A Space Manipulator with Its Base Attitude Controller

DYNAMIC INTERACTION OF A SPACE MANIPULATOR WITH ITS BASE ATTITUDE CONTROLLER

Eric Martin

B. Eng. (University of Sherbrooke), 1992 M. Eng. (McGill University), 1995

Department of Mechanical Engineering McGill University Montreal, Quebec, Canada

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements of the degree of Doctor of Phiiosophy

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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author' s ou autrement reproduits sans son permission. autorisation. To Dominique, Charles, Philippe and Jérémie for their patience and understanding during the drajting of this manuscript. Abstract

Space manipulators mounted on a free-floating base are structurally flexible mechanical systems. For some applications, it is necessary to control the attitude of the base by the use of on-off thrusters. However, thruster operation produces a rather broad frequency spectrum that can excite sensitive modes of the flexible system. This situation is likely to occur especially when the rnanipulator is moving a big payload. The excitation of these modes can introduce further disturbances to the attitude control system; therefore, undesirable fuel replenisliing limit cycles may develop. To investigate these dynamic interactions, the dynamics mode1 of an N-flexible-joint space manipulator is developed and used to describe a three-flexible-joint manipulator mounted on a six-degree-of freedom spacecraft dubbed in t his thesis "spatial systern". Since the attitude controller assumes the use of on-off thrusters, which are nonlinear devices, the describing-function technique, an approximate method for the analysis of nonlinear systems, is used to analyze four different control systems using an approximate two-mas8 system. This technique is adapted to be used in conjunction with the root-locus concept, thus providing a different picture of the problem and helping in the control system design. A pararnetric study is performed to compare the control schemes using those analytical tools, which is validated with simulations using the spatial system. It is shown that the three proposed variations of a classical control schemc are very effective to minimize such undesirable dynaniic interactions. Finally, Mecano, a commercial mechanical-system modelling and analysis software tool, is adapted for the study of control systems and used to study the effect of link elasticity on the robustness of the proposed control schemes. Résumé

Les robots manipulateurs montés sur une base flottante et opérant dans l'espace sont des systèmes mécaniques flexibles. Pour certaines opérations, il est nécessaire de commander la position de la base en utilisant des fusées de type tout-ou-rien. Cependant, l'opération de ces fusées produit un large spectre de fréquences qui peuvent exciter les modes vibratoires du système, cet te situation devenant plus probable lorsque le manipulateur transporte de grosses charges. L'excitation de ces niodes peut introduire davantage de perturbations au système de commande, continuant alors le cycle et augmentant du même coup la consommation de combustible, sans stabiliser la base. Afin d'étudier ce problème, le modéle dynamique d'un manipulateur à N articulations élastiques est développi! et utilisé pour décrire un robot spatial à trois articulations élastiques. Puisque la commande présume l'utilisation de fusées tout- ou-rien, qui sont des mécanismes non-linéaires, la technique des fonctions descriptives a été utilisée pour étudier quatre modèles différents à l'aide d'un système simplifié à deux masses. Cette technique a été utilisée conjointement avec la méthode du lieu g/'eométrique des racines pour ainsi analyser le problème d'un point de vue différent. Une étude paramétrique a été effectuée à l'aide de ces méthodes pour comparer les différents modèles de commande. Cette étude a été confirmée par des simulations du robot spatial décrit précédemment. Cette étude démontre que les trois variations proposées aux méthodes de commande utilisées à l'heure actuelle dans l'espace sont de très bonnes solutions de rechange étant donné qu'elles peuvent à toute fin utile éliminer le problème d'interactions dynamiques. Acknowledgement s

I would like to thank my research supervisors, Professors J. Angeles and E. Pa- padopoulos for their guidance, encouragement and support during the course of this research. Their help in reviewiiig the manuscript is also gratefully acknowledged. Working in conjunction with them was a very enriching experience and a great plea- sure. The support of this work by Quebec's Fonds pour la Formation de Chercheurs et l'Aide à la Recherche (FCXR) and by Canada's Natural Sciences and Engineering Research Council (NSERC) is gratefully acknowledged. Funding was also provided to the author through NSERC and FCAR graduate scholarships. The support of Quebec's Ministère des unaires internationales under the Quebec-Wallonia Scientific Collaboration Agreement is highly acknowledged. 1 also want to thank Dr. Dave Parry of Spar Aerospace Ltd. for providing us with detailed CANADARM data, and Mr. Yves Lombard of Samtech to let us use freely the finite-element software package Mecano. Many thanks are also due to the Centre for intelligent Machines (CIM) for al1 the computer facilities provided and for the pleasant research environment. Working at CIM was a wonderful experience; special thanks are due to CIM7s staff, secretaries, and dl fellow students. Finally, 1 am very grateful to my wife, Dominique Mathieu, for her love, support and understanding throughout rny thesis work. Thanks must also be given to my sons Charles, Philippe and Jérémie for allowing me to work on this research. Contents

Abstract

Résumé

Acknowledgements

List of Figures

List of Tables

Nomenclature xvi

1 Introduction 1 1.1 Robots in Space ...... 1 1.2 Literature Survey ...... 4 1.2.1 Dynamics and Controi of Space Robots ...... 4 1.2.2 Attitude Control of Spacecraft ...... 6 1.2.3 Control of Large Space Structures (LSS) ...... 11 1.2.4 Payload- Attitude Controller Interaction ...... 14 1.3 Objectives and Organization of this Thesis ...... 17 1.3.1 Problem Formulation and Objectives ...... 17 1.3.2 Research Tools ...... 18 1.3.3 Thesis Organization ...... 19 1.4 Contributions ...... 20

2 Modelling. Control and Analysis of Rigid Spacecraft 21 2.1 Introduction ...... 21 2.2 Modelling of Rigid Spacecraft ...... 21 2.3 Classical Attitude Control Scheme ...... 2.3.1 On-Off Thrusters Command ...... 2.3.2 State Estimation ...... 2.3.3 Model Integration ...... 2.4 Methods of Analysis ...... 2.4.1 Describing-Function Analysis ...... 2.4.2 Root-Locus Analysis ...... 2.4.3 Simulation ...... 2.5 Stability ...... 2.5.1 Definitions ...... 2.5.2 Application ...... 2.6 Summary ......

3 Modelling of Flexible Space Manipulator Systems 3.1 Introduction ...... 3.2 General Lagrangian Formulation ...... 3.3 Linearization of the Equations of Motion ...... 3.4 Description and Validation of the Spatial System ...... 3.41 Model Description and Parameter Identification ...... 3.4.2 Model Validation ...... 3.4.3 Determination of Frequencies and Damping Ratios ...... 3.5 Finite Element Formulation-Adaptation of a FEM Package for Con- trol Purposes ...... 3.5.1 General Background on Control Systems ...... 3.5.2 Derivation of the State-Variable Equations ...... 3.5.3 Application: Attitude Control ...... 3.6 Summary ......

4 Control Synthesis Using an Asymptotic State Estimator 4.1 Introduction ...... 4.2 Classical Rate Est imator-Pro blem Description ...... 4.2.1 The Interaction Problem with a Planar System ...... 4.2.2 The Interaction Problem with A Spatial System ...... 4.2.3 Problem Demonstration with Theoretical Analysis and Para- metric Study ......

vii 4.3 Attitude Control using an Asymptotic State Estimator ...... 112 4.3.1 Asymptotic State Estimator Theory ...... 112 4.3.2 Asymptotic State Estimator for Space Robotic Systems .... 116 4.4 Demonstration of the Proposed Estimator Design with Simulation Re- sults and Analysis ...... 123 4.4.1 Analysis of the Asymptotic State Estirnator Using the Two- Mass System ...... 123 1.1.2 Pararnetric Study for the Asymptotic State Estimator ..... 128 4.4.3 Simulation Results for the Asymptotic State Estimator .... 128 4.4.4 Effects of Mode1 Perturbation in the Asymptotic State Estimatorl30 4.5 Discussion and Conclusions ...... 132

5 Control Synthesis Using Compensation Techniques 135 5.1 Introduction ...... 135 5.2 Rate Estimator with Linear Compensation ...... 136 5.2.1 Description of the Rate Estimator with Linear Compensation 136 5.2.2 Determination of the Compensator Pole and Zero Locations . 140 5.2.3 Demonstration of the Proposed Estimator Design with Simu- lation Results and Analysis ...... 142 5.2.4 Simulation Results for the Rate Estimator with Linear Corn- pensatiori ...... 117 5.2.5 Verification of the Robustness of the Rate Estimator with Lin- ear Compensation ...... 149 5.3 Xew Rate Estimator with Linear Compensation ...... 152 5.3.1 Description of the New Rate Estimator with Linear Compen- sation ...... 152 5.3.2 Demonstration of the Proposed Estimator Design with Simu- lation Results and Analysis ...... 153 5.3.3 Simulation Results for the New Rate Estimator with Linear Compensation ...... 160 5.3.4 Verification of the Robustness of the New Rate Estimator with Linear Compensation ...... 160 5.4 Effect of Link Flexibility using A Planar Example ...... 162 5.5 Discussion and Conclusion ...... 166

viii 6 Conclusions and Recommendations for E'urther Work 168

6.1 Conclusions ...... + ...... 168 6.2 Recommendations for Future Work ...... 175

References

Appendices

Attitude Description using Euler Angles

Tkansformation of nth-Order Differential Equations 190

Modelling of the Two-Mass and Planar Systems 192 C.l Two-Mass System ...... 192 C.2 Planar System ...... 197

Transfer Function of the Linear Elements of the Simulation Models199 D.l Classical Rate Estimator ...... 200 D.2 Asymptotic State Estimator ...... 200 D.3 New Rate Estirnator with Linear Compensation ...... 203 List of Figures

1.1 The C.4NADARM mounted on the Space Shuttle...... 1.2 Two concepts of free-flying robots . (a) U.S. Flight Telerobotic Servicer (FTS), (b) Japan NASDA OSV ......

2.1 A six-dof rigid spacecraft ...... 2.2 Standard spacecraft control scheme...... 2.3 Switching logic in the error phase plane ...... 2.4 Controller block ...... 2.5 Parabolic phase plane trajectories: (a) for u < O; (b) for u > 0..... 2.6 Single-mas example ...... 2.7 Controller with hysteresis . (a) Relay nonlinearity, (b) Phase plane switching logic ...... 2.8 Single-mass example with an hysteretic controller ...... 2.9 Effects of a pure time delay on the switching logic ...... 2.10 Block diagram for the rate estimator only...... 2.11 Mode1 with a 1-auis classical rate estimator ...... 2.12 Mode1 with a 3-axis classical rate estimator...... 2.13 A nonlinear system ...... 2.14 Loci of the describing functions for the relays ...... 2.15 .4 nonlinear system analyzed with describing functions...... 2.16 Limit cycle detection ...... 2.17 Reliabiiity of limit cycle prediction: (a) poorly-reliable results; (b) highly-reliable results ...... 2.18 A general linear system ...... 2.19 Stability range determination using the Nyquist criteron ...... 2.20 Stability prediction using the describing-function method . (a) Stable system, (b) Unstable system ...... 2.21 Stability of limit cycles using the root-locus method...... 2.22 Stability prediction using the root-locus method. (a) Stable system, (b) Unstable system...... 2.23 Block diagram for the classical rate estimator with white noise included in attitude measurements...... 2.24 Simulation results for the rigid spacecraft: (a) spacecraft error phase plane; (b) thruster command history; and (c) fuel consumption. . . . 2.25 Dcscribing-function plot for a type- l instability...... 2.26 Describing-function plot for a type-2 instability...... 2.27 Describing-function plot for a stable system...... 2.28 Examples of stability determination. (a) Fuel-consumption curve of a stable system, (b) Fuel-consumption curve of an unstable system, (c) Spacecraft error phase plane when the motion diverges. (d) Space- craft error phase plane when the motion reaches a large limit cycle. .

3.1 A space manipulator system...... 3.2 A flexiblejoint model...... 3.3 A three-flexible-joint space manipulator...... 3.4 (a) Input torque profile over time; Joint angles using the hfatlab model: (b) O2 history; (c) O4 history; and (d) Os history...... 3.5 (a) Open-loop control, (b) Closed-loop control......

4.1 .A planar free-flying manipulator...... 4.2 Simulation results for the planar system with ,l?= 0.05: (a) spacecraft error phase plane; (b) spacecraft error phase plane (zoom); (c) spacecraft attitude phase plane; (d) thruster-command history; (e) fuel- consumption history; and (f) joint-angle history...... 4.3 Simulation results for the planar system wit h a = 0.3: (a) spacecraft error phase plane; (b) spacecraft attitude phase plane; (c) thruster- command history; (d) fuel-consumption history; and (e) joint-angle history ...... Simulation results for the spatial systern with P = 0.05: (a) @-axis error phase plane; (b) @-ais thruster-command history; (c) $-axis fuel-consumption history; (d) &axis error phase plane; (e) &axis thruster-command history; (f) qbaxis fuel-consumption history; (g) 8- axis error phase plane; (h) 8-axis thruster-command history; (i) &mis fuel-consumption history; (J) joint-angle O2 history; (k) joint-angle e4 history; and (1) joint-angle Os history...... 101 Simulation rcsults for the spatial system with ,O = 0.3: (a) +mis error phase plane; (b) $-ais thruster-command history; (c) +-mis fuel-consumption history; (d) #+cuis error phase plane; (e) &auis thruster-cornmand history; (f) &auis fuel-consumption history; (g) 0- axis error phase plane; (h) O-axis thruster-command history; (i) 0-auis fuel-consumption history; (j) joint-angle O2 history; (k) joint-angle O4 history; and (1) joint-angle es history...... 102 The two-mass system...... 103 Model with a classical rate estimator for the two-mass system. . . . . 104 Theoret ical analysis wit h the classical rate estimator for the low- payload case (b = 0.05): (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom)...... 106 Theoretical analysis with the classical rate estimator for the high- payload case (p = 0.3): (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom)...... 107 4.10 Block diagram for the asymptotic state estimator...... 114 4.11 Asymptotic state estimator for the spatial system...... 120 4.12 Model with an asymptotic state estimator for the spatial system. . . . 121 4.13 Model with an asymptotic state estimator for the two-mass system. . 124 4.14 Theoretical analysis with the asymptotic state estimator for the low- payload case (P = 0.05): (a) describing-function plot; (b) describing- function plot (zoom); (c) root-locus plot; and (d) root-locus plot (zoom).126 4.15 Theoretical analysis with the asymptotic state estimator for the high- payload case (P = 0.3): (a) describing-function plot; (b) describing- function plot (zoom);(c) root-locus plot; and (d) root-locus plot (zoom). 127

xii 4.16 Simulation results for the spatial system using the asymptotic state estimator and p = 0.3: (a) @-axiserror phase plane; (b) Qauis thruster- cornmand history; (c) @-axis fuel-consumption history; (d) &âuis error phase plane; (e) +axis thruster-command history; (f) +mis fuel- consumption history; (g) &ais error phase plane; (h) O-auis thruster- command history ; (i) 0-axis fuel-consumption history ; (j) j oint-angle B2 history; (k) joint-angle iî4 history; and (1) joint-angle Os history. .. 131 1.17 Siiliulatiori results for the spatial systern using the asymptotic state estimator with perturbed mass properties and B = 0.3: (a) 8-ais error phase plane; (b) B-axis thruster-command history; (c) 8-auis fuel-consumption history...... 132

Model with a rate estimator and linear compensation for the two-mas system...... 139 Block diagram for the rate estimator with linear compensation. ... 140 Theoretical analysis for the rate estimator with linear compensation and p = 0.05: (a) describing-function plot: (b) root-locus plot: and (c) root-locus plot (zoom)...... 144 Theoretical analysis for the rate estimator with linear compensation and 4 = 0.3: (a) describing-function plot; (b) root-locus plot: and (c) root-locus plot (zoom)...... 1-15 Simulation results for the spatial system using the rate estimator with linear compensation and ,O = 0.3: (a) 11-auis error phase plane; (b) iil- axis thruster-command history; (c) S-auis fuel-consumption history; (d) +axis error phase plane; (e) &ouis t hruster-command his tory; (f) @axis fuel-consumption history; (g) O-axis error phase plane; (h) 8- axis thruster-command history; (i) 0-ais fuel-consurnption history; (j) joint-angle e2 history; (k) joint-angle O4 history; and (1) joint-angle Os history...... 150 Simulation results for the spatial system using the rate estimator with linear compensation with modified zeros and ,!3 = 0.3: (a) O-axis error phase plane; (b) 8-axis thruster-cornmand history; (c) 8-axis fuel- consumption history...... 152 Model with a new rate estimator and linear compensation for the two-rnass system...... 153

xiii 5.8 Block diagram for the new rate estimator with linear compensation. . 154 5.9 Theoretical analysis for the new rate estimator with linear compensation and /3 = 0.05: (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom)...... 156 5.10 Theoretical analysis for the new rate estimator with linear compensation and /3 = 0.3: (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom)...... 157 .5.11 Simiilation rwiiltls for the spatial system using the new rate estimator with linear compensation and B = 0.3: (a) q-axis error phase plane; (b) 11-axis thruster-command history; (c) +-axis fuel-consumption history; (d) &axis error phase plane; (e) qkauis thruster-command history; (f) @-aisfuel-consumption history; (g) 8-axis error phase plane; (h) 8-axis thruster-command history; (i) 8-auis fuel-consumption history; (j) joint-angle O2 history; (k) joint-angle O4 history; and (1) joint- angle Os history...... 161 5.12 Simulation results for the spatial system usir.g the new rate estimator with linear compensation with niodified zeros and = 0.3: (a) 0-ais error phase plane; (b) 0-axis thruster-command history; (c) @-ais fuel-consumption history...... 162 5.13 Simulation results for the planar system with link flexibility using the classical rate estimator: (a) spacecraft error phase plane; (b) spacecraft attitude phase plane; (c) thruster-command history; (d) fuel- consumption history; and (e) joint-angle history...... 164 5.14 Simulation results for the planar system with link flexibility using the rate estimator with linear compensation: (a) spacecraft error phase plane; (b) spacecraft error phase plane (zoom); (c) spacecraft attitude phase plane; (d) thruster-command history; (e) fuel-consumption history; and (f) joint-angle history...... 165

D.l A nonlinear system...... 199 D.2 Mode1 with an asymptotic state estimator using transfer functions. . 202

xiv List of Tables

Spacecraft. payload and link parameter values ...... 70 Four possible sets of joint stiffnesses...... 73 Cornparison of natural frequencies in the plane of links 2 and 3.... 74 Cornparison of natural Frequency out of the plane of links 2 and 3... 74 Final joint angles and rates comparison ...... 78 Natural frequencies and damping ratios for various configurations and payloads ...... $1

Planar system parameter values ...... 97 Parameter values used the classical rate estimator for the planar example ...... 97 Pole-and-zero location for the classical rate estimator ...... 108 Results of the parametric study for the classical rate estirnator .... 111 Pole-and-zero locations for the asymptotic state estimator . 125 Results of the parametric study for the asymptotic state estimator . . 129

Pole-and-zero locations for the rate estimator with linear compensation . 146 Results of the parametric study for the rate estimator with linear compensation ...... 148 Parameter di.= for various configurations and payloads ...... 151 Pole-and-zero locations for the new rate estimator with linear corn- pensation ...... 155 Results of the parametric study for the new rate estimator with linear compensation ...... 159 Planar system with flexible link parameter values ...... 163 Parameter values used for the linear compensator ...... 165 Nomenclature

Al1 bold-face, lower-case, Latin and Greek letters used in this thesis denote vectors; al1 bold-face, upper-case, Latin and Greek letters denote matrices.

: zero vector : identity matrix

: coefficient matrix of state-space representation : amplitude of the sinusoidal input in the describing function rnethod

: coefficient matrix of state-space representation : amplitude of the force developed by the thrusters for the two-mass system : coefficient vector of state-space representation for a single-input system : coefficient matrix of state-space representation

: damping matrix : generalized damping niatrix of the space manipulator system

: generalized damping matrix of the one-link planar system : damping matrix : damping matrix in Mecano : centre of mass (CM) of the spacecraft : CM of the 2-th body : centre of mass : spacecraft CM position vector : position vector of the CM Ci of the 2-th body : damping coefficient of the two-mass system

xvi : COS(~X),x = $,4, e : ith-joint damping coefficient for the space manipulator system : COS(X), = q,4, e : damping coefficient of the one-link planar system : coefficient matrix of state-space representation : output matrix of the state equations : spacecraft attitude error : spacecraft attitude rate error : frame attached to the spacecraft CM (XOYoZo) : fuel consumption : fuel consumption (smoothed version of Fc(t)) : inertial frame (XIYJI)

: inertial forces in Mecano : interna1 forces in Mecano : vector of the external forces applied to the spacecraft : input function for the thrusters to validate the spatial system using Mecano : transfer function representing the linear elements for the two-mass system with the asymptotic state estimator : transfer function of the second-order compensator : transfer function representing the linear elements for the two-mass system wit h the rate estimator with linear compensation : transfer function of the second-order filter : transfer function representing the linear elements for the two-mass system with the new rate estimator with linear compensation : transfer function representing the plant of the system : transfer function representing the linear elements for the two-mass systern with the classical rate estimator : transfer function of the differentiator-filter

: (2,j) component matrix of Mi1 : (i,j)component of the 3 x 3 matrix Ha3 : generalized inertia matrix of the rigid spacecraft : spacecraft inertia dyadic (inertia matrix written in a basis-independent way. ) : ith-body inertia dyadic for the space manipulator system : moment of inertia of the spacecraft about the principal avis &: k = x,y, z, and K = X, Y,Z : moment of inertia matrix of the rotors : Jacobian matrix for the space manipulator system : performance index : stiffness matrix : generalized stiffness matrix of the space manipulator system

: generalized stiffness matriv of the one-link planar system : stiffness matrix : stiffness matrix in Mecano : spring stiffness of the two-mas system : ith joint torsional spring stiffness of the space manipulator system : spring stiffness of the one-link planar system : gain matrix in the asymptotic state estimator : ith component of 1 : gain vector in the asymptotic state estimator for the single-output systems : generalized mass matrix for the space rnanipulator systern : generalized mass matrix for the one-link planar system : assembled mass matrix in Mecano : total mass of the space manipulator system : mass of the base in the two-mass system : mass of the payload in the two-mass system : total mass of the two-mass system

xviii : ith body mass for the space manipulator system : number of joints in the space manipulator system

: describing function for the relay with a dead zone : describing function for the relay with a dead zone and hysteresis : amplitude of the moment produced by the thrusters about the I(o âuis; k=x,y,z, and K =X,Y,Z : vector of Coriolis and centrifuga1 forces for the space manipulator system : vector of nonlinear velocity terms in the spacecraft equations of motion : vector of nonlinear velocity terms for the one-link planar system : vector of the external moments applied to the spacecraft : external moment about the Ka axis; k = x, y, z, and K = -Y.Y, Z : external moment applied to the spacecraft of the one-link planar system : zero rnatrix : origin of the inertial frarne

: modal matrix composed of the eigenvecton of W : arbitrary point of the space manipulator system : inertial position vector of P : position vector of point P with respect to the i-th body CM : rotation matrix carrying Fr into F0 : vector of generalized coordinates

: vector of generalized coordinates for the one-link planar system : reduced vector of generalized coordinates : i-th component of q : Rayleigh's dissipation function for the space manipulator system : rate of fuel consumption : constant rate of fuel consumption at steady-state, obtained from F:(t) : maximum rate of fuel consumption for stability : iteration matrix of the implicit Newrnark method in Mecano : rnatrix rnapping Euler rates 5 to the angular velocity [w& : sin(x), x = @, #, O : state transformation rnatrix (x' = Tx) : total kinetic energy in the systern : kinetic energy of the rotors for the space manipulator system : kinetic energy of the manipulator for the space rnanipulator system : Laplace transform of the scalar input ü(t) : control input vector of the state-space representation : command of the thrusters, attaining values of +l, O, or -1 : input variable of a Single-Input Single-Output (SISO) system : command of the thrusters for the k-th suis : observability rnatrix : total potential energy for the space manipulator system : uncorrelated white-noise processes, for i = 1,2 : dynaniic matrix for the space manipulator system

: output of the nonlinearity in the describing-function method : state vector : estimate of the state vector : error in the estimate of the state vector : spacecraft CM coordinate for the one-link planar systern : Laplace transforrn of the scalar output Q(t) : output vector of the state-space representation : estimate of the output vector of the state-space representation : output variable of a SingleInput Single-Output (SISO) system : position of the base for the two-mass system : estimate of the position of the base for the two-mass system : position of the payload for the two-mas system : position of the centre of mass for the two-mass system Y~I : desired position of the base for the two-mass system Y f : relative displacement of the payload with respect to the base for the two-mas system

Ys : spacecraft CM coordinate for the one-link planar systern

ZC: : unit vector along the axis of rotation of the k-th joint

Qi : constants defined in the Mecano user-element , i = 1,2,3 p : ratio of the mass of the payioad over the mass of the spacecraft 7 : acceleration of the two-mass system 70 : nominal acceleration of the two-mass system

?'k : angular acceleration about the Ko axis of the spacecraft, for k = r), @, 0, and K = .Y, Y, Z I'r : amount of hysteresis included in the relay At9 : vector of the relative deformation at the joints Ti: minimum operating time of the thrusters 6 : set of Euler angles to describe the orientation of the spacecraft 6 : attitude or dead zone limits &) : deviation from an operating point C : damping ratio of the two-mas system

Ci : i-th damping ratio for the spatial system 6 : damping ratio of the second-order filter CP : damping ratio of the poles of G,(s)

Le : damping ratio of the differentiator-filter L : damping ratio of the zeros of G,(s) 0, : rotor-joint variable vector 0, : link-joint variable vector 8 : angular rotation about the 2-axis in the ZYX Euler-angle description ê : estimated or filtered attitude e : estimated attitude rate 00 : spacecraft attitude of the onelink planar system hi : angulm position of z-th link for the space manipulator system

82i- 1 : angular position of the motor of joint i for the space manipuletor system 0, : estimate of the attitude of the system CM gd : desired attitude 6

$f : deviation of 6, from the actual attitude of the spacecraft, 8 A : modal damping matrix A : negative inverse of the slope of the switching lines, or the velocity gain

Aefl : effective A, Aer = X - rd Ai : i-th nonzero eigenvalue of W C1 : equivalent reduced mas of the two-mass system n : total power developed by driving devices supplying control forces n, : sum of al1 power developed by external forces on the spacecraft P : position vector of point P with respect to C : variance of a stable motion anoise : variance of the white noise

T : joint torque vector for the space rnanipulator system

T : time delay of the sensor reading

7d : time delay due to the relay operation ri : ith-joint applied torque for the space manipulator system 4 : vector of external forces and moments applied to the spacecraft

@P : vector of extemal forces and moments applied to the spacecraft of the one-link planar system 4 : angular rotation about the Y-axis in the ZYX Euler-angle description @ : angular rotation about the X-axis in the ZYX Euler-angle description f? : modal decoupled frequency matrix

00 : spacecraft angular velocity vector

Wi : angular velocity vector of the i-th body w : frequency of the sinusoidal input in the describing function method 4 : natural frequency out of the plane containing links 2 and 3 of the spatial system

Ui : i-th natural frequency for the spatial system 4 : i-th natural frequency in the plane containing links 2 and 3 of the spatial system, z = I,2

Wf : cutoff frequency of the second-order filter cJn : resonmcc frcquency of the two-mass systcm

W~ : frequency of the poles of GJs) use : cutoff frequency of the differentiator-fil ter

UC : frequency of the zeross of G&) r.1 : roof function of the real argument (-) Chapter 1

Introduction

1.1 Robots in Space

In the last few years, robotics has begun to play a very important role in space exploration and exploitation. Space robots are expected to become an increasingly vital part of future space operations. Not only will they be used for the assernbly and fabrication of large space structures, but also for in-orbit service and repair activities. It is expected tbat the mission cost and hazards of human orbital presence will be reduced by minimizing the need for astronaut Extra Vehicular Activity (EVA). The control of space manipulators in this context brings about various chal- lenges. For example, unlike terrestrial robots, space manipulators are mounted on free-floating bases. Since robots are likely to carry large payloads compared to the rnass of the spacecraft, large disturbances may result at the base, thereby causing the end-effector to miss its target. Moreover, structural flexibility is present in space robots, as they are required to be lightweight and to have large workspaces. Currently, the only operational space manipulator available is the CANADARM, which is a six-degee-of-freedom 15 m long arm, weighing nearly 400 kg. This manipulator, shown in Fig. 1.1, was designed by a Canadian team in cooperation with NASA. It is prïmarily used for deploying or retrieving satellites and space modules Chapter 1. Introduction 2

Figure 1.1: The CANADARM mounted on the Space Shuttle. in orbit. .\ larger and more advanced version of the CANADARM is currently under design for the International Space Station and will be the contribution of Canada to this international project. This manipulator, called SSRMS, for Space Station Re- mote Manipulator System, will assist in the construction, operation and maintenance of the Space Station. These two teleoperated manipulator systems operate in Low Earth Orbit (LEO) and are limited to their own reach, thereby requiring very long links to have a large workspace. However, we can imagine for the future a completely autonomous robot mounted on a spacecraft that will be able to grab, dock and manipulate while in orbit. For example, it could go in a Geostationary Orbit (GEO) at 35,800 km from the Earth, pick up a satellite and bring it back to the Space Station for maintenance. Two interesting concepts are the U.S. Flight Telerobotic Servicer (FTS) shown in Fig. 1.2(a), and the Japan NASDA OSV of Fig. 1.2(b). Such free-flying systems are to be equipped with thrusters, manipulators, several visual sensors, a high-gain antenna, and a docking mechanism. .41so, they are to be teleoperated from Earth or from orbit through a satellite link. Moreover, there are currently two kind of spacecraft equipped with manipulators Cha~ter1. Introduction 3

Figure 1.2: Two concepts of free-flying robots. (a) U.S. Flight Telerobotic Servicer (FTS),(b) Japan NASDA OSV. that are built and being or ready to be tested in space. The ETS-7 spacecraft, built by NASDA in Japan, was launched on November 28, 1997. This spacecraft is cornposed of a chaser satellite and a target satellite used to carry out experiments to confirrn the basic technologies for rendezvous-docking and space robotics. Space robotics experiments are being conducted by teleoperation from the ground using the satellite chaser robot amand exchange of equipment in orbit using the space robot. On the other hand, the US Ranger TSX, designed in a joint effort between the University of Maryland and NASA, including other universities and industry, is scheduled to fly late in 2000 aboard the Space Shuttle. This free-flying spacecraft equipped with four manipulator arms used for dexterous manipulation, grappling and video-sensing will perform, among others, telerobotic servicing on the International Space Station. The data from the on-orbit experiments will be correlated with the data obtained from a cornpanion vehicle, the Ranger NBV, designed for buoyancy environment. As we can see, there are many possibilities for robots in space, which is why extensive research is currently being conducted to further improve and develop Chapter 1. Introduction 4 new technologies in this new field of robotics.

