ADAPTIVE REACTIONLESS CONTROL OF A SPACE …digitool.library.mcgill.ca/thesisfile117066.pdf · adaptive reactionless control of a space manipulator for post-capture of an uncooperative

ADAPTIVE REACTIONLESS CONTROL OF A

SPACE MANIPULATOR FOR

POST-CAPTURE OF AN UNCOOPERATIVE

TUMBLING TARGET

Thai Chau Nguyen Huynh

Department of Mechanical Engineering

McGill University, Montreal

February 2013

A Thesis submitted to the Faculty of Graduate Studies and Research

in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

c© Thai Chau Nguyen Huynh, 2012

ABSTRACT

Abstract

The increasing number of satellites and space debris in the Earth orbit has created

an urgent need to develop robotic technologies for on-orbit servicing and space debris

removal. The system solution investigated in this thesis involves a servicer (base)

with an attached robotic manipulator and this system must dock to, or get a hold

of the target satellite in order to conduct the required operations. In that context,

the problem of post-capture of a tumbling target with unknown inertial properties

and angular momentum, by using a space manipulator, is addressed. In particu-

lar, a critical aspect investigated is how to maintain minimum coupling between the

manipulator-target and the base in order to keep the base attitude unchanged while

estimating the unknown properties of the target – this without the use of attitude

control devices.

First, the kinematics and dynamics of a free-floating space manipulator are an-

alyzed where particular attention is given to the coupling motion between the arm

and the base. Then, the previously developed concept of reactionless manipulation is

introduced by presenting the reaction null-space control method, which is unique to

free-floating space manipulators.

Next, motivated by the reaction null-space control method, a new Adaptive Re-

actionless Control algorithm to produce arm motions with minimum disturbance to

the base is proposed. The algorithm aims to generate the reactionless manipulation

i

ABSTRACT

in the presence of uncertainty as a result of capturing an unknown target. In addi-

tion, the online momentum-based estimation method is developed for inertia param-

eter identification, after the space manipulator grasps an unknown tumbling target.

The combined control/identification scheme is intended to segue from the instant of

capture till the unknown parameters are identified and/or the available stabilization

methods can be applied properly. To verify the validity and feasibility of the proposed

concept, MSC.Adams/Simulink co-simulation platform is employed to implement a

planar base-manipulator-target model as well as the three-dimensional model of the

Engineering Test Satellite VII system. The numerical results show that the space

manipulator is able to perform reactionless motion while the inertial parameters of

the target are identified, for a range of target sizes relative to the base.

To make the proposed control scheme practical, a redundancy resolution tech-

nique is formulated within the adaptive reactionless control algorithm to integrate

a secondary manipulation task, while executing the reactionless motion in the post-

capture scenario. In particular, the gradient projection method is employed and

implemented to avoid the physical joint limits of the manipulator. The performance

of the algorithm is verified through simulations of the two aforementioned models,

thus demonstrating that minimum disturbance on the base and joint-limit avoidance

can be achieved after capture of an unknown tumbling target. Lastly, a preliminary

design for a planar test-bed that can serve for experimental validation of the proposed

algorithms is presented. The design utilizes an air-bearing table to create a near fric-

tionless surface which would support a base-3-DOF manipulator and a target satellite

mock-up. The design includes the dimensioning and sizing of the main components,

and selection of actuation, sensing and control hardware.

ii

RESUME

Resume

Une augmentation alarmante du nombre de satellites et de debris orbitaux autour

de la Terre a engendre un besoin grandissant de developper des technologies pour

l’entretien des satellites en orbite et afin de disposer des debris existants. Cette these

porte sur une solution constituee d’un satellite de service muni d’un manipulateur

robotise qui lui permet de s’arrimer au satellite cible et en maintenir le controle afin

de completer les operations requises. Plus specifiquement, cette etude porte sur la

phase apres la saisie d’un satellite culbutant avec un moment angulaire inconnu et

ayant des proprietes inertielles inconnues. Un aspect critique a l’etude est le maintien

d’un niveau minimal de liens dynamiques entre le satellite capture et la base du

manipulateur afin de preserver son orientation spatiale tout en estimant les proprietes

inconnues de la cible, et ce, sans autres systemes de controle d’attitude.

