicting the offset mechanism translating on a platform, can be moved in a horizontal plane. This permits time dependent displacement of the tether attachment point to implement the offset control strategy. A Nylon thread, 1 mm in diameter, connecting the subsatellite with the carriage, serves as tether. The tether can have a maximum length of 2.25 m. A larger platform with a height of 5 m to accommodate longer tethers has also been designed and constructed. Details of the dimensions and mass properties of the setup are given in Appendix VI. The experimental apparatus consists of three main parts: the sensor; actuator; and controller. 7.2.1  Sensor Role of the sensor is to measure the angular deviation of the tether from the 159  vertical position. To that end, a pair of optical potentiometers (Si series Softpot, U.S. Digital Corp.) were used. The Softpot optical shaft encoder is a noncontacting rotary to digital converter. Useful for position feedback, it converts the real—time shaft angle, speed and direction into the Transistor—Transistor Logic (TTL) compat ible two channel quadrature outputs plus a third channel index output. It utilizes a mylar disk, a metal shaft with bushing, and an LED. The unit operates from a single +5 V supply. Low friction ball bearing makes it suitable for motion control applications. The output pulses are monitored by a counter circuit, which generates an eight bit digital signal. The counter output can directly be read by the data acqui sition system (a commercially available card [107]). The resolution of the sensor de pends on whether one or two potentiometer output channels are used in the counter. Two different resolutions, 0.25° and  10,  can be obtained with the present counter  circuit. The index pulses are used to reset the counter to a reference value when the tether swings through the vertical position. This feature eliminates sensor drift that is normally associated with a resistance potentiometer. Attractive features of the system are : low friction in the bearing; no sensor drift; and accurate resolution to 0.25°. An important step in the sensor development was the design of a suitable mechanism to rotate the potentiometer shaft with the tether swing. The mechanism has to perform its task without interfering with the deployment/retrieval maneuvers and yet maintain the unavoidable friction at a low level. The design consists of a light, slotted, aluminum semi—ring as shown in Figure 7-i. The ring is so designed as to maintain dynamic balance during acceleration of the carriage. Any dynamic unbalance may introduce an inertia torque on the ring leading. to an error in the  160  Carriage  /  Potentiometer Bearing Slotted Ring  —‘  Tether  Subsatetlite  Front View  Figure 7-1  Side View  A device to measure the tether swing through rotation of the po tentiometer shaft.  161  potentiometer reading. One side of the ring is connected to the potentiometer shaft while the other is supported by the bearing thus permitting it to rotate about a fixed axis. The tether passes through the slot in the ring permitting detection of the swinging motion even during deployment and retrieval. A pair of such semi—rings are used to measure the angles in two orthogonal planes. 7.2.2  Actuator The actuation mechanism can be visualized as a large x-y table, where a car  riage representing the tether attachment point traverses a horizontal plane (Figure 7-2). The motion of the carriage is controlled by a pair of stepper motors. The carriage carried a reel mechanism, driven by another stepper motor, to deploy or retrieve the tethered payload. All the three motors were commanded by a digital computer that implements the control strategy. 7.2.3  Controller The controller was a 486/33 MHz IBM compatible personal computer. A typ  ical control loop consists of sensing the tether angles, computation of the corrective control effort, transmitting actuation commands for moving the tether attachment point, and saving the required information into an output file. The stepper mo tor commands are sent through translators, which supply required voltage to the motors for each pulse received from the controller. The pulse train for the de ployment/retrieval motor is generated by a commercially available card [108]. A trapezoidal velocity profile is selected for this maneuver. The pulses for the car riage motors are generated through the digital output port of the PCLabCard [107]. A linear velocity profile is chosen to implement the constant acceleration of the tether attachment point during each time—step. Photographs of the test facility are presented in .Figures 7-3 to 7-6. 162  I-.  oq  C)  S  CD  C)  I?3  CD  m Cl)-rr -<-A.,, m :i —1 m  Co  0-iQ  >co OQm  io  m  z  -o  QC -IH  0  H  m  Q  z  C-) I  Figure 7.3  Photograph of the test-rig constructed to validate offset control strategy: (a) aluminum frame; (b) inpiane motor; (c) carriage for out-of-plane motion; (d) wooden stand; (e) linear bearings; (f) tethered payloa d.  (.11  Figure 7.4  d&,. d  •4P  I  Carriage and sensor mechanism: (a) potentiometer on mounting bracket; (b) moveable aluminum semi—ring mechanism with slots for tether; (c) tether; (d) inpiane traverse with linear bearings; (e) payload.  I  Figure 7-5  Digital hardware used in the experiment: (a) translator module, deployment and retrieval; (b) translator module, offset motions; (c) power supply; (d) function generator. 166  I.  Figure 7.6  Photograph showing the recently constructed larger test—facility which can accomodate tethers upto 5 m in length. The present smaller set—up on which the experiments reported here were carried out can be seen in the foreground to the left.  7.3  Controller Design and Implementation The equations governing dynamics of the laboratory model of the tethered  satellite system were obtained. The linearized equations of motion can be written as: a= 3 +a a+cd; 1 a &  (7.1)  +a y+c 7 7 1 a 3 7= ,  (7.2)  where: ai  =  a  —g/L;  =  —2L/L;  c  =  —ilL.  Here: L is the instantaneous tether length; g is the acceleration due to gravity; a and ‘y are the tether angles in two orthogonal vertical planes; and da and th are the accelerations of the offset point for controlling a and y, respectively. The a and equations are independent of each other and hence the controllers can be designed separately. The controller design is based on the FLT and LQG procedures which are explained below.  7.3.1  FLT design The procedure for the FLT controller design was outlined in Chapter 5. For  the experimental model, it consists of state and control transformations: z  =  T() —  =  [tjj]x;  P(x)  Uj  ;  =  where: = aic+ac; 1 tj  168  (7.3) (7.4)  =  —arc 2 + a c, 3  =  ; —a c 3  =  and for i= 2,3,4 ui  =  O 2 t =  t(_l)l  t(_l) 3 a +;  P(z)  =  3+ )x 4 t a 3+x x 4 t a 1 43 + (t t a 2 i + 3 4  Q(x)  =  ct43 + . 44 t  The secondary controller,  Vt,  42 (u  3 ) 4 t a ; 1- 4  for the system is designed to place the closed ioop  eigenvalues at some desired locations. This leads to the following expression for v =  where:  zi kt ( 1 and  (z kt z2 _Z2d)+ 3 zid)+kt ( 2  Zid  i  =  _Z3d)+  z4_z4d), kt ( 4  (7.5)  1, , 4, are the coefficients of the required closed loop  polynomial and the desired values for zj, respectively. Controllers for a and 7 degrees of freedom are designed by considering x as {a, cl, ix, d}T and  {,  J,  ,  T, with u as q } 7  and d.y, respectively. The  coefficients of the desired closed loop polynomials correspond to rise—times of 0.5 and 0.6 s for tether and offset motions, respectively, and settling—times of 3.0 s for both the tether and offset motions.  169  7.3.2  LQG design Considering the dynamics of  ,  the linear equations for the LQG controller  design can be written as:  =  Acxc + brua;  (7.6a)  =  Ccxc,  (7.6b)  where:  =  ua  {cr da a da}T;  =  o o  Aa  ai  o  1 0  0 0 0 0  (0  0 1 0 0  0  Jo Ii  T. _r ya_1a (hJ  1000 0 1 0 0  Ccx  The controller design (i.e. LQR gains and KBF matrices) is arrived at using the subprograms available in MATLAB. The weighting matrices considered in the design are: 0 2.0 0 0 0 0 0 30.0 01.0 0  50.0  o  —  =  o  o  0 0 1.0 0 0 0 100.0 0  0 0 0 0.1  IL—LOS  0 0 0 100.0  and  11.0  0 =  L  0  0 1.0  The continuous time LQG controller is implemented as a sampled data system To that end, the dynamic controller equation is transformed into the discrete model, 170  which can be expressed as:  Xfa(k + 1) u.(k) For L  =  1 m; L  =  A fcr  Bf  Cfa  =  AfaXf(k) + Bfaya(k);  (7.7a)  CfXfy(k).  (7.7b)  0; and a sample time of 0.09 s, the controller matrices are:  —  —  =  0.52214 0.01194 —1.17623 0.22942  0.00177 0.62945 0.01768 —0.74557  0.43029 0.00341 —0.03394 0.11417  0.00243 0.36614 0.08718 0.64020  [5.49377  —1.41421  0.06212 0.00585 0.81021 0.14336  0.00639 0.06604 0.15977 0.80424  ; and  1.73788  —2.12424].  