1.2 Literature Survey

The problem of controlling a spacecraft with varying natural frequencies, as for example, a space robot taking various configurations and payloads, has not received much attention in the literature, especially ahen using on-off thrustcrs. Ho~vcvcr, many related subjects have been t horoughly studied. Many researchers are st udying the dynamics and control of spacecraft assuming rigid body motions and continuous control laws. On the other hand, a considerable amount of literature on the control of flexible spacecraft using both continuous and on-off control laws can be found. The problem of controlling large flexible space structures is also the focus of many researches. These three topics are treated separately in the next t hree su bsections. Finally, in Subsection 1.2.4, a literature survey is reported on the interaction between payload and attitude controller in space robotic systems.

1.2.1 Dynamics and Control of Space Robots

The kinematics, dynamics, and control of space robotic manipulators are much more complicated than their counterparts on Earth due to the dynamic coupling between the manipulators and their host spacecraft. Several control schemes have been proposed for such systems. Most of them assume that the rnanipulator rnoves sufficiently slow to neglect the flexibility in drives, shafts, links, and gear transmissions. These control met hods can be classified in three major categories. In the first one, the position and the orientation of the spacecraft are controlled by jet thrusters and reaction wheels, or a combination of both, to compensate for any manipulator dynamic forces exerted on the spacecraft, the base of the manipulator thus being of a free-flying type. In this case, the spacecraft is kept almost stationary, the control methods for ground-fked robots thus being applicable. The kinematic problem is consequently Chapter 1. Introduction 5 relatively simple. However, the use of these control methods is limited, due to the relativeiy high fuel requirements and the possibility to saturate the reaction-jet system (Dubowsky, Vance and Torres, 1989). To minimize these problerns, Nenchev, Umetani and Yoshida (1992) and Quinn, Chen and Lawrence (1994) studied motions of the manipulator arm that do not disturb the attitude of the spacecraft. With the same objective, Torres and Dubowsky (1992) developed an Enhanced Disturbanced Map (EDM) used to suggest paths for a given manipulator that result in low-attitude fuel consumption. In the second category, reaction wheels or jet thrusters are used to control the attitude only. The centre of mass of the spacecraft, however, is still free to translate in response to the force disturbances from the robot and its payload. This is an interesting approach, since electrically-driven reaction wheels can be used, t hereby reducing the fuel consumption while keeping the attitude fixed when necessary, as for antennae pointing towards the Earth. The control problem is more cornplicated than in the first category because the relative disturbance translation of the payload with respect to the spacecraft must be taken into account. This problem was addressed in various ways in (Longman, Lindberg and Zedd, 1987; Vafa and Dubowsky. 1987; Vafa and Dubowsky, 1990a,b; Lindberg, Longman and Zedd, 1990). In the third category, the free-floating case, no actuators are used to control the position and orientation of the spacecraft. Therefore, the spacecraft is free to move in response to the manipulator motion. This control scheme has the advantages that no fuel is required to control the spacecraft and that the risk of collision of the robot end-effector with an object about to be grasped, resulting from the attitude control thrusters suddenly firing, is eliminated. However, path planning becomes much more complicated than before, because the platform is floating and, therefore, as shown in (Lindberg, Longman and Zedd, 1990), the position of the robot end-effector is no longer a function of the present robot joint angles, but rather of the whole history of these joint angles. The inverse kinetics problem (instead of inverse kinematics Chapter 1. Introduction 6 for ground-fixed robots) is very complicated and generally has an infinite number of solutions. The correct solution depends on the joint-space history; Longman (1990b) developed one of these solutions for the free-floating case at hand. Moreover, system singularit ies called "dynamic singularitiesn are pat h-dependent and t heir location in the workspace depends on the system dynamic parameters (Papadopoulos and Dubowsky, 1993). Despite this complicated dynamics, Papadopoulos and Dubowsky

(1991b) suggested that nearly any controi algorithm that cm be used for fixed- based manipulators can also be implernented for free-floating space robots with a few additional conditions. In other studies, coordinated controllers were designed to control both the spacecraft and the end-effector, and allowing the cornrnand of a desirable manipulator configuration and the planning of a system motion with thc use of thrusters (Papadopoulos and Dubowsky, 1991a; Papadopoulos and Moosavian, 1994; Moosavian and Papadopoulos, 1997). In al1 previous control schemes reported, no analysis has been performed that includes the actual flexibility of space robotic systems. In fact, one might presume that the vibrations of the robot arm will only induce attitude oscillations for the spacecraft and that, after these vibrations are damped, the spacecraft attitude would be the same as directed from the reaction-moment compensation torques derived for a rigid-body mode1 (Longman, Lindberg and Zedd, 1987). However, Longrnan (1990a) showed that such a presumption is false and that the most common situation is that the structural vibrations of the robot arm will try to tumble the spacecraft. Longman developed a general formulation to determine the satellite attitude control torque required to counteract robot motion disturbances that include the effects of robot flexibility.

1.2.2 Attitude Control of Spacecraft

The control schemes outlined in Subsection 1.2.1 that require thmsting actions assume the use of reaction jets that provide forces and torques proportional to the Chapter 1. Introduction 7 commanded control input. Unfortunately, this is never the case in space, since such technology is still not applicable and only on-off thrusters can be used to control the position and attitude of the spacecraft (Anthony, Wie and Carroll, 1990). These on-off thrusters are nonlinear devices, the design of a control system becoming a very difficult probiem when flexible modes must be controlled. .4 detailed description of reaction thruster attitude control is available in (Sidi, 1997). The attitude control probleni of rigid spacecraft is first acldressed iri tliis subsection. Then, flexibility is introduced in the discussion. The control of flexible spacecraft with continuous control devices is first outlined, followed by a discussion on the control of flexible spacecraft using on-off thrusters. Although al1 these schemes assume an exact or approximate knowledge of the flexible modes, thus not being directly applicable to the case of space manipulators which have frequencies that Vary with payloads and configurations, some ideas could certainly be useful in future analyses and development of new control schernes.

Control of Rigid Spacecraft with On-off Thrusters

Currently, the cornmon approach to the design of control systems using on-off devices is to consider single-axis rigid-body motion and to define a switching logic for a single set of thrusters by the use of phase-plane analyses. The optimal-fuel problem for this kind of rigid-body motion appears in many textbooks, e.g., in (Bryson and Ho, 1975). Moreover, D'Amario and Stubbs (1979) developed a single-rotation-mis autopilot that accurately compensates for nonlinear angular-velocity coupling effects to achieve nearly maximum jet-torquing capabilities and facilate rapid attitude manoeuvre of a space vehicule. In the same period, Bergmann et al. (1979) developed a new spacecraft autopilot, capable of controlling vehicles of arbitraxy design with changing mass properties and jet failure status, using the phase-space control law concept. More recently, Agrawal and Bang (1995) proposed a switching function for single-axis spacecraft slew manoeuvre that provides robust control performance in the presence Chapter 1. Introduction 8 of modelling errors, while eliminating double-sided firings. Sirnilarly, Cristi, Burl and Russo (1994) developed an adaptive controller for rotational manoeuvres of a rigid body with considerable dynamic uncertainties. The inertia matrix is supposed to be unknown, which might be the case of a space manipulator designed to pick up objects of various sizes and weights. Several other researchers have addressed this problem over the years. One may mention the work of Carrington and Junkins (1984), and Wie and Barba (1983) as examples. .A cornprehensive review on the subject was performed by Singh, Kabamba, and McClamroch (1989). However, the actual space structures are likely to be flexible and their control using tbese nonlinear devices may interact with the structural modes creating instabilities that can be manifested as limit cycles (Millar and Vigneron, 1979).

Control of Flexible Spacecraft with Continuous Control Devices

Many researchers have dealt with the problem of controlling the attitude of a tlexible spacecraft assuming the use of continuous torques, as those provided by momentum gyro wheels or proportional valve thrusters not yet available for space applications (Anthony, Wie and Carrol, 1990). Thompson, Junkins and Vadali (1989) derived an open-loop near-minimum time manoeuvre with control of flexible modes for single- avis manoeuvres by shaping the control profile with two independent parameters. The more general case of accomplishing large angle, nonlinear, three-dimensional attitude manoeuvres in either near minimum-time or near-minimum fuel is addressed in (Bell and Junkins, 1994). Chenumalla and Singh (1994) proposed a control system design approach for large-angle rotational manoeuvres of elastic spacecraft based on nonlinear inversion and singular perturbation t heory. The spacecraft dynarnics is decomposed into a slow subsystem and a fast subsytem. A dual level control scheme was also proposed by Meirovitch and OZ (1980). This control scheme, for the positional, attitude and elastic motions, is demonstrated for nonoptimal and optimal proportional controls, as well as for nonlinear on-off control. This approach is particularly Chapter 1. Introduction 9 suitable for high-order systems and is free of control spillover into the uncontrolled modes of the truncated dynamical mode1 considered. Similarly, Lin and Lin (1995) proposed a three-part composite control strategy allowing simultaneous multi-mis attitude reorientation of a spacecraft with flexible structures by minimizing control energy, eliminating structure deflections, and reaching a specified final attitude over a given manoeuvre time. More recently, Tham et al. (1997) proposed two different control schemes, one using a classical method and the other one using H, control theory. These two methods were also used in (Kida et al., 1997) to design three different controllers that were tested succesfully during an on-orbit attitude and vibration control experirnent. Using quaternions, Kelkar and Jashi (1996) derived a nonlinear control law that shows global asymptotic stability in large angle manoeuvres with superior performance and reduced control energy requirements, as compared to more classical linear control laws. A nonlinear feedback control law based on Lyapunov control design methods was also derived in (Schaub, Robinett and Junkins. 1996), but utilizing the unique properties of a recently developed set of attitude parameters, the modified Rodrigues parameters. Finally, Silverberg (1986) derived control laws to achieve uniform damping of the modes of a flexible spacecraft. It was shown that these control laws are decentralized, independant of structural stifhess and proportional to the distribution of mass over the spacecraft.

Control of Flexible Spacecraft with On-off Thrusters

kIany researchers have also addressed the problem of controlling a flexible spacecraft using on4thrusters. Wie and Plescia (1984) designed an on-off pulse modulator attitude control system using the describing-function analysis for a spacecraft having large flexible solar arrays. They used the relative stability rnargin, with respect to the limit-cycle condition of a structural mode, as a measure of the robustness of the nonlinear control system. Using the same idea, Anthony, Wie and Carroll (1990) showed that the describing-hnction analysis can be utilized for practical Chapter 1. Introduction 10 control design problems such as flexible spacecraft equipped with pulse-modulated reaction jets. Hablani (1992a) developed a rnethod to optimize the pulse-width of the thrusters for fast active damping of flexible modes, without destabilizing the rigid-body modes. Adaptive bandpass filters were used to obtain an accurate measure of the mode frequency, which was known imprecisely before. In another paper, Hablani (1994) also showed that the integral pulse frequency modulator (IPFM) reaction jet controllers can perform, with precision: multiavis cracking of moving objects, attitude cont rol of flexible spacecraft under orbit-adjustment Forces, single- axis slews, and vibration suppression. hssuming that control moment gyros or other interna1 mechanisms were available for proportional fine control, Skaar, Tang and Yalda-Mooshabad (1986) developed an open-loop control scheme using only t hree switching times for rest-to-rest manoeuvres. These switching times were chosen to minimize residual elastic energy at the end of the reorientation. A similar open-loop scheme was proposed by Hablani (1992b). Similarly, Singh, Kabamba, and McClam- roch (1989) considered the time-optimal, single-axis, rest-to-rest rotational manoeuvre of an elastic spacecraft. They derived a tirne-optimal open-loop control law that is restricted to linear , elastic, undamped, nongyroscopic systems. Bedrossian et al. (1995) derived an optimal open-loop, feed-forward thruster pulsing strategy, which is independent of the plant model, to minimize structural loads. An interesting approach was provided by Williams, Woodard and Juang (1993) to avoid the repeated on-off transients that occur with traditional thruster attitude control. They minimize the possible excitation of spacecraft flexible mode by constantly firing the thrusters throughout the manoeuvre and controlling the slewing of the spacecraft by gimbaling the thruster using proportional control laws. The innovative technique of input shaping to control flexible structures has received extensive attention during the past few years. This open-loop control met hod, introduced by Singer and Seering (1990),is meant to avoid residual vibrations at the end of the manoeuvre, and usually requires variable-amplit ude act uat ion force, which Chapter 1, Introduction 11 cannot be attained with on-off thrusters. Some heuristic methods for extending input shaping to the case of on-ofF actuators have been developed (Rogers and Seering, 1996). Singhose, Derrezinski and Singer (1996) proposed a technique for designing command profiles for flexible spacecraft equipped with on-off reaction jets that is significantly more robust to modelling errors than similar techniques previously reported. This technique was fiirther improved in (Singhose, Bohlke and Seering, 1996) for fuel efficiency. In (Singhose, Singh and Seering, 1997), the input shaping command was designed considering a pre-specified amount of actuator fuel. Al1 these methods did not consider the amplitude of transient-induced oscillations, w hich was the case in (Singhose, Banerjee and Seering, 1997). Finally, the case of controlling multimode flexible spacecraft was investigated in (Singhose, Pao and Seering, 1997). It was shown that the method has a reasonable level of robustness to errors in the low modes, but the robustness to errors in the second mode varies greatly and is poor for large ranges of system parameters. Input shaping techniques were tested in a zero-g environment aboard the Space Shuttle Endeavour in March 1995. These techniques were found to be simple and effective strategies for reducing vibrations in flexible space structures (Tuttle and Seering, 1997). The techniques can be used alone or with closed-loop control techniques.

1.2.3 Control of Large Space Structures (LSS)

Control system design for large flexible space structures is a challenging problem. For various practical reasons, the control system must be designed with a reduced order mode1 of the plant. The interaction between the control system and the modes that have not been taken into account in its design (the residual modes) is known as the spillover effect (Preumont, 1988). There is a considerable amount of publica- tions dealing with this problem, as can be seen in the literature reviews by (Balas, 1982) and (Hyland, Junkins and Longman, 1993). The controllability of an isolated mode of a flexible structure is straightforward to veri&, as is the controllability of Chapter 1. Introduction 12 exactly repeated modes. Williams (1991) derived simple expressions for the degrees of controllability of near-repeated modes of flexible structures. The more popular problem of controlling LSS using continuous control devices is outlined in this section, followed by a discussion on the more complicated problem of controlling LSS using on-off thrusters.

Control of LSS with Continuous Control Devices

Most researchers consider that continuous torque levels are available to control the given space structure. Many authors addressed this problem using a two-level hier- archicai control strategy where the lower level consists of local controllers based on a reduced-order-modal model of the structure, and the higher level is a stabilizing compensator to account for any instabilities caused by controller-structure interactions with unmodeled dynamics (Das and Balas, 1989; Grewal and Modi, 1996). Wie, Hu and Singh (1990) investigated the effects of multibody dynamic interactions on attitude control and momentum management of the International Space Station. This study indicates a need for an adaptive and robust control or gain scheduling for the large payload manoeuvres on the Space Station. Liang and Balas (1990) derived an adaptive control scheme based on a reduced-order model, while Bishop, Paynter and Sunkel (1991) investigated an adaptive control approach that provides closed-loop stability after a large inertia change, without considering the flexibility of the Space Station. More recently, Griffin and Maybeck (1997) showed that the moving- bank multiple model estimator/controller (MMAE/M MAC), which consist of a parallel bank of Kalman filters, used to estimate the position and velocity of the bending modes of the structure, and linear quadratic control techniques, is a well-suited method of estimating variations in the vector of undarnped natural frequencies and quelling vibrations in the structure. Bennett et al. (1993) demonstrated the potential for the nonlinear adaptive control of multiaxis, large-angle slewing of Chapter 1. Introduction 13 spacecraft with elastic structural interactions, using an explicit structure of the dynamics arising from Lagrange's equation, thus simplifying the derivation of nonlinear control laws. On the other hand, Balas (1978) used a phase-locked loop prefilter to virtually eliminate observation spillover, thus allowing for very good control performance using a straight-brward linear feedback control law for controlling N modes of a flexible system, while Preumont (1988) applied nonlinear saturation state feedback to the control of vibrations and reduced the spillover effects by reducing the magnitude of the control in the neighbourhood of equilibrium. This alleviation procedure can be used with any type of state feedback (e.g. linear). In a recent paper, Joshi, Maghami and Kelkar (1995) proposed a class of dynamic dissipative compensators which ro- bustly stabilize the plant in the presence of unmodeled dynamics and parametric uncertainties. Both attitude control and vibration supression can be achieved using a controller that uses both attitude and rate measurements. Balas and Doyle (1994) synthesized controllers suit able for lightly damped, flexible structures with noncol- located sensors and actuators using the structural singular value (p) control design technique in the presence of uncertain flexible modes in the controller crossover re- pion.

Control of LSS with On-off Thrusters

Some authors also adressed the more complicated problem of controlling the flexible modes using on-off thrusters. Chu et al. (1990) proposed a step by step design procedure for the preliminay design of the Space Station. The robustness to plant variations is considered using gain scheduling. The paper illustrates how traditional analysis and design tools can be succesfuily applied, with insight derived from recent research on control-structure interaction, to prelimintlry control system design for a large space structure such as the International Space Station. On the other hand, Vander Velde and He (1983) implemented the phase-plane approach to design Chapter 1. Introduction 14 a control system for a flexible space structure of any order, while using any nurnber of thrusters, based on an approximation to an optimal control formulation. Barbi- eri and Ozgüner (1988, 1993) analyzed the problem of minimum time, rest-to-rest slewing of a flexible structure by means of phase-plane techniques. They proposed a state-dependent optimum control law for a linear one-bending mode mode1 of a flexible structure. Finally, Foster and Silverberg (1991) introduced a near-minimum fuel method for the on-off decentralized control of flexible structures where the fuel minimization is achieved by turning on an actuator when both the local velocity is in the neighbourhood of its maximum and when the local displacement is in the neighbourhood of its minimum. Unfortunately, al1 the methods reviewed in Subsections 1.2.2 and 1.2.3 assume a precomputed exact or approximate knowledge of the flexible modes. For a space robotic system, or for a multitask servicer as the Space Shuttle. the natural frequencies are always changing with the robot configuration or the payload carried.

Therefore, these control methods are not applicable, addit ional research t hus being needed.

1.2.4 Payload- Attitude Controller Interaction

The Space Shuttle Reaction Control System (RCS) is similar to the one that flew in .2pollo missions. It evolved under the assumption that the Orbiter is sufficientiy rigid to allow the use of rigid-body mechanics in the description of Orbiter response to RCS activity (Sackett and Kirchwey, 1982; Cox and Hattis, 1987). No special provision was taken to include structural flexibility in the RCS design that is briefly described in (Hattis, 1984; Sackett and Kirchwey, 1982; Nakano and Willms, 1982). Only possible high-frequency structural vibration, as in traditional spacecraft, were considered, which is vaiid for many payloads. The RCS consists of 38 large prîmary reaction control system (PRCS)thrusters used for translation and coarse rotation control, and 6 small vernier reaction control system (VRCS)thrusters used for fine Chapter 1. Introduction 15 rotation control. The qualitative differences between these two sets of thrusters are illustrated in (Hattis, 1984). However, at the time of payload deployment, with or without the CAN.4DARM system, flexibility becomes important. The structural modes can have rather low frequencies and can be excited by the RCS activity. Sackett and Kirchwey (1982) looked at the performance degradation of the RCS due to the deployment of a flexible payload by various means. They grouped these dynamic interaction possibilities in order of increasing severity:

1. control effects - fiexibility either induced additional firings or oniitted some of t hese;

2. structural motion and load response to typical, aperiodic jet-firings;

3. structural resonance due to periodic jet-firing caused by rigid-body flight control system (FCS) response to disturbance accelerations;

4. closed-loop instability, where RCS firings cause flexure, which passes through the inertial measurement units (IMU) and state estimator causing RCS firings. which reinforces fiexure and continues to eventually reach a limit cycle.

After conducting extensive simulations, Sackett and Kirchwey concluded that the judicious selection of control parameter values and careful operational procedures. based on a knowledge of the payload structural characteristics, can reduce dynamic interactions and load problems. Included are restrictions on parameters such as angular rate limits, attitude deadband, and pulse size, the use of the PRCS may thus become prohibitive. Operational restrictions are sometimes imposed, such as limiting the rate of rotation manoeuvres, use of free drifts, or carefully planning the timing of jet-firings. In some cases, payload design has been affected by dynamic interaction considerations (Cox and Hattis, 1987). Penchuk, Hattis and Kubiak (1985) used the describing-function met hod to analyze the problem of a payload deployed by means of a tilt table with a pivot near the Chapter 1. Introduction 16 aft end of the Space Shuttle. Stability maps were obtained and compared to simulation results to validate the describing-function analysis. In (Redding and Adams, 1987), a new attitude controller based on fuel-optimal manoeuvres was developed for the Space Shuttle, while Kubiak and Martin (1983) developped a new design for the RCS to reduce the impact of large measurement uncertainties in the rate signal during attitude control. In both cases, the performance of the RCÇ is increased significantly for rigid-body motion. However, Kubiak and Martin did not deal with the flevibility problem and only mentioned that by diminishing the required firings, the iikelihood of structural problems diminishes. This interaction pro blem was studied using a single-mode, linear t ranslat ional mechanical system to approximate the dynamic behaviour of a two-flexible-joint manipulator mounted on a three-degree-of-freedom (dof) base with a constant system damping ratio (Martin' Papadopoulos and Angeles, 1995), and with a variable one (Martin, Papadopoulos and Angeles, 1996~). A model-based state-estimator and design guidelines were suggested to minimize such undesirable dynamic interactions, as well as thruster fuel consumption. These results were validated in (Martin, Pa- padopoulos and .Angeles. 1996a; Martin, Papadopoulos and Angeles, 1996b) using a more realistic planar mode1 with rotational degrees of freedom. Two new estimators that are not model-based were suggested in (Martin, Papadopoulos and Angeles. 1998) to solve the problem of dynamic interactions using the mode1 of a spatial system with a 6-dof base derived in (Martin, Papadopoulos and Angeles, 1997). The future International Space Station is also likely to have control-structure interactions. Quinn and Yunis (1993) and Cheng and Ianculescu (1993) demonstrated the potential control-structure interaction problems of large flexible multibody stnic- tures as the planned International Space Station configured with solar dynarnic modules. Moreover, Pinnamaneni and Murray (1986) analyzed the problem of dynamic interactions by working with phase-plane rigid-body limit cycles by extracting the Chapter 1. Introduction 17 frequency contents of the corresponding thrust profile as a function of the limit cycle period and deadband. After performing a finite-eiement vibration analysis of the Space Station to obtain the significant flexible modes, they chose the limit-cycle period and deadband such tliat dominant modes are not excited.

1.3 Objectives and Organization of this Thesis

1.3.1 Problem Formulation and Objectives

As mentioned in Section 1.1, space manipulators are structurally flexible mechanical systems. When the free-flying base of the manipulator is controlled by the use of on-off thrusters, which produce a rather broad frequency spectrum that can excite sensitive modes of the flexible system, dynamic interactions are likely to occur. The excitation of t hese modes can introduce further disturbances to the attitude control system; therefore, undesirable fuel replenishing limit cycles may devclop. In those cases, thrusters fire without stabilizing the base and too much fuel is consumed for almost nothing. Since fuel is an unavailabie resource in space. the consequences of such interactions can be extremely problematic. In the case where the natural frequencies are dependent upon the payload and the configuration of the system, as for a free-flying robot, the current method for solving these problems is to perform extensive simulations to examine the possibilities for dynamic interactions. If these occur, corrective actions are taken, which would include adjusting the RCS parameter values, or simply changing the operational procedures (Sackett and Kirchwey, 1982; Penchuk, Hattis and Kubiak, 1985). Hence, classical attitude controllers must be improved to reduce these dynamic interaction possibilities. In this thesis, it is intended to mode1 the foregoing problems for a general space manipulator mounted on a free-flying base controlled by on-off thrusters, and to develop control methods to reduce these undesired effects. Approximate CANADARM flexible modes are used to make the mode1 behaviour more realistic; however, the Chapter 1. Introduction 18 analysis is not restricted to this robotic system. The dynamic interactions are modelled for the worst case that can occur, i.e. when the system is limit-cycling. Since the describing-function method has been shown to be helpful for such nonlinear systems (Wie and Plescia, 1984; Anthony, Wie and Carroll, 1990; Penchuk, Hattis and Kubiak, 1985), this method is dapted for the complicated system under study and used to help in the control system design. The main objective of this thesis is to developed control methods that are intented to reduce the undesired etfects of dynamic interactions. The proposed control methods are developed only for attitude-keeping but should be applicable for translational motions.

1.3.2 Research Tools

Most of the derivations of this thesis are executed in the Maple symbolic computation environment. The dynamics equations are thus derived using Maple which can save them in a C-format, to be compiled with the "cmex" function of Matlab to create an S-function. Finally, the S-functions are fed into Szmulznk in order to implement control algorithms and perform the required simulations. Simulations are also produced using Mecano, a module of Sarncef. Sarncef is a finite element sotfware package developed at the Laboratoire de Techniques Aéronau- tiques et Spatiales (LTAS) of Université de Liège, Belgium. Mecano was chosen for Our studies due to its ability to accurately model systems with flexible elernents. Since the package has built-in elernents allowing the modelling of ngid bodies and flexible links and joints, the modelling of flexible space robotic systems was straightforward. However, it can be noted that Mecano is only a numerical software and thus, it cannot provide any symbolic code of the model that could be used in analysis and control system design. Mecano was also used to validate the dynamics model derived using Maple and run inside Simulink. Generai 3-D manoeuvres were performed and the models were adjusted until the simulation results coincided within a very small error. Chapter 1. Introduction 19

The Open Inventor animation package was also used to develop a graphical animation of the model in order to demonstrate the results of cornputational simulations. The graphical animation was run on an SGI Indigo 2 workstation, with a 4400 processor and yields smooth animations of the manoeuvres.

1.3.3 Thesis Organization

The mathematical model of a rigid spacecraft is developed in Chapter 2. This model is used to discuss the basic attitude control theory using on-off thrusters. The classical attitude control scheme used throughout the thesis is described and some analysis methods to study nonlinear systems are introduced. Finally, the stability definitions used in this thesis are introduced. In Chapter 3, a general formulation for the modelling of space robots is introduced and the model of a three-flexible-joint manipulator mounted on a six-dof spacecraft is derived and validated using Mecano. In addition' the equations of motion of the space manipulator are linearized about an operating point and general expressions for the manipulator natural frequencies are derived. It is also shown how to use the commercial software package Mecano for control purposes. The control of space free-flyers with flexible manipulators mounted on them is studied in Chapt er 4. The conventional control scheme is analyzed t heoretically and a parametric study is performed to show the influence of various parameters. A model-based asymptotic state estimator is described and studied in detail, both theoretically and using simulations to show the increase in performance that can be at tained using t his part icular est imator . In Chapter 5, two new estimators that are not model-based are introduced. The first one is very efficient in addressing the problem of dynamic interactions, while the second one can also stabilize systems with a diverging motion. This estimator shows a similar performance to the model-based asymptotic state estimator derived in Chapter 4. The effect of link Bexibility is also described using Mecano with a Chapter 1. Introduction 20 planar example. Finally, Chapter 6 concludes the t hesis by summarizing the results of t his study, and then outlining some recommendations for future work.

1.4 Contribut ions

Ti, the best of the author's knowledge. the following dcvcloprncnts rcportcd in this thesis are original and have not been reported elsewhere. The main contributions of this work are:

a Development of a model-based asymptotic state estimator that reduces significantly the problem of dynamic interactions;

0 Development of a rate-estimator with second-order linear compensation and guidelines for the pole-and-zero placement such that the dynamic interactions are reduced for various payloads and configurations;

0 Development of a new-rate-estimator with second-order linear compensation that can stabilize systems with diverging motion, thus allowing more freedom in the choice of parameter values for optimal attitude control performance.