Tout d’abord, la cinematique et la dynamique d’un manipulateur en apesanteur

sont analysees avec une attention particuliere sur les liens mecaniques entre le ma-

nipulateur et le satellite de service a sa base. Ensuite, un concept deja etabli pour

la manipulation sans reaction est introduit par une methode de controle du noyau de

reaction, unique aux manipulateurs spatiaux.

Sur cette fondation, une nouvelle methode de controle adaptative et sans-reaction

est proposee qui produit des trajectoires de manipulation qui minimise les perturba-

tions a la base. L’algorithme permet de generer ces trajectoires tout en faisant face

a l’incertitude quant aux proprietes inconnues de la cible.

iii

RESUME

De plus, une methode d’estimation en ligne basee sur le moment cinetique est

developpee pour identifier les parametres inertiels a la suite de l’arrimage. Ces

deux methodes, de controle et d’estimation, peuvent etre utilisees de concert jusqu’a

l’identification des parametres inconnus ou jusqu’a ce qu’une methode de stabilisa-

tion puisse etre appliquee. Afin de verifier la validite et la faisabilite de la methode

proposee, une plate-forme de co-simulation entre MSC.Adams et Simulink est utilisee

pour simuler un systeme planaire en plus d’un systeme tridimensionnel representant le

“Engineering Test Satellite VII”. Les resultats demontrent que le manipulateur peut

executer des mouvements sans-reactions tout en identifiant les parametres inertiels,

pour un eventail de dimensions du satellite cible.

Afin de rendre la methode proposee plus pratique, une technique de resolution des

redondances est formulee au sein de l’algorithme de controle adaptatif sans-reaction

pour permettre d’accomplir une tache secondaire en simultane. Par exemple, une

projection du gradient est utilisee pour eviter les limites physiques du manipulateur.

La performance de cette combinaison est verifiee par des simulations qui demontrent

que les perturbations a la base peuvent etre minimisees tout en respectant les lim-

ites du manipulateur apres la saisie d’un satellite culbutant. Finalement, un concept

preliminaire pour une plateforme experimentale planaire est presentee. Cette plate-

forme cree une surface presque sans friction a l’aide d’une table a coussin d’air qui

peut supporter un manipulateur a trois degres de liberte et un satellite-maquette. Ce

concept preliminaire inclut les principales dimensions, la selection des actionneurs,

capteurs, et controleurs.

iv

ACKNOWLEDGEMENTS

Acknowledgements

I would like to express my gratitude to my supervisor, Prof. Inna Sharf, whose

guidance, stimulating suggestions and continuous encouragement helped me in the

course of this research. I also appreciate her assistance in improving my academic

writing skills with the valuable feedback through editing my written work. Without

her support and confidence in me, I cannot imagine finishing this work.

I also thank Prof. Peter Caines for a valuable discussion and suggestions regard-

ing the adaptive control design. I would like to acknowledge the useful comments and

criticism received from anonymous reviewers of my journal papers and the technical

support from research scientists at Canadian Space Agency. I want to thank specifi-

cally Dr. Dimitar N. Dimitrov for his useful discussions of my work and for providing

the detailed model of the Engineering Testing Satellite VII.

I want to thank my friends at CIM who have created a pleasant atmosphere for

research and shared the hard-working time during the program. I am grateful to

M. Persson for his kind help in translating the thesis abstract in French. Special

thanks to C. Wong and Dr. W. Tao for sharing their computers to run my numerical

simulations.

Finally, deep thanks to my parents, N. Tan and H. Quy, for their love and

encouragement; and to T. Nguyen: thank you for your love and patience during the

past three years. Thanks also to everyone I could not include.

This work was financially supported by the McGill MEDA award and Prof.

Sharf’s research grants.

v

CLAIM OF ORIGINALITY

Claim of Originality

The main contributions of this dissertation toward developing solutions for autonomous

capture of objects in space with a satellite-based robotic manipulator are summarized

as follows:

• Derivation of a new formula to compute the reactionless motions for a

free-floating space manipulator, without the use of attitude control devices.

The reactionless joint rates are computed based on their deviations from

the current joint rates and the observed response of the base attitude.