The controller for regulating ‘y is designed in a simillar way and leads to exactly the same design.  7.3.3  Controller implementation The two controllers were implemented as sampled data systems with the zero  order hold. The FLT compensator is described by algebraic equations. Therefore, the control inputs can be computed by Eqs.(7.3-7.5) at the sampling point. The LQG control inputs are calculated by the discrete time model given in Eq.(7.7). As pointed out before, in the present experimental setup, stepper motors are used to move the tether attachment point. These motors are commanded to move to a specified position by providing required number of pulses in each time—step. Therefore, the acceleration requirement, i.e. the control input, is converted into a displacement requirement. The displacement can easily be converted into the num 171  ber of pulses required, from the knowledge of pulses per revolution and parameters of the transmission mechanism (chain drive in the present case). The conversion of acceleration requirement to displacement is accomplished by integrating the accel eration profile, which is considered constant over a time—step. This leads to: dcz(k + 1)  =  o!a(k) + Ja(k)zt;  (7.8a)  da(k + 1)  =  dcx(k) + dcx(k)L?&t + cicr(k)(t)2,  (7.8b)  where: da(k) and da(k) are the magnitudes of cia and da at the kt1’ sampling point, respectively; and t is the sampling period. In case of the LQG controller, the control input da(k) is directly computed from the knowledge of the system outputs a(k) and da(k). In the present study, a(k) is measured directly by the potentiometers and da(k) is obtained from Eq.(7.8b). For the FLT controller, in addition to these outputs, magnitudes of à(k) and da(k) are required. da(k) can be computed using Eq.(7.8a). Since no sensor is used to measure  a, backward finite difference method is used for its evaluation, i.e. a(k)  =  As the equations of motion for a and  a(k)  ‘  —  a(k —1)  (7.9)  degrees of freedom are identical in form, the  7-controller can be implemented using exactly the same procedure. The next important step in the controller implementation is the selection of the sampling time t. It has to be sufficiently large so that the computations re quired in the control ioop can be completed in the time—step. In the present setup, the time required to complete a control loop, i.e. to perform sensing, computation, actuation and data storage is less than 5 ms. The sampling rate is based on the re quirement that the sampling frequency should be higher than the Nyquist frequency. 172  For better performance of the system, the sampling time should be less than one sixth of the minimum period of the system [109], which is 1.45 .s in the present case. From these considerations, the sampling time (st) is taken to be 90 ms. The computer program to implement the controller was written in MicroSoft C language. The real time implementation starts with initialization of the system parameters and special purpose computer cards, and calibration of the potentiome ters for reference vertical position. Application of a disturbance activates the control loop. Each loop consists of measurement of the angles; computation of the offset position, velocity and acceleration; checking the steady state and safety conditions; commanding the motors to move; writing the data into the output file; and updating the variables for the next time—step. The controller operation can be stopped by pressing an arbitrary key at any time. This can be used for emergency exit of the controller. The fow chart for the controller operation is shown in Figure 7-7.  7.4  Results and Discussion The experimental validation of the offset strategy in conjunction with the  FLT and LQG controllers provided considerable insight into the feasibility of the approach. The concept of controlling the attitude motion of a tether, using off set acceleration as the input, augmenting the tether dynamical equations with the identity cia  u, and providing both angle and offset variable feedback proved to  be feasible. Small discrepancies observed between the experimental and numerical simulation results may be attributed to the following factors: • Damping effects can not be completely eliminated in a real system. Here the contributions come from friction in the sensing mechanism (Figure 7-1), linear bearings of the carriage and aerodynamic effects. 173  IN ITIALIZATION: Define constants and deployment/retrieval parameters Initialize port addresses and system variables Choose controller (FLT or LOG  Measure angles and compute angular velocities I Compute offset parameters at the end of At  Send pulses to inplane and out-of-plane motors Measure time  No  Figure 7-7  Flow chart showing the real time implementation of the attitude controller.  174  • The stepper motors are controlled in the open ioop fashion. Therefore, any missing pulse can not be compensated by the controller. • The step—size for the motors is 1.8°. The required angular rotation of the motor in each time—step may not be a multiple of 1.8°. This introduces error in the implementation of the controller. Considerable amount of information was obtained through a series of care fully planned experiments. Only a sample of results is presented here. The system was purposely subjected to a very large disturbance of 10° in both the directions (a and ‘y) to assess the controller’s effectiveness under demanding situations. In practice, an external excitation would seldom result in motions larger than a few degrees. Performance of the FLT and LQG controllers is discussed in the following two subsections. The acceptable steady state error limits are set at ± 1 cm for offset positions and ± 0.5° for the tether angles. The maximum allowable offset positions are limited to ± 20 cm.  7.4.1  FLT control  Figure 7-8(a) compares numerical and experimental results during the sta tionkeeping phase at a tether length of 0.5 m. The uncontrolled response is fairly well predicted by the numerical simulations, however, small attenuation is observed in the amplitude of the experimental response. As pointed out before, this may be attributed to friction in the ring mechanism used to measure the angles. Compar ison of results during the controlled phase shows acceptable correlation. Note, the steady state errors are within the limits. The initial disturbance is damped within a. very short time (< 4 s). The offset motions required to regulate a and 7 are also within the specified limits. Subsatellite and carriage positions during the controlled experiment are shown 175  Numerical 100, (O) = 10  Experimental  (0) = FLT Controller  Stationkeeping: L=0.5 m  I U ncont rol ledI  10.0 0.0 -10.0  Uncontrolledi  10.0 0.0  -10.0  10.0 cx°  IControlledl  0.0 -10.0 10.0  Control led)  0.0 -10.0  V  8.0  da o.o Th\/ -8.0  8.0 j  d 0.0 -8.0  0.0 Figure 7-8  Time, s  10.0  Plots showing the comparison between numerical and experimental response results during the stationkeeping phase: (a) uncontrolled and controlled system. 176  Numerical  Experimental  a(O)=1O°, y(O)=1O°  Stationkeeping: L=O.5 m  FLT Controller  10.0 5.0 yO  0.0 -5.0 -10.0 -5.0  0.0  5.0  10.0  xo  5.0 [cj  I  I  I  I  -5.0  0.0  0.0  7 d -5.0 -10.0  cm[ ,  .5.0  da Figure 7-8  Plots showing the comparison between numerical and experimental response results during the stationkeeping phase: (b) subsatellite and carriage positions. 177  in Fig.7-8(b). The projection of the subsatellite position in a horizontal plane has two orthogonal components, L sin a cos  and L sin y COS a. For small angles they  may be represented, approximately, as La and L’y. Therefore,  vs. a plot rep  resents the scaled position of the subsatellite projected on a horizontal plane. In the present simulations, disturbance is given to the angular positions only. Initially, librational velocities, as well as position and velocity of the carriage (tether attach ment point)are zero. Under this situation, the subsatellite should oscillate in one plane during both the uncontrolled and controlled conditions. The numerical simu lation (solid line in -a plot) substantiates this observation. The small discrepancy in the experimental results is again due to the sensor noise mentioned before. The plots for carriage position show similar trends. Figure 7-9(a) shows response of the system during stationkeeping at a longer tether length of 1 m. The numerical and experimental responses for the uncontrolled as well as controlled system show excellent agreement. As expected, the frequency of angular motion is lower compared to that for L  =  0.5 m. The offset motion required  to regulate the inpiane and out-of-plane librations is rather modest (within +6 cm and —4.5 cm). Experimental results for the subsatellite and carriage positions are quite close to the straight line predicted by the numerical simulation (Figure 7-9b). Similar trends continued even at a higher length of 2 m (Figure 7-10). As emphasized earlier, retrieval is the most critical phase in the tether op eration. To show effectiveness of the controller, two different retrieval rates are considered with maneuver completed in 15 s and 5 s. In each case, the tether length was reduced from 2 m to 0.5 m in accordance with a trapezoidal velocity profile. The velocity was linearly increased (constant acceleration) from zero to the max imum value in O.ltr, held constant (zero acceleration) at the maximum value for  178  Numerical =  100, y(O)  =  Experimental  100  Stationkeeping: L=1 m  FLT Controller 10.0  cx  0  0.0 -10.0  0  lUncontrolled!  