To a les important level, the two following contributions were developed to facilitate the development of the above-mentioned contributions:

a Adaptation and interpretation of the describing-function concept in the analysis of space robotic systems controlled with on-off thrusters via the root-locus method;

a Development of a methodology to ailow the use of the finite element package Mecano for control purposes. Chapter 2

Modelling, Control and Analysis of Rigid Spacecraft

2.1 Introduction

As mentioned in Chapter 1, this thesis deals with the attitude control of spacecraft on which a manipulator is mounted. However, before looking into the details of such complex systems, the basic tools and definitions required throughout this thesis are first introduced in this chapter. First, the mathematical mode1 of a rigid spacecraft is developed in Section 2.2. This model is used in Section 2.3 to address the spacecraft control problem with on-off thrusters in order to describe a classical attitude control scheme used as a basis throughout the thesis. Since on-off thrusters are nonlinear devices, Section 2.4 discusses analysis methods for nonlinear systems. Finally, stability definitions used in this thesis are introduced in Section 2.5.

2.2 Modelling of Rigid Spacecraft

The dynamics model of a rigid spacecraft is obtained in this section using a La- grangian approach, neglecting microgravity effects and orbital mechanics. If c denotes the position of the spacecraft centre of mass (CM)C and wo its angular velocity, Chapter 2. Modeliing, Control and Analysis of Rigid Spacecraft 22 the kinetic energy expression of the system is given by

where no and Io are, respectively, the mass of the spacecraft and its inertia dyadic about its centre of mas. Note that the inertia dyadic of a body is a concept that defines an inertia matrix in a basis-independent way. The sum of al1 powers developed by driving devices supplying controlled forces is given by

where fs and n, are the resultant force and moment applied to the spacecraft with thrusters, expressed in the spacecraft frame 6 shown in Fig. 2.1. The -Yo,kb and Za axes of F0 are assumed to be parallel with the spacecraft principal aues. Moreover. if vector c is expressed in the inertial frame, Fr, it is expressed in Fo using the coordinate transformation

[CIO = QT(WI~ (2.3) where QI(6)is the rotation matrix carrying the inertial frame into an orientation identical to that of the spacecraft frame. 6 being a set of Euler angles chosen to describe the attitude of the spacecraft. The angular velocity wo can be expressed in terrns of the Euler rates b as [wa]o= ~o(6)b (2.4) where the matrix So(b) is chosen such that wo is expressed in 30. Matrices QI@) and So(S) are obtained in AppendLv A for a given set of angles S. Therefore, using Eqs. (2.3) and (2.4),Eq. (2.2) becomes

and the kinetic energy expression, Eq.(2. l),can be written as Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 23

Figure 2.1: -1six-dof rigid spacecraft.

In the realm of the Euler-Lagrange equations, we use q = [cT,6*lT, where c = [x,y, zIT and 6 = [@, #, BIT, as generalized coordinates. Then, applying the Euler- Lagrange equations

where the potential energy I' and the Rayleigh dissipation function R are bot h equal to zero, the equations of motion thus being obtained as

where Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 24 and QI(&)and So(6)are given in Eqs. (A.3) and (A.15), respectively, for a ZYX- description. Moreover, q6, se, c2@ and s2Q stand, respectively, for cos $J, sin $J, cos 2$ and sin 2@, with a sirnilar notation for and 8, and 101, Iovand Io, are the moments of inertia of the spacecraft about the XO,Y. and Zo principal axes, respect ive1y. As mentioned in Subsection 1.3.1, in this thesis our main concern is the control of the attitude of the spacecraft and not its position. Therefore, we have f, = O and Eq.(2.8a) reduces to c = O. Moreover, it is useful to simplify this dynamics mode1 for the case in which the rotation of the spacecraft is considered about only one ais, say Zo, assumed to be parallel to 2,. For this particular case and the ZIT-description Euler-angles, we have

,s) = qj = O rad

Thus, Eq.(2.8b) reduces to

Ioze= 12: as expected, for the rotation of a rigid body about one of its principal aues. Equa- tion (2.9) will be considered throughout Chapter 2 as an example to describe the basic theory about attitude control using on-off thrusters.

2.3 Classical Attitude Control Scheme

It is assumed in this thesis that the spacecraft attitude and position are controlled by jet thrusters. The technology currently available does not allow the use of proportional thruster valves in space, and thus, the classical PD and PID control laws cannot be used. Therefore, spacecraft attitude and position are controlled by the use Chapter 2. ivlodeiiing, Control and Analysis of Rigid Spacecraft 25

7 1 Output 21 estimate Controller Plant Output state - Estimator -

Figure 2.2: Standard spacecraft control scheme. of on-off thruster valves, that introduce nonlinearities. The classical way of dealing with such devices is to use a controller based on phase plane methods. Tliis controller requires the attitude and rate of the spacecraft as inputs. Since these two signals are not always available, a state estimator is needed to find an estirnate for the required States. Thus, the general control scheme for the spacecraft is illustrated with the block diagram of Fig. 2.2. In the following subsections. the controller and state-estimator blocks are considered separately to establish the models needed to analyze the problem at hand.

2.3.1 On-Off Thrusters Command

Simple Standard Controller Form

The usual scheme to control a spacecraft with on-off thrusters is by the use of the error phase plane, which have the spacecraft attitude error e as abscissa and the error rate é as ordinate. The on-and-off switching is determined by switching lines in the phase plane and can become very cornpleu, as, for example, the pliase-plane controller of the Space Shuttle (Sackett and Kirchwey, 1982). A simple switching logic used in this thesis is depicted in Fig. 2.3. The phase plane is divided into three regions, separated by two switching lines. The region between these two lines is a dead zone, whereby the thrusters are off. The right area is a zone where the thrusters are on in a given direction, while the left area is a zone where they are on in the opposite direction. The dead-zone limits [-6, 61 are determined by the attitude limit requirements, Chapter 2. Modelling, Control and Analysis of Rjgid Spacecraft 26 Rate error (é)

Figure 2.3: Switching logic in the error phase plane.

Figure 2.4: Controller block. the dope of the switching lines being given by the desired rate of convergence towards the equilibrium and by the rate limits. The equations of the switching lines are

where A is the negative inverse of the dope of these lines. Note that, for a constant 6, the smaller A. the higher the slope of the switching line and, therefore, the larger can be the rate errors. This control logic is sketched in block-diagram form in Fig. 2.4. It is cornposed of a relay nonlinearity with a dead zone. The input to this relay is the left-hand side of the switching lines, Eqs.(2.l0a & b), and the output is the command of the thrusters U, either +1, O or -1, based on the amplitude of the incoming signal. Chapter 2. Modelling, Coatrol and Analysis of Rigid Spacecraft 27 This control scheme is understood more easily with the analysis of a simple example, which is the subject of the subsection below.

Phase Plane for the Rotation of a Rigid Spacecraft

-4s an example, let us consider the rotation of a spacecraft about one of its principal axes with its dynarnics represented by Eq.(2.9), as derived in Section 2.2, namely?

where Jiz is the amplitude of the moment produced by the thruster and u is the command of the thruster as defined above. kforeover, Eq.( 2.11) can be written as

By noting that O = (d0/d0)(d0/dt)= 0(dé/d0), we caii write Eq(Z.12) as

Integration of Eq. (2.14) yields

where C is a constant. Let us assume that the initial conditions are @(O) = Bo and &O) = eo. Using these initial conditions, C is &en by

and Eq.(2.15) can be written as Chapter 2. Modeiiing, Control and Aaalysis of Rigid Spacecraft 28

(a) (b)

Figure 2.5: Parabolic phase plane trajectories: (a) for u < 0: (b) for u > 0.

If the desired attitude of the spacecraft is Bd = 0, the attitude error and the rate error can be defined as

Using Eqs.(2.18a Srb), along with eo = -Bo and èo = -Bo, Eq.(2.17) cm be written as 1 10: 1 lor .* e = (ea + --6:) - -- for u # O 2 Nzu 2 &ue This equation represents the equation of a parabola in the error phase plane for a particular set of initial conditions (eo, eo), the orientation of the parabola being determined by the direction of the thrusters firing, u. For a given inertia Io, and moment N,, the trajectories for a negative u are depicted in Fig. 2.5 (a) and the trajectories for a positive u in Fig 2.5(b). Therefore, the trajectories of the spacecraft in the error phase plane are a combination of parabolic paths when the thrusters are on and horizontal linear paths when the thrusters are off (constant-velocity coasting). Given a particular set of initial conditions in the error phase plane, Fig. 2.6 depicts the thruster control of the rigid spacecraft. The convergence rate is determined by Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 29

(è) t

Figure 2.6: Single-mass example. the inclination of the swi tching lines, a limit cycle being practically unavoidable in a zero-gravity environment and no disturbance forces. The srnaller the vertical dimension of this lirnit cycle. the smaller the fuel consumption. since the thrusters are firing for a very short period (during motions A-B and C-Dl. Next. to justify the fact that the limit cycle is unavoidable, we analyze the final limit cycle further. When a thruster is turned on, it will remain on for at least a minimum operating time ATmin.However, if we consider a perfect theoretical relay with no such mininium operating time, then we can imagine an impulse that will stop the spacecraft. Because of physical limitations, this is impossible in any practical system, which is why it was stated that the final limit cycle is unavoidable. In practical systems, hysteresis and time delays are present. The effects of these two parameters are described in the next subsection with the use of the same example. Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 30 Effects of Hysteresis and Time Delays

In practical relay systems, hysteresis must be included to avoid a phenornenon called chatter, which is well described in (Flügge-Lotz, 1968). When a thruster is chatter- ing, it turns on-and-off continuously, for a very short period ATmin. This behaviour can reduce considerably the useful life span of thrusters, while the addition of hysteresis can reduce the severity of this problem. Also, when relays are rnodelled, a tirne delay must be included since there is actually a delay between the time at which an open (or close) comrnand is sent to the valves and the time at which the valves open (or close). In this subsection, the effects of adding hysteresis and a pure time del- in the system are discussed. First, if hysteresis is included in the controller, the relay nonlinearity of Fig. 2.1 changes to that depicted in Fig. 2.7(a). The corresponding effect on the switching logic in the error phase plane is to add two switching lines. as shown in Fig. 2.7(b). This means that the turn-on switching lines (1 and 3) are not the same as the turn-off ones (2 and 4). Referring to Fig. 2.7(b). the equations of the switching lines are:

Since the turn-off switching lines are closer to the origin than the turn-on ones, the thrusters stay on for a longer period because they stop only when the turn-off switching line is reached instead of the original line, as when there is no hysteresis. Therefore, the more hysteresis A we add to the system, the larger the final limit cycle. A final limit cycle for the rigid-spacecraft example is presented schernatically in Fig. 2.8.

Now a pure time delay ~d is added to the hysteretic controller, such that when the controller sends the command to turn on or off the thrusters, the command is Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 31

Figure 2.7: Controller with hysteresis. (a) Relay nonlinearity, (b) Phase plane switching logic.

Figure 2.8: Single-rnass example with an hysteretic controller. Figure 2.9: Effects of a pure time delay on the switching logic. executed rd seconds later. The effect of the tirne delay in the error phase plane is to change the effective A, ,Aefl = X - rd, rnaking the switching lines more steep. as shown in Fig. 2.9. For the rigid spacecraft, the new switching line equations are (Graham and hIcRuer, 1961)

1 1V& 2. e+(X-~d)e = --- rd(2x- rd)+ b - il, e

2.3.2 State Estimation

-4 very important aspect in control system design is the design of state estimators, also known as state obsewers. Observers provide estimates for the states that are not readily available from measurements but are still required for feedback control, by using the states available by sensors. In the case at hand, the required states are the attitude and the attitude rate Chapter 2. Modelling, Control and Anaiysis of Rigid Spacecraft 33

Figure 2.10: Block diagram for the rate estimator only. of the spacecraft. Using current space technology, both States can be obtained by sensor readings. However, it can happen that only the attitude is available and the velocity must be estimated with the use of a state estimator, as for the state estimator presented in Fig. 2.10, which is similar to the one used on the Space Shuttle for on- orbit operations (Penchuk, Hattis and Kubiak, 1985; Sackett and Kirchwe- 1982:

Hattis, 1982). It uses the current angular acceleration yk of the system about one of the principal axes of the spacecraft, say Zo (ë, = y0 = !Vz/I0,), and the delayed attitude signal as input. Integrating the angular acceleration ëCimposed to the whole system by the thruster action, an estimate for the attitude rate 6, and the attitude 8, of the spacecraft can be obtained. However, since a spacecraft is more likely to have some flexible appendages like solar panels, antennae. manipulators, these estimates eC and 8, will not be the actual attitude or rate of the spacecraft. If we subtract this signal 8, frorn the sensor reading 8, we can define

where êI can be viewed as an estimate of the deviation of the attitude signal 0 frorn the attitude of the spacecraft 0=. Now, differentiating t his ''flexible-attitude" estirnate 9, and passing it through a filter to elirninate high-frequency noise, a "flexible-rate"

4 A estimate can be obtained. Then, adding the rate estirnate of the spacecraft, 8,,

C we finally obtain an estirnate for the angular velocity of the spacecraft 6. The differentiation of a noisy signal is usually not recommended because this Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 34 amplifies the noise level in the signal. However, in this case, only the flexible part needs to be differentiated. This means that, for a rigid system, no direrentiation is necessary. Therefore, this kind of state estimator can give very good results for cases of low flexibility. On the other hand, the attitude signal û is low-pass filtered to eliminate high frequency noise. The resulting signal O is used as input to the controller, together with 6. Because of their simplicity, second-order filters are used, as for the attitude controller of the Space Shuttle. The attitude fiiter can thus be represented wit h the following transfer function Gl(s):

The cutoff frequency uj niust be chosen to filter high frequencies such that the filter does not slow down the response of the system by reducing its bandwidth. Since. for any particular system, the exact frequency content of the noisy signals is not known, we will use wj as a parameter in Our study to examine its influence on the system performance. The damping term CI in Eq. (2.21) is chosen to be 0.707, which gives good performance, since it is relatively fast, with small overshoot (-4%) (Ogata, 1990). The differentiator-filter SC,,(s) is chosen similarly as

The cutoff frequency for the differentiator-filter is chosen as w,. = 0.2513 rad/s and the damping ratio as = 0.707. These two values correspond to the approximated ones used on the Space Shuttle, as explained in (Penchuk, Hattis and Kubiak, 1985; Sacket t and Kirchwey, 1982).

2.3.3 Mode1 Integration

This subsection introduces the complete model of the system considering the attitude controller described in Subsection 2.3.1 and the state estimator presented in Subsection 2.3.2. The complete model is shown in Fig. 2.11 and dlbe henceforth referred to as 'Lclassicalrate estimator" . For t his model, only the attitude is available Chapter 2. Modelling, Control and Andysis of Rigid Spacecraft 35

Figure 2.11: Mode1 with a 1-auis classical rate estimator.

for feedback. Since the attitude rate is also required, it is estimated with a differentiator cornbined with a filter G&). The attitude signal, in turn. is passed through a filter Gl(s). Finally, G,(s)represents the plant transfer function. In the mode1 of Fig. 2.11, only the attitude about one of the principal avis of the spacecraft is controlled. However, more generally, it is required to coritrol the attitude of the spacecraft about its three principal axes. such that any orientation of the spacecraft can be attained. In this thesis, we assume that the attitude of the spacecraft can be defined by a set of three Euler angles v, 4 and B. Iloreover, assuming small angular mot ions, Eq. (2.8b) reduces to

which is a set of three decoupled equations. Therefore, for small Euler angles, the motion of the spacecraft can be controlled by considering the same attitude controller of Subsection 2.3.1 and the estimator of Subsection 2.3.2 for each of its principal axis. The general control system required to control the orientation of a spacecraft in space is thus as sketched in Fig. 2.12, where the plant dynamics is described by the Chapter 2. Modelling, Control and Andysis of Rigid Spacecraft 36

/ attitude * state controller

4attitude 1, state controller plant estimator dynamics

CI 4 8 & attitude state Lm controller

Figure 2.12: Mode1 with a 3-auis classical rate estimator. lully nonlinear coupled dynamics of Eq.(2.8b), and the state estimators are designed according to Fig. 2.11.

2.4 Methods of Analysis

As mentioned in Section 1.3.1, this thesis deals with space manipulators mounted on a spacecraft controlled by on-off thrusters. We have seen in Subsection 2.3.3 the classical way of controlling such system with on-off thrusters. This section describes various methods that are used in the following chapters to analyze these nonlinear systems. The first method is based on the notion of the "describing function" , a function that approximates nonlinear components by linear "equivalent" ones. The second mithod makes use of the describing function of a nonlinearity, but in conjunction to the root-locus technique. Finally, a numerical simulation is included, Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 37 which is a useful tool to obtain the response of a very complicated system.

2.4.1 Describing-Function Analysis

The describing function is a tool used to find the approximate response of nonlinear systems using methods derived for studying the frequency response of linear systems. The main use of this method is the prediction of limit cycles in nonlinear systems. although it has a number of other applications, such as predicting the existence of subharmonics, jump phenornena, and the response of nonlinear systems to sinusoidal inputs. However, here, only the prediction of limit cycles is discussed in detail. First, it is important to define what kind of nonlinear systems can be analyzed with the describing-function method. Simply stated, any system that can be transformed into the configuration in Fig. 2.13 can be studied using describing functions. However. to use the describing-function technique in its simplest form (single-input describing function), in a system that has only one nonlinear component, tliree basic conditions must be observed, as stated in (Slotine and Li, 1991):

1. the linear element has low-pass properties, and therefore. for a sinusoidal input z = .4 sin(s>t),only the fundamental component wl(t)in the output w(t) needs to be considered,

2. the nonlinear component is time-invariant , and

3. the mean value of the nonlinearity is zero, which is the case for most common nonlinearities.

If a limit cycle is present in the system, the system signals must al1 be periodic, and hence these periodic signals can be expanded as the sum of many harmonies. Moreover, if the linear element in Fig. 2.13 has low-pass properties, which is true for most physical systems, then the higher frequency signals will be filtered out and the output y(t) will be composed mostly of the lowest harmonie. Therefore, for the Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 38 Nonlinear Linear Element Elernent

Figure 2.13: A nonlinear system. case where a limit cycle is present, it is appropriate to assume that the signals in the whole system are baçically sinusoidal. Using this assumption, a describing function can be found that represents the nonlinear component. Considering a sinusoidal input x(t) = .-!sin(wt), the output w(t)of the nonlinearity is often a periodic function. and can be expanded in a Fourier

00 w(t) = + 1[a, cos(nwt) + b,, sin(riwt) n= 1 1 where

Since the mean value of the nonlinearity is zero (third assumption above), one has a0 = O. Furthermore, due to the first assumption which states that the linear element has low-pas properties, only the fundamental component needs to be considered. Therefore, w(t) = al cos(wt) + bl sin(&) (2.25) which can be written as

where Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 39 and 4(A,w) = tan-' (al/bi) (2 -28)

If the describing function N(A,w) of a nonlinearity is defined as a function that maps the input x(t) to the output w(t),we have

Considering the complex representation of a sinusoid, Eq.(2.29) can be written as

Thus, the describing function of the nonlinearity is given by

with

The nonlinearity representation, N(A,w), is tabulated in many books for al1 typical nonlinearities. -4 good reference on the subject is (Gelb and Vander Velde, 1968). As an example, for a relay containing a dead zone, as shown in Fig. 2.1, the describing function N is solely a function of the amplitude A of the predicted limit cycle and is given by

where d is the deadband limit. .As well, the relay containing a dead zone aiid hysteresis of Fig. 2.'i(a), is represented by Chapter 2. ModeIling, Control and Analysis of Rigid Spacecraft 40 where the first term represents the real part and the second terrn, the imaginary part of the describing function. .Again, -4 is the amplitude of the predicted limit cycle, J the deadband limit and A the amount of hysteresis included in the relay. It is noted that, for a relay nonlinearity, since the parameters S and 4 are fixed by design, the describing function is simply a function of the amplitude -4 of the limit cycle and no longer a function of the frequency w. Figure 2.14 introduces the -l/lVd(.A)and -1/Nh(.4) locus, which are plotted in the complex plane.

-2 -1.5 - 1 -0.5 O 0.5 Real

Figure 2.14: Loci of the describing functions for the relays.

Now that we have a describing function for the nonlinearity, we are ready to analyze the system of Fig. 2.15 for the existence of limit cycles. G(jw)is the frequency response of the linear element of the system and is simply obtained by substituting s by jw in the transfer function G(s)and N(A,w) is the describing function of the nonlinear element derived assuming t hat t here exists a self-sust ained oscillation of amplitude d and frequency w in the system. The variables in the loop must satisfy the following relations Chapter 2. Modeiiing, Control and Analysis of Rigid Spacecraft 41 Describing Linear Func tion Element

Figure 2.15: .A nonlinear system analyzed with descri bing functions.

Thus, we have y = G(jw)N(A,w)(-y). Because y # O, this implies:

wbich can be written as

The amplitude -4 and the frequency ut of the limit cycle in the -stem niust satisfy Eq.(2.38). If the above equation has no solution, then the nonlinear systern has no limit cycle, and the describing function cannot be used to approxiniate the nonlinearity. Since it can be difficult to solve Eq.(2.38) algebraically, a simple solution method consists of plotting both sides of Eq.(2.38) in the cornplex plane by varying -4 and

YI, to observe whether the two curves intersect or not. The intersection point gives us the value of A and w and, therefore, the approximate limit cycle x = -4 sin(wt) is completely determined. As an example, let us consider the case when the describing function N is a function of the gain A only. So, Eq.(2.38) becomes Chapter 2. Modelling, Control and Analysis of Rigid S pacecraft 42

Figure 2.16: Limit cycle detection.

The frequency response function G(jw)can be plotted by varying LI in the cornplex plane as in Fig. 2.16. The same can be done for the negative inverse describirig function (-l/N(A))by varying A. If the two curves intersect, then there exist limit cycles and the values of -4 and w corresponding to the intersection points are the solutions of Eq.(2.39). If the curves intersect n times, then the system has n possible limit cycles and the one actually reached depends upon the initial conditions. Since the describing-function method is approximate, it is not surprising that the analysis results are sometimes not very accurate. Without going into details, we can state a general nile mentioned in (Slotine and Li, 1991): If the G(jw) locus is tangent or alrnost tangent to the -1/N(A, w) locus, as in Fig. 2.17(a), then the conclusions from a describing-function analysis might be erroneous. Conversely, if the -1/N(.-L, w) locus intersect the G(jw)locus almost at right angles, as in Fig. 2.l?(b), then the results of the describing-function analysis are usually accurate. An intersection point of the two loci within the complex plane does not guarantee stability of the predicted limit cycle. Such a limit cycle can be unstable and it will never be observed. For brevity, we can state here a simple Limit Cycle Criterion based on the extended Nyquist criterion as stated in (Slotine and Li, 1991), namely, Limit Cycle Criterion: Each intersection point of the cunie G(jw) and the cvrve -1/N(A) correspondg to a lzmit cycle. Assvning o stable G(jw), if points near the Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 43

Irn - Re

Figure 2.17: Reliability of lirnit cycle prediction: (a) poorly-reliable results; (b) bighl-reliable results.

intersection and along the increasing-A side of the curve -1/N(.4) are not encircled by the cume G(jw), then the correspondzng Izmit cycle is stable. Otherwise, the limit cycle is unstable. For example, Fig. 2.17(b) depicts a stable limit cycle. On the other hand, if no intersection of the G(jw)and - 1/N(.-L)loci exists. the stability of the system is assessed using the normal Xyquist criterion with respect to any point on the -l/iV(A) locus rather than the point (-1,O) (Atherton. 1975). This statement will be proved as follows: Let us consider the system of Fig. 2.18. Its performance can be assessed by looking at the location of the closed-loop poles, which are the roots of the denorninator of the transfer function T(s)mapping the input R(s) to the output Y (s), namely,

From Fig. 2.18, it is apparent that

Y (s) = KG (s) [R(s) - H (s)Y (s)] = KG(s)R(s)- KG(s)H(s)Y(s) Chapter 2. Modelling, Control and -4nalysis of Rigid Spacecraft 44

Figure 2.18: A general linear system. which can be written as

and thus

The closed-loop poles of the system are thus given by the roots of the characteristic equation F(s)of the system, namely,

where p(s) and q(s) are polynomials in S. CVe can further define the function &(y)

The normal Nyquist criterion consists of plotting the function &(jw), for O+ < J 5 +w, in the complex plane, and look for the number of encirclements of the point ( - 1, O). When none of the poles of Fd(s)are in the right-half plane, then the Nyquist criterion states that the number of encirclements of the point (-1, O) is equal to the number of mots of F(s) in the right-half plane. Thus, if we have one or more such encirclement, the system is unstable. Now, let us extend this criterion by defining a function F&) as Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 45

Figure 2.19: Stability range determination using the Yyquist criteron.

The scaling of the function Fd(s) by 1/K does not affect its general shape, but just scales it. Hence, we can still apply the Nyquist criterion, but by counting the number of encirclements of the point (-1/K, O) instead of the point (- 1. O). This new formulation has the great advantage that the stability of the system can be determined as a function of a gain parameter K, as for the root-locus methods that will be discussed in the nert subsection. For example, if we consider the plot of Fig. 2.19, the system is stable if the point (-l/K,O) is to the left of the G(jw) curve, since it is not encircled. Thus, if the point (-a, O) denotes the intersection point of the G(jw) locus with the real axis, then the system is stable if

Now, if we corne back to the describing-function method, the characteristic equation is given by Eq.(2.37), namely Chapter 2. ModeUing, Control and Analysis of Rigid Spacecraft 46

Figure 2.20: Stability prediction using the describing-function met hod. (a) Stable system, (b) Unstable system.

Let us define the function F&) as

whence the stability of the system is assessed by considering the riumber of encirclements of any point of the -I/N(A, w) loci by the G(jw) loci. The nonlinear system will thus be stable if none of the points of the -I/N(.-l, w)locus is encircled by the G(jw)locus. For example, assuming that G(jw) is stable, Fig. 2.20(a) depicts a stable system while Fig. 2.20(b) an unstable one.

2.4.2 Root-Locus Analysis

In the previous su bsection, the describing-function technique was introduced to analyze nonlinear systems. This method is basically the nonlinear system counterpart of the Nyquist method used extensively in linear control system analysis and design. Another widely used method for linear system control design is the root-locus, which Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 47 is also a graphical method. In the root-locus rnethod, the poles of a closed-loop system are plotted as functions of one of the system parameters or gains. When one such locus goes into the right-half plane, the root of the characteristic equation of the system becomes positive for the given value of the varying parameter, and the system goes unstable. In this subsection, this rnethod is expanded to nonlinear systems by considering the same hypothesis as for the describing-function method of Subsection 2.4.1. This novel adaptation and interpretation of the describirig-furictioii concept in the analysis of space robotic systems controlled with on-off thrusters is one contribution developed in this thesis.

Root-Locus Concept

Again, let us consider the system of Fig. 2.18, its characteristic equation being given by Eq. (2.45). As mentioned in Subsection 2.4.1, the stability of the system is assured if none of the roots of this equation is in the right-half plane. Equation (2.45) can be further expressed as

~(4f KP(s) = 0 (2.52)

From Eq.(2.52), it is apparent that when K = 0, the roots of the characteristic equotion are the poles of the open-loop transfer function G(s)H (y). Moreover, when K = w, we clearly see by dividing both sides of Eq. (2.52) by K that the roots of this equation are the roots of p(s),which are the zeros of the open-loop transfer function G(W(4 The root-locus method thus consists of placing the open-loop poles and zeros of the system in the complex plane, and by producing the locus of the closed-loop poles, which are the roots of Eq.(2.52), by varying the parameter I( from O to W. When one locus goes in the right-half plane, the system is classified as unstable. it is thus possible to determine in which range of the parameter K the system remains stable, and when it goes unstable, exactly as we did using the Nyquist hfethod in Subsection 2.4.1. Chapter 2- Modelling, Cont rol and Analysis of Rigid Spacecraft 43

Application to Nonlinear Systems

Here, the t hree basic conditions stated in Subsection 2.4.1 for the describing-function method are assumed again. In other words, we assume that the describing-function technique can be applied to the system at hand and thus, Eq.(2.37) can be considered to be the characteristic equation of the system, namely,

which can be written as 1 + N(A,w)G(s) = O

where p(s) and q(s) are again two polynomials in s, the numerator and the denom- inator of G(s),respectively. Equation (2.55) has the same form as Eq.(2.52). if we use as the varying parameter K, the describing function of the nonliiiear element of Our system N(.A, w). The varying parameters thus become the amplitude -4 and the frequency w of the assumed limit cycle. For the case where the describing function is only a function of the amplitude A, which is the case for a relay nonlinearity, and thus for the case under study in this thesis, the locus of the closed-loop poles in the complex plane is obtained by varying solely this parameter. We thus have

Moreovert let us assume that the i-th root of Eq.(2.56) is

when A has the value Ai. Thus, from Eq.(2.56), we have

We can further assume that for the given Ai, the root s, is on the jw-axis. Hence, Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 49 and Eq. (2.58) becomes q(jwi) + N(Ai)p(jwi)= 0 (2.60) which can be written as

Equation (2.61) thus has the same form as Eq.(2.37) and thus corresponds to the intersection of the two loci G(jw)and -1/N(.-l). Therefore, when a branch of the root-locus plot crosses the ju-axis at a certain wi, for a given value .-I,,then the system exhibits a limit cycle of amplitude adi and frequency w,.The stability of this limit cycle can be assessed by considering points of the loci near the intersection point. For the root-locus method, the Limit Cycle Criterion stated in Subsection 2.4.1, can be rewritten as Limit Cycle Criterion: Each intersection point of a locus of the root-locus plot with the jw-axzs corresponds to a limit cycle. Assuming a stable G(s),if points near the intersection and along the increasing-A side of the locw are in the fefl-half plane. then the corresponding limit cycle is stable. Otherwise, the limit cycle is unstable. For example, points B of Fig. 2.21 correspond to an unstable limit cycle. and points C correspond to a stable one. On the other hand, if none of the locus intersects the jw-axis, the system is stable if all the loci are in the left-half plane, and it is unstable if one or more of the loci is completely in the right-half plane, which means that there exists at least one pole of the closed-loop system with its real part being positive for any value of .A. For example, assuming that G(jw)is stable, Fig. 2.22(a) depicts a stable system while Fig. 2.22(b) an unstable one.