• Development of Adaptive ReactionLess Control (ARLC) algorithm to gen-

erate joint motion commands to the manipulator arm in order to minimize

disturbance to the base attitude, in the presence of dynamics uncertainty

as a result of capturing an unknown target.

• Integration of a momentum-based inertial parameter identification algo-

rithm with the ARLC scheme to estimate the inertial properties of the

target while executing adaptive reactionless motion in online manner.

• Reformulation of ARLC scheme for integration with the redundancy res-

olution technique to satisfy a secondary manipulation task. In particular,

employing the resulting formulation to ensure joint-limit avoidance within

the ARLC scheme.

• Formulation of the minimum kinetic energy attainable by a free-floating

space manipulator with constant angular/linear momentum.

vii

CLAIM OF ORIGINALITY

• Development of a co-simulation platform using MSC.Adams and Simulink

to validate the proposed control algorithms for two base-manipulator-target

models: a planar model with a 3-DOF manipulator and a 3D model with

a 7-DOF manipulator emulating the ETS-VII arm.

• Database of simulation results validating and demonstrating the effective-

ness of the proposed control algorithms in the post-capture of an unknown

tumbling target.

• Conceptual design of an experimental test-bed for testing autonomous cap-

ture missions, with the air hockey table concept employed to emulate the

free-floating condition in space, a base-3-DOF manipulator-target system

and the required sensing and control hardware.

viii

TABLE OF CONTENTS

TABLE OF CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Claim of Originality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

CHAPTER 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1. Kinematics and dynamics of space manipulators . . . . . . . . . . 9

1.3.2. Control of space manipulator . . . . . . . . . . . . . . . . . . . . 10

1.3.3. Target capture by space manipulator . . . . . . . . . . . . . . . . 12

1.3.4. Inertia parameter identification . . . . . . . . . . . . . . . . . . . 15

1.4. Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

CHAPTER 2. Kinematics, Dynamics and Reactionless

Control of Space Manipulator . . . . . . . . . . . . . . . . . . . 19

ix

TABLE OF CONTENTS

2.1. Kinematics of Space Manipulator . . . . . . . . . . . . . . . . . . . . 19

2.2. Momentum Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3. Generalized Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . 27

2.4. Dynamics Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5. Reaction Null-Space Motion Control . . . . . . . . . . . . . . . . . . . 33

2.6. Remarks on Kinetic Energy of Space Manipulator . . . . . . . . . . . 35

2.7. Numerical Co-simulation Platform . . . . . . . . . . . . . . . . . . . . 36

2.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

CHAPTER 3. Adaptive Reactionless Control Algorithm . . . . . . . . . . . 39

3.1. Formulation of Adaptive Reactionless Motion . . . . . . . . . . . . . . 41

3.2. Recursive Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1. Parameter adaptation of time-varying systems . . . . . . . . . . . 45

3.2.2. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3. Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1. Planar Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.2. 3D Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

CHAPTER 4. Adaptive Reactionless Control for Parameter Identification . . 67

4.1. Momentum-based Parameter Identification . . . . . . . . . . . . . . . 69

4.2. QR Decomposition Recursive Least Squares . . . . . . . . . . . . . . . 72

4.3. Persistent Excitation of ARLC Motions . . . . . . . . . . . . . . . . . 75


4.4.1. Planar Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.2. 3D Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

CHAPTER 5. Adaptive Reactionless Control with Joint-Limit Avoidance . . 85

5.1. Formulation of ARLC for Joint-Limit Avoidance . . . . . . . . . . . . 86

5.2. ARLC with Joint-Limit Avoidance . . . . . . . . . . . . . . . . . . . . 89

x

TABLE OF CONTENTS

5.2.1. Redundancy resolution . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.2. Integration with ARLC scheme . . . . . . . . . . . . . . . . . . . 91


5.3.1. 2D Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.2. 3D Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

CHAPTER 6. Analysis and Preliminary Design of Experimental Test-bed . . 103

6.1. Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2. Base-3-DOF Manipulator-Target . . . . . . . . . . . . . . . . . . . . . 106

6.2.1. Base design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.2. Arm design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.3. Target Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3. Air-bearing Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4. Sensing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5. Computer and Control Interface . . . . . . . . . . . . . . . . . . . . . 115