10.0 0.0 -10.0  10.0  a°  Controlledi  0.0  -10.0 10.0  Controlled  0.0 -10.0 8.0  da 0.0  \\/‘•  -8.0 8.0  i1  d 0.0 ..==========..=.===.====  -8.0 0.0 Figure 7-9  Time,s  10.0  A comparative study between numerical and experimental results for the system during stationkeeping at 1 m: (a). uncontrolled and FLT controlled system. 179  Numerical cx(O) = 100, ‘y(O) FLT Controller  =  Experimental  100  Stationkeeping: L=1 m  10.0 5.0 0.0 -5.0 -10.0 -10.0  5.0  d  -5.0  0.0 a°  5.0  10.0  cm  0.0  -5.0 .1....  0.0  -5.0  5.0  d Figure 7-9  A comparative study between numerical and experimental results for the system during stationkeeping at 1 m: (b) subsatellite and carriage positions. 180  Numerical (0) = 100, y(0) FLT Controller  =  Experimental  10  Stationkeeping: L=2 m  I  10.0  a°  0.0 -10.0 10.0 0.0 -10.0 10.0  cx°  0.0  IControlledi \\\_  -10.0 10.0  Controlledi  0.0 -10.0  15.0  da  0.0 -15.0 15.0  1 d  0.0 -15.0 0.0  Figure 7-10  Time, s  10.0  A comparative study between numerical and experimental results for the system during stationkeeping at 2 m: (a) uncontrolled and FLT controlled system. 181  Experimental  Numerical a(O)=1O°, y(O)= 100  Stationkeeping: L=2 m  FLT Controller I  ,  •  I  I  I  10.0 5.0 0.0 -5.0 I  I  -5.0  0.0  .  I  10.0  5.0  x° cm]  I  I  10.0  1 d  0.0  -10.0  icr I  I  -10.0  0.0  10.0  d Figure 7-10  A comparative study between numerical and experimental results for the system during stationkeeping at 2 m: (b) subsatellite and  carriage positions. 182  tr, and finally linearly reduced to zero in 0.lt,.. Here tr is the specified time to 8 O. complete the retrieval. The maximum velocities are 0.11 rn/s and 0.33 rn/s for the retrieval times of 15 s and 5 s, respectively. Figure 7-11(a) shows experimental response results for the retrieval com pleted in 15 s. As expected, with the tether length becoming smaller, inertia of the system reduces causing the tether oscillations to grow to conserve the system angular momentum. During the uncontrolled operation, the system response (c and 7) grows to a maximum amplitude of ±  290  as the retrieval ends. Subsequently,  the frictional damping causes the amplitudes to slightly diminish as expected. The offset procedure effectively controls the motion within 8 s, i.e. long before the re trieval is completed. Note, excursion of the tether attachment point is maintained within the specified limit of ± 20 cm. Even at a faster retrieval rate (t  =  5 s), the  offset strategy continues to be effective (Figure 7-llb).  7.4.2  LQG control The offset controller, designed using the LQG algorithm (Section 7.3.2), was  also implemented on the same tethered system. The uncontrolled and controlled responses, for the stationkeeping case at L  =  lm, are compared in Figure 7-12(a).  As before, the results show good agreement with the numerically predicated per formance. The initial disturbance is damped within 5 s. Considering the frictional effects, the subsatellite and carriage positions also show good correlation with the numerically predicted results (Figure 7-12b). The experimental results for the retrieval phase are presented in Figure 7-13. As before, a trapezoidal velocity profile was employed with the retrieval from 2 rn to 0.5 rn completed in 10 s. Instability of the uncontrolled system is shown rather dramatically. Gain scheduling was used during the retrieval phase with the gains 183  =  110, ‘y(O)  =  110  Retrieval: 2.0 m to 0.5 m in 15 s  FLT Controller 30.0  degrees  0.0  rolIed’”11VV\/\. t n  -30.0  .  10.0  c°  [ptrolldJ  0.0 -10.0 10.0 0.0 -10.0  20.0  da  1  0.0 -20.0  d  20.0 0.0  -20.0 2.0  L 0.Qoo Figure 7-11  Time, s  20.0  Uncontrolled and controlled experimental results for retrieval of the subsatellite: (a) retrieval time of 15s.  184  x(0) = 110, y(0) FLT Controller  =  110  Retrieval: 2.0 rn to 0.5 m in 5 s  30.0 0.0  -30.0 10.0  cx°  0.0  -10.0 10.0  0.0 -10.0  20.0  d  0.0 -20.0 20.0  d ‘‘  0.0  -20.0 2.0  L 0.0  0.0  Figure 7-11  Time, s  20.0  Uncontrolled and controlled experimental results for retrieval of the subsatellite: (b) faster retrieval in 5s.  185  Numerical  Experimental  x(O)=1O°, y(O)=1O° LQG Controller  Stationkeeping: L=1 m  10.0 0.0 -10.0  0  -  UncontroIled  10.0  0.0 -10.0 10.0  a°  Control led  0.0 -10.0 10.0  IControlledi  0.0 -10.0 8.0  d cx  1  0.0 -8.0  8.0  d 0.0 -8.0  0.0 Figure 7-12  Time, s  10.0  A comparative study with the LQG controller during stationkeep ing at 1 m: (a) uncontrolled and controlled performance.  186  Numerical ct(0) = 1 00, ‘y(O) = LQG Controller  -—•-••-••--••-  Experimental  100  Stationkeeping: L=1 m  10.0  .—.  —.  5.0 .--;.‘  0.0  ,s._._4.__  I  .. .  -5.0  c—  ,.  •,•  -.-..-. / .—. .-,  .—-—  -10.0 -10.0  -—  -5.0  0.0  5.0  id.0  c° 10.0 cm  d  0.0  -10.0  I  I  -10.0  0.0  .  crn[ 10.0  d Figure 7-12  A comparative study with the LQG controller during stationkeep ing at 1 m: (b) subsatellite and the tether attachment point posi tions.  187  a(O) = 110, ‘y(O) LQG Controller 30.0  =  110  Retrieval: 2.0 m to 0.5 m in lOs  degrees  ---•-••-‘‘  JVVVVVV\ 0 c 11  0.0 -30.0  Controlled 10.0  a0  0.0 -10.0  10.0 0.0 -10.0 20.0  da  1  0.0 -20.0 20.0  d  i1  0.0 -20.0 2.0  L —----  0 0j  Time, s Figure 7-13  20.0  Experimentally observed system response during retrieval with the LQG controller.  188  adjusted at 0.75 m and 1.5 m. The LQG controller is quite successful in regulating such a large initial disturbance within 12 .s. The steady state errors are less than 5 % of the initial disturbance for the angles and less than ± 1 cm for the offset position. Finally, performance of the LQG controller in regulating the spherical pendu lum type motion was evaluated (Figure 7-14). This required application of a rather severe set of initial disturbances both to the angular velocity and position. The con troller continues to be remarkably effective as shown by the inward spiral motion. The corresponding carriage position also converges to the steady state value. The time to damp the disturbance was found to be around 15 .s. A video of the experimental set—up, presenting in some detail the sensor, actuator and the controller, was taken. It shows, rather dramatically, effectiveness of the offset control strategy in damping a variety of severe disturbances.  7.5  Concluding Remarks Experiments carried out employing a unique ground based test—facility sug  gests that the offset control strategy, using the FLT as well as LQG algorithms, can effectively damp rather severe disturbances during both stationkeeping and retrieval phases. The controllers continue to be effective even during a faster retrieval rate of 0.33 rn/s. Note, the above mentioned performance is attained within the specified limit of the offset motion. Correlation between the experimental and the numerically predicted results is also good considering the frictional effects encountered in the real-life situation. A video captures, quite effectively, the remarkable performance of the offset control procedure.  189  Spherical Pendulum Motion LQG Controller  Stationkeeping: L=1 .5 m  10.0 5.0 0.0 -5.0 -10.0  -5.0  0.0  5.0  10.0  cm  10.0  d o.o  cm[  -10.0 -10.0  0.0  10.0  d Figure 7-14  Subsatellite and carriage positions during control of the spherical pendulum using the LQG regulator.  190  8.  