2.4.3 Simulation

The third method used to analyze a nonlinear system is by simulation, a very convenient way of obtaining results, since one just has to set up an adequate model Chapter 2. Modelling, Control and Aoalysis of Rigid Spacecraft 50

Figure 2.21: Stability of limit cycles using the root-locus method.

Figure 2.22: Stability prediction using the root-locus met hod. (a) Stable system, (b) Unstable system. Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 51 and simulate its behaviour numerically. However, performing inany simulation runs, such as those required in a parametric study, can be very time consuming and give little insight on the physical system, as compared to algebraic met hods. Therefore, in this thesis, simulations are used only to verify important results obtained with the describing-function and root-locus analyses. The simulation model will be implemented with Simulink, a Matlab package that accommodates model definition and dparnic simulation. Morcovcr, simulation allorvs us to includc white noise in the attitude measurements. When white noise is included, the classical rate est imator of Fig. 2.23 can be used instead of the one of Fig. 2.10. Since we are not dealing with any specific system, the amount of noise is not known, because this is dependent upon the quality of the sensors. For this reason, we assume some reasonable values for the parameters of the noise. White noise is assumed: this means that the noise is normally distributed with a variance O,,,. and a zero mean p,,,, = O. The Simulink white noise generator block was used to gnerate the noise into the models. The noise variance was selected to be 20% of the variance of a stable systeni motion, which is quite a large noise level, since typical values are of the order of 10%. For deadband limits from -6 to 6, the variance of the stable motion becomes

The variance of the noise is thus chosen as

2.5 Stability

2.5.1 Definitions

In this subection, a stability definition is given and used to describe the possible behaviours of the system, as modeiled in Subsection 2.3.3. This stability definition Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 52

diff/filter

Noise filter

Figure 2.23: Block diagram for the classical rate estimator with white noise included in attitude measurements. is based on the rate of fuel consumption of the system, which is erplained in more detail below. The fuel consumed by the thrusters is proportional to their opening time. There- fore, we can write

where S is a specific constant dependent upon the type of thruster fuel used and the characteristics of the thrusters, and u is the thrusters command, either +1, O or - 1. Since S is constant, the fuel consumption Fc(t)is defined in this thesis as

and the units of Fc(t)will be simply called "fuel units". As well, the rate of Fuel consumption Rf(t)can be defined as

Since F,(t) is a nonsmooth function, Ri(t) is not defined at the nonsmooth points. However, a smooth function F:(t) can be defined that best fits the fuel consumption curve and then, the rate of fuel consumption Rj(t) is thus defined everywhere as a smooth function. In order to minimize the order of the polynomial required to best fit Fc(t), it was decided to use a linear fit for the steady-state part of F'(t) only, and thus, Rj(t) is simply the dope of this line. For example, the control system of Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 53

Fig. 2.11 was simulated using A = 5 s and 6 = Io, with the planar mode1 of the rigid spacecraft developed in Section 2.2. The inertia Io, was chosen as a typical inertia of the Space Shuttle about its 2-axis, namely

while the thruster moment n, was chosen as

y, being chosen as 0.5'/s2. Simulation results are depicted in Fig. 2.24. From Fig. 2.24(a), we see that a rigid body limit cycle is attained. which results in a sporadic firing of the thrusters, as seen in Fig. 2.24(b). The discontinuous fuel consumption Fc(t) is displayed in Fig. 2.24(c) toget her with the linear smoot hing FC(t) for the steady-state part. The corresponding rate of fuel consurnption Rj is observed to be Rj = 0.0015 fuel units/s (2.69)

e (rad) (4 Figure 2.24: Simulation results for the rigid spacecraft: (a) spacecraft error phase plane; (b) thruster command history; and (c) fuel consumption.

In al1 cases studied in this thesis, three main different classes of behaviour were observed. In the first class, the system eventually reaches a limit cycle sirnilar to a rigid body limit cycle, where the system states remain contained between the switching lines. The resulting rate of fuel consumption is thus minimal and comparable to Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 54 that of a rigid body, as the one obtained in Fig. 2.24, and the system is considered stable. In the second class, the position of the spacecraft diverges from the equi- libriurn and thus the Fuel consumption increases with time, leading to a large rate of fuel consumption. Since this rate is much larger than that of the desired rigid body case, the system is considered unstable, in accordance with the divergence of motion. For the third possible class of behaviour, the spacecraft motion follows a limit cycle of large amplitude due to the excitation of the systern flexible modes. As a result, the limit cycle is not contained inside the switching lines and the thrusters fire continuously, thus leading to a nonzero rate of fuel consumption comparable to the one obtained when the motion diverges. This type of behaviour will also be considered unstable, since it is not desirable from the fuel consumption point of view. Finally, some intermediate behaviors were also observed where the general motions were similar to a rigid-body limit cycle, but the thruster activity was quite important. sometimes due to sensor noise, or to the high flexibility of the systeni. Since the fuel consumption of this type of behavior is also important, those systems were classified as unstable if their rate of fuel consumption Rj is greater than four times the one we would obtained for a rigid body, which is 0.0013 fuel-units/s for the example of Fig. 2.24. Thus, the limit value for stability is chosen as

Rjlim= 0.0060 fuel units/s (2.70)

These observations lead to the following stability definitions: Stability Definit ions:

1. a stable system (S) is a system whose motion reaches a limit cycle similar to a rigid-body limit cycle, thus being contained between the switching lines, and resulting in a rate of fuel consumption Ri srnalier than Rjlim.For examples used in this thesis, Rj,,_ = 0.0060 fuel-units/s;

2. an unstable system (U) is a system whose resulting rate of fuel consumption Rj is larger than R;,,_. Moreover, most of the unstable systems fa11 into one Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 5 5 the two following categories:

(a) a type-1 instability (Ul) describes an unstable behaviour for which the motion diverges, which results in a rate of fuel consumption larger than

%im ; (b) a type-2 instability (U2) describes a system whose motion reaches a limit cycle that is not contained inside the switchinq lines as for a rigid body limit cycle, thus resulting in a rate of fuel consumption larger than

4im.

2.5.2 Application

The stability definitions of Subsection 2.5.1 are applied by using either the describing- function method or simulation. For the describing-function method, three typical describing-function plots are introduced below to show the applicability of the stability definitions. Type4 Instability The plot shown in Fig. 2.25 is typical of a type-1 instability Since al1 the points of the -l/Nd(rl) locus are encircled clockwise by the G(jw)locus, they are al1 unstable according to the Nyquist criterion and the amplitude A of the osciliations will increase indefinitely. Therefore, the two intersecting points in Fig. 2.25 represent unstable limit cycles that will never practically be observed. Type-2 Instability In this case, the plot of Fig. 2.26 represents a type-:! instability. Since al1 points to the left of the G(jw)locus are not encircled clockwise by this locus, they represent a stable zone, while the points to the right of the same locus are unstable points. Two types of behaviour are observed in this plot. If we take a point on the -l/Nd(A) locus that corresponds to 6 < A < 1.016, then the system is stable since none of those points are encircled by the G(jw) locus. Hence, the amplitude of the oscillations will decrease till A = 6, the deadband limit, which, as defined in Eq.(2.32), is the Chapter 2. Modelling, Control and Analysis of Rigid Spacecraft 56

- 2 -1.5 - 1 -0.5 O 0.5 Real

Figure 2.25: Describing-function plot for a type- l instability. minimum value of -4 where -1/!Vd(.4) is defined, thus corresponding to a stable motion according to the stability definitions of Subsection 2.5.1. However, if the point on the -l/&(.L) locus corresponds to 1.016 < A < 13.936, then this point is encircled by the G(jw) locus and is thus unstable with an increasing amplitude. If the amplitude becomes greater than d4 = 13.936, then this point becomes a stable point and the related amplitude will decrease. Therefore, the iiitersecting point corresponding to A = 13.936 predicts the presence of a stable limit cycle of amplitude -4 = 13.936, which is not contained between the two switching lines, since A > 6. Thus, depending on the amplitude A, the plot of Fig. 2.26 predicts stable motions and unstable ones. The actual behaviour of the system will depend upon the initial conditions. However, since the possibility of a type-2 instability is present, the system is classified as being unstable of type-2. Moreover, the stable case is unlikely to bappen since the amplitude A of the limit cycle must also be greater than the deadband limit, by definition. The zone of stability is therefore very smalt. Chapter 2. Modelling, Control and Analysis of Riad Spacecraft 57

-2 - 1.5 - 1 -0.5 O 0.5 Real

Figure 2.26: Describing-function plot for a type-2 instability.

Stable System A stable system is represented by the describing-function plot show in Fig. 2.27. Since no point of the -l/Nd(.-l) locus is encircled clockwise by the G(jw) locus. al1 the points are stable, and the system exhibits a stable behaviour according to our stability definition. No large-amplitudz limit cycles are present in this system because there is no intersection of the two loci and, therefore, a small unavoidable limit cycle due to ATminwill be reached, as explained in Subsection 2.3.1. In the case where simulations are performed, the fuel consumption Fc(t)given by Eq.(2.65) can be obtained easily, by integrating the absolute value of the thrusters command u, and the linear fit F,'(t) of the steady-state part of Fc(t)can be obtained. The stability definitions are simply applied by comparing the timcderivative of F,'(t), R;, with the limit rate of fuel consumption R;,,_ given in Eq.(2.70). For example, Fig. 2.28(a) represents a stable system, while Fig. 2.28(b) an unstable one. For most systems, the type of unstable behaviour can be determined by exarnining the error phase plane of the spacecraft to see if the motion diverges, as in Fig. 2.28(c), or if -2 -1.5 - 1 -0.5 O 0.5 Real

Figure 2.27: Describing-function plot for a stable system. it reaches a large limit cycle that is not contained between the switching lines. as iri Fig. XB(d).

2.6 Summary

In this chapter, the dynamics model of a rigid spacecraft was obtained using a Lagrangian approach, neglect ing microgravi ty effects and orbi ta1 mechanics. This model was also simplified for small-angle rotations to obtain uncoupled dynamics. The planar case for rotation about only one principal avis of the spacecraft was considered as a basis to describe a classical attitude control scheme using on-off thrusters, similar to the one used on the Space Shuttle. This control scheme makes use of the error phase plane to determine the opening and closure of the thruster valves. A rate estimator that gives good performance for rigid body is also part of this controller. Since on-off thrusters are used, which are nonlinear devices, some techniques to analyze nonlinear systems were also discussed. The describing-function Chapter 2. Modeiiing, Control and Analysis of Rigid Spacecraft 59

0L O 100 200 300 400 Tirne (s) Time (s) (d)

Figure 2.28: Examples of stability determination. (a) Fuel-consumption curve of a stable system, (b) Fuel-consumption curve of an unstable system, (c) Spacecraft error phase plane when the motion diverges, (d) Spacecraft error phase plane when the motion reaches a large limit cycle. Chapter 2. Modelling, Control and Anaiysis of Rigid Spacecraft 60 technique was discussed in detail. This technique was adapted for the analysis of space robotic systems controlled with on-off thrusters via the root-locus method. This novel adaptation of the describing-function concept will be employed in Chap ters 4 and 5, and results in an improved physical insight in the design of new control schemes that eliminate the dynamic interaction problem. Finally, the stability definition used in this thesis was introduced. The definition is based on the rate of fuel consumption Rj(t) of the system. -4 threshold value

R;,.lm (t) was chosen based on the fuel consumption for a rigid spacecraft. Rates of Fuel consumption below this threshold correspond to stable systerns. while rates of fuel consumption above it describe unstable systems eit her for diverging rnot ions or large limit cycling due to flexibility of the system excited by the operation of the on-off thrusters. Detailed examples for the application of this definition were discussed . Chapter 3

Modelling of Flexible Space Manipulator Syst ems

3.1 Introduction

In Section 1.1, five different space manipuiator concepts were introduced. Al1 these concepts have the common feature that they operate in a zero-gravity environment. their base being floating and hence subjected to the motion of the manipulator. Therefore, the modelling and control of such systems is more complex than for fixed- base rigid robots. Moreover, structural and joint flexibility is important in such robots, as they are required to be lightweight, move large payloads and have large workspaces. A general formulation for the modelling of flexible space robots is introduced in this chapter. To simplify the analysis, this dynamics modelling is restricted to flexible-joint manipulators mounted on a six-dof spacecraft. This approximation is legitimate since free-flyers, like those of Fig. 1.2, are likely to have reduced structural flexibility but will have joint cornpliance which becomes important when payloads are massive. In the case of the CANADARM, both structural and joint flexibility are present, but the latter is more important than the former. For simplicity, it can Chapter 3. Modelling of Flexible Space Manipulator Systems 62 be assumed that al1 flexibility in this system is lumped at the joints. A general formulation using Lagrange's equations is described in Section 3.2 for the modelling of flexible-joint manipulators. In Section 3.3, the equations of motion of the manipulators axe linearized about an operating point and general expressions for the manipulator natural frequencies are derived. The model of a three-flexible- joint manipulator mounted on a six-dof spacecraft is described and validated using a commercial finite-element package in Section 3.4. Finally, in Section 3.5, it is shown how to adapt this finite-element package for control purposes using its general user element .

3.2 General Lagrangian Formulation

In this section, the dynamics model of an N-flexible-joint space manipulator is obtained using a Lagrangian approach. Since the time scale of robotic motions is assumed to be relatively small compared to the orbital period, while control Forces and torques are relatively large, orbital mechanics effects are neglected. The kinematics of the free-flying space manipulator is expressed using the spacecraft centre of mass (CM) C as a reference point to describe the translation of the system, as depicted in Fig. 3.1. The inertial position vector of an arbitrary point P of the system, p, can be written as p=c+p (34 where c L the position vector of C and p is the position vector of point P with respect to C, as shown in Fig. 3.1. The position vector p can be expressed, in turn, as P = ci + pi (3.2) where c, is the position vector of the CM Ci of the i-th body with respect to the spacecraft CM, and pi is the position vector of point P with respect to the i-th body Chapter 3. Modelling of Flexible Space Manipulator Systems 63

CM. The position vector ci can be expressed as

with vectors { rc );-' and { lk )f indicated in Fig. 3.1. The velocity of point P on the manipulator is thus expressed as

with i- 1 ci = wox + 1~k x (rk - lk)- Wi x li (3.5) k= 1 where wo is the angular velocity of the spacecraft, and wi is the angular velocity of the coordinate frame attached to the i-th link of the manipulator, which can be expressed as

iri which eZk is the rate of the k-th joint and zt is the unit vector dong the avis of rotation of the same joint, as shown in Fig. 3.2. Since 92k is measured with respect to link k - 1, there is no need to consider the rotor rate ûZk-l in the calculation of

Assuming lumped flexibility at the joints.1 links are considered rigid and Eqs.(3.4) and (3.5) can be substituted in the kinetic energy expression of the manipulator, given by 1 .. where M is the total mass of the system, and To,Tl, and T2are defined as Chapter 3. Modelling of Flexible S pace Manipulator S ystems 64

w Figure 3.1: -4 space manipulator system.

mechanical

' Figure 3.2: A flexible-joint model. Chapter 3. Modelling of Flexible S pace Manipulator Systems 65 with mi and 1, being the i-th body mass and inertia dyadic with respect to the centre of mass of the corresponding body. Now, vectors

are defined as the rotor-joint variables and the link-joint variables, respectively, as per Fig. 3.2. Here, subscripts a and n stand for the actuated and non-actuated variables. These two vectors are not the same, due to joint flexibility. Therefore, the kinetic energy of the rotors can be written as

where J is the moment-of-inertia matrix of the rotors. For a free-flyer, rnicrogravity effects are very small compared with control forces. and hence, they are neglected. Thus? the potential energy is only due to joint elasticity, and can be written as where Kr is the corresponding stiffness matrix, defined as

while 98 is the vector of the joint deformations, defined as

Viscous friction forces due to damping can be taken into account using Rayleigh's dissipation function R, given by

where Cris the corresponding damping matrix, defined as Chapter 3. Modelling of Flexible Space ~~anipulatotSystems 66 The sum of al1 powers developed by driving devices supplying controlled forces is given by .T n=e,~+n, (3.16) where r is the vector containing al1 torques applied by the motors at each joint and ns is the power developed by the forces and moments applied to the spacecraft with thrusters? as derived in Section 2.2 for the rigid spacecraft, narnely,

In the realrn of the Euler-Lagrange equations, we use q = [q;f,O:] ', where T qT = [CT,6T, ~fl] , as generalized coordinates. Then, applying the Euler-Lagrange C J equations - R)

for i = 1, ,6 + 2N, and defining the following quantities

03x3 03x3 03%~ 03x3 03x3 03x.v

0~x30~x3 Cr 0~x30~x3 Kr

the equation of motion can be written as

where M, is a (6 + N) x (6 + N) positive-definite mass matrix, n is a (6 + iV)- dimensional vector containing the nonlinear velocity terms, and 4 is the 6-dimensional vector of resultant external force and moment applied to the spacecraft. C hapter 3. Modelling of Flexible Space Manipulator Sys tems 67 3.3 Linearization of the Equations of Motion

In this section, the equations of motion of the manipulator are linearized about an operating point and general expressions for the manipulator natural frequencies are derived. Undesired oscillations in terms of limit cycles and settling times are mostly important when the joints are locked; therefore, we are interested in the case where the joints are locked in a specific configuration 8:. In such a case. e, = O and e, = 0. Thus, Eqs.(3.20a St b) become

Equation (3.21b) gives the expression for the torques required to brake the joints. and Eq.(3.21a) represents the dynamics of the system. This equation can be linearized about an operating point, for example, the configuration c = c' = const., 6 = 6' = const., and 8: = 8: = const. Defining bq, = q, - q;, the linearized equations can be written as

The natural frequencies of this system are simply given by the square roots of the eigenvalues of the dynamic matrix W, which is defined as

with M,(q;) being a positive-definite matrix, and hence, nonsingular. Moreover, six eigenvalues are zero and correspond to spacecraft rigid modes, while the last iV-th ones correspond to joint flexibility. Chapter 3. Modelling of Flexible Space Manipulator Systems 68 3.4 Description and Validation of the Spatial Sys- tem

The analysis of the previous sections is used here to develop equations of motion for a three-flexible-joint space manipulator rnounted on a six-dof spacecraft used sub- sequently in this thesis. This system will be referred to as the "spatial system". The geometric and mass propcrties of this systcm arc idcntificd ir. Subscction 3:l.I. based on the CANADARM-Space Shuttle system, and the model is validated in Sub- section 3.4.2 using a commercial finite-element package. Finally, the mat hematical model is used in Subsection 3.4.3 to obtain the natural Erequencies and damping ratios of this spatial system for various typical configurations of the manipulator. and various payloads.

3.4.1 Mode1 Description and Parameter Identification

In this subsection, the three-flexible-joint manipulator mounted on a six-dof spacecraft of Fig. 3.3 is considered. This model has basically the same architecture as the CANADARhI-Space Shuttle system when considering only its three first joints. Yoreover, in order to choose realistic parameter values for the system, the pararn- eter values of the CANADARM-Space Shuttle system are used, while lumping link flexibility at the joints and considering zero joint angles for its last three joints. The dynamics model of this three-link system is obtained using the techniques described in Section 3.2 and employing the Maple symbolic environment. The dynamics equations are then converted to C code and compiled with the "crnex" function of Matlab to create an S-function. Finally, the S-functions are used into Szmulznk in order to implement control algorithms. The numerical values used for the spacecraft, payload, and links of the system are displayed in Table 3.1. These values, expressed in body-fixed frames, as defined in Fig. 3.3, are based on the CANADARM parameters (Spar Aerospace Ltd., 1996). Chapter 3. Modelling of Flexible Space Manipulator Systems 69

Figure 3.3: .A three-flexible-joint space manipulator.

The inertia dyadic of the payload is calculated assuming a homogeneous cylindrical payload with dimensions which are functions of its total mass. The ratio of the mass of the payload m4 over the mass of the spacecraft mo is denoted by $. The most challenging task in parameter estimation was the determination of the spring stiffnesses and damping coefficients in the three joints of the system, since the link flexibility must be lumped at those joints. Moreover, the overall joint stiffness of the three first joints of the CANADARM were not provided in (Spar Aerospace Ltd., 1996); only the gearbox stiffnesses were given. In order to determine this stiffness at each joint for the model, Eq.(3.23) was used to solve an inverse problern. For this system, the manipulator has three links and thus 1'V = 3, and matrices M,(q;) and Kgare 9 x 9 matrices. The matrices MiL(q;)and Kgthus have the form

where O is the 3 x 3 zero matrix and al1 other submatrices are of dimension 3 x 3. Chaptcr 3. Modelling of Flexible Space Manipulator Systems 70

Table 3.1: Spacecraft, payload and link parameter values.

[o. :a] Chapter 3. Moddling of Flexible Space Manipulator Systems 7 1 Thus, the dynamic matriu W has the form

and its eigenvalues are obtained as

where 19,, is the 9 x 9 identity matrix. Equation (3.26) can be further expressed as

413x3 0 H13Kr 0 413x3 H23Kr ] (3.37) O O H33Kt - &x3 or det(-A13,3)det(-A13,3)det(H33Kr - = O

Thus

Note that the term X6 corresponds to the 6 rigid modes of the system, while the three nonzero natural frequencies are given by

det(Hs3Kr- = O

Now, let Ai, for i = 1,2,3, be a nonzero eigenvalue of W; t hen

det(HJ3K, - = 0, i = 1,2,3

Moreover,

whence C hapter 3. Modelling of Flexible S pace Manipulator Systems 72 Thus,

hllkl - Xi h12k2 h3k3

(H33Kr-Xi13x3)= h12kl h22k2-Xi h23k3 7 i=l12,3 (3.34) I h13k1 h23k2 h33k3-Xi 1 After some algebra, Eq.(3.31) becomes

where

uf = 0.3734 Hz, wg = 3.6913 Hz, LJ; = 0.4779 Hz (3.36) for the CANADARbl in its home configuration, i.e., with al1 its joint angles being zero, without payload. Here, the superscript p and o stand for plane and out-of- plane. The frequencies wl and 4 correspond to the first two modes of vibration in the plane containing links 2 and 3, while r~i'corresponds to the natural mode for the vibration out of this plane. Therefore,

XI = (u;)*= 5.504 rad 2 /s 2 (3.37a)

2 2 Xz = (WY)~- 9.016 rad /s (3.37b) 2 X3 = (d)*= 537.921 rad2 /s (3.37~) Chapter 3. Modelling of Flexible Space Manipulator Systems 73

Table 3.2: Four possible sets of joint stiffnesses.

- Sol. # kl (Nm/rad) 1 k~ (Nm/rad) k3 (Nm/rad) 1 298 139 1 189 724 407 230

Thus, using = O3 = = 0' and P = 0, Eq.(3.35) gives a set of three nonlinear equations for three unknowns, kl,ki, and ka. Note that these equations are al1 cubic. but their only cubic term appears in al1 three equations with the same coefficient; hence. these equations can be reduced to one cubic and two quadratic equations. The Bezout number (Salmon, 1964) of these equations, which gives the upper bound for its number of real and complex roots, is then 3 x 2 x 2, and hence, up to 12 roots are to be expected. Solving this system numerically. four different real, positive solutions for the k,'s, which are recorded in Table 3.2, were obtained. Arnong those solutions, the first set was chosen, since these values give a better agreement with the frequencies of the CANADARM for other configurations. Finally. since we are not interested in matching exactly the performance of the CXNADARXI. but our objective is simply to pick-up realistic parameter values, the spring stiffnesses are rounded off to yield

Using these values, the three natural modes of the system are calculated and compared with those of the CANADXRM-Space Shuttle system for four typical configurations of the CANADARM (Nguyen and al., 1982). The actual calculated frequencies, dong with the expected ones, are recorded in Table 3.3 for the two modes in the plane of links 2 and 3, and in Table 3.4 for the mode out of this plane. The angles O&-,, for i = 1, 2 and 3, are the joint rotor variables, namely the angle Chap ter 3. Modelling of Flexible S pace Manipulator Systems 74

Table 3.3: Cornparison of natural frequencies in the plane of links 2 and 3. 4 4 0; 0; 8; Exp. -4ct. Err. Exp. Act. Err. (del9 (deg) (ded (Hz) (Hz) (% (Hz) (Hz) (%) O O O 0.373 0.373 O 3.691 3.420 -7 O 90 O 0.355 0.374 5 3.557 3.419 -4 4.9 70.4 -122.7 0.712 0.653 -8 1.126 1.331 18 121.5 9.3.8 -104.8 0.541 0.559 3 1.529 1.447 -5

Table 3.4: Cornparison of natural frequency out of the plane of links 3 and 3.

id; 1 0; 1 0: 1 0; 1 Exp. 1 Act. 1 Err.

between -Yi and when measured in the positive direction of Z,, as shown in Fig. 3.3. We can observe that the error for the first mode in the plane of links 2 and 3 is less then 8%, which is acceptable, and the highest error for the second mode is 18%, which is reasonable for our purposes. However, if we look at the mode out of the plane, we see that for the second configuration, the error becomes large. This can be readily explained by considering that, in this configuration, our system does not have any stifhess in its joints that would allow vibrations perpendicular to the plane containing the manipulator, while for the CANADARM, link flexibility could admit such vibrations. This configuration should therefore be avoided in future studies in order to obtain realistic results. Now, in order to determine the damping coefficients 4 for each joint we introduce the modal matrix P, which is composed of the eigenvectors of the dynamic matr~x Chapter 3. Modelling of Flexible Space Manipulator Systems 75 defined in Eq.(3.23). If the following change of variable is introduced

the dynarnic equations, Eq.(3.22) can be written in the form

where cz2 = P-~M;~K,P

Matrix n2is decoupled, but in general, A is not so. To obtain a decoupled set of equations, we have two options:

0 Assume that the off-diagonal terrns are small and neglect them

.\ssurne that the damping matrix is proportional to the stiffness matrix (C, = pK,). Then, the darnping rnatrix A becomes decoupled.

The first possibility cannot be employed in this system since the off-diagonal terms are of the same order of magnitude as the diagonal ones. Thus, we choose the second option, i.e., letting C, = pKg, the damping matrix A then becoming

Matrix n2contains explicitly the natural frequencies of the system, namely, Chapter 3. Modelling of Flexible Space Manipulator Systems 76

Moreover, from Eq.(3.43) we have

Thus, Eq.(3.40) can be written in decoupled form as

If we compare Eq.(3.46b) with the standard second-order form,

we obtain 2Gwt = w:, k = 1,2, 3 For k = 1, we have p=- 131 and substituting p in Eq.(3.48) for k = 2 and k = 3, we obtain

If W* or WJ > ul, then proportional damping results in C2 or > Ç1. This is in agreement with the data published in (Singer, 1989) for the CANADARII-Space Shuttle system, where it can be observed that the oscillations corresponding to the second and higher modes die out very quickly, and thus, the first mode dominates the response of the system. Therefore, the hypothesis of proportional damping is reasonable. Moreover, from the original assumption of proportional damping (Cg = pK,), we have

Ci = /A i = 1,2,3 (3.51) The domping ratio for the CANADARM without payload is known to be ap proximately 0.05. Therefore, using Eq.(3.49) with wi = 2r(0.3734) rad/s, we obtain p = 0.043, which can be rounded to p = 0.05, thus leading to Chapter 3. Modelling of Flexible Space Manipulator Systems 77 3.4.2 Mode1 Validation

In this subsection, the model of Subsection 3.4.1 and developed with the aid of the general formulation of Section 3.2 is validated using Mecano, a module of Samcef. Samcef is a finite element sotfware package developed at the Laboratoire de Tech- niques Aéronautiques et Spatiales (LT-4s) of Université de Liège, Belgium. Mecuno was chosen for Our studies due to its ability to accurately niodel systems with flexible elements. Since this package has built-in elements allowing the modelling of rigid bodies and flexible joints, the modelling of the system depicted in Fig. 3.3 was quite straightforward. However, it can be noted that Mecano is only a numerical software and thus, it cannot provide any symbolic code of the model that could be used in control system analysis and design. The Mecano and Matlab models were simulated by considering the third configuration of Table 3.3, namely 0; = Y. 0j = 70°, and 0; = -120". with the data of Table 3.1. and with a typical 15 000 kg payload (P = 116). The system was then excited by providing the following torque profiles to the thrusters:

where f (t) is shown in Fig. 3.-L(a),y is chosen as 2 deg/s2, nl: (k = x, y and 2) is the torque applied about the k-axis attached to the spacecraft and IOk (k = x. L/ and z), the corresponding moment of inertia, under the assumption that I, y and z are principal axes of inertia of the spacecraft. The simulations were run for a period of 10 seconds and the generalized coordinates and velocities were recorded over that period. Figures 3.4(b)-(d) show the joint-angle histories obtained using the Matlab model. The results obtained using Mecano are oot provided, since t hey are identical. The final value, at the end of the simulation, of the joint angles and joint rates are recorded in Table 3.5. Cornparison of Matlub to Mecano results shows differences that Vary in absolute value between 0.00 and 0.14%. These smdl errors, not apparent from the plots of the variable histories, are probably due to the different numerical integration schemes of the two software packages. Matlab uses explicit schemes, e.g., Chapter 3. Modelling of Flexible Space Manipulator Systems 78

Table 3.5: Final joint angles and rates cornparison.