6.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

CHAPTER 7. Conclusions and Recommendations for Future Research . . . 117

7.1. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2. Recommendations for Future Research . . . . . . . . . . . . . . . . . 120

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xi

LIST OF FIGURES

LIST OF FIGURES

1.1 Two generations of Canadarm: (A) Canadarm1 viewed from the

Space Shuttle; (B) Canadarm1 and Canadarm2 working side by side

[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Space vehicles equipped with a robotic arm: (A) Remote manipulator

system (JEMRMS) [2]; (B) Engineering Test Satellite VII [3] . . . . . 3

1.3 Canadarm2 moves SpaceX Dragon spacecraft into place for attach-

ment to the ISS [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Space Infrastructure Servicing [5] . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Capture mission phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Base, manipulator, target system and on-board devices . . . . . . . . . 7

2.1 General Model of Space Manipulator Robot. . . . . . . . . . . . . . . . 20

2.2 Two decompositions of total angular momentum . . . . . . . . . . . . . 24

2.3 Co-simulation platform using MSC.Adams and Matlab/Simulink . . . 37

2.4 Space manipulator models built in MSC.Adams . . . . . . . . . . . . . 37

3.1 ARLC as part of capture mission . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Adaptive reactionless control scheme. . . . . . . . . . . . . . . . . . . . 48

3.3 Planar base-3-DOF manipulator-target model . . . . . . . . . . . . . . 52

3.4 Reaction null-space system response for model verification . . . . . . . 54

3.5 Joint rates, Lt = 0, without ARLC. . . . . . . . . . . . . . . . . . . . . 56

xiii

LIST OF FIGURES

3.6 Base angular rate, Lt = 0, without ARLC. . . . . . . . . . . . . . . . . 56

3.7 Joint rates, Lt = 0, with ARLC. . . . . . . . . . . . . . . . . . . . . . . 57

3.8 Base angular velocity, Lt = 0, with ARLC . . . . . . . . . . . . . . . . 57

3.9 The relationship between Lm and Lt, Lt = 0, with ARLC . . . . . . . 58

3.10 Joint rates, Lt 6= 0, mt = 250 kg . . . . . . . . . . . . . . . . . . . . . . 59

3.11 Base response: angular velocity and attitude, Lt 6= 0, mt = 250 kg. . . 60

3.12 ARLC joint rates error, Lt 6= 0, mt = 250 kg. . . . . . . . . . . . . . . . 60

3.13 The relationship between Lm and Lt, Lt 6= 0, mt = 250 kg . . . . . . . 61

3.14 Joint accelerations, Lt 6= 0, mt = 250 kg. . . . . . . . . . . . . . . . . . 61

3.15 Five configurations (snapshots) of the arm after capturing a spinning

target, Lt 6= 0, mt = 250 kg . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.16 ARLC with Lt 6= 0, mt = 500 kg . . . . . . . . . . . . . . . . . . . . . . 63

3.17 ARLC with Lt 6= 0, mt = 125 kg . . . . . . . . . . . . . . . . . . . . . . 63

3.18 Base angular velocity, mt = 500 kg, 3D case. . . . . . . . . . . . . . . . 64

3.19 Joint rates φ1, φ2, φ3, mt = 500 kg, 3D case. . . . . . . . . . . . . . . . 65

3.20 Joint rates, φ4 – φ7, mt = 500 kg, 3D case. . . . . . . . . . . . . . . . . 65

4.1 Incorporation of ARLC and momentum-based parameter identifica-

tion into the capture mission . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Estimation of medium target, mt = 250 kg . . . . . . . . . . . . . . . . 78

4.3 Estimation of target mass, mt = 500 kg (Target 1) and mt = 125 kg

(Target 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Estimation of I33, mt = 500 kg and mt = 125 kg . . . . . . . . . . . . . 80

4.5 Estimation of position of the center of mass, mt = 500 kg and mt =

125 kg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6 Estimation of target mass, mt = 500 kg, 3D case . . . . . . . . . . . . . 81