8.1  CLOSING COMMENTS  Concluding Remarks  Using a relatively general model, the thesis develops a methodology and as sociated computational tools for studying planar dynamics and control of tethered satellite systems. Versatility of the model is illustrated through its application dur ing all the three phases of a typical mission involving deployment, stationkeeping, and retrieval. The focus is on the system control using two distinctly different types of actuators: thrusters located at the subsatellite; and movement of tether attach ment at the platform. Both linear as well as nonlinear controllers, using the actuators and their hybrid combinations, are developed applying the Linear Quadratic Gaus sian (LQG) regulator and the Feedback Linearization Technique (FLT). It may be recalled that the FLT accounts for the complete nonlinear dynamics of the system. As can be expected, the governing equations of motion are highly nonlinear, nonautonomous and coupled. They were used to assess uncontrolled dynamical per formance of the system as affected by the important system parameters. However, widely spaced frequency for the rigid and flexible degrees of freedom was often taken advantage of through decoupling during the controller design. Thus controllers for the rigid and flexible parts of the system were designed separately using the coupled sets of linearized equations. Of course, effectiveness of the designed controllers was assessed through their application to the original nonlinear and coupled system. The thesis presents innovations in several areas. More important contributions of the thesis, which have not been reported in the literature, include the following: (i) the system model that accounts for the tether flexibility and motion of the tether attachment point at the platform end; 191  (ii) control of the system’s attitude dynamics using the offset scheme in presence of tether flexibility; (iii) application of the Feedback Linearization Technique, which accounts for the complete nonlinear dynamics, to tethered systems; (iv) vibration suppression along both the longitudinal and transverse directions using the offset strategy; (v) simultaneous attitude and vibration control of the tethered systems; (vi) ground based experiment to substantiate effectiveness of the FLT and LQG based offset control synthesis. It should be emphasized that the objective here was to establish a methodology to understand dynamics and control of such a complex system. It was not intended to acquire large amount of information useful in the system design. Of course, such information can be generated quite readily as the dynamics and control programs are operational. Even then the amount of information obtained is rather extensive. The thesis presents only some typical results useful in establishing trends. Based on the analysis following general conclusions can be made: (a) Offset of the tether attachment point leads to coupling between the platform and tether dynamics. The tether pitch libration significantly affects the plat form dynamics, however, the effect of platform motion on the tether dynamics is relatively unsignificant. (b) As can be expected, static deflection of the tether as well as frequency of both transverse and longitudinal oscillations are affected by the mass density and elastic properties of the tether material. Higher modes of the longitudinal vibrations have rapidly decaying characteristics due to the structural damping, which has only the second order effect on the transverse tether vibrations. 192  (c) The tether dynamics is susceptible to instability when the deployment/retrieval rate exceeds the critical value. (d) The FLT based controller using the rigid nonlinear model of the system is quite successful in regulating attitude dynamics of the system with a flexible tether. The controller structure is relatively easy to implement. A single con trol algorithm is applicable to all the three operational phases of deployment, stationkeeping and retrieval. (e) The FLT based controller is found to be better than linear, time invariant reg ulators designed using the graph theoretic approach and implemented through gain scheduling. (f) A hybrid strategy, relying on the thruster control at longer tether lengths and the offset control for shorter tethers, appears quite promising for regulating the tether pitch libration. Results show that a pair of passive dampers can be used to control the unstable elastic degrees of freedom. (g) Besides controlling the tether pitch motion, the offset strategy can be used to regulate, simultaneously, both longitudinal and transverse oscillations of the tether duing the stationkeeping. Results for a 20 km tether showed the controller to be remarkably effective in damping the motions with the tether offset maintained much below the specified ± 20 m limit. (h) Substantiation as to the effectiveness of the FLT and LQG based offset control strategies using a rather unique test facility represents a significant contribu tion to the field. Results show that the concept of controlling the motion of a tethered payload through specification of the acceleration at the point of attachment is not only effective but can also be implemented in practice.  193  8.2  Recommendations for Future Work  The present thesis represents a modest contribution to the challenging field of tether system dynamics and control. There are several avenues open for future exploration which are likely to improve our understanding of the field. A few of them, more directly related to the present study, are indicated below: (i) Extension of the present investigation to three dimensions, i.e. generalization of the model to account for inplane as well as out-of-plane dynamics represents the logical next step. (ii) Offset control of the tether vibrations during a retrieval maneuver needs to be explored. If successful, it will make the offset strategy more attractive and versatile. (iii) In presence of an offset along the local horizontal, the tether dynamics sig nificantly affects the platform motion. This presents an exciting possibility of controlling the platform libration through an offset strategy. Simultaneous control of the platform and tether dynamics through offset would represent an important contribution to the field. (iv) It would be useful to assess effect of the free molecular environment forces on the tether dynamics and control. (v) Multibody tethers have been proposed for several scientific experiments, in cluding monitoring of Earth’s environment as in the case of Mission to Planet Earth. The present model can be extended, through a recursive formulation, to account for such configurations. 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[108] Model 5030, Software Guide, Manual No. 40041—009—bA, Industrial Computer Source, San Diego, California, U.S.A., 1992 [109] deSilva, C. W., Control Sensors and Actuators. Prentice Hall, Englewood Cliffs, New Jersey, U.S.A., 1989, . 30 5 pp.  [110] Lakshmanan, P. K., Dynamics and Control of an Orbiting Space Platform Based Tethered Satellite System, Ph.D. Thesis, Department of Mechanical Engineer ing, University of British Columbia, Vancouver, Canada, 1989, pp. 216-220.  203  APPENDIX I:  MATRICES USED IN THE FORMULA TION  For concise derivation and efficient computer implementation of the governing equations of motion, a number of matrices are defined to express the system energies. The matrices, dependent on modal integrals and attitude angles, are also reported here. Matrices used in the Kinetic Energy (Eq. 2.25) } 1 {K  =  }+} 0 LL{Ak 0 w 2 L t{Bk + msL{Dk }+} 1 2 L 8 m wt{Dk  ] 2 [K  =  ] 1 ] + ms[Ap 2 [Ak  ] 3 [K  =  tl; 3 ] + wt[Bk 1 L[Ak ] + msct[Uk][Ap 1 ] E R 1  ] 4 [K  =  -ms[A ] 1 ] T[Ap p + -[Ck ] 2  ] 5 [K  =  mswt[A [ T ] 1 j Uk][Ap + L[Ck ] + wt[Hk 3 ] 4  ] 6 [K  =  mswz[ApljT [Uk]T[Uk] [A ] + j,2[ck] + Lwt[Hk 1 ] 3  } 7 {K  +  ; 3 E R  )<Ntl; 3 E R  E RNtltl;  +] msL{D } 1 [Ap k  msLwt } 2 , [Ap {Dj +1  204  E RNtZ;  E RNtltl;  E RNtZtl;  } 8 {K  =  L2{ck}T + Lwt{Hk 1 [Uk][Apij T } 1 T + msLwt{Dk } 1  + mgLw{Dk [Ukj[Apl1 T } 2  E RI.  Matrices used in the Potential Energy (Eq. 2.28) ] 1 [F  =  M[U]  ] 2 [F  =  ma[U]  —  —  3M{V]  ; < 3 E R  3ma[ Vpo]  E  2p  31 —  —  }T 0 6pt cos(at){f{F.p(yt)}dyt} {I/  2p{f{F(yt)}dyt}{Utp}T + 6 pt sin(at){f{F(yt)}dyt}{17po}T [Api]T[Ttp] 8 + 2m  {  {F(L)}  }  } 4 {F  =  L 8 2m  ] 5 [F  =  ms[Api]T [Ap ] 1  } 6 {P  =  3 + pL)L{Wtp} (2m  =  (at) 2 cos  —  —  —  6ms[Api]T{17to}{Wpo}T  {Wto} T ] 1 GmsLcos(crt)[Ap  3ms[Api]T{Wto}{Wto}T [A j 1  —  E RNtZ;  E RNtlTtl;  3 + pL)L cos(crt){T’Vpa} 3(2m  E R.  205  ; 3 E RNt1X  E  Inertia Dyadics and Their Derivatives  The terms associated. with the inertia matrices and their derivatives, used in derivation of the governing equations of motion, are presented here.  =  ‘Pz Pzy 1 zz 4  t 1  0  0  0 0  jt  2 t 1 y  ‘ty  ‘tz  [4]  [ t 1 ]  =  -  ‘Pzy ‘Py yz 4  ‘Pzz ‘Pyz ‘Pz Ib+Ic  0  0  0 0  Ic  ‘bc ‘b  =  ‘bc  where: + {B}T[Iblj{B} + 2{1b }{B}; 2  =  {C}T[Ici]{C};  Ic  {B}T[1oc ] 2 {C};  1 b c  =  —{Ib } 1 {C}  ] b 1 [  =  Pt f{Fb}{Fb}TdYt; ptJyt{Fp}Tdyt;  } 2 {Ib [Icr]  { & 1 ci} l 2 [‘bc  —  =  Pt  {Fq}{F,}Tdyt;  jL  =  =  j  vt{F}Tdyt; pt  jL{i}{i}Tdyt.  