1 II Matlab 1 Mecano 1 Error (%) 1

the Runge-Kutta 45th order method was used here, while Mecano uses the irnplicit method of Newmark. Moreover, in Mecano, it was not possible to give a step input as the one of Fig. 3.-L(a),but only an initial steep ramp input, followed by a constant part, which were determined in terms of the time integration step chosen. Since al1 the variables of the system look the same qualitatively using both models, it was concluded that the Matlabproduced results are valid and that the dynamics modelling code is correct.

3.4.3 Determination of Fkequencies and Damping Ratios

In this subsection, Eq.(3.23) is used to obtain the natural frequencies and damping ratios of the system of Fig. 3.3, using the parameter values introduced in Subsec- tion 3.4.1, for various typical configurations of the rnanipulator and various payloads. The five configurations of the manipulator shown below are considered: Chaptcr 3. Modeiiing of Flexible Space Manipulator Systerns 79

Figure 3.4: (a) Input toque profile over time; Joint angles using the Matlab model: (b) Oz history; (c)O4 history; and (d) history. Chapter 3. ModeUing of Flexible Space Mani~ulatorSystems 80 as well as the payload values:

where p = 0.3 represents a typical maximum payload that can be carried by a space manipulator, while P = 1.0 is an extreme case used to show the robustness of the proposed attitude controllers. The results are displayed in Table 3.6. The first natural frequency wl and damping ratio CI will be used in Chapters 4 and 5 to determine the parameter values of a two-rnass system on which studies wvere performed.

3.5 Finit e Element Formulat ion-Adapt at ion of a FEM Package for Control Purposes

The general Lagangian formulation of Section 3.2, used to to model flexible space robots, is valid only if we assume that Aexibility can be lumped at the joints and the links are considered rigid. This limitation was int roduced to simplify the dynamics model in order to concentrate the effort on control system design. However, it is interesting to test the developed control rnodels on a system with significant link flexibility with the purpose of gaining confidence on their robustness in practical systems. -4convenient way of dealing with systems possessing inherent link flexibility is the finite-element method. Here again, Mecano was chosen due to its ability to accurately model systems with flexible elements. Besides Mecano, the finite element package Samcef includes modules for the dynamics of rotating structures, thermal analysis, fracture mechanics, analysis of structures in composite materials, analysis of cable strutures, viscoplasticity analysis, and structural optimization. However, this package has no built-in functions allowing the use of control techniques to analyze the systems at hand. Since the main objective of our research is to develop control methods that are intended to reduce the undesired effects of dynamic interactions, the use of control techniques is mandatory. This problem was overcome by progamming Chapter 3. Modelling of Flexible S pace Manipulator Systerns 81

Table 3.6: Natural frequencies and damping ratios for various configurations and pay loads. Chapter 3. Modelling of Flexible S pace Manipulator Systems 82 Our own control subroutines via the Samcef user-element. This section is intended to introduce the new control tools developed for Mecano to model the use of on- off thrusters, time delays, as well as second-order filters and state-estimators. This material is a contribution developed to facilitate. the study and development of the main contributions found in this thesis.

3.5.1 General Background on Control Systems

Generally, - see Section 2.3 - a control system is composed of three main parts: the plant, the controller, and the observer or state-estimator. The plant denotes the physical system under control. The plant usually provides output signals measured by sensors, and admits input signals generated by control systems, in order to modify its performance. The role of the controller is to synthesize the control strategy, i.e., to derive the command signals for the plant actuators in response to the sensor outputs. The controller may be implemented using analog devices. but digital control has become prevalent, its elernents including digital electronic cards and cornputer software. Finally. when specific outputs are required by the controller, but not readily available by sensors, an observer or state-estimator is included to obtain an estimate of the required output. based on the available signals and a model of the plant. An important part of a control systern implementation is the design of the controller and the state estimator. First, a good mode1 of the plant is required in order to simulate properly the dynamics of the plant with the aid of software. This dynamics, like those of the controller and the observer, are represented mathematically by a set of differentiai equations. In modern control theory, these differential equations are usually represented with a system of first-order ordinary diflerential equations (ODE) of the fom x = f(x,u) (3.54a) while the output takes the form Chapter 3. Modelling of Flexible Space Manipulator Systems 83 where u and y are, respectively, the input and output vectors of the system, while f and g are, in general, nonlinear functions of the state variables x of the system and the input u. In classical control theory, where tinear systems are involved, the transfer function concept is also widely used. A transfer function is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable with al1 initial conditions assumed to be zero. This transfer function represents a relation describing the dynamics of the system under consideration. There are basically two types of controller structures. The first one, the open-loop controi, consists of a system where we give a specified input u, which may be based on a desired output of the system and a good mathematical model, but does not use any feedbtick of the actual states of the system. This kind of control actions, shown in Fig. 3.5(a), may not work in the preserice of parameter variations in the system, or in the presence of disturbances. To overcome this problem, closed-loop control has been developed. In this case, show in Fig. 3.5(b), the output y of the plant has a direct influence on the control input u, since this control input can be based on the error e between the desired output yd and the actual output. Therefore, closed-Ioop control is relatively insensitive to parameter variation or disturbances; in addition, the overall dynamics of the system can be modified as desired. However, the stability of the system cm be affected by the control scheme chosen, the implementation of the controller being more complicated and expensive than its open-loop counterpart.

Mecano, which is based on the finite element rnethod, was not developed with control system design in mind. Its main feature is the modelling of a mechanical systems-the plant- including flexible bodies. Extemal inputs can be supplied to the system to study its dynamic behaviour under different excitations defined by the user. Mecano can be readily used for the analysis of open-loop control when the input does not involve any dynamics. However, the analysis of closed-loop systems becornes Chapter 3. Modelling of Flexible Space Manipulator Systems 84

r Output u Output state estimate k Controlier - Plant I - -Estbator -

(b) Figure 3.5: (a) Open-loop control, (b) Closed-loop control. cumbersome and calls for additional work. Happily, Mecano has a general fortran- based user-elelnent that can be used to describe any component that could not be modelled using the built-in elements. This user-element can be used to describe the dynamics that may be contained in the controller or the observer. This problem dlbe ascertaincd in the next subsection to see how this gencral user-element of Mecano can be used to simulate a control system involving feedback control and state-estimators.

3.5.2 Derivation of the State-Variable Equations

Mecano can integrate multiple second-order equations while control systems are usually in state-space form. To bridge the gap, the state-space description is transformed in an input-output transfer function form, from which the input-output nth-order differential equation is obtained. Using a special technique, the derivatives of the input are eliminated and the resulting algebraic differential system is transformed again via a state-space description to the required by Mecano system of second-order differential equations. Chapter 3. Modelling of Flexible Space Manipulator Systems 85 Basic Formulation of the Mecano User-Element

In order to apply the Fortran-based user-element of Mecano, the system of equations must be cast in the form of a vector second-order equation, namely,

Since the assembled mass matrix MT can be singular, in Mecano, the time-integration of the States of the system is performed using Mewmark's implicit one-step method. Hence, the output signals of the user-element Fortran subroutine are the iriertial forces fine.the interna1 forces fint and an iteration matrix S, which are given by

where ai, for i = 1,2,3, are constants defined in the Mecano user-elernent, and KT and CT are, respectively, the stiffness and the damping matrices. defined as

Transformation of State-Space Equations in nth-Order Differential Equa- t ions

As mentioned in the previous subsection, control systems generall y involve transfer functions and first-order equations to represent the dynamics of the system. Most of the time, controllers and estimators are designed using linear differential equations. Therefore, first-order equations, as those of Eqs.(3.54a & b), can be written in the usual state-space form as Chapter 3. Modelling of Flexible S pace Manipulator Systems 86 Then, a representation of this system in transfer-function form is readily available, namely, where 1 denotes the n x n identity matrix, G(s)being generally a rnatrix composed of transfer functions, and is, hence, called the transfer-function matrix. Therefore, considering a block diagram representing a control system, we can first write the statcspace equations in transfer-function form; then, by block-diagram transformation (Dorf, 1995), the transfer function mopping a desired input variable ü to a desired output variable 0 can be obtained. Generally, for a single-input single-output systern (SISO), this transfer function takes the form

where U(s) represents the Laplace transform of the input variable and Y(s) that of the output variable. Note that the order of the numerator is, in general, srnaller or equal to n. In cases, where the order of the numerator is equal to m, with m < n, we have bi = O for i = m + 1, - . n. Frorn Eq.(3.62), a nth-order differential equation is readily obtained, namely,

To avoid differentiations of the input when only ü is known, a special technique is introduced, rnaking use of the following transformation

where x is solution of the following differential equation

The coefficients e,, i = 0, , n, of Eq.(3.64) are determined in Appendix B as Chapter 3. Modelling of Flexible S pace Manipulator Systems 84 Therefore, if an equation of the form of Eq.(3.63) is to be integrated for a given input fi, then Eq.(3.65) cari be integated and the desired time response Q(t) is obtained using Eq.(3.64) with the coefficients ei, i = 0, . . . , n. of Eqs. (3.66a & b).

Transformation of nth-Order Differential Equations in a Set of Second- Order Equations

Moreover, since Mecano can only handle second-order equations, we must write Eq.(3.65) as a set of second-order equations. We cmintroduce the change of variables

with 1.1 defined as the roof Junction of its real argument (e), Le.. in our case, as the integer that is closest to 42from above. Differentiating Eq.(3.67), we obtain

For i = 1,2, -- *, ([n/21 - l),we have

which leads to (rn/21 - 1) equations. Moreover, substitutiiig the zi, fi and 3 expressions in Eq.(3.65), and assuming an even n, we obtain Chap ter 3. Modelling of Flexible S pace Manipulator Systems 88 w here

z =

In cases where n is odd, Eq.(3.72) is exactly the same, except that we have an = 0, and M becomes semi-definite. This semi-definiteness is not a problem, since, in the irnplicit integration sçherne used in ilfecano, there is no need to invert M; only the iteration matrix S is inverted. Therefore, in order to apply the user-element of Mecano, we have

3.5.3 Application: Attitude Control

In this subsection, we show how to use Mecano to simulate the model depicted in Fig. 2.11 by introducing the foregoing techniques.

Plant Dynamics

In that system, the plant dynamics block could represent the model of a spacecraft with a space manipulator mounted on it, where we try to control the attitude of the Chapter 3. Modelling of Flexible Space Manipulator Systems 89 base in a planar motion. This plant is modelled using the standard-elements library of Mecano. Moreover. although Mecano can give the velocity and the acceleration of the base, we assume that only the attitude is available by sensors and, therefore, only the angular displacement of the spacecraft is considered. A time delay r is implemented by writing a Fortran subroutine that is called by the user-element subroutine and that stacks values of the attitude for previous time steps in a vector array. The desired value of the attitude for the current time step is obtained by linearly interpolating the value in that vector çorresponding to the attitude at T seconds before.

State Estirnator - Attitude Filtering

NOW,in order to implement the state estimator of Fig. 2.10, a transfer function mapping the filtered attitude signal 0 to the attitude signal 0 must be derived, as well as a transfer function mapping the estirnated attitude rate 8 to the attitude signal û and the command of the thruster u. For the attitude, we readily have

Using the definit ion of Gj(s) given by Eq. (2.2 1) , we have

Comparing this equation with Eq.(3.63), and using n = 3, since we have a second- order differential equation, we obtain

which is in a form readily integrable by Mecono. Note that 0 is not an external input to the user-element, but the position of the node' describing the CM of the

lin Mecano, like in any finite-element formulation, nodes are the points of attachment of two elements, and the points in an assemblage of elements for which displacements are sought. Chapter 3. Modelling of Flexible Space Manipulator Systems 90 spacecraft represented as a ngid body in Mecano. Therefore, the user-element must be cornposed of 2 nodes, the actual CM of the spacecraft, and a second node that will be the desired estimated attitude ê obtained by solving the differential equation of Eq. (3.76).

State Estimator - Attitude Rate Estimator

For the attitude rate, we have, by defining r(s)as the Laplace transform of &(t)and by examining Fig. 2.11,

where y0 is the üngular acceleration impinged to the system by the thrusters.

king the definition of the second-order filter GS,(s),as given in Eq.(2.22). we have

Moreover, we define

to obtain, from Eq.(3.79), Chapter 3. blodelling of Flexible S pace Manipulator Systems 91

Since this equation is linear, it can be obtained by superposition of the following two equations

and then, Eqs.(3.82a 9r b) are in the E'orm of Eq.(3.63) which can be integrated with Mecano using the previously developed met hodology. Comparing Eq. (3.82a) wit h Eq.(3.63), we have

Since the input to Eq.(3.82a) is 4 and only 0 is assurned available. we must resort to the special technique introduced in the previous subsection. Let us consider the equat ion xl + 2CSe~jeIi+ u,,x~2 = (3.84)

Cornparing this equation with Eq.(3.65), we have

The desired time response ql (t) is thus obtained by integrating Eq.(3.84) in a user- element and then using Eq. (3.64) wit h eo, el and ez given by Eq. (3.66a & b) , namely,

Now, comparing Eq. (3.82b) with Eq. (3.63), we have

Resorting to the same special technique just introduced above, we can consider the equat ion 2 x2 + 2&c~se~2+ wicx2 = wSe~u (3.88) Chapter 3. Modelling of Flexible S pace Manipulator Systems 92 which gives d = u:e7e (3.89)

The desired time response ~(t)is thus given by integrating Eq.(3.88) and by using Eq. (3.64) with eo, el, and ez given by Eq.(3.66a 9r b), namely,

Therefore, the estimate of the attitude rate is giveii by

Cont roller

The last item required to simulate the whole system of Fig. 2.11 is a user-element for the controller. From that figure. we have that

o=&-é-~é (3.92)

= e + Xe (3.93) where e and e are, respectively, the error on the attitude and that of the attitude rate, as defined in Section 2.3.1. The parameter o thus represents the left-hand side of the switching-line equations. The required command of the thruster u is thus simply the output of a relay with a dead zone whose input is o. Thus, we have the algorit hm

This algorithm can be readily implemented in a user-element. Chapter 3. Modelling of Flexible Space Manipulator Systems 93 3.6 Summary

This chapter dealt with the modelling of flexible space manipulator systems. -4 general formulation to obtain the dynamics model of an N-flexible-joint space manipulator using a Lagrangian approach was described. The dynamics equations obtained using this approach were linearized in Section 3.3 to obtain the dynamic matrix W used to derive the natural frequencies of the svstem. The dynamics model of the spatial system, a three-flexible-joint manipulator mounted on a six-dof spacecraft with the same architecture as the C.ANADAR'V1- Space Shut tle Systern, was obtained using the above mentioned formulation. Realis- tic physical parameters for this system were identified using the C.-\N.-\DARM data provided by Spar Aerospace Ltd. (1996). While the mass and geornetric properties of the system were readily available. it was required to solve an inverse problem in vibration to derive the spring stiffnesses at the joints. The damping coefficients were obtained assuming damping proportional to stiffness. The system is decoupled using a change of variable based on the modal matrix P. This decoupling was required for the equations of motions in order to obtain the standard second-order form and thus be able to determine the damping coefficients based on the known damping ratio. The dynamics model of the spatial system was validated in Subsectiori 3.4.2 using Mecano, a finite-elernent software package. Finally, this software package was adapted to model control systems using the Mecano user-element. In order to use this element, the dynamics of the system to be modelled must be cast in a second-order form. -4 general technique was proposed to transform n-order differential equations in the required form, thus ailowing the modelling of various control schemes after block-diagram manipulations. The classical rate estimator of Chapter 2 is used as an example to show the usefulness of the proposed adaptation. Since Mecano was not originally developed for the modelling and analysis of control systems, this adaptation required intensive work, which was worthwhile due to the significant increase in capabilities we obtained for t his finiteelement software. Chapter 4

Control Synthesis Using an Asymptotic State Estimator

4.1 Introduction

The general theory of attitude control using on-off thrusters was introduced in Chap- ter 2, along with sorne methods useful in the analysis of this kind of rionlinear systems. In Chapter 3, a general formulation for the modelling of space robots was introduced and the mode1 of a three-flexible-joint manipulator mounted on a six-dof spacecraft was derived. In this chapter we study the control of space free-flyers with flexible manipulators. The classical rate estimator is used to demonstrate the problem of dynamic interactions using both a planar and a spatial system. An asyrnptotic state estimator is introduced, which shows significant performance improvement cornpared with the classical rate estimator. Section 4.2 introduces the problem of dynamic interaction of flexible modes wi t h on-off thrusters, using two examples: a planar system and a spatial system. In the same section, the classical rate estimator described in Chapter 2 is analyzed theoretically using a two-mass system, and a parametric study is performed to show the influence of the various parameters. These results are confirmed using simulations Chapter 4. Control Synthesis Using an Asymptotic State Estimator 95

Figure 4.1: .A planar free-flying manipulator. with a spatial system. An alternative controller, which makes use of an asymptotic state estimator, is then described in Section 4.3. Finally. this controller is studied in detail in Section 4.4, both theoretically and using simulations to show the increase in performance t hat can be at tained using t his particular estimator.

4.2 Classical Rate Estimator-Problem Descrip- tion

In this section, the problem of dynamic interaction of flexible modes with on-off thrusters is described with the aid of two examples using the conventional control scheme introduced in Subsection 2.3.3 and shown in Fig. 2.11, namely the classical rate estimator. This control mode1 is similar to the one used on the Space Shuttle. In order to better understand the problem, the first example deals with the planar free-flying manipulator of Fig. 4.1. For the second example, the t hree-flexi ble-joint manipulator mounted on a 6-dof spaceraft described in Section 3.4.1 is considered. This spatial system will be used in this chapter and in Chapter 5 to demonstrate the usefulness of the new proposed control schemes. Chapter 4. Control Synthesis Using an Asymptotic State Estimator 96 4.2.1 The Interaction Problem with a Planar System

The mode1 of the one-flexible joint manipulator mounted on a three-dof planar base of Fig. 4.1 is obtained using the general formulation of Section 3.2 and is described in AppendLv C.2. The classical rate estimator of Fig. 2.11 was simulated using this planar system as the plant with a configuration described by &(O) = 02(0) = 45", and with

(i) a 4 500 kg payload, which corresponds to P = 0.05;

(ii) a 27000 kg payload, which corresponds to /3 = 0.3, and thus, is the maximum payload that the CANADARM was designed for.

The plant parameter values. derived from the Space Shuttle-CANADARM system, are displayed in Table 4.1 for the two payload ratios 3, while the control system parameters are displayed in Table 1.2. The frequency expression of Eq(3.23) was used to obtain the required spring stiffness k and damping coefficient c of the flesible joint such that the first natural frequency of the Space Shuttle-CANADARSL system was matched for a specific configuration. without payload. Csing the parameter values of Table 4.1, the required k and c values were found to be

(4. la)

(4.1 b) for the CANADARM in a configuration that corresponds to the one of Fig. 4.1. with el = e2 = O deg. In this configuration, its first natural frequency is w, = k(0.373) rad/s, and the damping ratio is = 0.05, without payload. Simulation results for an initial spacecraft angular error of 0.05 rad are shown in Fig. 4.2 for the case where P = 0.05, and in Fig. 4.3 when /3 = 0.3. From Fig. 4.2(a), (b) and (c), it can be seen that the control system is effective in stabilizing the spacecraft inside the desired attitude limits. From Figs. 1.2(d) and (e), we see that ody a few firings of the thrusters are necessary, which results in a low fuel Chapter 4. Control Synthesis Using an Asymptotic State Estimator 97

Table 4.1: Planar system parameter values.

Body li (m) ri (m) mi (kg) Ii (kg m') O 1.9 90 O00 9 490 533

payload ( p) 2.25 (38)'1" Dmo 28.38mo(3/3)'13 l+p (@= 0.05) 16.3 0.6 4 826 66 582 l+p (0 = 0.31 17.7 0.1 27326 748 791

Table 1.2: Parameter values used the classical rate estimator for the planar example.

consumption of 55.7 fuel units for the 2000-s run. The resulting rate R; of fuel consumption is 0.0010 fuel-units/s, which is below the limit of 0.0060 fuel-unitsfs, and thus the system is classified as stable. However, if we look at Figs. 4.3(a) and (b), we see that starting with the same initial error for the attitude of the spacecraft, 0.05 rad, the controller cannot stabiliae the system inside the deadband lirnits. h large limit cycle is reached due to the dynamic interactions. which results in a continuous operation of the thrusters and a high fuel consumption of 1703.6 fuel units, as can be observed in Figs. 4.3(c) and (d). Moreover, in this case, the rate R; of fuel consumption is 0.9298 fuel-units/s, which is much higher than the lirnit of 0.0060 fuel-units/s, and thus, the system is classified as unstable. Therefore, we see that the performance of the controller of the Space Shuttle is very good when the natural frequency of the system is high, namely, with a small payload, but fastly deteriorates when this natural frequency decreases because of a bigger payload. The controller of the Space Shuttle could thus fail should the CANADARM gasp a bigger payload than it should be able to handle by design. To avoid this, the operational capabilities of the arm suffer. Before going to the next sections and see how the controller can be modified to improve signifiantly its Chapter 4. Control Synthesis Using an Asymptotic State Estimator 98

e (rad) e (rad) û (rad) (4 (b) (4

1.5 60

1 505 0.5 40, Fûel a, u O -0.5 Units , -1 iO

Figure 4.2: Simulation results for the planar system with P = 0.05: (a) spacecraft error phase plane; (b) spacecraft error phase plane (zoom); (c) spacecraft attitude phase plane; (d) thruster-command history; (e)fuel-consumption history; and (f) joint-angle history. Chapter 4. Control Synthesis Using an Asymptotic State Estimator 99

e (rad) 9 (rad) (4 (b)

Figure 4.3: Simulation results for the planar system with = 0.3: (a) spacecraft error phase plane; (b) spacecraft attitude phase plane; (c) thruster-comrnand. history: (ci) fuel-consumption history ; and (e) joint-angle history. performance, in the subsection below we examine the spatial system of Fig. 3.3.

4.2.2 The Interaction Problem with A Spatial System

In this subsection, the pro blem of the dynarnic interactions demonstrated in Sub- section 1.2.1 is addressed by considering the dynamics mode1 of the spatial system developed in Section 3.4. The spatial classical rate estimator of Fig. 2.12 is first simulated using the parameters of Table 4.2 for the controller about each avis of the spacecraft, and a 4 500 kg payload for the configuration #5 of the manipulator, namely, el = 1200, d3 = 900, e5 = 1050 the results being displayed in Fig. 4.4. The system, initially at rest with $, 4 and 0 = O rad, is commanded to reach an orientation in which its three Euler angles @, 4 and 6 are 0.05 rad. The initial attitude errors for the three axes of the spacecraft are C hapter 4. Control Synt hesis Using an Asymptotic State Estimator 100 thus 0.05 rad. Figures 4.4(a), (d) and (g) depict, respectively, the error phase plane for the $, 4 and 0 Euler angles. It can be observed that the controller can bring the system into the desired final orientation without a problem. Figures 4.4(b), (e) and (h) depict the thruster history, and Figs. 44(c), (f) and (i) illustrate the resulting fuel coasumption, which is quite low for the three cases, namely, 46.0, 60.1 and 53.3 fuel units for the li>, Q and 0 controller, respectively. Moreover their rate of fuel consumption Rj is 0.0002 fuel-unitsjs for the three cases, which is below the previously established limit of 0 .O060 fuel-units/s, t hus corresponding t O a stable system. Finally, Figs. 44), (k) and (1) display the three joint-angle histories. CVe can see that the oscillations in the joints remain low for this simulation run. The same system is now considered for the same coiifiguration of the manipulator and the same desired final orientation of the base, but using a 27000-kg payload. which corresponds to ,8 = 0.3. The results, displayed in Figs 4.5(a) to (1), show the failure of the controller to bring the spacecraft into the desired orientation. The thrusters fire continuously, thus resulting in a high fuel consumption of 1718.8. 444.4 and 1336.6 fuel units, respectively, for the Sr, & and B controller, which is high compared to the stable case. Moreover, for the three aues, the system is classified as unstable since the rate of fuel consumption Rj is. respectively. 0.9533, 0.3013 and 0.8667 fuel-units/s for the #, # and 0 controllers. Finally, from Figs. 4.5(j) to (l), it is observed that high oscillations in the three joints of the manipulator result from this unstable motion. We can see, once again, that the Space Shuttle controller would have failed, should it have been required to handle a massive payload in some configurations. Thus, the use of the CANADARM in its full capabilities would have been prevented.

4.2.3 Problem Demonstration wit h Theoretical Analysis and Parametric Study The interaction problem identified in Subsections 4.2.1 and 4.2.2 is analyzed here using the describing-function and the root-locus methods, for the two-mass system Chapter 4. Control Synt hesis Using an Asymptotic State Estimator IO 1

Units 20

-0.5

10 U+ -1 hl-

Fuel Units (4)

Fue 1 Units (0)

Figure 4.4: Simulation results for the spatial system with ,û = 0.05: (a) $-auis error phase plane; (b) @-aist hnister-command history ; (c) 1C>-auisfuel-consump t ion history; (d) +axis error phase plane; (e)+axis thruster-command history; (f) +cuis fuel-consumption history; (g) 8-ais error phase plane; (h) @-misthruster-cornmand history; (i) 0-axis fuel-consumption history; (j) joint-angle B2 history; (k) joint-angle B4 history; and (1) joint-angle Os history. C hapter 4. Control Synt hesis Using an Asymptotic State Estimator 102

e,: (rad) (4

Figure 4.5: Simulation results for the spatial system with ,t?= 0.3: (a) +ais error phase plane; (b) thruster-command history; (c) $-ais fuel-consumption history; (d) #-axis error phase plane; (e) &axis thruster-command history; (f) $-axis fuel-consumption history; (g) 0-axis error phase plane; (h) O-axis thruster-command history; (i) 0-axis fuel-consumption history; (j)joint-angle O2 history; (k) joint-angle O4 history; and (1) joint-angle Ob history. Chapter 1. Control Synthesis Using an Asymptotic State Estimator 103

Figure 4.6: The two-mas system.

of Fig. 4.6. Using those methods, a parametric study of system stability is perfornied for the same system, the results being validated using simulations of the spatial system of Section 3.4.

Analysis of the Problem Using the Two-Mass System

The dynamics model of the two-mas system of Fig. 4.6 is derived in Appendix C.1 in various forms. The model parameters. Ml, hi2,k and c are chosen such that the natural frequency unand the damping ratio C, of this system match the first natural

mode ;JI and damping ratio of the spatial systern of Section 3.4. for a given configuration and payload. It will be shown that this simple system can predict well the performance of the more realistic spatial system. Moreover, we introduce the same control algorit hm of Section 2.3.3 to analyze this system, namely, the classical rate estimator. This algorîthm is adapted for the two-mass system in Fig. 4.7. where y is the acceleration of the system under the thruster action, and can be written as

+yo being the nominal acceleration that can be provided by the thrusters when ,û = 0. Chapter 4. Control Synt hesis Using an Asymptotic State Estimator 104

Il I It i I t uY 1% f II - . I If acc. Me Ga i II II I II I II Ys I II 1 II I

II II delay ( dif f/filter ' II

Figure 4.7: Mode1 with a classical rate estimator for the two-mass system.