4.7 Estimation of position of the center of mass, mt = 500 kg, 3D case . . 82

4.8 Estimation of moment of inertia, mt = 500 kg, 3D case . . . . . . . . . 82

xiv

LIST OF FIGURES

5.1 Adaptive reactionless control scheme with joint-limit avoidance . . . . 92

5.2 ARLC motion without joint-limit avoidance, Lt 6= 0, mt = 250 kg . . . 93

5.3 ARLC motion with joint-limit avoidance, Lt 6= 0, mt = 250 kg . . . . . 94

5.4 Joint angles and potential function for 2D case, Lt 6= 0, mt = 250kg . 95

5.5 Momentum exchange between Lt and Lm, mt = 250 kg . . . . . . . . . 96



5.8 Base angular velocity, 3D test case, Lt 6= 0, mt = 500kg. . . . . . . . . 99

5.9 Joint rates, 3D test case, mt = 500 kg . . . . . . . . . . . . . . . . . . . 100

5.10 Joint angles, 3D test case, mt = 500 kg . . . . . . . . . . . . . . . . . . 101

6.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Free-flying test-bed at Canadian Space Agency . . . . . . . . . . . . . . 106

6.3 Top view of the base-manipulator-target system . . . . . . . . . . . . . 107

6.4 Joint rates profile and base response for Model1 . . . . . . . . . . . . . 108

6.5 Four configurations of Model1 during simulated motion . . . . . . . . . 108

6.6 Satellite base with on-board devices . . . . . . . . . . . . . . . . . . . . 109

6.7 Link and motor-link interconnection . . . . . . . . . . . . . . . . . . . . 111

6.8 Joint rates and torque profiles for ARLC with joint-limit avoidance

(Model3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.9 Animation snapshot for dimensioning air-table, 125×125 cm2 wire-

frame test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xv

LIST OF TABLES

LIST OF TABLES

3.1 Planar 3-DOF manipulator simulation model parameters . . . . . . . . 52

3.2 3D, 7-DOF manipulator simulation model parameters . . . . . . . . . . 53

3.3 Denavit-Hartenberg parameters of 7-DOF manipulator . . . . . . . . . 53

5.1 Joint limit constraints for 7-DOF and 3-DOF manipulators. . . . . . . 92

6.1 Base-3-DOF manipulator testing models . . . . . . . . . . . . . . . . . 107

6.2 Selection of motor for manipulator joints . . . . . . . . . . . . . . . . . 110

xvii

NOMENCLATURE

NOMENCLATURE

ai position vector pointing from joint i to the mass center of link i

nan position of center of mass of last link with respect to Σn frame

bi position vector pointing from mass center of link i to joint i+ 1

E identity matrix

Ii inertia tensor of link i with respect to its mass center

nIn inertia tensor of last link with respect to its mass center represented in Σn

frame

J∗ generalized Jacobian matrix for space manipulator

L angular momentum of whole system with respect to inertial frame

Lm angular momentum of space manipulator before capture with respect to

inertial frame

Lg system angular momentum with respect to mass center of whole system

Lt angular momentum of target with respect to inertial frame

mi mass of link i (i = 0 represents the base)

M total mass of system

pr position vector of end-effector from origin of inertial frame

P linear momentum of whole system with respect to inertial frame

rg position vector of mass center of whole system

ri position vector of mass center of link i

IRn rotation matrix from frame Σn to ΣI

T kinetic energy of system

xix

NOMENCLATURE

v0 linear velocity vector of base

vr linear velocity vector of end-effector

ζ arbitrary vector with units of angular rates

φi joint rate of revolute joint i

φimax mechanical limit of ith joint

φ arm joint rates

τ arm joint torques

ω0 angular velocity of base

ωi angular velocity of link i

ωn angular velocity of last link

(.) skew-symmetric matrix of vector (.)

ΣI inertial coordinate system

Σn last link body fixed frame

xx

1.1 BACKGROUND

CHAPTER 1

Introduction

1.1 Background

For several decades now, space activities not only have had a strategic value in

space exploration missions, but also a high commercial value. In addition to the

highly popularized activities of the International Space Station, many commercial

satellite stations have been operating on-orbit for different purposes, such as, weather

forecasting, television broadcasting and telecommunication. These services play an

important role for human life on Earth. With increasing utilization of space and in

particular, the increase in space activities in near-Earth orbits, a new application of

robotic manipulators and related technologies has emerged over the past decade –

the on-orbit servicing. Similar to other industries on Earth, serving satellites even

after they lift off the ground, is a must for the development of space industry. Since

human safety considerations make such missions too risky for astronauts, robotics

technology is the only means to fulfill these missions.