The time derivative of [Ip] is zero. The derivatives of the elements of  [‘]  used in  the equations of motion, are as follows: =  + 2{B}T[Ibl){B} + {B}T[.ibl]{B} J{C} 1 +} {{B} 2 }{B} + 2{C}T[I 2 1b + 2{lb  +{C}T[ici]{C}; 206  f{ItzB}1 2  o{X}(tT[1t])  t  t}Ttjt})  ô{X}  ã{X}  I  ({w}T[I]{w})  =  sin(2czt){ItyZB } + sin (ct){ItZB } 2 (t){Itz} 2 cos  {  —  %  Sifl(2t){Ityz}  }  where:  {‘tZB}  [ 2 = ] 1 {B}+2{Ib IbWiT.  { { } {  =  J{C}; 1 2[1c  =  ]{C}; 2 —[IbC  =  {J&  ] b 1 [  =  } 1 Ub  =  1tzc}  ItYZ  [i1  =  }T_  lT{B}; 2 [Ibc  F fL{{DF}{F}T -f {F}{DF p}T}dy + {F}{F}T]; 1 Lo pt{  JL{  +  +  F fL{{DF}{F}T + {F}{DF}T}dy + {F}{F}T]. [JO  Modal Integral and other Constant Matrices  {Ako}={Pt}ER3;  207  o  0  pJoL{DF}dy  ]= 1 [Ak  o  ] 2 [Ak  =  ] 1 [A  tl; 3 ER  0 ptJ{DF#}dyt  o  0  Pt f{F(yt)}Tdyt  0  o  p f{F(yt)}Tdyt  o  0  {F(L)}T  0  o  ,(L)} 1 {F T  tl; 3 eR  E  (01 0 (pt/2J  }z 0 {Bk  ]= 1 [Bk  I  1 k i 1  o  0  _ptf{F(yt)}Tdyt  Pt fJ{F(ut)}Tdyt  0  >tl; 3 ER  Ptf{DF}{DF}Tdut  o  0  Pt fj’{DF,}{DF}Tdyt  —  —  ] 2 [Ck  o  RNtZXNt1  p(yt)}{F(yt)}Tdyt 1 pt ,f{F  0  0  Ptff{F(vt)}{F(vt)}Tdyt  =  208  E RNt1)<Ntl;  0  ] 3 [Ck  =  0  } 4 {Ck =  { { {}  J’ó’{DF}dyt  Pt f{Fi,(yt)} 2  } 5 {Ck =  } 1 {Dk  pt f{F(yt)}{DF}Tdyt 2  }  }  ]  ER’tltl;  E RNtZ;  € RNt1;  ; 3 € R  0  }= 2 {Dk  } 1 {Hk =  } 2 {Hk =  O €R ; 3 1 {oi  { {  foL{yt{DF#}  f yt{F}dyt 0  —  }  {F}}dyt  }  E RNtZ;  €  pt f{F(yj)}{DF,}Tdyt  ] 3 [Hk  0  =  209  ]  € RNtltl;  0 1 4 [Hk  —Pt  0  pt  [T]  0  1 0 0  =  0  cos(at — Qp) sin(czt — ap)  —  sin(at  cos(at  —  —  crp) czp)  100 0 1 0 001  [Uj=  [Uk]  000 0 0 —1 010  =  e  1  0 {U} —sin(at—ap) cos(at ap) —  0 0 0  ] 0 [V  {W}  0  0  (ap) 2 cos  —sin(ap)cos(ap) (ap) 2 sin  0 cos(at  I  h  J  sin(ap) cos(ap)  —  1  sin(at  — cxp) ; —  cxp)  J  (0 } 0 {1i7  {1’t,}  e  =  cos(ap)  =  ;  (—sin(ap)  J  10  ‘I  cos(at)  =  I  —sin(at)  {F(yt)} ={.(y,L)}  ;  J  Nt  e  RNt;  210  RNtltl;  1 ,(y)} ={‘I’ct,L)}’ 1 {F,,  E Rl;  OF ôF {DF}+  ERNt;  8F ôF,b {DFti}=-ã---+---  eRl.  Here:  1 N  =  number of modes used to represent the longitudinal tether vibration;  Nt  =  number of modes for transverse oscillation of the tether;  Nti= Nt+Nj.  Time Derivative of Matrices  } 0 {Ak  ={O};  ] 1 [Ak  o  0  f{D2F}Tdyt + {DF(L)}T  0  o  ] 2 [Ak {E}  JL{D2F}Tdy  -  ]; 1 [Ak  ={0};  211  {DF(L)}T  0  0  0  —[Ak ( 2 3,Nl + 1: N)]  )] 1 [Ak ( 2 2,1 N  0  -  [E]  =  -  1 f{{D2 , }{DF} F,, T+1 {D p }{D2F}T}dy Ft 1 =ptL [Ck 0 0 ,f{{D2F}{DF}T + {DF}{D2F}T}dyt T ,+{D 4 (L)}{DF(L)} F  {F}{DF,, }T}d y + 11 ] 2 [Oh  =L 0 0 + {F}{DF#}T}dyt  + {F,}{D2F}T}dy  ] =2pL 3 [Oh 0  212  0 J’{{DF}{DF}T +  I f{D2F}dyt + {DF(L)}  } =2pL 4 {Ck  1  )  0  }; 4 } .L{Ck 5 {Ok  1  0  }=ptL 1 {1rk  I f yt{D2F}dyt + L{DF,(L)} 1 {iIk}=ptL 2  —  {F(L)}  J  0 + L{F(L)}  f{{F#} +  J  0 ] =ptL 3 [Ilk  —  J’{{DFçj,}{DF}T + 1 ,}{D2F {F } T}dyt T —{F,(L)}{DF,,h(L)}  f{{DF,ç}{DFq}T + {1\t}{D2Fc }T}dyt 5  0 213  0 ] ptL 4 [Hk  f{{vF}{F}T + {F}{DF}T}dy  }T}dy 4 f{{F}{DF#}T + {DF}{F T +{F(L)}{F(L)} 0  o [1}(aà) 0  0  0  —sin(at—cp)  cos(atap)  o —cos(at—ap)  sin(cxtap)  where: ,}= 4 {D2F  8{DF.} ôyt  +  ‘‘  ,} 1 ô{DF  } 1 {D2F =  yt 9 c  Ô{DF4  ô{DF} +  ÔL  e RN1.  Derivatives of Matrices used in the Potential Energy  [Pi]cxp  =  ]ap 2 [P  =  —  P [3lap -  3M[Vpr,jap; —  3ma[Vpo]ap; 2pt{J’{Fp(yt)}dyt}{Wtp}’p  —  }’ 0 Pt cos(ct){f{F,,,(yt)}dyt }{iV 6  2{fL{F}d}{U}T + 6pt  214  -I- 2ms[Api]T[Ttp]ap  {P4}ap  =  —  6ms[ApijT{Wto}{Wpo}’;  {O};  ]cxp= [0]; 5 [P } 6 {P  [P2]at  =  8 + ptL)L{Wtp}ap (2m  =  [0];  =  [O];  —  8 + ptL)Lcos(ct){1’Vpo}a; 3(2m  2pt{f{F,t,(vt)}dyt}{Wtp}’ + 6pt sin(at){f{F(yt)}dyt}{17po}T  —  lt  {  —  Pt cos(at){f{F(yt)}dyt}{Wpa}T f{F(yt)}dyt}{utp} + 6 8 [A 1 ]T [Ttp]at + 2m  {P4}at  =  —  jat 5 [P  =  -  {P6}  =  —  8 [A 6m 1 ]T{TVp7to}at {l:vpa}T;  {Ti7jo}a + 6msL sin(czt)[Ap T ] 1 L cos(a)[Ap 8 6m {Wto}; T ] 1  [Ap 3 3m T ] 1  [  ]; 1 [Ap  T+ o}at{Wo}  8 + ptL)Lsin(at){T’’po}, 8 + ptL)L{ Wtp}at + 3(2m (2m  where:  =p’  ()ai?’  o [Ttp]crp  [Ttp1c  =  0 0  0 sin(at cos(at  —  —  czp) ap)  =  215  0 cos(at cip) sin(at ap)  —  —  —  o [Vpo ]czp  {Wtp}ap  0  =  1  { Utp}ap {Utp}at  —  —  —  —  {Wtp}ap;  =  1  =  0 cos(at I. sin(  cp) cxp)  —  —  {Utp}ap;  =  1° {T’Vpo}crpz  ‘I  —sin(ap) ;  —cos(ap)  J  (o =  0 cos(2ap) sin(2c)  —  0 sin(a ap) ; cos(c cp) J  =  I. {Wtp}at  —  o  0 sin(2cp) cos(2a)  sin(a) —cos(at) —  216  APPENDIX II:  ELEMENTS OF MATRICES ‘M’ AND ‘F’  The matrices M(q, t) and F(q, v, t) appearing in Eq. (2.37) can be partitioned as follows  M(q,t)  =  Mr(11)+MSMr(1,1)  Mr(l,2)+MSMr(2,1)  Mr(2,1)+MSMr(2,1)  Mr(2,2)+MSMr(2,2)  Mfr(:, 1) + MSMf  (:, 1)  Mfr(:, 2) + MSMf  (:, 2)  .Mrf(1:)+MSMf(1:)  MTf(2,:) + MSMf(2:) Mf+MSMf Fr(1) + FSMr(1) F(q,,t)  tl); 2 E R(  =  Ff + FSMf  where Ntz= Nt+Nz;  M(1, 1)  = ‘P2  Mr(1,2)  =  + ma{dp}T[UkIT[Uk]{dp};  } 2 {dp}T[Uk]T[Ttp]T{L2{BkO}+msL{Dk + [[Bk ] +ms[Uk][Api]j{X}}; 1  {Mrf(1,  :)}  Mr(2,1)  =  {dp}T[Uk]T[Ttp] [K ]; 2 Mr(1,2); 217  E R(2t1)x(2ti);  Mr(2, 2)  =  IT[Uk]T[Uk][Apl]{X} 1 2 + ms{X}T[Ap L 8 It + m +1 [Uk][Ap T } 2 2msL{Dk ] {X};  {Mrf(2,  :)}  =  {X}T {[Hk IT + ms[Apl]T[Uk]T[Apl]] 4 }T + 1 2 + {Hk [Ap T } 2 msL{Dk ] ;  {Mfr(:, 1)}  =  {Mrf(1, :)}T;  {Mfr(:,2)}  =  {Mrf(2,:)}T;  Mf  =  ]; 4 2[K  Fr(1)  =  (Mr(1 1) + Mr(12))8 + mQ{dp}T[U]T{Jp} ]{X}}L 1 + {dp}T[uk]T[Ttp]T{L{AkO} + ms{Dk } + [Ak 1 H- ma{4} ’[Uk]{4} + 2mawpz{dp}T[Uj$,]T[Uk]{dp} 9 + {dp}T[Uk]T[Ttp]T{L2{AkQ} + 2LLwt{Bk } 2 } + msLwt{Dk 0 ]] {J} + [L[Akl] + wt[Ekl]] {x}} 3 + [[Ak ] + [K 2 + {dpT[uk]T[tp]T + {a}T[Uk]T[TtIT}  { -  +  {{}T  {K} + [K ]{5} + [K 2 ]{X}} 3  + wp{dp}T[uk]T}[rtp]P{{Kl} + [K ]{i} + 2  ]{x}} 3 [K  lap{dp} + 3 2 ({dp}T[P [ T {X} ] ap{dp} P + {P }’{d} + 6{lp}’p[Ip]{lp}); 6  Fr(2)  =  (M(1 1) + Mr(1, 2))  +  }}[tp]{4} 2 ltwt + 2msLLwt + {L2{BkO} + msL{Dk  + {L2{BkO} + msL{Dk } + [[Bk 2 ] + ms[Uk][Api]] 1  218  {x}}  [Tt]{J}  + ({x}T[Hk ]{x} + {{Hkl}T + ms{Dkl}T[Uk][Apl]}{X}) L 3 + {2LL.Bk 0 +  }} 2 msL{Die  [T]{4}  }}wpz 2 + {dp}T[Uk]T[Ttp]T{2LL{BkQ} + m$L{Dk + {.1apT[uk]T[Ttp]T + {dp}T[Uk]T[ipJT}  {  } + msL{Dk 0 L{Bj }}wpz 2  +T [i] [[Bk {4} ] 1  +  ms[Uk][Api]j {X}  + {4}T[Tt]T [[Bk] + ms[Uk][Api]] +  [  {ã  {;t} + {ã}T[Tt]T[Ek]{x}  }T[Uk]T[Tt]T + {dp}T[Uk]T[ip1T]  ] 1 [[Bk  ms[Uk][Api]j {X}p  + {dp}T[Uk]T[Ttp]T[Bkl]{X}wpz + {dp}T[Uk]T[Ttp]T [[Bk ] + ms[Uk][Api]] {}wp 1 +  ]T + ms[Apl]T[Uk]T[Apl]] {} + 4 4 [{X} T {5} ] iIk T [[Hk {i(}  L[Hk I Tj {X} ] + L[Hk 1 j+3 3 + {k}T [4mswt[Apl]T [Uk]T[Uk] [A +  2 {ik  + msL{Dk }T[Apl]}{iC} + L{X}T [ilk 2 ] {X} 3  ] 1 T [Uk] [A } 1 + {L{Hkl}T + msL{Dk }T [Uk] [A 2 ] 1 + 2msLwt {Dk }T[Uk][Api]}{X} 2 + {L{ilkl}T + 2msLwtz{Dk -  +  {{}T + wpZ{dp}T[UkjT}[Ttp]{{Kl} + [K ]{X}} 3 ]{} + [K 2  ]czt{X} 5 }cxt + {X}T[P 4 ]at{dp} + {X}T{P 3 ({X}T[P  219  + {P6}cxt{dp} {Ff}  =  —  (Vto)ct -I- 6{lt}t[It]{lt}); 2 3mgL  (Mfr(:1)+Mfr(:2)) + 2 [K ] T[Tt]{i} + {{Ck {Dk + T ] ms[Ap } }+1 5 +  ]T[] + [Ak 2 [[K ]T[Ttp}] 2  {{a} +wP[Uk]{dP}}  + [K [Ttp][Uk]{p}wp + [[Ok T ] 2 ] + [Ks]] {} 2 +  ]+ 3 [L[ak  [ilk ] 4 ] {X} + msLwtz[Apl}T{Dk }} 2  + {i{Ok }+ 5 -  -  ]T[TJ{{4} +wP[Uk]{dP}} 3 [K ]T]{X} 6 ] + [K 6 [[K  -  T } 8 {K -  -  a{x}  c9Us OMe 1 j[P ] 3 {dp} + 4 {P + } + ã{X} + 2R —  +  â{X}  () f  (tr[Itl) 8{}  1 fo){  °  --  }dt  8{ó}  MSMr(1,1) =_MfôM} {CM}; 1.. &p T  M} {CMczt}; 3 _M1t  {  M(1,  :)}  =  —  MsM(2, 1)  =  —  , 2) 2 MSMr(  =  —  MI  ÔRSM  MI  ÔRSM  MI  RSM 9 f  1 p 1 1.  at  at  } } }  1  j; }dYt  T  MSMr(1,2)  a{X}  T  [CMx]; T  {CMcxp}; T  {CMcxt};  220  {} T ] 5 [K ({W}T[I]{W})  JT]{X} 5 [[] + [P  ({lt}T[It1{l})  }  {  :)}  =  —  Mf  ÔRSM  l  &t  }T[cMx];  {MSMf(:1)}  =  {MsMf(1,:)}T;  {MSMf(:2)}  =  {MsMf(2,:)}T;  [MsMff]  =  1 _M[’ RSM 9 [CMxJ;  =  (MSMr(11)+MSMr(12))  FSMr(l)  +  {  d (8RSM ]{X}) 1 f-M{1sM}- ptL (L{Dki} + [A p Jj dt 1. I ÔRSM  —pt.  I  —  -  p  I  ,  + L[A ]{X} + 1  (LL +  MI ÔRSM T{cM} I  J  p  J aR } T {_M{SM} I  GMe +2R3  (  [p 2 ÔRSM T  api 9 I c  -  ]{X}) 1 PtL (L{Dki} + [A  }  ]{d} 2 } + 2[P  ]{R  I  }}) 6 + [P {X} + {P T ] 3 +  GMeI {RSM}T{[Pl]ap{RSM} + 2[P ]cxp{dp} 2 2R }op}); 6 + [P ]{X} + {P 3  )= (MSMr(21)+MSMr(22)) 2 FSMr( T I d (ORSM at  jJ  I ÔRSM  —pt I —  -  at  MI ÔRSM I  at  -M{1SM} I  -  i{X})} 1 PtL(L{Dki} + [A + L[A ]{X} + 1  (LL + ) T  J  {M{ T II aRSM at }  ]{X}) 1 } pL (L{Dki} + [A -  221  }  }  GM /IORSM }T{ + 2R oat + [P {X} + {P T ] 3 }}) 6 GMeI + 2R {FSMf}  =  + {P }at}); 6  {MSMf(:1)+MSMf(:2)}  )]  rd fOR\  -  —  pt  {_M{1SM}  roRSM1  [  .  