For the classical rate estimator of Fig. 4.7, the transfer function of the linear elements Gr,&) (Fig. 2.13) is derived in Appendix D.l and is given by

where G,(s), Gl(s) and G,.(s) are defined in Eqs.(C.32), (2.21) and (2.Z), respectively. The plant transfer function is represented by G,(s),while G/(s)and G,,(s) are the transfer functions of second-order filters. Finaily. For the root-locus analysis. the delay T is represented as a third-order Padé approximation, namely,

We consider again the same two cases introduced in Subsection 4.2.2 for the spatial system, namely, a configuration of the manipuiator where

and two different payloads

i) p = 0.05 ii) P = 0.3 Chapter 4. Control Synt hesis Using an Asyrnptotic S tate Estimator 105 From Table 3.6, the first natural mode and damping ratio of the system for these values are

i) wl = 2~(0.096)rad/s, CI = 0.015

ii) w2 = 21r(0.053) rad/s, G = 0.008

Therefore, we consider the two-mass system with two sets of parameters:

i) /j=O.OS, w, = 2n(0.096) rad/s, C =0.015

ii) ,Ll = 0.3, ui, = 2r(0.053) rad/s, C = 0.008

These two sets will be used throughout this thesis and referred to as low-payload case for the first set, and high-payload case for the second set. Moreover, for this study, we choose X = 5 s, wl = 0.47 rad/s

The plots obtained using the describing-function technique described in Subsec- tion 2.1.1 are displayed in Fig. 4.8(a) for the low-payload case and in Fig. 49(a) for the high-payload case. Similarly, the root-locus plots obtained using the root-locus analysis of Subsection 2.1.2 with the amplitude .4 of the predicted limit cycle as the varying parameter. are displayed in Figs. 4.8(b) and (c), and in Figs. 4.9(b) and (c), respectively for the low-payload case and the high-payload case. From Fig. 4.8(a), it is clear that the system is stable since none of the points of the -1/Li(=l) locus is encircled by the G(jw) locus. The sarne conclusion is obviously obtained by looking at Figs. 4.8(b) and (c), since none of the closed-loop poles lies in the right-half plane. Obviously, Figs. 4.8(a) and (b-c) correspond to two different pictures of the same problem. Table 4.3 lists the numerical values of the open-loop poles and zeros depicted in Figs. 4.8 and 4.9. In this table, the open-loop poles are directly related to the responsible physical component by considering the tranfer function G,&) of Eq.(4.3). On the other hand, by looking at Fig. 4.9(a), we see that the system is U2- unstable, according to the stabiiity definition of Section 2.5, since the - l/N(A) and Chapter 4. Control Synthesis Using an Asyrnptotic State Estimator 106

Figure 4.8: Theoretical analysis with the classical rate estimator for the low-payload case (13 = 0.05): (a) describing-function plot; (b) root-locus piot; and (c) root-locus plot (zoom). Chapter 4. Cont rol Synthesis Using an Asymptotic State Estimator 107

Figure 4.9: Theoretical analysis with the classical rate estimator for the high-payload case (13 = 0.3): (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom). Chapter 4. Controi Synt hesis Using an Asymptotic S tate Estirnator 108

Table 4.3: Pole-and-zero location for the classical rate estimator.

rZeros G(jw)loci intersect, the intersection point corresponding to a stable limit cycle. The sarne conclusion is drawn when looking at Figs. -4.9(b) and (c). where two loci go in the right-half plane, thus corresponding to positive closed-loop poles. and hence. to an unstable system. It is very interesting to compare Figs. 4.8(b) and (c) with Figs. 4.9(b) and (c). We know that the effect of increasing the payload, or equivalently, of fi, is to lower the natural frequency of the system. and thus, the poles and aeros corresponding to this frequency move towards the real axis in the root-locus plot, namely, those in the upper right part of Figs. 4.8(c) and 4.9(c). From Fig. 4.8(c), it is apparent that these poles and zeros do not interfer with those of the filters Gr(s) and G,.(s). However, as ,O increases, the locus completely changes and the dynamics of the plant interfers with the dynamics of the filters, as shown in Fig. 4.9(c), the result being that two loci cross the imaginary axis into the right-half plane. From this analysis, it becomes clear that any controller designed to avoid such interference would improve the system stability. This analysis using the root-locus plots is very important in understanding the problem at hand and in finding solutions to eliminate it. Chapter 4. ControI Synt hesis Using an Asyrnptotic State Estimator 109

Parametric Study for the Classical Rate Estimator

A parametric study, using the describing-function and the root-locus methods, is next performed. System stability is investigated as with the two foregoing examples. A11 results have been validated using simulations with the spatial systern, including white noise in the sensor readings. In general, simulation results confirm those obtained by the descri bing-function and root-locus methods. In this pararnetric study, the negative inverse of the slope of the switching lines A and the cutoff frequency wfin the second-order filter Gf(s)used to eliminate high frequency noise in the attitude measurements were varied. We have considered two different values of A, A = 3 s and X = 5 s, since for these values, the performance of the system is highly dependent upon the configuration of the manipulator and the payload carried. The cutoff frequency wf was chosen, in turn, as wl = 0.2513 rad/sl q = 0.47 rad/s, and wj = 0.6911 rad/s. Howewr. the attitude limit 6 and the acceleration level y0 provided by the thrusters were kept constant. since, as shown in (Martin, 1995), their effects are less important than the parameters A and q. Moreover, three different configurations of the rnanipulator are considered. namely,

i)O1 =O0, 03=Oo1 &=O0

ii) 0, = O", O3 = 135', O5 = -90'

iii) 01 = 12Ool O3 = 90°, O5 = 105O

From Table 3.6, we see that the first configuration usually corresponds to the lowest frequencies of the system for a given payload, while the third configuration corresponds to the highest frequencies, and the second configuration corresponds to intermediate values. Finally, we have considered four different payloads,

The cases of ,O = O and P = 1 are extreme (no payload and payload with the same mass as the one of the spacecraft, not likely in a real system). The value ,û = 0.3 Chapter 4. Control Synt hesis Using an Asymptotic S tate Est imator 110 corresponds to the maximum value that was considered for the CANADARbf-Space Shuttle system, while P = 0.1 corresponds to a more typical payload. The results of this study are summarized in Table 4.4. In this table. "UVmeans an unstable system, "Ul" a type-1 unstable system, "U2" a type-2 unstable system, and "Sn a stable system. We see, from the describing-function (DF) analysis that the system is always U1-unstable for X = 3 s and wj = 0.2513 or 0.47 rad/s. For these parameters, the flexibility in the system does not influence its stability, since the results are the same for any congiguration and payload. The same conclusion is drawn when X = 5 s and wj = 0.2513 rad/s. However, for A = 3 s and wl = 0.6911 rad/s, and for X = 5 s and wj = 0.47 or 0.6911 rad/s, the system is stable for low payload-to-spacecraft mass ratio ,8, but, as the ratio increases, the problem of dynamic interaction occurs and the systems becomes US-unstable, which means that a large limit cycle is obtained resulting in a high fuel consurnption. We can also observe that the stability of the system increases when X increases. and further increases by increasing i~!. Moreover, from Table 4.4, we see that when the theoretical analysis predicts a type4 unstable system (ül),theri this results is verified for al1 cases using simulation, with white noise added in the sensor readings, as discussed in Subsection 2.4.3. Similarly, when a stable system is predicted from the t heoretical snalysis, t his result is confirmed by simulations in al1 cases. Finally, when the theoretical analysis prc dicts a type-:! unstable system (U2), then this result is verified by simulation in 86% of the cases. For the remaining 14%, we observe the following. In some cases, the describing-function method rnay predict limit cycles of small and large amplitudes, like in Fig. 2.26. According to our definition for stability, the former correspond to a stable system, while the latter to an unstable system. Which one will result rnay depend on the initial error conditions. Since it is unknown a priori which kind of stability will result, we prefer to classify these cases as unstable, although a stable system may actually result. Thus, we see that the analytical study of the two-mass Chapter 4. Cont rol Synt hesis Using an Asymptotic State Estimator 111 system can predit very well the stability of the more realistic spatial systern. In fact, simulations vaiidated the theoretical predictions in 96% of the cases of Table 1.4.

Table 4.4: Results of the pararnetric study for the classical rate estimator.

S: Stable system U1: Type4 unstable system U: Unstable system U2: Type-2 unstable system C hapter 4. Control Synt hesis Using an Asymptotic State Estimator 112 4.3 Attitude Control using an Asymptotic State Estimator

We found in Subsection 4.2.3, using the root-locus method, that the main problem of the classical rate estimator is the interference of the first natural frequency poles and zeros with the poles and zeros of the second-order filters required in the classical rate estimator. The poles and zeros of the filters must be close to the origin since the attitude signal must be differentiated to obtain an attitude rate estimate. and high frequency noise should be eliminated to avoid any amplification in the differentiation process. We concluded that an estirnator that would elirninate this interference problem would increase the performance of the system. One possibility is to use a classical asymptotic state estimator, as described in many books, for example, in Chen (1984). In this estimator, the poles and zeros are placed "optimally" to give good tracking of the states while reasonably eliminating high-frequency noise in the sensor measurements. Since this estirnator is model-based, there is no need to directly differentiate the attitude signal to obtain an estimate of the attitude rate. thus reducing the need of having poles and zeros close to the origin of the comples plane. The asymptotic state estimator is described in Subsection 4.3.1, together with the pole-placement problem. This asymptotic state estimator is adapted for the spatial system in Su bsection 4 A.2 and a pole-placement criterion is proposed. This development is the major contribution found in this thesis.

4.3.1 Asymptotic State Estimator Theory

Description of the Asymptotic State Estimator

Let us consider a linear time invariant plant mode1 written in state-space form as Chapter 4. Control Synt hesis Using an Asyrnptot ic State Es timator 113

where y contains al1 the continuous measurements of the plant obtained with sensors, and u is the input to the plant which, in Our case, includes the thruster forces and moments. We can obtain an estimate for the state vector x with the use of y?u and knowledge of A and B. By defining x as the estimate of x, an estimated value of the measurernents y is given by

With knowledge of A, B and u, an estimate of x is available by feeding u into a software-implemented cornputer mode1 of the plant, namely,

Under perfect knowledge of A and B, Eqs(4.6) and (4.7) would give the actual states of the system. However, this is never the case; the way to solve this problem is to feed the error y - y back into every eqiiation of Eq.(4.7), i.e..

where L is a suitable gain matrix. By defining %=#-x as the error in the estimate, E can be determined by subtracting Eq.(45a) from Eq.(4.8): I=A%+L(y-Ci<) (4.10)

Using Eq.(4.5b), Eq.(4.10) can be written as

Therefore, the gain matrix L can be chosen such that the error equations, Eq. (4. Il), are stable, providing that {C, A) is an observable pair. A system is said to be corn- pletely observable if every initial state x(0) can be deterrnined from the observation Chapter 4. Control Synt hesis Using an Asymptotic S tate Estimator 114

Figure 4.10: Block diagram for the asvmptotic state estimator. of y(t) over a finite time interval (Ogata, 1990). In other words, if a system is observable, then al1 its states x can be estimated using an estimator which makes use of finite output measurernents y and inputs u to the system. Then, the error X will tend toward zero as t increases. Yow that a reasonable estimate x is available, the desired estimated outputs can be obtained with

where E is chosen such that y,,, contains al1 the required states. The block diagram describing the asymptotic state estimator is illustrated in Fig. 4.10. The determination of the gain matrix L is an important part of this state-estimator design. If the frequencies corresponding to the poles in the system of Eq.(.l.ll) are too large, then we obtain a very good estimate. but the filtering is not sufficient. Conversely, if these frequencies are too small, the filtering effect is very good but we obtain a poor estimate. Therefore, these gains must be determined carefully to obtain a reasonable performance of the state estimator. This problern is now tackled.

Pole Placement for the Asymptotic State Estimator

.As rnentioned above, the determination of the gain matrk L is an important part of the state-estimator design. The well known method of pole placement based on the Kalman Filter is a very good tool to attain the optimum pole selection. However, for this method, the covariance matrices of noise in the measurernents and Chapter 4. Control Synt hesis Using an Asymptotic State Estimator 115 the plant perturbations must be known. In this thesis, since we are not dealing with any particular system, we decided not to use this method, which would have required some noise level estimation, thus focusing more on a particular system. In Subsection 4.3.2, the poles of the asymptotic state estimator will be chosen such that the estimator behaves well with any flexible space robotic system. Some pole locations will be determined based on the optimality criterion for choosing observer poles as proposed by Kailath (1980) and briefiy described here. This criterion is valid for single-input single-output systems. Suppose that the plant perturbations and noise measurernents can be represented as uncorrelated whi te-noise processes VI(t) and u2 (t), respectively, wi th spectral iri- tensity of unity and r, respectively. Then, the single-input single-output version of the plant mode1 of Eqs.(4.5a & b) can be written as

x = Ax + bu + bul (t)

lJ= CTX + ~?(t)

where A is an n x n matrix, and b, c and x are n-dimensional vectors. If we set up an observer as before, Le.,

the error 1 is not only dependent upon the initial error %(O) as before, but also on the noise vi(t) and uz(t),which are driving forces of the previously autonomous error dynamics i= (A - icT)n+ bul (t) - 1v2(t) (4.15) Again, by choosing the gain vector 1 sufficiently large, we can make the expected or mean value of 2, tend toward zero as fast as desired. However, here we have to worry about the fluctuation of X(t), since choosing large values of 1 will accentuate the driving noise term 1v2(t). One criterion is to choose L so as to minimize the mean-square error, Le., to minimize C hapter 4. Cont rol Synthesis Using an Asymptotic State Estimator 116 Here, we address this minimization problem according to Kailath (1980). Suppose that {A, b) is controllable and {cT,A) is observable. Then, the roots of the characteristic polynomial det(s1- A + lcT) of the optimal observer caii be found as the left- half plane roots of the 2n-degree polynomial

where a(s)=det(sl-A), g(s)=cT.-\dj(sl-A)b (4.18)

For example, when the observation noise is high, r + W. the optimum observer poles are obtained as the stable roots of the original characteristic polynomial a(s) along with the unstable roots reflected across the jw mis. On the other hand. wtien r + O, we can have n - m poles going off to infinity in a Butterworth configuration, with the ot her rn poles going to the zeros of g(s).

4.3.2 Asymptotic State Estimator for Space Robotic Sys- tems

Adaptation of the Asymptotic State Estimator to the Spatial System

The system model of Eq.(3.20a), derived in Section 3.2 and used as the plant in our control system is obviously nonlinear. However, while using this nonlinear system to describe the plant dynarnics, its linearized dynamics derived in Section 3.3 can be used to develop the required asymptotic estimator. Let us consider the linearized dynarnics model of Eq. (3.22) and reproduced here:

The state vectors xi and x, can be defined as Chapter 4. Control Synthesis Using an Asymptotic Staie Estimator 117 whence

We can thus write

or with

where p = 6 + :V, :V being the number of links of the manipulator. In this thesis, only the attitude of the spacecraft is controlled and thus, no thuster forces f, are considered, but only thruster moments n,. Thus, # = [O*, nJIT and J, reduces to

for the N-link manipulator systern at hand.

For a fair cornparison with the current Space Shuttle attitude control system, Ive assume that only the three Euler angles +, 4 and 0 are available by sensor readings, and thus, the estirnator must provide an estimate of these Euler angles and their corresponding rates 6,4, and 8. Therefore, Chapter 4. Control Synthesis Using an Asymptatic State Estimator 118 and

Yout

bloreover, using the dynamics mode1 of the spatial system of Section 3.4. with its numerical values, its nm x n observability matrix V is obtained as

The rank of this matrk is found to be 12. Since this value is lower than the order of the system, which is n = 18, the system is not completely obsenrable. However, since we are not interested in estimating al1 the states. but only the attitudes and the attitude rates. the available information should be enough to estimate the desired state. To examine this conjecture, we introduce the transformation

and bring the system of Eqs.(4.23a & b) and (4.25) in the form

where

where x, represents the observable states and Xno the nonobservable ones. Thus, Chapter 4. Control Synthesis Using an Asymptotic State Estirnator 119 xo = Aoxo+ Bon, (4.31b)

Y = Cao (4.31~)

hgain, int roducing numerical values, the matrices of the system of equat ions (4.3la- c) can be obtained using the command ctrbf of Matlab. Moreover, it is obtained that? from the observable state xo, the following output estimate can be derived:

-a.- [ 0 8, è4 8 4 4 $, éJ j61T (4.32) and thus, there is no problem to recover the desired variable y,,,, defined in Eq(4.26). Yow, an asymptotic estimator cari be designed based on Eqs.(?.Slb Sr c) to estimate the state vector x,, namely x,. Hence,

no = A& + Bon, + L(y - y,) (4.33) with y = c,x,

Following the procedure of Subsection 43.1, we obtain

k, = (A, - LCo)Xo

where no is the error on the estimate defined as

The gain matr~vL can thus be chosen such that the error equations Eq.(4.35) are stable. Finally, the desired state youtcan be obtained from

where we have decomposed T-l as

T-' = [T,, T, ]

The block diagram implementation of the asymptotic state estimator for the spatial system is depicted in Fig. 4.11, and the complete rnodel, including the attitude Chapter 4. Cont rol Synthesis Using an Asymptotic State Estimator 120

Figure 4.11: Asvmptotic state estimator for the spatial system. controller and the plant dynarnics in Fig. 4.12. Note that the plant dynamics is represented by the nonlinear mathematical model given in Eq.(3.20a) and not by the linearized model, in order to better represent the actual system.

Pole Placement for Space Robotic Systems

Now, let us consider the optimum observer criterion, discussed in Subsection 4.3.1, with the simplified two-mass system of Fig. 4.6, which is a single-input single-output system. For the A matrix and b and c vectors given in Eqs.(C.33) and (C.35), we have

a(s) = det(s1 - A) = s2(s2 + 2&s + da) (-4.39) and Y g(s) = cTadj(sl - A)b = -[(1+ ,8)s2 + 2Cu.s + un] (4.40) 1+8 Therefore, the 2n-degree polynomial

can be written as

If r = oo (high observation noise), then the negative roots of Eq.(4.42) are those of Chapter 4. Control Synthesis Using an Asymptotic State Estimator 121

/ attitude controller

plant asppotic dynamics state estirnator A

&_I attitude 1, A controller

Figure 4.12: Mode1 with an asymptotic state estimator for the spatial system. Chapter 4. Control Synthesis Using an Asymptotic State Estimator 122 namel y,

On the other hand, if r = O (no observation noise), then the negative roots of Eq. (4.42) are those of (1 + P)s2 + 2

(~i.î)r=~ (4.46)

Comparing Eqs.(4.44) and (4.46),we see that among the four roots required for the pole placement. two optimum roots would be close to the poles associated with the natural frequency of the system, whatever the noise level of the system. In fact . when there is no payload (/3 = O), the optimum roots are exactly the same for high and low observation noise. However, for a very high payload, say 9 = 1. the magnitude of the roots for low observation noise would be away of those for high observation noise only by 30%, namely,

For the typical payloads studied in Subsection 4.2.3, ,d = 0.05 and J = 0.3, the magnitude of the roots for the low observation noise case would be, respectively,

Is~+~~~=~= 0.98~~ for O = 0.05 (4.48a) 1~~,~1~=~= 0.88~~ for ,3 = 0.3 (4ASb) which are close to the magnitude of the optimum roots for high observation noise, namely I s1,2 l,,- = un- Finally, according to the optimality criterion, for high observation noise, two roots would be placed at the origin, while these two roots would be placed in a Butterworth configuration for low observation noise. Knowing these facts, a criterion was elaborated for the pole placement for the asyrnptotic estimator developed in this thesis. The poles placed using this criterion Chapter 4. Control Synthesis Using an Asymptotic State Estimator 123 were found to give very good performance for a wide variety of systems without knowledge of any specific component, as would be required for an optimum design. Next, we summarize this developed criterion as it is applied to the multiple-input multiple-output systems studied in t his thesis. Pole Placement Criterion: Let n be the number of flexible modes in the system under study, and m be the nurnber of states to be estimated with the observer. The n + rn poles required for the state estimator poles should be placed as

a one set of complex conjugate poles at each set of complex conjugate natural frequency of the system, for a total of n poles;

m poles in a Butterworth configuration at a desired frequency y,,,. 4.4 Demonstration of t he Proposed Estimator De- sign wit h Simulation Results and Analysis

The describing-function and root-locus methods are now used to analyze the simple two-mass system of Fig. 4.6, with the control system of Fig. 1.13, which is simply the asymptotic state estimator of Fig. 4.12 adapted to the two-mass system. The methods are used to conduct the parametric study of Subsection 4.2.3, employing t his time the asyrnptotic state estimator, instead of the classical rate estiniator. These results are validated using simulations results with the spatial systern of Section 3.4, including white noise in sensor readings.

4.4.1 Analysis of the Asymptotic State Estimator Using the Two-Mass System For the mode1 of Fig. 4.13, the transfer function of the linear elements Ga&) is derived in Appendix D.2, and is given by Chapter 4. Control Synthesis Using an Asymptotic State Estimator 124

Figure 1.13: Mode1 with an asymptotic state estimator for the two-mas system. where,

with

and the plant transfer function GP(s)is defined in Eqs.(C.32). Li, for i = 1,2,3 and 4, are the gain components of 1. We consider again the same two cases of Subsections 4.2.2 and 42.3, namely the low-payload case (@= 0.05) and the high-payload case (fi = 0.3). Again, we consider the case where X = 5 s and wf = 0.47 rad/s. Chapter 4. Control Synthesis Using an Asymptotic State Estimator 125

Table 4.5: Pole-and-zero locations for the asymptotic state estimator.

1 I l/s2

Zeros

The plots obtained using the describing-function and root-locus methods are displayed, respectiwly, in Figs. 4. M(a- b) and (c-d) for the low-payload case, and in Figs. 4.15(a-b) and (c-d) for the high-payload case. From Figs. -I.l-l(a) arid (b), it is clear that the systern is stable since none of the points of the -l/N(.-I) loci is encircled by the G(jw) locus. The same conclusion is drawn by looking at Figs. 4.14(c) and (d), since none of the closed-loop poles goes to the right-half plane. The numerical values of the open-loop poles and zeros are listed in Table -4.5, the open-loop poles being directly related to the responsible physical component by considering the denorninator D(s) of the tranfer functions gl 1 (4, g12(s), 921(8) and g2*(s) defined in Eq.(4.49). In that table, PP stands for pole placement, as per the criterion stated in Subsection 4.3.2. Moreover, by looking at Figs. 4.15(a) and (b), we see that the system is also stable for ,O = 0.3, according to the stability definition of Section 2.5, since none of the points of the -1/N(A) locus are encircled by the G(jw)locus. Accordingly, none of the loci of Figs. 4.15(c) and (d) go to the right-half plane, thus corresponding to a stable system. Therefore, we see that the same system of Subsection 4.2.3, that Chapter 4. ControI Synt hesis Using an Asymptotic State Estimator 126

Figure 4.14: Theoretical analysis with the asymptotic state estimator for the low- payload case (P = 0.05): (a) describing-function plot; (b) describing-function plot (zoom): (c) root-locus plot; and (d) root-locus plot (zoom). Chapter 4. Control Synthesis Using an Asymptotic State Estirnator 127

Figure 4.15: Theoretical analysis with the asymptotic state estimator for the high- payload case (P = 0.3): (a) describing-function plot; (b) describing-function plot (zoom); (c) root-locus plot; and (d) root-locus plot (zoom). Chapter 4. Control Synthesis Using an Asymptotic State Estimator 128 ------was unstabie using the classical rate estimator, is now stable if the asymptotic state estimator is employed. We already see the improvement in performance that can be achieved if this control system is used instead of the current attitude controller of the Space Shuttle.

4.4.2 Parametric Study for the Asymptotic State Estima- tor

The results of this study are summariaed in Table 4.6. First. from this table. we see that the perforniance of this estimator is much better than that of the classical rate estimator studied in Subsection 4.2.3, with the results of Table 4.4. In fact, the describing-function (DF) analysis predicts stability for al1 cases, while simulations are unstabie only for three cases, al1 with the extreme value /l = 1.O. This estirnator thus results in very good performance and hence, can reduce considerably the problem of dynamic interaction in flexible systerns. Finally, we can note t hat the rate of success of the predictions using the simple two-mass system is 96%, which is surprising due to the simplicity of its dynamics compared to the one of the spatial system.

4.4.3 Simulation Results for the Asymptotic State Estima- tor

In this subsection, we evaluate the asymptotic state estimator using the spatial system. Again, the parameters of the controller about each avis of the spacecraft are listed in Table 4.2, and a 27 000-kg payload is considered with the configuration #5 of the rnanipulator, namely,

The results are displayed in Fig. 4.16 for initial attitude errors, for the three ais of the spacecraft, of 0.05 rad. Figures 4.16(a), (d) and (g) show, respectively, the error phase plane for the 11, 4 and 0 angles. It can be observed that the controller can bring Chapter 4. Control Synthesis Using an Asymptotic State Estimator 129

Table 4.6: Results of the parametric study for the asymptotic state estimator.

S: Stable system U1: Type-1 unstable system U: Unstable system U2: Type-2 unstable system Chapter 4. Control Synthesis Using an Asymptotic State Estimator 130 the system into the desired final orientation without any problem. Figures 4.16(b), (e) and (h) depict the thruster history, and Figs. 4.16(c), (f) and (i) the resulting fuel consumption, which is low for the three cases, namely, 41.2, 40.2 and 33.9 fuel units for the @, # and 0 controllers, respectively. Moreover their rate of fuel consumption Rj are, respectively, 0.0006 fuel-unit+, 0.0003 fuel-unit+ and 0.0003 fuel-units/s (below the limit of 0.0060 fuel-units/s), thus corresponding to a stable system. Fi- nally, Figs. 4.16(j), (k) and (lj dispiay the three joint-angle histories. LVe cari observe that joint oscillations rernain low for this simulation run. We thus see that the same spatial system that was unstable when considering the classical rate estimator (Fig. 44,similar to the one used on the Space Shuttle, is now stable. In fact, the fuel consumption and the general performance of the controller are similar to those we would obtain for a rigid spacecraft.

4.4.4 Effects of Mode1 Perturbation in the Asymptotic State Estimator

We observed in Subsection 4.4.2 that the model with an asymptotic state estimator was almost always stable. However, for this model, a dynamic representation of the plant is required; it was assumed previously that this was perfectly known. In this section, the effect of perturbing this dynamic representation is addressed. To perturb the system, we will assume that the mass properties used to construct the model of the plant, matrices A and B, are off by a factor p that varies between -10% and 10%. This factor was chosen randomly in Matlab for each mass property of the system. These perturbations seem realistic, since the mass properties are usually well known in space systems. The spatial system simulated in the previous subsection is now simulated using this perturbed model of the plant to be used in the estimator, noise in the measurements being also included. The results of this simulation show that the controlled system remains stable. The fuel consumptions have slightly increased for the same 2 000-s run; t hey are now 42.8,4 1.2 and 36.5 fuel Chapter 4. Control Synthesis Using an Asymptotic State Estimator 131

Fuel Units ($1

Units 20

Fuel Units (0)

0. "kpi d6 (deg) (deg)-los1 B0.8 - 1 os2 89.7

Figure 4.16: Simulation results for the spatial system using the asymptotic state estirnator and p = 0.3: (a) 1C>-axis error phase plane; (b) $-axis thruster-commnnd history; (c) $-suis fuel-consumption history; (d) +axis error phase plane; (e) #- mis thruster-command history; (f) #-axis fuel-consumption history; (g) 8-axis error phase plane; (h) 0-axis thruster-command history; (i) 0-axis fuel-consumption history; (j)joint-angle history; (k) joint-angle t4 history; and (1) joint-angle Os history. Chapter 4. Control Synthesis Using an Asymptotic State Estimator 132

Figure 4.17: Simulation results for the spatial system using the asymptotic state estimator with perturbed mass properties and /3 = 0.3: (a) @-miserror phase plane: (b) O-auis thruster-command history; (c) 6-axis fuel-consumption history. units for the @, 4 and 0 controllers, respectively. Their rate of fuel consumption Rj is, respectively, 0.0012, 0.0002 and 0.0002 fuel-unitsls. For example, the results for the 19 controller are displayed in Fig. 4.17. Therefore, we see that the system remains stable even with perturbed mass properties and additive noise. The asymptotic state estimator is thiis very robust and considerably improves the stability zone of the actual controller used on the Space Shuttle. Using this state estimator, the likelihood of actually exciting the flexible modes of the manipulator is small, even when assuming errors in the mass properties. Since al1 these mass properties are known precisely before sending an object into space. the performance cannot he worse than the one predicted in this section. Therefore, the asymptotic state estimator provides very good control characteristics.