PtL(L{Dki} +  T MFORSM] {CMF} + ptL[Apl]T{i?sM}  [  -  PtL(L{Dki} +  ]{X}) 1 {A  GMe I roRSM {2[Pl]{RSM}+2[P T ] ] 2 {dP} + 2R  U  + [P {X} + {P T ] 3 }}) 6 GMeI  2R  1{RsM}); 3 [P  where:  8 I 8u  •‘  1 a{X} ) I ou 8 t 8{B}  =  —  /8{C} )‘ 8 8U {ous’a{Bfl  r EAI  L(8F, 1OFIT  I 8 Yt Lb I —-j L/r8FT  + (EA/2)  f  O  (oU } 3 O{C}  ]{X})} 1 [A  + L[A ]{X} + 1  +  j  T {_M{SM} raRSM1 -  -  (j-__) \  L(8F,T  EAI !—.--$ .‘  I  ôt j  (OF 2 \  {C}) i_a—)dut;  (OFT {B}j-_? {C}}dt  8FT +(EA/2)  1  dutj{B}  -—  (OF 3 \ {C}) j-_-dt;  222  }  T 1 t8F  Id  jaj  j  ({)}T)(11  i  ayt J  T  dãF  {O}+{_{_}}  T  {C});  a •‘ 1aF to{E}J ia1’ (  (  o  •‘  jo{E} {RSM}  /(ÔFT  z(i.aj =  —  \1âF’ {C})t_j;  {ma{dp} + (msL + 1 /2)[Ttp] 2 ptL { T } Dk + [Tt]T [ms[Api] +  {1SM}  =  -  {ma[ip]{dp} + ma[Ttp]{4} + (ms + ptL)L{Dk } 1 +  +  {CMap}  =  {CMcxt}  =  [CMx]  =  {CMF}  =  [Ak ] 2 ] {X}};  {wt}  ]{X} + [ms[Api] + [Ak 2 [Ak ]] 2  x {[Ttp]{RSM}};  ma  + [Uk][Ttp]{RSM};  —  —  —  [ms[Api] + [Ak ]]; 2  -{ma(àt  —  2 [T a) ]{d} + 2ma[i’tp]{4,} 2 223  + ma[Ttp]{äp} + ((ms + pL)L + pL2){D} + [L[Aki] +  ]{}} 2 ] {X} + 2[Ak L[Ak ] 1  + [Uk] [T] {RSM } + wt [Uk] [i] {RSM } + w [Uk][Ttp]{isM};  IÔRSM’I a  1  =  d 8 SM1 I R a.  )=  ma {dp}; [T j 1  ma ma }; +[T {dp j{ãp} 1 [T ]  ma  t  ma [TtpJap{dp}  àtàp){Ti]ap{dp}  ; ][Ttp]{R }czp 5 wtz[Uk + wtz[Uic][Tp]ap{RSM} + M  SM1 [t R 3 aa  d 1ÔRSM1 ôt  —  =  =  ma ma —[Tl]{dp}—[Tl]{dp}+[Uk]{sM};  ma  1 [ÔRSM  i ac  )  ma j[Ti}{dp} + [Uk][Ttp]{RSM};  —  p)[Ti]at{dp}  ma y[Ttp]at{dp}  =  + wt [Uk] [Ttp]at {RSM } +  1  raRSM1  [  —  =  Wtz [Uk]  [ms[Api] + [Ak 1j; 2  —  d [aisM1 dtl 9j jM  224  [Tt] {RSM }at;  1 IÔRSM  L  ax j  Wta  IÔRSM1  1.  aa j  RSM’l 8 (  —  =  =  —  raRsM1  L  ax j  —  [Uk] [Tt]  ] 2 [T  1 —  1 yj{(msL + 1 /2)[Ttp]{Dk + [T]’ {ms[Api] + [Ilk 2 PtL } ]] 2  {x}};  1 j{(msL + } 1 / 2 ptL 2)[Ttp]{Dk + [Ttp]at [ms[Apil + [Ilk ]] 2  {x}};  1 [Tt]T [ms[Api] + [Ak ]]; 2  =  ro  ] 1 [T  ÔRSM] [ ax j  =  =  [Ti]ap  =  [o  —cos(ot—ap)  [0 10 [0  0  0  (at  —  1  0 cos(at—cxp) —sin(at—ap)j  lo  =  ] 1 [t  0 —sin(at—ap)  COS(tap)  sin(t—cp)  —  I  —sin(at—ap)  1  I;  _cos(at_ap)]  a) [T }; 2  ]; 2 [T  ]. 2 [Ti]at _[T  The matrices used in the above equations were defined in Appendix I.  225  APPENDIX III: NONLINEAR AND LINEARIZED EQUA TIONS OF MOTION FOR THE RIGID SUBSYSTEM  Nonlinear Equation The nonlinear equations of motion for the rigid degrees of freedom can be expressed as [Mu LM21  f&,jfF j 1 _f M j 1&tJ 1F JLTatJ’ 2  M 1 2 22 M  III1  where: M 1 1 =mamc(d + d ) + ‘; 2  1 cos(c) + M 1 2 =mlmc (d  sin());  ; 12 M 2 1 =M /3 + m 3 2 L 8 M 2 2 =pL  —  =(M + M )S 1 2 + mamcdp’ydpz + 2mamcdpzdpzwpz 1 11 F + +  { {  (mrn + pL)L +  2 pL  —  mlwmc} (d sin(a)  —  }  (msmc + ptL + 2 m m c)Lwt (d cos(a) + d sin(c))  wj sin(cz))ma4z/M 1 + (_msL cos(a) + m + 3uma(D  —  2 cos(2ap) D) sin(2p) + 6/.LmaDtDt  1 (D sin(a) + 2m  —  + 6iimi cos(t) (Dt +1 3 p z  —  2 cos(c)) Dt  sin(ap)  Ipy) sin(2ap)  —  + Dt cos(czp)) cos(2cp) 226  cos(a))  +3itmp(R% —R%)sin(2ap) ; 2 + 6IImPRSMYRSMZ cos(2ap) + ii + up 2 21 F =(M + M )8 2 2 + mlmc cos()Jz LL 3 + (pL2L + 2m  —  + m1mc (d sin(cz)  L/M)wt 2 2mim —  2 cos(a)) d  + mi(1 + mc)4zwp sin() —  maptLL4z cos(a)/M  —  —  2mi (D sin(a)  + 6mi sin(ot) (Dt cos(ap)  —  2 u 4 (msL + pL /3)sin(2czt) 3 +3 P1  =  () {  —2mDt, sin(a) (i  maptLLwp  —  cos(a) + —  sin(cx))/M  2 cos(cz)) Dt  .Dt sin(ap)) up  —  —  3 cos2(ap)) + 2maDt 2 cos(a) (i  —  3 sin2(cp))  1 sin(2czt)}; + 3ma sin(2ap) (D cos(a) Dt sin(cx)) + 3m 1 mpm 2 {_2 (RSMY sin(a) RSMZ cos(a)) + 6 RSM sin(a) cos (ap) M —  P2  =  (  )  —  —  (ap) + 3 sin(2ap) (RsM cos(a) 2 6RSMZ cos(o) sin  RSM  =  (—madpy  RSMZ  =  (_madpz  —  —  1 cos(a)) /M; m ml sin(a)) /M;  1%, tRSM + d; Dt  RsM + dpz;  ; 3 m+m ma =ptL + 0 1 =m m L + ptL 3 /2; 2 2 =m m 3 + ptL; mc =1  —  maiM;  +ptL+m; 0 M=m+m  227  —  RSMZ sin(a))  };  GMe 2R  Linear Equation The nonlinear equations of motion for the attitude dynamics of the tethered system are linearized about some arbitrary trajectory for the platform and tether pitch, and offset motion along the local horizontal. The offset along the local vertical is kept The governing equations of motion can be represented as  fixed at  [ML 11 21 [ML  ] 12 ML j 22 ML  Jl & J  i KL 1 1 12 + FKL j 22 KL LKL21  [CL 11 GL  1 f’i 12 GL 22 j 1 & CL  1 j I cz 1 + f MD 2 f f MD ‘  2  (111.2)  where: 11 ML  =  ma(D + D) + ‘; 2 sin(a)); 1 (D cos(ã) + D m  12 ML  ; 12 21 =ML ML 2 + ptL L 5 22 =m ML /3; 3  11 CL  =  12 GL  =  21 CL  =  8 1 2m  22 CL  =  L + ptL 2m L 3 L; 2  11 KL  =  1 (D sin(ä) m  2maDpzDpz; —  Ô (D sin(ä) 1 2m sin(ã)  —  —  —  2 L (D cos(ã) + D 2 2 cos(a)) + 2m D  2 cos(ã)) + 2miDpz sin(a); D  2 cos(ä)) D  —  228  L 2 m  cos(ä) + D 2 sin(ä))  —  2 sin()) + miÔ 2 (D cos(ä) + D ptL 2 (D cos(ã) +  L8 (D sin(ä) 2 + 2m —  cos(a)) + 6ma u(D 1  —  12matDpyDpz sin(2ãp)  —  —  D) cos(2äp)  2mip (D cos(a) + sin(aP)) 2 D  + 6miILcos(ät)(Dpcos(ãp)  —  sin(ã))  sin(ã))  6j(Ip  —  Ip)cos(2ap)  + 12,Ulpyz sin(2ap); 12 KL  —  1 m  2 + ptL —  —  21 KL  =  sin(ã)  cos(ä) + D 2 sin(ã)) sin(ä)  L9 2 2m  1 sin(ät) (D 6m  1 (D m 11 sin(ä) —  2 Ô 1 m  2 sin(a)) cos(a)) + m L (D cos(ã) + D 2  —  —  2 (Dm Ô 1 m 1 cos(ä) +  —  2 cos(a)) + 2mi (D cos(a) -I- Dpz sin(ä)) D  sin(ãp)  cos(ap));  +  cos()) + mliipz sin(ã)  —  cos(ä) +  sin(ä))  —  2mi8bpz cos(a);  2 sin()) + 2mt (Dpy cos(ä) + D —  22 KL  =  —  mi(Dpy sin(ä)  cos())  —  2 (D cos(a) + + miÔ —  i (D cos(ã) + 1 2m  + 6mi cos(ät)  cos(p));  sin(äp) +  1 s1n(t) 6m  miiipz sin(ã)  sin(ã))  cos(äp)  —  =maDpy; 1 MD =  —  sin(ä)) + 2miODpz cos(a)  2 + ptL /3) cos(2at); 3 + 6(msL  2 MD  sin(ä))  ml cos(a); 229  sin(ãp))  1 GD  —  maSDpz; 2  2 GD  =  8 sin(a); 1 2m  1 KD  =  6 sin(ã) 1 2ma6Dpz + m —  —  L cos(ä) + 2ma8Dpz 2 m  2 cos(ã) -I- mÔ cos(a) + 2m pL LÔ sin(ã) 2  + 6maitDpy cos(2p) 2 KD  =  6 sin(ä) + m 1 m (2 1  1 FL  =  11 + )8 (M 12 M  6maDpz sin(2p)  1 cos(ä) + 6m 2m ,u cos(ãt) cos(äp); 1  —  —  —  2) cos(ä)  —  p sin(äj) sin(äp); 1 6m  )ö + Ipö + mi (D cos(a) + 2 + ma(D + D L 2 + maDbpz + m  sin(ã)  + 2maSDpzDpz+ (p.L 2  —  —  2 cos(ä)) D  mlÔ2)(Dpij sin(ä)  cos(ã))  L6 (D cos(ä) + D 2 2 sin(a)) + 3ma(D + 2m 1 + 6mafLDpyDpz cos(2ãp) + 2m + 6mlI4cos(ãt)(Dpsin(äp) + —  2 FL  sin(a))  sin(ä) cos(p))  —  —  —  ) sin(2äp) 2 D cos(ä))  P(Ipy 3  6,.iIp cos(2ãp);  12 M 1 + )8 (Mi 1 (D cos(a) + +m  sin(ä)) + (msL 2 + ptL /3) 3  L)8 2 LL + ptL 3 + mliipz cos(ä) + (2m mÔ (Di sin(ä) +2 —  1 2m  sin(ã)  6m sin(ãt) u + 11  —  —  cos(a)) + 2miÔi3pz sin(ã) cos(a))  cos(äp)  —  /3) sin(2ät); 3 2 + pL L 3 + 3(m  230  sin(äp))  —  Ip)sin(2a)  a =a  —  ,. 3 a  Here, ã,, ä define the reference trajectories for platform and tether pitch angles, 3, and a are the differences between the actual and reference values respectively; cz 3, and D for the platform and tether pitch angles, respectively; D 3, are the reference  values for the offsets along the local vertical and local horizontal, respectively; and 2 is the offset required by the controller along the local horizontal direction. D  Equations for the Offset Control The dynamic model for the offset control of the tether attitude is based on the linear equation which is decoupled from the platform motion. However, since the platform dynamics is strongly coupled with the tether attitude in the presence of nonzero offset, the platform pitch controller is based on the complete nonlinear equation. These can be presented as p, x, th, t)  3, + map(cxp, x, t)& =  aa  =  M;  2 + b+ cD D 4 , 2 D + a3c + €L 2 +a  where: 11 map(ap,x,t) =M 1 f,(ap,àp,x,th,t) =F 1 a  =  —  —  —  2 1 /M 1 2 M ; 22  2 M 22 1 M / 2 F ;  _?_{mi (_D 3, sin(ä) + 3 D , cos(&)) 2 ( 1 +m  —  2)(D cos(ä) + D 3, sin(ä))  + 6i.tmi cos(ãj) (D cos(äp)  -I-6fLmatcos(2at)}; 231  —  3,2 sin(dp)) D  (111.3) (111.4)  a  =  sin(ä) + mi(2.t  —  mct —  a3  =  a4  =  b  =  —  —  ) cos(ä) 2 Ô  1 sin(ãt) sin(ap)}; 6m  mat 1 1 m1Ssmn(a); 2 ____t mat _  —  —  _--_{mi(Dpi, cos(ã) + 2 + mi(  —  2t) (D sin(ä)  sin(ä)) + maj + thatÔ cos(ã))  —  1 sin(at) (D cos(äp) + 6,um  —  2 sin(p)) D  + 3umat sin(2at)}; —  m cos(ã)  mat ; 2 L 3 /3 + m 3 mat =ptL LL. 8 L H- 2m 2 nat =ptL  The terms used to define the platform pitch equation, Eq.(III.3), were given in the sub—section titled “Nonlinear Equation”, Eq.(III.1).  232  APPENDIX IV: CONTROLLER DESIGN USING GRAPH THEORETIC APPROACH  Some details of the mathematical background and the controller design proce dure which assigns specified eigenvalues to the linear time invariant tethered system are described here.  Mathematical Background Definitions  *  A directed graph (also called digraph) 13  =  (V, E) consists of two finite  sets • V, the vertex set, a nonempty set of elements called the vertices of 13; and • E, the directed edge set, a (possibly empty) set of elements called the directed edges of 13; such that each e in E is assigned an ordered pair of vertices (u, v). If e is a directed edge (also called edge) in the digraph 13, with associated ordered pair of vertices (u,v), then e is said to join u to v. u is called the origin or the initial vertex of e,  and v is referred to as the terminus or the terminal vertex of e. If a number (or an edge weight) is assigned to each edge of a digraph, then the graph is called the weighted digraph. In G  =  (V, E), if the element e of E is assigned an unordered  pair (u,v), then G is called a graph.  *  Clark, J., and Holton, D. A., A First Look at Graph Theory, World Scientific Publishers, Singapore, 1991, Chapters 1, 7. 233  A directed walk in the digraph D is a finite sequence  W  =  vjelv  ..  whose terms are alternately vertices and edges such that for i  =  1,2, , k, the  1 and terminus v. If u and v are the starting and ending edge e has the origin v_  vertices, respectively, of the walk W then the u open depending on whether u  =  v or u  —  v walk (i.e. W) is called closed or  v, respectively. If W does not contain an  edge then it is called a trivial walk, else it is nontrivial. A nontrivial closed walk in a digraph D is called a cycle if its origin and inter nal vertices are distinct. In detail, the closed walk C  =  1 e 0 v  vk_lekvk(vk  =  vo)  is a cycle if: (i) C has at least one edge; and , 2 , v 1 (ii) v , v 0  ,  v_ are k distinct vertices.  Since C contains distinct vertices, it also has distinct edges. The integer k, the number of edges in the cycle, is called the length of C. A cycle C, of length Ic, is denoted by Ck. A set of vertex disjoint cycles is referred to as a cycle family. The width of the cycle family is the total number of vertices in it. A graph C is called connected if every two of its vertices are connected. A tree is a connected graph whose number of edges is one less than the number of vertices. A spanning tree (or complete tree) of a connected graph C is a subgraph which is a tree that involves all the vertices of C. A digraph is said to have a root r if r is a vertex and, for every other vertex v, there is a path which starts in r and  ends in v. A digraph D is called a rooted tree if D has a root from which there is a unique path to every other vertex. 234  Mapping of linear state-space model into diagraphs  *  Consider the Linear Time Invariant (LTI) equation (IV.1)  x—Ax+Bu,  where z  e  R is the state vector; u E R is the control input vector; A E  and B E RnXm. For the state feedback situation the control law can be expressed as (IV.2)  u=Fx,  where F  e  R>< is the controller matrix. The closed-loop system of Eqs. (IV.1)  and (IV.2) can be mapped into the digraph (GS) defined by a vertex set and an edge set as follows: • the vertex set consists of m input vertices denoted by Ui, U2, state vertices denoted by 1, 2,  •,  ...,  urn, and n  n;  • the edge set results from the following rules: —  if the state variable  occurs in the x-equation, i.e.  exists an edge from the vertex —  if the input variable  Uk  j  to the vertex i with ajj as its weight;  occurs in the z-equation, i.e.  exists an edge from the input vertex  —  Uk  0, then there  to the vertex i with the weight  finally, if the state variable x occurs in the ui-equation, i.e.  f  then there exists an edge from the vertex i to the vertex  with the  weight *  0, then there  Uk  0,  fj.  Reinschke, K. J., Multivariable Control: A Graph Theoretic Approach, Lecture Notes in Control and Information Sciences, Edited by: M. Thoma and A. Wyner, Berlin, Springer-Verlag, 1988, Chapters 1,2. 235  Here, ajj is the  row and th column entry of the matrix A; and bk and fk are  the entries of B and F, respectively. For illustration, consider a system defined by the matrices  A =  [ai  a32  a33  a34];  41 a  42 a  43 a  44 a  B  =  [];  F  =  [fi  f2  f f i•  The digraph C for this dynamic system can be represented as in Figure IV-1. Obviously, the diagraph C 5 contains less information than Eqs. (IV.l) and (IV.2). Actually, G 8 reflects the structure of a closed-loop system with state feed back. As far as the small scale systems are concerned, it seems unnecessary to investigate their structures separately. In case of the large scale systems, however, one should start with a structural investigation. Thus, particularly for higher order systems, the diagraph approach is extremely useful. A typical feature of large scale systems is their sparsity. This structural . The digraph reflects only the non5 property becomes evident in the digraph C 2 state edges and nm control vanishing couplings of the system. So, instead of n  edges one has really to take into account only a small percentage of this number in 5 gives an immediate impression of most sparse systems. Moreover, the diagraph C the information flow within the closed ioop system. So for higher order systems, the digraph approach to controller design is advantageous from computational point of view. For lower order systems, which is the case in the present study, this design method gives a simple closed-form expression for the controller that can be readily included in the dynamic simulation program or implemented in real system. This feature allows for efficient simulation with a gain scheduling control for the highly 236  ———  —  —  —  —  —  7 3 b  I”  i 31  3 a 3  4 \f  a  fi  1 a  2 a 3  —-  2  Figure IV-1  Digraph representation of a linear, time-invariant system.  time varying dynamics of the tethered systems.  Controllability A class of systems characterized by the structure matrix pair [A, B] is said to be structurally controllable (or s-controllable) if there exists at least one admissible realization (A, B)  e  [A, B] that is controllable in the usual numerical  sense. The n x (n -I- m) structure matrix pair [A, B] is s-controllable if and only if its associated digraph G([Q]) contains a set of  i  ( m) disjoint cacti, each of them  rooted in another input vertex and, together, touching all the state vertices. Here 237  i_ ([A]  LQJ  -  [F]  [B]’\ [0])’  ( 1V3.)  and the ‘cactus’ associated with G([QJ) is the spanning input-connected graph consisting of a simple path with an input vertex as the initial vertex, p  (  0)  vertex disjoint cycles and p distinguished edges each of which connects exactly one cycle with the path or with another cycle. Note, the digraph  G([Q])  associated with the structure matrix  [Q]  is the same as  the diagraph GS corresponding to Eqs.(IV.1) and (IV.2) defined earlier.  Characteristic polynomial of a square matrix Let the characteristic polynomial of the closed-loop system be represented by +Pn_ls+Pn.  The coefficients p, 1  i  (IV.4)  n, can be determined by the cycle families of width i  within the graph G( [Q]). Each cycle family of width i corresponds to one term in p. The numerical value of the term results from the weight of the corresponding (_].)d if the cycle family cycle family. The value must be multiplied by a sign factor  under consideration consists of d vertex disjoint cycles. In particular, pi results from all cycles of length 1, with the common sign factor as —1; and P2 arises from all cycles of length 2, each with a sign factor —1, as well as all disjoint pairs of cycles of length 1, each pair with a sign factor +1, etc.  Controller Design Algorithm  Based on the preliminaries discussed above, the state feedback controller 238  which assigns the closed-loop poles at the desired locations in the complex plane can be obtained by the following procedure. Step 1: Consider the digraph G([Qj), Eq.(IV.3), and choose a subgraph consist m disjoint cacti as indicated before. Enumerate these cacti  ing of  arbitrarily from 1 to ñì.. There are fiz.! possible different enumerations. Step 2: Choose an mx n feedback structure matrix  [P’] whose th— 1 nonvanishing  elements correspond to feedback edges leading from the final vertex of the path of cactus jto the root of cactus  j + 1 (j  —1).  =  Choose a unit structure vector {gj such that the input vertex associated with the column structure vector [Bg], is the root of cactus 1. The matrix [F] and the vector [g] assure the s-controllability of the single input pair [A + BP, Bg]. Moreover, for almost all admissible (A, B) [A,B],  P  E  e  [P’j, g E [g], the pair (A + BF,Bg) is controllable in the  numerical sense. Step 3: Choose admissible (A,B,P’,g) E [A,B,P,g] such that the pair (A + BF, Bg) becomes controllable. Set up the system of equations P  where: p  =  {Pi, P2,  ,  [pl]Tf  (IV 5)  —  }T  is the vector of characteristic polyno  mial coefficients defined by the desired pole locations of the closed loop system; p 0 contains the characteristic polynomial coefficients determined by the cycle families in G(A + BP); and ] 1 [F  =  [P,)]’ w  =  1,2,••• ,n;  j  =  ,n.  , i.e. Pw,j)’ is the (sign weighted) sum of weights [F ] The element of 1  239  of all those cycle families of width w within the diagraph of Eq.(IV.3) that contain one feedback edge (with weight 1), from state vertex  j  to  the input of cactus 1. [F’] has rank n, if and only if (A + BP’, Bg) is controllable. Hence,  Eq.(IV.5) can be solved for  f for  almost all (A, B) E [A, B].  Step 4: The overall feedback matrix F=P+gfT  (IV.6)  provides the desired eigenvalue placement. For single input systems, the controller can directly be designed from Eq.(IV.5) with  P=  [0] and g  =  {0}.  Design of the Controller The state feedback controller is designed to regulate the rigid degrees of freedom, i.e.  p  and  .  The governing linearized equations of motion for the rigid  system are used to that end (Appendix III, Eq.(III.2)). The platform and tether pitch controllers are designed based on the decoupled equations of motion. For the offset control, to regulate the offset motion, the tether pitch equation is augmented by the identity = Ut.  The equations governing the tether dynamics can be written in the state space form as  ±=Ax+bu, 240  (IV.7)  where: x  =  {at, 0 0  }T;  at,  a3 1 41 a  0 0  1 0  0 1  a32 a42  a33 43 a  a34 a44  b=  0 0  The entries in A and b matrices are obtained from Eq.(III.2), Appendix III. The digraph of this system with the state feedback controller is shown in Fig. IV-1. The controller, u  =  fx, can now be designed using Eq.(IV.5) with p° and [F ] as given 1  below:  2’  o  — —  —a33 a44 a33a44 a3 a42 a34a43 4 a 34 4 —a32a43 + a42a33 + a3a4 1. 32 4 a 1 a 32 4 1  ( J  —  —  —  —  —  —  rpll  —  I  —  0 0 3 —b 4 —b  —b 3 4 —b 44 b a 3 b 34 a 4 43 + b a 3 —b 33 a 4 —  b 4 a 3 4 43 a 3 —b 42 a 3 b 41 a 3 —b  b 3 a 4 4 33 a 4 +b 32 a 4 b j 3 a 4 -I- b  —  —  b 4 a 3 2 4 b 3 a 2 41 -I- &ia3i a 3 —b 0 0 —  Similarly, the state model of the platform dynamics can be represented by the ma trices 0 aj  1 a22  and  lo  b=c  The corresponding matrices for the controller design are  { } 22  =  and  [p1] =  241  [—2  I  LINEARIZED EQUATIONS OF MOTION  APPENDIX V:  AND CONTROLLER FOR THE FLEX IBLE SUBSYSTEM  Linearized Equations of Motion  The governing equations of motion (Appendix II) are decoupled separating the rigid and flexible subsystems. Neglecting the nonlinear terms from the decoupled equations, the linear model for the flexible system can be represented by the vector equation K + Mdydpy + Gddpy + Kdydpy Z MZ + GZ + 2 + Mdzdpz -I- Gdpz + Kdzdpz + 1z  =  (IV.1)  QzTL,  where: Z  =  {Beq}  =  2 M  =  2 C  =  {{B  =  Beq}T, CT}T;  equillibrium value of {B}; ; 2{K } 4 ] 5 + [K +  2 K  —  —  F{F(Ydl)} {Q}  [  T ] 5 [K  ()1 [[i  d]] 3 [AC + {O} )} {O} T FCdz{F(Ydl {F(Ydt)}j {O} j T Cdz{F(Ydt)}  rrig[Apl]T[Uk] [A ] .+ 1  1  I  +  +  ]T 5 T + [P [K ] ] -f- [p 5 ]— 6 6 + Ô[1k ] — [K 4  242  ‘  2 (— +  L  )[I] — i 4  +1  (2  [  {F(Ydt)}j  + 2i)[Ici] + [AC]  CdjL{DF(Ydl)} T T Cdt{F(Ydt)}  ]  Cdl{F(Ydl)} T cdL{DF(yd)}T]  IT[Ttp]; 2 [K  Md  =  ]T[Tt] 3 ]T[Ttp] + [K 2 [Ak ]T[Ttp][Uk] — [K 2 +  =  r I  L  110 {0} {F(Ydt)}j L0  {0}  Cdl 0  I  ]T[Ttp][Uk] + [Ak 2 ë[K [Ttp][Uk] T ] 2 +S  J  [ =  {0}  {0}  r  iz  1  {0}  {0}  L  Kd  [0]  [0]  l{F(Ydj)}  Gd  + [AB]  1  {0} {0} {F(Ydt)}j  {0} } F(Y ) 1 {0}  {  —  0 1 Cdt]’  [Ttp][Uk] T [K3]  —Cdl 0  + [P ] 3  0 Cdi]’  } + {{Ck msL[Apl]T{Dk } } + ms[Apl]T{Dkl}}L 5 +2 {Hk } 2 -  T + {Ok5} + {K } 8  2 + 4){’b I (Ô } 2 {}  -1  }  +  + mSLÔ[Apl]T{Dk } + {P 2 } 4  {  1 F.Ø(Ydl)} {0}  r ,L(ãF. (ÔFI1T  [AB]=EAIJ0 t  t 6  J  {0} {F(Ydt)}J  {  Lcdl  ,  YCj  dY];  —‘-  I ij_} [AC}=EA’J L o  T  IÔF 18FT  {Be})t?t  1  dYtj;  (8FlT ] [AC d 8  8ytfJ  -j  ditj.  Here: Cdl and °dt are the damping coefficients of the longitudinal and transverse 243  dampers located at distances dl and dt, respectively;  and w 0 are as defined  in Eqs.(2.32) and (2.33); {B} and {C} are the vector containing longitudinal and transverse modes of the tether; and the coefficient matrices are defined in Appendix I.  Numerical Values of the Coefficient Matrices The numerical values for the coefficient matrices in Eq.(IV.1) are given be low. They correspond to the stationkeeping case with a tether length of 20 km, and mass and elastic properties as given in Chap.4.  =  532.7 519.6  519.6 514.0 [0]  [t1 [Diag.(98.0)]  [0.405 0.4051 [0.405 0.729] [G]  Gt =0.1  K  X  —1.023 0.512 —0.341 0.256 —0.205 0.171 —0.146 0.128 —0.114 0.102  13.08 [3.08  3.081 5.54] [0]  c  e  T  122 R E ;  [0]  —0.401 0.434 —0.318 0.246 —0.199 0.168 —0.144 0.127 —0.113 0.102  [0] [Diag.{K}]  244  12 2; R E X  =  Md  dy 0  =  =  —42.982 —41.928 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  —549.0 —524.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  0.0 0.0 0.2046 0:0 0.0682 0.0 0.0409 0.0 0.0292 0.0 0.0227 0.0  0.0211 0.0845 0.1902 0.3382 0.5284 0.7609 1.0357 1.3527 1.7120 2.1136  Kt  Md  =  0.0 0.0 —88.231 0.0 —29.410 0.0 —17.646 0.0 —12.604 0.0 —9.803 10.0  —1.2733 —1.2165 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  Gd  245  Kd  —0.2215e —O.2116e 0.0 0.0• 0.0 0.0 0.0 0.0 0.0 0.0  =  1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  —  0 C  —  2 2  1.0 1.0 0.0 0.0 0.0 =  0.0 0.0 0.0 0.0  2’ 0.0 0.0 0.2221e 3 0.4443e 3 0.6664e 3 0.8886e 3 0.lllle—2 0.1333e 2 0.1555e 2 0.1777e—2 0.1999e 2 0.2221e 2 —  —  —  —  —  —  —  —  where: (L) 1 b 0 C  =  0  ...  ..  0  •..  (a1,o)  •..  0 Yt=  [Diag.(a)] =[D]; with  D= 0, =  a,  =  a(i),  Vij; if a is a scalar; if a is a vector.  246  0 (8N /8t) 0 Yt  Design Parameters and Controller Matrices The numerical values for the weighting matrices used for the controller design  and final controller matrices are given below. - Controller (LQG/LTR) 1 B q  =  1.0e5;  =  R  =  1.0e5; 3.Oe 2 0.0 0.0 0.0  0.0 3.Oe —4 0.0 0.0  —  —  —  =  —  0.0 0.0 0.0 1.Oe 4 —  1.5e2; —1.811e 7.277e —4.460e —7.135e  —  Afb  Bfb  0.0 0.0 1.Oe 2 0.0  ( I j  I  —  —  —  —  2 1 2 1  1.0 9.480 00 —9.199  —1.811e 2 1.029 —4.460e 2 —1.001 —  —  —1.8110e 2 —1.0165e—3 —4.4601e 2 —8.1649e 4  Cfb= {0.7126  —  —  —  9.1993  1.0002  247  10.2202 };  0.0 1.053e+1 1.0 —1.022e + 1  - Controller (LQC) 1 C 0.0 1.Oe—2 0.0 0.0  0.0 0.0 1.Oe—1 0.0  0.0 0.0 0.0 1.Oe 3  1.Oe+9 0.0 0.0 0.0  0.0 1.Oe+7 0.0 0.0  0.0 0.0 1.Oe+1 0.0  0.0 0.0 0.0 1.Oe 1  1.Oe+9 0.0  0.0 1.Oe+9j’  1.Oe 1 0.0 0.0 0.0 —  —  R=1.Oe+8;  —  —  Afc  1.  r—1.2363e 3 I —2.3062e 4 0.0 1.5779e—5 —  I  L  r  Bfc=  I  I  L  —  5.5654 3.3292e 3 0.0 0.0 —  Cf= {1.5779e—5  1.0 —1.6433e —3 0.0 1.4645e 3 —  0.0 2.8471e —4.4732e —4.1623e  —  —  —  5 3 5  0.0 7.3920e 1.0 —8.2105e  —  —  0.0 0.0 1.Oe  —  5  1.4646e—3  —3.1623e  248  —  5  —8.2105e  —  3 };  3 3  APPENDIX VI: LABORATORY TEST SETUP  The experimental setup used three step-motors with corresponding translator modules and two optical potentiometers. The motors consist of permanent magnet rotors while the stators contain a stack of teeth with several pairs of field windings. The windings can be switched on and off in sequence to produce electromagnetic pole pairs that cause the rotors to move in increaments. The sequential switching of the windings is accomplished by a translator module. The translator module has logic circuitry to interpret a pulse train and “translate” it into the corresponding switching sequence for stator field windings (on/off/reverse state for each phase of the stator). The details of the motors and translators are well documented in reference [110]. Relevant mass, geometry, and other system parameters used in the analysis are listed below: Mass of Carriage  =  70.43 kg  Mass of inplane traverse  =  2.23 kg  Mass of the pendulum  =  0.1108 kg  Deployment/Retrieval reel diameter  =  2 x 10_2 m  Pulley diameter  =  4.4 x 10—2 m  Maximum tether length  =  2.25 m  =  (± 0.7 m) x (± 0.7 m)  =  200 pulses/rev  Maximum offset motion in X  —  Y plane  Number of pulses required to move the motors  A pair of Softpot optical shaft encoders (Si series, U.S. Digital Corp.) is used to measure the angular deviation of the tether from the vertical position. The 249  Softpot is available with ball bearings for motion control applications or torqueloaded to feel like a potentiometer for front panel manual interface. Characteristic features of the Softpot are given below: • 2-channel quadrature, TTL (Transistor  —  Transistor Logic) squarewave out  put; • r 3 d channel index option;  • tracking capability  —  0 to 10,000 RPM;  • ball bearing option tracks to 10,000 RPM; •  —  40 to + 100°C operating temperature;  • single + 5v supply; • 100 to 1024 cycles/rev; • small size; • low cost.  250  

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