4.5 Discussion and Conclusions

The dynamic interaction problem was described using the classical rate estimator. Using a planar one-flexible-link system, it was shown how the reduction of the system natural frequency, by increasing the payload, cm deteriorate the control system performance, resulting in high fuel consumption and large-amplitude vibrations. This performance degadation resulting from the increase in the payload mass was also Chapter 4. Control Synthesis Using an Asymptotic State Estimator 133 demonstrated using the spatial system. This problem was st udied t heoretically using the describing-function and root-locus techniques wit h the simple two-mass system. It was shown that the effect of increasing the payload, or, correspondingly, reducing the natural frequency, is to bring the plant poles and zeros closer to the real-auis, close to the origin of the complex plane. Since this control scheme eniploys second- - order filters with low cutoff frequencies to eliminate high-frequency noise, it has poles and zeros close to the origin. It is shown that the interference of the poles and zeros of the plant with those of the filters, for high payloads, amplifies the problem of dynamic interaction. Since this scheme uses a differentiator to estimate the attitude rate of the spacecraft, it is not possible to modify the pole-and-zero locations without the adverse effects of noise amplification. In order to remove the problematic poles and zeros close to the origin due to the use of low cutoff frequency second-order filters. an asymptotic state estimator was introduced. Since this estimator is model-based, there is no necd to directly differentiate the attitude signal to obtain an estimate of the attitude rate, thereby reducing the need of having poles and zeros close to the origin to eliminate high- frequency noise. In this estimator, the poles and zeros are placed more "optimally" to give good tracking of the states while reasonably reducing high-frequency noise in the sensor measurernents. This estimator also has the advantage of reducing the lag in the system by eliminating the use of second-order filters. The lag in the system is basically a time delay. and thus, as discussed in Subsection 2.3.1, it has the effect of tilting the switching lines. If the lag in the system is important, it can result in an effective switching line with a positive dope resulting in diverging motions and type4 unstable systems. By reducing the lag in the system, the asymptotic state estimator can thus stabilize systems with a diverging motion. With this reasonning, the increase in performance that can be attained using Chapter 4. Control Synt hesis Using an Asymptotic S tate Estimator 134 the asymptotic state estimator compared to the classical rate estimator can be explained. These conclusions were tested using the spatial system for a large number of simulations for various configurations and payloads. It was also shown that even when açsuming errors in the rnass properties in the dynamics mode1 used by the estirnator, this control scheme still gives very good performance. Finally, t his study has shown that the two-rnass system is very effective in predicting the stability of the more complicated spatial system and can thus be used for quick stability predictions and for control system design. The development of the asymptotic state estimator represents the major contribution in this thesis. This estimator is significantly more efficient than the actual classical rate estimator. Chapter 5

Control Synt hesis Using Compensation Techniques

Introduction

The problem of dynamic interactions was introduced in Chapter 4 with the use of two examples, where a conventional on-off thruster attitude controller was considered, namely the classical rate estimator. The use of an asymptotic state estimator was proposed to significantly improve the stability and performance of the system. However, this estimator being model-based, its use could be compromised in systems with low onboard computational power. In this chapter, we propose two new estirnators that do not require a model of the plant. The first one, introduced in Section 5.2, is very efficient in addressing the problem of dynamic interactions and of resulting large limit cycles. The second estimator, which can also stabilize systems with a diverging motion, is introduced in Section 5.3. This estimator shows similar performance to the model- based asymptotic state estimator. These two est imators thus represent two important contributions develop in this thesis. Chapter 5. Control Synthesis Using Compensation Techniques 136 5.2 Rate Estimator with Linear Compensation

5.2.1 Description of the Rate Estimator with Linear Corn- pensat ion

We consider here the use of second-order linear compensation to improve the performance of the attitude control system. This controller, which is designed using the simple two-mass system and implernented with the spatial system. results in significant performance improvernents. First, we consider the classical rate estimator of Fig. 4.7 for the two-rnass system. which is similar to the one used for the Space Shuttle. -4s shown in Section 4.2, this attitude control scheme results in a very good performance when the natural frequency of the system is high, or the system behaves as a rigid body. However, by decreasing the natural frequency in the systern, Say by increasirig the mas of the payload (B), the performance quickly deteriorates, resulting in large vibrations and high fuel consumption, not desirable in space. Comparing the root-locus plots of Figs. 4.8 and 4.9, obtained for the same controller but with different payloads, namely p = 0.05 and ,8 = 0.3, we see that the poles and zeros corresponding to the natural mode are concurrent when /3 = 0.05, but separate when B = 0.3. This fact is readily explained by looking at the plant transfer function, namely'

We see that for low 8, the zeros and poles of GP(s)are very close, but become different by increasing P. Moreover, when P is increased, the natural frequency of the system is obviously reduced, while the effect on the root-locus plot is that the corresponding poles and zeros move towards the real auis. As these poles and zeros move towards the real auis, their dynamics begins to interfer with the dynamics of the poles and zeros associated with the estimator, basically that of second-order filters. From Fig. 4.9(c), we see that the locus stemming from the zero to the pole associated with {un,C) goes into the right-half plane, and thus the system is unstable, as verified Chapter 5. Control Synt hesis Using Compensation Techniques 137 with simulations for the spatial system in Fig. 4.5. Therefore, one way to improve the performance of the system is to add poles and zeros using linear compensation such that these loci do not move into the right-half plane when B is increased. In Subsection 2.4.1, the characteristic equation F(s) of the system of Fig. 2.18 was found to be

where p(s) and q(s) are polynomials is S. Because s is a cornplex variable, Eq.(5.2) may be written in polar form as

Therefore, it is necessary that

In general, p(s) and q(s) may be written as

Then, the magnitude requirement, Eq. (5.5), can be writ ten as

This condition means that any point sl of the root-locus plot must satisfy Eq.(5.7) for a given K = KI,namely,

where Is1+ri li=l,...,m is the magnitude of the phasor from -zi to SI,and Is1+pi Ik1,--,n

is the magnitude of the phasor from -pi to SI, in the cornplex plane. Now,, if one Chapter 5. Control Synthesis Using Compensation Techniques 138 zero z,+ 1 and one pole p,+l are added to the system on the real axis, the magnitude requirement , Eq. (5.5), now becomes

IS + ~~11~+ z~IIs + z~/*** IS + zrnl Is+ tn+,l K = 1 (5.9) Is+piIIs+ P~II~+P~I Is+ pnI Is+~n+iI

Obviously, the point sl of the original root-locus plot will not satisfy Eq. (5.9) for the same KI, unless p,+ 1 and r,+ 1 coincide. and t hus,

Thus, by adding zeros and poles. we see that the locus in the complex plane has to change to satisfy the magnitude condition. Therefore, it is possible to modify the locus to obtain a desired shape that would correspond to desired cliaracteristics. For example, the locus that moves into the right-half plane in Fig. 1.9(c) can be "attracted" to the left by adding two zeros at the left of this locus, close to the real auis. In order to have a realizable filter, two poles are also added. Therefore, we can consider a second-order compensator of the form

Moreover, the root-locus plot of Fig. 4.9 was obtained using the transfer function of the linear elements of the classical rate estimator, namely,

Here, Grate(s)is the open-loop transfer function of the control system, namely p(s)/q(s) in the derivation above. From Subsection 2.1.2, we recall that for the nonlinear system at hand, the gain K is in fact the describing function N(A) of the relay nonlinearity. As mentioned above, it is possible to alter the root-locus plot by adding poles and zeros, Say by considering GJs) of Eq.(Lll). The new set of open-loop poles and zeros is thus given by

Gcomp (s) = Gc (s)Grate(s) (5.13) Chapter 5. Control Synthesis Using Compensation Techniques 139

Figure 5.1: Mode1 with a rate estimator and linear compensation for the two-mas systern. namely,

A realization of this open-loop transfer function GcOmp(s)is included in Fig. 5.1. This control system is derived from the classical rate estimator of Fig. 4.7 by passing both feedback signals il and through the linear cornpensator GJs). By a proper selection of the poles and zeros of G,(s), the root-locus plot of the classical rate estimator can be altered such that none of its loci goes into the right-half plane, thus stabilizing the previously unstable system. The issue of pole-and-zero selection will be discussed in Subsection 5.2.2. Obviously, this compensator G,(s) can also be included in the classical rate estimator of the spatial system. The new state estimator for each control avis of the spacecraft is included in Fig. 5.2, the complete block diagram of Fig. 2.12 being still applicable. This new control system will be termed henceforth "rate estimator with linear compensation". Chapter 5. Control Synthesis Using Compensation Techniques 140

* acc. 8.

I diff/filter comp.

Figure 5.2: Block diagram for the rate estimator with linear compensation.

5.2.2 Determination of the Compensator Pole and Zero Lo- cations

An important aspect of the control system of Fig. 5.1 is the location of the poles and zeros of the second-order linear cornpensator GJs) of Eq.(S.ll), namely, the determination of G, w,, C, and w,. For this task, it is very important to keep in mind that the compensator is cascaded to a second-order filter, which is used to eliniinate high-frequency noise. Therefore, the zeros of G,(s) should not be too close to the origin to avoid noise amplification of the signal. On the other hand. the poles of G,(s) should not be too Far froni the origin for the sarne reason. 'uloreover. by our own experience with many test cases, it was observed that the imaginary part of the zeros of G,(s) should be below the lower imagina- part of either the poles of the second-order filter or the poles of the plant transfer function. After many trials, a set of guidelines was elaborated to automatically select the poles and zeros of G,(s). The experience gained studying the asymptotic state estimator was also very helpful in deriving those guidelines. These guidelines do not always lead to the optimum design of the compensator for a particular case. but were found to give good performance in most cases. hnother important aspect was the sensitivity of the stabiliy of the system to parameter variations, like those related to the motion of the space manipulator. These guidelines were found to give good reasonable performance for a wide range of motions of the manipulator. Chapter 5. Control Synthesis Using Compensation Techniques 141 Guidelines for compensator design

i) Determine the real and imaginary part dT and di of the poles associated with the second-order filters Gl(s) and G,. (s) :

ii) Determine the imaginary part di, of the poles associated with the first natural mode w, of the system: di,, = Js

iii) The imaginary part di,, of the zeros of the cornpensator G,(S)is chosen as the lower value between di,, di,,, and 0.ï5di,. The constant 0.73 \vas found to be the higher value that eliminates the interference between the poles and aeros in most cases. If the poles of the estimator are too close to the zeros of the first natural mode, the conipensator was not effective to stabilize the system. Thus, 4%= min{di,, di,, ,0.75di, }

iv) The real part d,,, of the zeros of the compensator GJs) is chosen as half the higher value of d,, and d,,, , namely,

This value was found to give good results in most cases. If the zeros of the compensator were closer to the poles of Gl(s) and GSe(s),the interference problem with the poles of the first flexible mode would not be resolved. On the other hand, zeros closer to the origin would reduce the filtering effects of the second-order filters.

v) The imaginary part di, of the poles of the compensator GC(s)vanishes, di, = 0. Chapter 5. Control Synthesis Using Compensation Techniques 142

vi) The real part d,, of the poles of the compensator G,(s)is chosen as the mean value of dr, and dr,, , namely,

This criterion was chosen based on our experience with the asyrnptotic state estimator.

vii) The parameters G,uz, C, and wp of the second-order linear compensator GJs) of Eq.(5.11) are thus given by

5.2.3 Demonstration of the Proposed Estimator Design wit h Simulation Results and Analysis

In this subsection, the describing-function and mot-locus methods are used to study the two-mass system of Fig. 1.6 using the control system of Fig. 5.1, narnely, the rate estimator with linear compensation. These rnethods are used to repeat the paramet ric study done for the classical rate estimator and the asymptotic state estimator. These results are vaiidated using simulation results with the spatial system of Section 3.4, including white noise in sensor readings.

Analysis of the Rate Estimator with Linear Compensation Using the Two- Mass System

For this closed-loop system, the transfer function of the linear elements G,,,,(s) of the control system of Fig. 5.1 was obtained in Subsection 5.2.1 as

where GP(s),Gf(s), G,.(s) and G&) are defined in Eqs.(C.32), (2.21), (2.22) and (5.Il), respectively. The plant transfer function is represented by Gp(s) , while Gl(s) Chapter 5. Cont rol Synthesis Using Compensation Techniques 143 and G,&) are the transfer functions of the second-order filters, and GJs) is the transfer function of the linear compensator. For the root-locus analysis, the delay r is represented as the third-order Padé approximation given in Eq.(4.4. We consider again the same two cases previously studied, the low-payload case (9= 0.05) and the high-payload case (P = 0.3). Applying the guidelines of Subsection 5.2.3, the poles and zeros of the compensator G,(s) are obtained, for both cases, as

The plots obtained using the describing-function technique and the root-locus method are displayed, respectively, in Figs. 5.3(a) and (b-c) for the case where LI = 0.05, and in Figs. 5.4(a) and (b-c) for the case vhere 9 = 0.3. The added poles and zeros of the compensator G,(s) are located with arrows in Figs. 5.3(c) and

(c).These two figures can be analyzed together with Figs. 48(c) and U(c)to compare the root-locus plots before and after the addition of the compensator poles and zeros. From Fig. 5.3(a), it is clear that the system is stable since none of the points of the - l/iV(A) locus is encircled by the G(jw) locus. The same conclusion is also obtained by looking at Figs. 5.3(b) and (c), since none of the closed-ioop poles goes into the right-half plane. The open-loop poles and zeros are listed in Table 5.1: using the transfer function G,,,,(s) of Eq. (5.15), it is possible to associate each pole with its responsible physical component. Sirnilarly, by looking at Fig. 5.4(a), we see that the system is also stable for ,d = 0.3 according to the stability definition of Section 2.5, since none of the points of the -1/N(A) locus are encircled by the G(jw)locus. Accordingly, none of the loci of Figs. 5.4(b) and (c) goes into the right-half plane, thus corresponding to a stable system. Therefore, we see that the same system of Subsection 4.2.3 that was unstable using the classical rate estimator is now stable if linear compensation is added. We already see the improvement in performance that cm be achieved if this Chap ter 5. Control Synthesis Using Compensation Techniques 144

Figure 5.3: Theoretical analysis for the rate estimator with linear compensation and ,O = 0.05: (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom). Chapter 5. Controi Synthesis Using Compensation Techniques 145

Figure 5.4: Theoretical analysis for the rate estimator with linear compensation and p = 0.3: (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom). Chapter 5. Cont rol Synt hesis Using Compensation Techniques 146

Table 5.1: Pole-and-zero locations for the rate estirnator with linear compensation.

Poles zZeros simple implementation is considered wit h the current attitudc controller of the Space Shuttle.

Parametric Study for the Rate Estimator with Linear Compensation

The results of this study are summarized in Table 5.2. We see, from the describing- function (DF) analysis, that this new controller improves the range of parameters for which the system is stable, when compared with Table 4.4 for the classical rate estimator. We see that al1 cases that were previously U2-unstable are now stable, which means that this estimator is effective in reducing the instability due to t,he dynamic interactions resulting in large limit cycles. However, when the system is U1-unstable, in most cases, it is still U1-unstable when employing compensation. Moreover, from Table 5.2, we see, again, that the simple two-mass system is very effective in predicting the stability of the more compiicated spatial system. In fact, 100% of the U1 predictions of instability are verified by simulation, while 82% of the Chapter 5. Control Synthesis Using Compensation Techniques 147

predictions of stability are verified by simulation. Finally, 67% of the U2 predictions are verified, for an overall succes of 89% in the predictions.

5.2.4 Simulation Results for the Rate Estimator with Lin- ear Compensation

In this siihsacbion: the spatial system is simulated using the rate estimator with linear compensation. The same control system parameters about each avis of the spacecraft previously studied are considered here. Theçe parameters are recorded in Table 4.2, and a 27000 kg payload is considered with the configuration #5 of the manipulator, namely,

The results of the simulation are displayed in Fig. 5.5 for the following compensator G, (s) definition, accordirig with the design guidelines, naniely.

Figures 5.5(a), (d) and (g) depict, respectively, the error phase plane for the v, p and B Euler angles. It can be observed that the controller can bring the system into the desired final orientation without any problem. Figures 5.5(b), (e) and (h) display the thruster history, and Figs. 5.5(c), (f) and (i) the resulting fuel consumption, which is quite low for the three cases, namely, 125.8, 115.9 and 95.1 fuel units for the @, 4 and 9 controllers, respectively. kloreover, their rate of fuel consumption Rj are, respect ively, 0.001 1 fuel-unit+, 0.0006 fuel-unitsls and 0.0017 fuel-units/s, which are below the limit of 0.0060 fuel-units/s, thus corresponding to a stable system. Finally, the three joint-angle histories are displayed in Figs. 5.5(j), (k) and (1), frorn which it can observed that the oscillations in the joints remain low. Again, we observe that the same system that was unstable using the classical rate estimator is now stable when adding linear compensation to the controller. This new controller Chapter 5. Control Synt hesis Using Compensation Techriiques 148

Table 5.2: Results of the parametric study for the rate estimator with linear corn- pensation.

S: Stable system U1: Type-1 unstable system U: Unstable system U2: Type-2 unstable system Chapter 5. Control Synthesis Using Compensation Techniques 149 is thus effective in reducing the dynamic interaction due to flexibility, although it is simpler than the asymptotic state estimator studied in Chapter 4, since it does not require explicit knowledge of the system dynamics and the associated parameters.

5.2.5 Verification of the Robustness of the Rate Estimator with Linear Compensation

In this subsection, the robustness of the estimator is investigated by looking at the poles and zeros of the compensator G,(s) for different co~figurationsof the manipulator and payloads. The effect of a payload or of a change in manipulator configuration is to modify the natural frequencies of the system. According to the guidelines for the compensator design stated in Subsection 5.2.3. a change in the natural modes can only affect the imaginary part di,, of the zeros of the compensator G,(s), which is given as

Since w,, is fixed to w,, = 0.2513 rad/s and q = 0.2513. 0.47 or 0.6911 rad/s. we have

Thus, according to Eq.(5.16), d*,, is chosen based on the natural mode of the system only if 0.75di, < min {dif,di,, ) (5.18)

For the manipulator configurations and payloads studied in the parametric study, the term dip = dw w, is recorded in Table 5.3. For that matter, the first natural modes and damping ratios listed in Table 3.6 were considered. Looking at Table 5.3, we see that di, < 0.237 rad/s in only four cases. Thus, for ,û = O and 0.1, the C hapter 5. Control Synthesis Using Compensation Techniques 150

Fuel Units (4)

r IO-' 1.5 toO+

2 t 805 1 0.5 Fuel ee O Ue 0 it- Units (rad/s) -' -0.5 q -2 , -1 - 20. -3 1 -4 -1.5' 4.04 -0.02 O 0.02 0.04 O 500 1000 1W 2000 O0 SOo lO00 l& 2000

4 1202 (deg) "O 1 19.0 -1052

Figure 5.5: Simulation results for the spatial system using the rate estimator with linear compensation and 0 = 0.3: (a) @-axis error phase plane; (b) $-auis thruster- command history; (c) $-axis fuel-consumption history; (d) &axis error phase plane; (e) baxis thruster-command history; (f) #-axis fuel-consumption history; (g) 9-axis error phase plane; (h) 9-axis thruster-command history; (i) 0-auis fuel-consurnption history; (j) joint-angle history; (k) joint-angle t4 history; and (1) joint-angle Os hist ory. Chapter 5. Control Synthesis Using Compensation Techniques 151

Table 5.3: Parameter d,, for various configurations and payloads.

Config. 1 Config. 3 Config. 5 ] O 2.337 3.116 3.506 1

poles and zeros of the compensator are the same for al1 configurations. They remain also the same when = 0.3 and 1.0 for configuration #5. Hence, we see that for most configurations and payloads, the poles and zeros of the compensator do not need to be updated. Moreover, for @ = 0.3 and 1.0, even if the poles and zeros are not updated throughout some manipulator motion, the systern will still remain stable. as can be seen from the simulation results of Fig. 5.6. These results were obtained by considering the same simulation case of Subsection 5.2.4, but using the zeros associated with configuration #1 and adding white noise to the sensor measurements. Thus, the set i~,= 0.151 rad/s and G = 0.587 are used instead of the usual set ut = 0.199 rad/s and C, = 0.447. As can be seen in Fig. 5.6 for the 0-auis,the system is stable with a rate of fuel consumption Rj = 0.0007 fuel-units/s and a total fuel consumption of 38.1 fuel units. For the t) and 4 aues, stable results are also obtained with a fuel consumption of, respectively. 54.1 and 45.1 fuel units and fuel-consumption rates of 0.0004 and 0.0005 fuel-units/s, respect ively. Thus, we see that the poles and zeros of the compensator G,(s) chosen according to the guidelines of Subsection 5.2.2 are robust. Even with variations of the payload and manipulator configuration, the system remains stable without requiring any update of the compensator t hroughout the motion. C hapter 5. Cont rol Synt hesis Using Compensation Techniques 152

x 1o4 1.5- 1 40- 1. 1 30. O* 0.5 Fuel

$ , 41 -, P ue 0 Units 20. (rad/s) -2 * -0.5 (4 1, -1 - -3 * -1.5 O -0.02 O 0.02 0.04 0.06 O 500 lm lm 2066 O KK) IO00 1!W 2000

Figure 5.6: Simulation results for the spatial system using the rate estimator with linear compensation with modified zeros and B = 0.3: (a) O-ais error phase plane; (b) 0-mis thruster-command history: (c) O-ais fuel-consumption history.

5.3 New Rate Estimator with Linear Compensa- tion

5.3.1 Description of the New Rate Estimator with Linear Compensation

It was shown in Section 5.2 that the use of linear compensation was effective in reducing the problem of dynamic interaction due to the fiexibility. However, it was not possible to stabilize a system in which the motion diverges (Ul-unstable), as it can be done using the asyrnptotic state estimator. Such improvement is desirable. since it will allow more freedom in the selection of the attitude controller parameters. For example, those parameters could be the negative inverse of the dope of the switching lines A, the deadband limits 6, or the magnitude of the force developed by the thrusters. In order to obtain similar improvements as those of the asymptotic state estimator, we tried to move the poles and zeros of the classical rate estimator such that they would mimic those of the asymptotic state estimator. Such improvements were finally achieved, their corresponding realization being now discussed. It was found Chapter 5. Control Synthesis Using Compensation Techniques 153

Figure 5.7: Mode1 with a new rate estimator and linear compensation for the two- mass system. that instead of actually filtering the attitude rneasurement , only the difference between this signal and the estimated attitude of the spacecraft e, should be filtered. as done for the attitude rate estimate. The resulting control system is depicted in Fig. 5.7 for the two-rnass system. We observe that the input to the Cl(s) filtcr is = y, - y, instead of y1 as for the classical rate estimator. Then, Q, is added to the output of the Gf(s)filter to provide a filtered position used as input in the compensator block G,(s), thus providing the feedback signal dl. On the other hand, i, is obtained exactly as in the classical rate estimator. For the spatial systeni, the control system of Fig. 2.12 can still be considered, but using the estirnator of Fig. 5.8 for each principal avis of the spacecraft. From now on, this control system will be termed "new rate estimator with linear compeusation" and represents the third major contribution developed in this thesis.

5.3.2 Demonstration of the Proposed Estimator Design with Simulation Results and Analysis

In this subsection, the two-mass system of Fig. 4.6 controlled using the new rate estimator with linear compensation of Fig. 5.7 is studied using the describing-function Chapter 5. Control Synthesis Using Compensation Techniques 154

' A%-ëe '/s acc. b

,-1 ,-1 diff/filter 1 ' comp.

Figure 5.8: Block diagram for the new rate estimator with linear compensation. and root-locus methods. The parametric study done for the previously studied control systems are repeated using these two methods and the results are validated using simulation results with the spatial system of Section 3.4. For al1 the simulation cases. white noise was included in the sensor readings.

Analysis of the New Rate Estimator with Linear Compensation Using the Two-Mass System

In order to study the control system of Fig.5.7, it is required to reduce the system to a two block diagram by block diagram transformation: one block to represent the relay nonlinearity and one bIock to describe the linear elements of the system. This transfer function of the linear elements of the new rate estirnator with linear compensation Gne,(s) is derived in Appendix D.3 and is given by

where G,(s), Gj(s), G,,(s) and G,(s)are defined in Eqs.(C.32), (2.21), (2.22) and (5.1 1) , respectively. The plant transfer function is represented by Gp(s) , while Cf(s) and G,.(s) are the transfer functions of second-order filters, and G,(s) is the transfer function of the linear compensator. For the root-locus analysis, the delay T is represented as the third-order Padé approximation described in Eq.(4.4). The same two cases previously studied are revisited here, namely the low-payload case (P = 0.05) Chapter 5. Control Synthesis Using Compensation Techniques 155

Table 5.4: Pole-and-zero locations for the new rate estimator with linear compensa- t ion.

LZeros and the high-payload case (P = 0.3). Moreover. applying the guidelines of Subsec- tion 5.2.2, the poles ans zeros of the compensator G,(s)are obtained for both cases as ul = 0.199 rad/s, CL = 0.447, up= 0.361 rad/s, <, = 1.0

The plots obtained using the describing-function and root-locus methods are displayed, respectively, in Figs. 5.9(a) and (b-c) for the low-payload case, and in Figs. 5.10(a) and (b-c) for the high-payload case. From Fig. 5.9(a), it is clear that the system is stable since none of the points of the - l/N(A) locus is encircled by the G(jw) locus. The same conclusion is obtained by looking at Figs. 5.9(b) and (c), since none of the ciosed-loop poles goes to the right-half plane. The open-bop pole can be directly related to the responsible physical component by considering the tranfer function G,(s) of Eq.(5.20). These open-loop poles, together with the open-loop zeros, are listed in Table 5.4. Chapter 5. Control Synt hesis Using Compensation Techniques 156

Figure 5.9: Theoretical analysis for the new rate estimator with linear compensation and j3 = 0.05: (a) describing-function plot; (b) root-locus plot; and (c) root-locus plot (zoom). Chapter 5. Control Synthesis Using Compensation Techniques 157

Figure 5.10: Theoretical analysis for the new rate estimator with linear compensation and .O = 0.3: (a) describing-function plot; (b) root-locus plot; and (c)root-locus plot (zoom). Chapter 5. Control Synt hesis Using Compensation Techniques 158 Similarly, by looking at Fig. 5.10(a), we see that the system is also stable for p = 0.3, according to the stability definition of Section 2.5, since none of the points of the -1/N(..L) locus are encircled by the G(jw) locus. Accordingly, none of the locus branches in Figs. 5.10(b) and (c) move to the right-half plane, the system thus being stable. Therefore, as expected, the same system that was unstable using the classical rate estimator and stabilized using linear compensation in Subsection 5.2.3 is still stable using this new rate estimator with linear compensation. In fact, this new rate estimator was designed to stabilize systems that are ül-unstable using the cIassical rate estimator. Its usefulness will be shown in the next subsection with the parametric study. Parametric Study for the New Rate Estimator with Linear Compensation The results of this study are sunimarized in Table 5.5. If these results are compared with those of Table 5.2, we see that this new rate estimator is effective to stabilize systems that were previously U1-unstable. The control systern is now always stable, excepted when X = 3 s and wf = 0.2513 radis. Without being as effective as the asymptotic state estimator, this new rate estimator increases significantly the range of parameter values for which a system is stable, but with a reduced complexity, since no mode1 of the system is required in the estimator. The control designer thus has more freedorn in the selection of the different control system parameter values, i.e., the negative inverse of the dope of the switching lines A, the deadband limits 6, or the magnitude of the force developed by the thrusters. Finally, we can also observe that the two-rnass system is very effective in predicting the stability of the more complicated spatial system. Out of 63 stability predictions, 61 were confirmed by simulations for a percentage of succes of 97%. From the remaining nine U1 predictions, three were confirmed by simulations while the six other simulations gave stable results. The overall succes rate in the predici- tion is thus 89%, which is very good if we consider the sirnplicity of the two-rnass system compared with the spatial system. S S S S S s 0.1- S S S S S C'O , s ,

, S S S S S . S O 1169'0 S S S . S S S 0'0 n s s I s s s .O*T Chapter 5. Control Synthesis Using Compensation Techniques 160 5.3.3 Simulation Results for the New Rate Estimator with Linear Compensation

The spatial system is now simulated using this new rate estirnator with linear compensation using the control system of Fig. 2.12 with the estimator of Fig. 5.8. The parameters of the controller about each axis of the spacecraft are listed in Table 4.2, and the same pavload and configuration of the manipulator are considered here. The results are displayed in Fig. 5.11, for initial attitude errors for the three axes of the spacecraft of 0.05 rad. Figures 5.1 l(a), (d) and (g) display, respectively? the error phase plane for the v, 4 and 0 Euler angles. It can be observed that the controller can bring the system into the desired final orientation wit hout a problem. Figures 5.ll(b), (e) and (h) display the thruster history, and Figs. Xl(c), (f) and (i) the resulting fuel consumption, which is low for the thrce cases. namely. 45.9. 43.8 and 43.0 fuel units for the @, 4 and 0 controller, respectively. Moreover their Fuel consumption rates R j are, respectively, 0.0005 fuel-unitsls. 0.0004 fuel-unit+ and 0.0003 fuel-units/s, which are belor the limit of 0.0060 fuel-units/s, thus corresponding to a stable system. Finally. it is observed that the oscillations in the joints remain low for this simulation run by looking at Figs. 5.11(j), (k) and (1). Thus, we see that this new rate estimator still gives good results. Moreover, it can be noted that the fuel consumption is reduced compared to the rate estimator with Iinear compensation. In fact, the fuel consumption is of the same order as the one obtained in Subsection 4.4.3 for the asymptotic state estimator.

5.3.4 Verification of the Robustness of the New Rate Esti- mator with Linear Compensation

As in Subsection 5.2.5, the robustness of the design of the compensator G,(s) for this estirnator is investigated for various configurations and payloads. Since the guidelines of Subsection 5.2.2 are also used here, the results of Table 5.3 are still Chapter 5. Control Synthesis Using Compensation Techniques 16 1

r taJ 1.S

1 1

O 0.5 hiel ,, eicl u+ 0 Units HI (rad/s)-' -0.5 - -2 () IO' -1 - -3 -1.5 -0.02 O 0.02 0.01 O 500 1000 ~sOo 2000 O0 500 Io00 1500 2000

pd (rad) t (s) t Cs) (4 (b) (4

Figure 5.11: Simulation results for the spatial system using the new rate estimator with linear compensation and P = 0.3: (a) +axis error phase plane; (b) $- axis t hruster-command history ; (c) t,b--auis fuel-consumpt ion history ; (d) +suis error phase plane; (e) gaxis thruster-command history; (f) qkaxis fuel-consumption history; (g) 6-axis error phase plane; (h) 0-axis thruster-command history; (i) O-axis fuel-consumption history; (j)joint-angle O2 history; (k) joint-angle O4 history; and (1) joint-angle O6 history. Chapter 5. Control Synthesis Using Compensation Techniques 162

Fuel Units (0)

ee (rad) t (4 t (4 (a) (b) (cf Figure 5.12: Simulation results for the spatial system using the new rate estimator with linear compensation with modified zeros and ,û = 0.3: (a) O-axis error phase plane: (b) 0-axis thruster-command history; (c) 8-axis fuel-consumption history. valid. Thus. again, we can claim that the estimator will remain stable even with a change of payload or manipulator configuration. This can be confirmed by simulating the same system that was used in the previous subsection, but using the wrong set of zeros for G,(s). Again, we choose the zeros associated with the configuration #1 for /3 = 0.3, namely = 0.151 rad/s and < = 0.587 instead of dI = 0.199 rad/s and C = 0.447. The results about the 8-axis are displayed in Fig. 5.12. where we can see stable results with a total fuel consumption of 26.8 fuel units and a rate of fuel consumption of 0.0007 fuel-units/s. For the 111 and 4 axes, the fuel consumptionç are, respectively, 35.9 and 30.8 fuel units, their rate of fuel consumption being, respect ively, 0.001 1 and 0.0007 fuel-units/s. Again, we see that the guidelines derived in Subsection 5.2.2 are robust and provide poles and zeros for the compensator G,(s) that yield good performance for a wide range of payload or manipulator configuration. No update of the compensator parameters are required throughout a motion of the manipulator. 5.4 Effect of Link Flexibility using A Planar Ex- ample In this section, the effect of link flexibility on the performance of the proposed control systems is investigated using Mecano. The planar system considered in Subsec- tion 42.1 and shown in Fig. 4.1 is revisited here. In this section, the rigid link is Chapter 5. Cont rol Synthesis Using Compensation Techniques 163

Table 5.6: Planar system with flexible link parameter values.

BO~Y I lj (m) ri (m) mi (kg) Ii (kg m2) (Nm2) 1 O 1 1.9 90000 9 490 533

1 payload (p) 2.25(3P)'I3 Brno 28.3Pm0(3B)*I3 l+p (P = 0.3) 17.7 0.1 27326 748 791 3.5e6

replaced by a flexible one and the stiffness at the joint is increase such that both systems have the same first natural frequency for a given configuration. The flexural stiffness EI of the link is derived from the Space Shuttle-CANADARM system (Spar Aerospace Ltd, 1996) and the same configuration and parameter values used in Sub- section 4.2.1 are considered here with a payload ratio B of 0.3. These parameters are reproduced in Table 5.6 while the joint stiffness k and damping coefficient c used are

Simulation runs are performed using the finite element package hlecano with the techniques for control systern modelling introduced in Section 3.5. The plant was readily modelled using Mecano built-in functions. First, the simulation was executed using the classical rate estimator of Fig. 2.11 with the parameter values of Table 4.2. The results are illustrated in Fig. 5.13 for an initial spacecraft angular error of 0.05 rad. As expected, the system is unstable and the results of Fig. 5.13 are qualitatively the same as those of Fig. 4.3, the only difference between the two systerns being the way flexibility is modelled. From Figs. 5.13(a) and (b), we see that a large limit cycle due to the dynamic interaction is reached, resulting in a continuous firing of the thmsters and a total fuel consumption of 1744.4 fuel units for the 2000-s run, as seen in Figs. 5.13(c) and (d). In this case, the rate of fuel consumptioa R; is 0.9418 fuel-units/s, which is far greater than the limit of 0.0060 fuel-units/s for a stable system and thus, the system is classified as unstable. From Fig. 5. U(e), we Chapter 5. Control Syathesis Using Compensation Techniques 164

e (rad) t3 (rad) (4 (b)

Figure 5.13: Simulation results for the planar system with link flexibility using the classical rate estimator: (a) spacecraft error phase plane; (b) spacecraft attitude phase plane; (c) thruster-command history; (d) fuel-consumption history; and (e) joint-angle history. can observe oscillations in the joint angle of the manipulator that va- between 30' and 60°, which is too large. In order to show the stability of the proposed control systems, even in the preseiice of link flexibility, the rate estimator with linear compensation is chosen to reduce the complexity of the modelling part in Mecano. It was relatively straighforward to add the two linear compensator, derived in Section 5.2, to the mode1 used to simulate the classical rate estimator. Using the guidelines for compensator design of Subsection 5.2.2, the frequencies and darnping ratios of the poles and zeros of the second-order linear compensator are given in Table 5.7. The results of the simulation are displayed in Fig. 5.14; as expected, the system can be classified as stable. The controller is effective in reducing the initial error of 0.05 rad and bringing the attitude of the spacecraft between the attitude limits, as can be observed in Figs. 5.14(a) to (c). Figure 5.14(d) shows that only a few firings Chapter 5. Cont rol Synt hesis Using Compensation Techniques 165

Table 5.7: Parameter values used for the linear compensator.

e (rad) e (rad) 9 (rad) (4 (b) (4

60 50, 45.5 40. . Fuel ,, oz 4s Units 20, (d%) 44.5 1O -lS1 O O 500 1000 1500 2000 O 500 Io00 lsoo 2000 O 500 1000 t500 2000

Figure 5.14: Simulation results for the planar system with link flexibility using the rate estimator with linear compensation: (a) spacecraft error phase plane; (b) spacecraft error phase plane (zoom); (c) spacecraft attitude phase plane; (d) thruster- command history; (e) Fuel-consumption history; and (f) joint-angle history. are necessary, resulting in a fuel consumption of only 54.4 fuel units for the same 2000-s run, as shown in Fig. 5.14(e). The resulting rate of fuel consumption R j is only 0.0010 fuel-unitsls, which is below the limit of 0.0060 fuel-units/s for stability. Moreover, Fig. 5.14(f) shows that the oscillations in the joint remain small through the whole motion. Therefore, we see that even if distributed link flexibility is considered, the control systems derived in this thesis still provide good performance and are effective in eliminating the dynamic interactions. It is thus a good approximation to lump Chapter 5. Control Synthesis Using Compensation Techniques 166 the link flexibility at the joints for the design of such control systems, especially if we remember that joint flexibility is more important than link flexibility in space robots due to the large payloads. Moreover, this section also showed the potential of Mecano to simulate systems that involve feedback control laws, even if this finite element package was not originally designed for such applications.

5.5 Discussion and Conclusion

Two novel attitude control schemes that make use of linear compensation techniques were introduced in this chapter to effectively resolve the problem of dynamic interactions. Although they are less effective compared to the asymptotic state estimator of Chapter 4, they have the advantage of not being "model-based", while still giving very good performance. Both models introduced in this chapter eliminate the dynamic interaction problem using a second-order conipensator. This compensator wvas designed using the root-locus method by selecting proper poles and zeros to attract to the left the locus that goes to the right-half side of the complex plane. Based on the experience gained with the asymptotic state estimator and msny test cases. proper pole and zero locations were found to stabilize the system. Wttile selecting these parameters, it was kept in mind that the compensators are cascaded to second- order filters required to eliminate high-frequency noise. Thus, special care was taken to design the compensator without increasing significantly the noise content in the feedback signals. A set of guidelines, readily implementable in an algorithni was proposed to achieve the required pole and zero placement. It was shown that these guidelines are quite robust and do not need to be updated throughout a manipulator motion. The first mode1 introduced, the rate estimator with linear compensation is very effective in eliminating the problem of dynamic interaction. However, it cannot stabilize systems with diverging motions, as was the case with the asymptotic state estimator. The new rate estimator with linear compensation was introduced to Chapter 5. Cont rol Synt hesis Using Compensation Techniques 167 overcome that problem. This control scheme was shown to give similar performance to the asymptotic state estimator, without its complexity. It was concluded that the two-mas system can reliably predict, in most cases, the stability of a system using the describing-function method, without the need of lengthy simulations on more realistic and corn plex systems. Finally, Mecano was used to study the effect of link Rexibility on the robustness of the proposed algorithms. X planar one-link-manipulator system was used to show that similar results are obtained if the link flexibility is considered or. on the other hand, lumped at the joints. The rate estimator with linear compensation still provided stable results, even in the presence of long flexible links. It was thus concluded that lumping link flexibility at the joints is legitimate in designing control systems intended to eliminate the dynamic interaction problem. Link Rexibility does not destroy the stability of the system, as one could fear. Chapter 6

Conclusions and Recommendat ions for Furt her Work

6.1 Conclusions

This thesis examined the possible dynamic interactions between the attitude controller of a spacecraft and the flexible modes of a space manipulator mounted on it. In order to illustrate the classical attitude control theory using on-off thrusters, the model of a rigid spacecraft was derived using the Lagrangian formulation in Chapter 2. A planar version of this model was obtained assuming small-angle rotations to uncouple the system dynamin. This simplified system, rotating about one of its principal axes, was used as a basis to show the operation of a conventional attitude control scheme similar to the one used by the Space Shuttle. This on-off attitude controller , based on phase-plane met hods, evolved under the assumption that the spacecraft is sufficiently rigid to allow the use of rigid-body mechanics in the description of the spacecraft response to the reaction control systen activity. It assumes that only the attitude of the spacecraft is available through sensor readings Chapter 6. Conclusions and Recommendations for Further Work 169 and the attitude rate required for feedback is obtained by differentiating the attitude signal. This differentiation process can be problematic due to the problem of noise amplification, thus requiring low-frequency second-order filters. Since the attitude controller assumes the use of on-off thrusters, which are nonlinear devices, the system cannot be adequately analyzed through the application of linear analysis methods. This problem was solved using the describing-function method outlined in Chapter 2. This nonlinear system analysis technique is weil suited for systems showing limit-cycle activities, like the ones under study in this thesis. This technique was adapted to the analysis of space robotic systems controlled with on-off thrusters via the root-locus method. This novel adaptation of the describing-function concept allows the analysis of a system from a different per- spective and has shown to give better physical insight in the design of new control schemes. Finally, the stability definition used in this thesis was also introduced in Chapter 2. This definition is based on the rate of fuel consumption Rj(t) of the system. -1limit value R;,i,(t) was chosen based on the fuel consumption for a rigid spacecraft. Rates of fuel consumption below this limit correspond to stable systems, while rates of fuel consumption above the limit describe unstable systems either for diverging motions or large limit cycling due to Aexibility of the system excited by the operation of the on-off thrusters. Detailed examples for the application of this definit ion were included. In Chapter 3, the modelling of flexible space manipulator systems was considered both using a symbolical derivation together with numerical integration, and using a commercial finite-element software package. A general formulation to obtain the mode1 of an N-flexible-joint space manipulator using a Lagrangian ap- proacli was described and used to obtain the dynamics mode1 of a three-flexible- joint manipulator mounted on a sir-dof spacecraft with the same architecture as the CANADARM-Space Shuttle System. To simplify the analysis, this dynamic modelling was restricted to flexible-joint manipulaton. This approximation is legitirnate Cha~ter6. Conclusions and Recommendations for F'urther Work 170 since free-flyers are likely to have reduced structural flexibility but will have joint cornpliance which becomes important when payloads are massive. In the case of the CANADARM, both structural and joint fleribility is present but the latter is more important than the former. For simplicity, it can be assumed that al1 flexibility in this system is lumped at the joints. Realistic physical parameters for this system were identified using the CANAD.4RM data provided by Spar Aerospace Ltd. (1996). As mentioned above, the link Hexibility was lumped to the joints. This was attained by solving an inverse problem in vibration to derive the spring stiffnesses at the joints. The spring stiffnesses were chosen to match the first three natural modes of the CANADARM-Space Shuttle system for a given configuration of the manipulator. Similarly, damping coefficients at the joints were chosen to match the damping ratio of the first mode by assuming damping proportional to the stiffness, at each joint. Mecano, the finite-elernent package used in the course of this work, was used to validate this dynamics model. This FEM package was chosen due to its ability to accurately simulate flexible dynamical systems. However, one of the drawback of this software is its inability to simulate control systems. Thanks to its general user-element, it was possible to overcome this problem by developing our own elements to simulate the control systems studied in this thesis. A general technique was derived to transform n-order differential equations into a set of second-order different ial equations required by the user-element . The classical rate estimator of Chapter 2 was used as an example to show the usefulness of the proposed adaptation as well as the required block-diagram manipulations. This work is very useful in extending the capabilities of Mecano and was positively received by the Mecano development team in Liège, Belgium. In Chapters 4 and 5, the dynamic interaction problem was described using the ciassical rate estimator and three new control schemes were introduced to soive this problem and even more. Using a planar one-flexible-link system, it was shown how the reduction of the system natural frequency, upon increasing the payload, can Chapter 6. Conclusions and Recommendations for Furt her Work 171 deteriorate the control system performance, thus resulting in high fuel consumptions and large vibrations. This performance degradation, resulting from the increase in the payload mass, was also demonstrated using the spatial system. Throughout those chapters, a simple two-mass system was used to study theoretically the problem at hand and to design new control schenies to eliminate it. The spring stifhess and damping coefficient of this simple system were chosen to match the first mode of the spatial system in order to reproduce the relative rriotiori of the payload (carried by the manipulator) with respect to the spacecraft. C'sing the describing-function and root-locus methods, the simple two-rnass systern was proven to be useful to understand why the classical rate estimator becomes unstable upon increasing the payload. Clear understanding of the physics of the problem, made possible the design of the three proposed control schemes in this thesis. Using the two-mass system, a parametric study was performed to investigate the significant parameters of the system and to serve as a basis of cornparison for the four control systems. A11 the cases studied t heroret ically using the describing-function niethod were also tested by simulating the spatial system. It was found that in most cases. the stability prediction obtained with the simple system was correct and that this system is very effective in predicting the stability of the more complicated spatial system. Therefore. it can be used for quick stability predictions. without the need of lengthy simulations on more realistic and complex systems. Moreover, and most of all, this simple systern, together with the describing-function and root-locus methods, is a very good tool for the attitude control design of flexible systems. Using the two-mass system, it was shown that the effect of increasing the payload. or accordingly, reducing the natural frequency, is to bring the poles and zeros related to the plant closer to the real-ais, close to the origin of the complex plane. Since this control scheme employs second-order filters with low cutoff frequencies to eliminate high-fiequency noise, it has poles and zeros close to the origin. It is shown that the interference of the poles and zeros of the plant wit h those of the filters, for high Chapter 6. Conclusions and Recommendations for Further Work 172 payloads, amplifies the problem of dynamic interaction. Since this scheme uses a differentiator to estimate the attitude rate of the spacecraft, it is not possible to modify the pole and zero locations without the negative effects of noise amplification that would result. In order to remove the problematic poles and zeros close to the origin due to the use of low cutoff frequency second-order filters, our first approach was the use of an asymptotic state estimator. Siuce this estiriiator is riod del-lasd, tliere is liu iieed tu directly differentiate the attitude signal to obtain an estimate of the attitude rate. thereby reducing the need of having poles and zeros close to the origin to eliminate high frequency noise. In this estimator, the poles and zeros and placed "optimally" to give good tracking of the states while reasonnably reducing high frequency noise in the sensor rneasurements. This estimator also has the advantage of reducing the lag in the system by eliminating the use of second-order filters. Since the effect of the lag is to tilt the switching lines, as for a time delay, diverging motions may result if the lag is such that the slope of the "effective" switching lines become positive. By reducing the lag in the system, the asyrnptotic state estimator can thus stabilize systems with a diverging motion. The performance of this estimator is vev impressive. For the 72 payload and configuration sets studied in this thesis, only three were unstable using this estimator. Moreover, these three unstable cases were al1 for an extreme payload with a mass equal to the mass of the spacecraft, unlikely to occur in any real system. A11 these cases were simulated using the spatial system, including white-noise in the sensor measurements. It was also shown that even when assuming errors in the mass properties in the dynamics model used by the estimator, this control scheme still gives very good performance. Moreover, the fact that the estimator requires an accurate dynamic model of the plant is not a large drawback, since accurate inertial properties of a spacecraft can be obtained prior to its launching into space. However, in the case of a space robotic system, the dynamics can become very complicated in the presence of flexibility in the links and joints of the robot, dong with the payload Chapter 6. ConcIusions and Recommendations for Furt her Work 173 flexibility. The computational effort required for this complicated system becornes significant, and the use of models running in real-time can become difficult and will be dependent upon the available hardware. However, a simpler approximate model may be sufficient to achieve good performance. More research is therefore needed on the possible implementat ion of t his type of controller. For cases where the onboard computational power would not be sufficient and would prevent the use of the asymptotic state estimator, two novel attitude control schemes that make use of linear compensation techniques were introduced to effectively resolve the problem of dynamic interactions. Although they are not as effective as the asymptotic state estimator introduced in Chapter 4, they have the advantage of not being "model-based", while still giving very good performance. They both make use of second-order compensators to eliminate the dynamic interaction problem. The poles and zeros of the compensator were designed using the root-locus method in such a way as to attract to the left the locus that goes into the right-half side of the complex plane. Based on the experience gained with the asyrnptotic state estimator and plenty of test cases, proper pole and zero locations were found to stabilize the system. The poles and zeros were specifically selected 1" kecp thp noise level in the feedback signals as low as possible. This was mandatory since the coni- pensators are cascaded to second-order filters required to elirninate high-frequency noise. A set of guidelines, readily implementable in an algorithm was proposed to achieve the required pole and zero placement. Moreover, the robustness of these guidelines was discussed; it was shown that the poles and zeros of the compensators do not need to be updated throughout a manipulator motion. The first model introduced, the rate estimator with linear compensation. is very effective in eliminating the problem of dynamic interaction. However, it cannot stabilize systems with diverging motions, as is the case for the asymptotic state estimator. The new rate estimator with linear compensation was introduced to overcome that problern. This control scheme was shown to give similar performance Chapter 6. Conclusions and Recommendations for Further Work 174 to the asymptotic state estimator without its complexity. Since this estimator not only solves the dynamic interaction problem, it can thus, as for the asymptotic state estimator, allow for more freedom in the choice of the control-system parameters. This could be the negative inverse of the dope of the switching lines, A, the deadband limit, 6, or the magnitude of the force developed by the thrusters. It should be mentioned that the stability of the system is also dependent upon the iriitial coiiditioiis. Iii tlie aiiiiulatiùiis rey ortecl iii tliis tliesis, tlie systeiii wuasauiiied to have large initial errors; it was attempted to restore it to within the deadband limits. Since the initial errors are large, there is a high probability of exciting the flexible modes of the manipulator. In a more practical situation, the spacecraft would already be within the attitude limits and it would most likely be disturbed due to manipulator motion, hence requiring reaction control. In t his case. the correcting action would take place when the error is small. Therefore, the thrusters would fire for a short period, as for a rigid body-limit cycle, which would not likely excite the flexible modes. However, the rapid commanded motion of the manipulator could incur a larger disturbance; t herefore, the likelihood of rendering the systeni unstable becomes more significant. Hence, the use of the three proposed control schemes would allow for faster motions that are most iikely expected in future space exploitation. Finally, Mecano was used to study the effect of link flexibility on the robustness of the proposed algorithms. .A planar one-link-manipulator system was used to show that sirnilar results are obtained if link flexibility is considered or, on the other hand, lumped at the joints. The rate estimator with linear compensation was still providing stable results, even in the presence of long flexible links. It was thus concluded that lumping link flexibility at the joints is legitimate to design control systems intended to eliminate the dynamic interaction problem. In fact, link flexibility does not destroy the stability of the system. Chap ter 6. Conclusions and Recommendations for Furt her Work 175 6.2 Recommendations for Future Work

In this thesis, the dynamic interaction of the flexible mode of a space manipulator system with its base attitude controller was studied. To extend the resiilts obtained, and develop new contributions to this fast growing field of knowledge, some sugges- tions for future activities are outlined below:

1. Use an improved controller with optimum switching functions and velocity limits drift channel instead of the simple switching lines used in this thesis.

2. Investigate the use of a Kalman filter whenever noise properties are known.

3. Study the implementation of the asymptotic state estiniator for real-time control in the case of complicated dynamic models with limited computer time available.

4. Use Mecano to study the robustness of the proposed coritrol algorithms with a very detailed model of the plant dynamics. This could be a six-flexible-link- and-joint manipulator carrying a flexible payload and mounted on a spacecraft .

5. Extend the proposed control schemes for the .Y, Y and Z translational motion of the spacecraft using on-off thrusters.

6. Study the orbital rnechanics effects in the formulation of the dynamics model.

7. Study the possibility of selecting the limit value for the rate of fuel consumpt ion RjIimto insure stability based on an analytical approach instead of the heuristic method used in this thesis.

8. Investigate the possibility of using pulse-widt h modulation techniques to control the attitude of the spacecraft, together with the proposed control schemes.

9. In this research work, simulation routines were very helpful in improving the new algorithms, and evaluating them. However, experimental studies could Chapter 6. Conclusions and Recommendations for Further Work 176

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Attitude Description using Euler Angles

In this thesis, the attitude of the spacecraft is described by a set of Euler angles using a ZYX description. In this description, we first perform a rotation of 9 about the 2-axis of the inertial frame, FI,to obtain 3p. Then, a rotation of 4 about the Y'-axis of 3p is conducted to obtain 3p. Findly, we perfom a rotation @ about the resulting X"-axis to obtain the spacecraft frame, FO.The rotation matrix that brings the inertial frarne FIinto the orientation of the spacecraft body-fixed frame F0 is given by

Moreover, the angular velocity wo can be expresxd in tenns of the Euler angle Appendix A. Attitude Description using Euler Angles 188 where 6 is defined as 6 = [$, 4, BIT. In order to obtain matrix So(6), we first recall the well know expression for the angular velocity matrix fl (Angeles, LM'), namely,

where Q is a skew-symetric matrix expressed in FI,since QIand Q~are also expressed in that frame. The angular velocity vector w is the vector of a,i.e.,

The expression for in Facan be obtained using similarity transformations, namely

Using, Eq.(A.l), we bave

QI = QZQYQPI + QZQY~QP + QZQY&

Therefore, Eq.(A.7) cm be written as

Moreover, we have

Q&Q;~ [%]I~QYJQ~. = (~$11 ~Tyt[WZ] p ) (A. 10) and Appendix A. Attitude Description using Euler Anglea 189

Therefore,

[W]O= Q& Q& [w~]~~+ Q;,,[w~~]~~ + [w~~~]~

Hence,

Finally, we have O -s$ Appendix B

Transforrnat ion of nt h-Order Different ial Equat ions

Let us consider a nth-order differential equation of the form

To avoid differentiations of the input when only ii is known, the following transformation is used g = eox + eix(') + - - + e,-lz(n-l)+ e,ü (B-2) where x is solution of the following differential equation

Therefore, determining the coefficients ei, i = 0, , n, of Eq.(B.2), the output variable g(t) can be obtained as a linear combination of the time response x(t) of Eq.(B.3), and its &th time derivative, for i = 1, - - , (n- 1). In order to determine these coefficients, let us consider the i-th derivative of y, for i = 1, - - O, n, Substituting Eqs.(B.4) into Eq. (B.l), we obtain

Differentiating Eq. (B.3) m times, we derive

Therefore, using a suitable value for m, Eq.(B.5) reduces to

which can be written as

By equating terms of the same powers, we obtain Appendix C

Modelling of the Two-Mass and Planar Systems

C.1 Two-Mass System

The equations of motion for the two-mass system of Fig. 4.6 can be readily derived as

where Ml is the mass of the base, M2the mass of the payload, y1 the position of the base, y2 the position of the payload, k the spring stiffness, c the damping coefficient, f (t) = Bu, with B the magnitude of the force developed by the thrusters, and u is the command of the thrusters, taking values +1, O or -1. The overall motion of the system can be decomposed into a rigid-body motion of the system centre of mass CM, and a flexible-body motion around the centre of mas. The equations goveming these two motions are now derived. The position of the centre of mass is given by Appendix C. Modelling of the Two-Mass and Planar Systems 193 and hence, which can be written as Mt& = Ml& + M2ÿ2 where the total mas of the system is

Adding Eqs.(C.la) and (C.lb), we obtain

Inserting Eq. (C.6) into Eq. (C.4), we obtain

This equation governs the motion of the system centre of mas. Conversely, subtracting Eq. (C.1 b) from Eq. (C. la), we obtain

By defining

Yj = Y1 - Y2 and differentiating Eq. (C.9), we obtain

YI = Y1 - Y2 (C.10) yr = y1 - y2 (C.11)

The first two terms of Eq.(C.B) can be written as Appendix C. Modelling of the T-Mass and Planar Systems 194 Substituting Eqs.(C.4) and (C. 11) into Eq.(C.12), we finally obtain

Using Eq.(C.7), Eq(C.13) cm be written as

(C. 14)

Therefore, substituting Eqs.(C.9), (C.lO) and (C.14) into Eq. (C.8), we obtain

By defining the equivalent reduced mass p as

(C. 16)

Equation (C.15) can now be written as

which can also be transformed into the usual form

(C. 18) where the natural frequency un is Qven by

(C.19) and the damping ratio C is defined as

(C.20)

In summary, the system equations of motion, Eqs.(C.l), can now be written as Appendix C. Modelling of the Two-Mass and Planar Syetems 195 with un and C defined in Eqs.(C.19) and (C.20). Equation ((2.21) represents the rigid- body motion, while Eq. (C .22) represents the flexiblebody motion wit h a resonance frequency un* Fkom Eq.(C.l9), we obtain the system stifhess as

and substituting Eq.(C.23) into Eq.(C.20), c can be written as

Therefore, using Eqs.(C.23) and (C.24), k and c can be chosen to match a specific resonance frequency w,, and a specific damping ratio ifor given masses Ml and b12. The transfer function mapping the input u into the base position yi, is now derived. That is y&) = Gp(s)u(s) (C.25)

Using Eqs.(C.2) and (C.9), we can write

Taking the Laplace transform of Eqs.(C.21) and (C.22),and assuming zero initial condit ions, we obtain M~S'Y,(S)= BU(s) and

Substituting Eqs.(C.27) and (C.28) into Eq.(C.26), we obtain Ap~endixC. Modellinn of the TweMass and Planar Systems 196

the transfer function GP(s) relating the input u to the base position y1 becomes

where w, and C axe defined in Eqs.(C.19) and (C.20).

In order to wnte the system equations (C.l)in state-space form, we define: XI =

YI, x2 = yll 23 = y2 and x4 = y2. Making use of Eqs.(C.23), (C.24) and (C.l6), and defining the state vector as x = [xl,x2, x3, alT, Eqs.(C.l) are written as

where

and T b=[070 O O]

The available acceleration of the base 70 is given by

7 being the acceleration of the CM. The required outputs are either yl, or y1 and yi, and are obtained using, correspondingly, Appendix C. Modellig of the Two-Mass and flanar Systems 197 C.2 Planar System

In this appendix, the equations of motion of the planar system are developed using the rnodelling techniques of Section 3.2. Here, we consider the planar one-flexible- joint manipulator on the Sdof spacecrait of Fig. 4.1. We assume that only on-off thnisters are available to control the attitude of the spacecraft, and that the moments produced axe O, rh,, or -th=. Moreover, we assume that uo çoiitrol oii tlie loçd x0 and y, direction of the spacecraft is exerted. If we assume that the joint is braked in a specific configuration 07, then the vector of generalized coordinates is q, = [x,, y,, q, &lT, where s, and y, are the spacecraft CM coordinates with respect to an inertial frame, Bo is the spacecraft attitude, and O2 is the angular position of link 1. Applying the techniques of Section 3.2, the equations of motion are obtained as

where al1 matrices are of 4 x 4 and al1 vectors are Cdimensional. Moreover,

where the components Mij of Mp are AppendW C. Modding of the Two-Mm and Planar Systems 198 MM = Il + mil: + mlrollcos O2 Md4=Il +mil: (C.38) and the components ni of n, are

nl = -[miro cos Bo + mlll cos(@o+ oz)]$ -2nlll cos(l)o + &)do& -mlll co~(t9~+ n2 = -[miro sin 80 + mlll sin(Bo + e2)]00 -2mill sin(& + e2)e0è2 -mlll sin(Oo + &)& n~ = -mlroli sin 0~81- 2mlroll sin 82$e2

n4 = mlrollsin 0~8: (c.39)

Note that a payload cmbe added at the end of the link without any modification of the previous dynarnic model. The parameten of the link, ml, Il, ri and 11, just have to be adjusted accordingly. Appendix D

Transfer Function of the Linear Elements of the Simulation Models

The transfer function of the linear elements of the four rnodels studied in this thesis are derived here. This transfer function is represented by G(s) in Fig. D.1. By examining Fig. D.l and defining C(s) as the Laplace transform of o(t),we can wnte

C(s) = -Y (s)

C(s) = -G(s)U(s) where u is the output of the relay nonlinearity, taking values +1, O or -1. Therefore, if an equation similar to Eq.(D.l) can be obtained, this means that the mode1 is reduced into a suitable fom for describing-function analysis.

Figure D.1: A nonlinear system. Appendix D. Transfer hnction Derivation 200 Da1 Classical Rate Est imat or

Examining Fig. 4.7 and defining Z(s) as the Laplace transform of y, we can write

= - [~-T,G,(SI (G, (s) + ASG,.(s)) + -s (1 - G.. (s)) ] O (s) by defining

Equation (D.2)is of the same fonn of Eq.( D.l) and therefore, G,,,(s) is the transfer function of the linear elements of the mode1 with a classical rate estimator.

D.2 Asymptotic State Estimator

The dynamics of the plant in the mode1 of Fig. 4.13 is now represented in state-space form. This dynamics is written in transfer-function form in Appendix C, this transfer function being denoted by G,(s) , w hich is deûned in Eq. (C .32). For the state estimator part, we have

where Appendix D. Transfer F'unction Derivation 201 We also have

In transfer function form, using again Z(s) as the Laplace transform of y, Eqs.(D.4) and (D.6)cm be written as

where

Using the expressions of A, b, c, E and 1, which are defined in Eqs.(C.33), (C.35), (4.12) and (D.5)respectively, we obtain

wit h

and

The block diagram of Fig. 4.13 can therefore be represented as the one of Fig. D.2. Appendi D. 'Ikansfer Function Derivation 202

Figure D.2: Mode1 with an asymptotic state estimator using transfer functions.

Examining Fig. D.2, we can write

wit h the definition

where g,l (s), glz(s),g21 (s) and gZ2(s)are defined in Eq.(D.8). Equation (D.9) is of the same form of Eq.( D.1) ; therefore, G,(s) is the transfer function of the linear elements of the mode1 with an apymptotic state estimator. Appendix D. Transfer F'unction Derivation 203 D.3 New Rate Estimator with Linear Compen- sation

Examining Fig. 5.7, we can write

C(3) = -Il (s) - A& (s)

1 1 e-"GP(s)U(s) - T7~(s))s + s (s)]

= -Gc(s) [(G,(s)+ XSG.. (s)) (e-"G,(s) - f 7) + (f + A)] Li (8)

with the definition

Equation (D.1 1) is of the same form of Eq. ( D. 1); therefore, Gn, (s) is the transfer function of the linear elements of the mode1 with a new rate estimator and linear compensation.