The use of robotics in space has been of vital importance in orbital operations

for several decades. Their main missions to date have been to maneuver payloads, to

replace/assemble components on the International Space Station (ISS) and to sup-

port astronauts during space walks. The first ever robotic arm installed on the space

shuttle is the Shuttle Remote Manipulator System (SRMS), or Canadarm1 (Fig. 1.1).

1

CHAPTER 1. INTRODUCTION

(A) (B)

Figure 1.1. Two generations of Canadarm: (A) Canadarm1 viewed fromthe Space Shuttle; (B) Canadarm1 and Canadarm2 working side by side [1]

After many years of successful operation and numerous demonstrations of the prin-

ciples of robotic manipulation in space, Canadarm1 was retired on the final shuttle

mission in July 2011. Currently, there are three robotic arms operating on the ISS:

the European Robotic Arm built by European Space Agency (ESA), Canadarm2

built by Canadian Space Agency (CSA), and the Japanese Remote Manipulator Sys-

tem (JEMRMS) developed by Japan Aerospace Exploration Agency (JAXA). JEM-

RMS is being used on the Japanese science module of the ISS for moving equipment

around the module (Fig. 1.2A). With the success of two consecutive generations of

Canadarm, Canada has maintained its world leadership in space robotic technologies

and operational experience for over three decades.

For the purpose of advancing robotic technologies and research on robotic on-

orbit servicing, the first free-flying spacecraft with a robotic arm, the Engineering Test

Satellite VII (Fig. 1.2B), was launched by JAXA in 1998 for conducting a number of

basic experiments related to docking rendezvous, ground teleoperation, exchange of

Orbital Replacement Units (ORUs), dynamic coupling control and capturing a target

[3]. A more recent mission flown as the Orbital Express Demonstration Manipulator

System (OEDMS) and funded by the Defense Advanced Research Projects Agency

(DARPA) was aimed to study the safety and to demonstrate the cost effectiveness of

autonomous satellite servicing in-orbit.

2

1.1 BACKGROUND

(A) (B)

Figure 1.2. Space vehicles equipped with a robotic arm: (A) Remote ma-nipulator system (JEMRMS) [2]; (B) Engineering Test Satellite VII [3]

Over the past decade, however, the risks associated with increased space utiliza-

tion have become a serious concern for space communities around the world. It is

now widely acknowledged that the increasing number of space debris, including mal-

functioning and out of control satellites in the near-Earth and geostationary orbits,

can seriously affect and even pose danger to human and unmanned space activities.

An example of a recent catastrophic accident in space in 2009 is a collision that took

place between a deactivated Kosmos 2251 and an operating Iridium 33, with a high

impact as the two craft collided at a relative speed of more than 10km/s and were

both completely destroyed. To reduce the risks of such collisions, in addition to the

new satellite manufacture policies issued by NASA, one system solution that has been

under investigation by several research groups and space agencies around the world

is a servicing vehicle equipped with a robotic arm to be used for space debris re-

moval and on-orbit servicing. The specific tasks such a system could be launched and

employed for are to refuel, repair, stabilize and de-orbit space objects or debris [6].

Success of such missions relies on the availability of proven and robust algorithms for

autonomous robotic capture of free-floating uncooperative objects in space and this

has spurred considerable amount of research on various aspects of this problem [7].

3


Figure 1.3. Canadarm2 moves SpaceX Dragon spacecraft into place forattachment to the ISS [4]

Over the past several years, there have been some remarkable achievements and

substantial progress on the capture task of free-floating objects. The successful com-

pletion of OEDMS mission in 2007 confirmed the ability of a robotic arm to au-

tonomously dock and rendezvous to a cooperative object [8]. Recently, Canadarm2

has completed a historic mission by successfully berthing the Dragon spacecraft with

the ISS (Fig. 1.3). SpaceX Dragon is the first commercial vehicle in history to suc-

cessfully attach to the ISS and this mission opened a new era for commercial on-orbit

activities. However, the aforementioned missions were accomplished with the perfect

cooperation from the target to be captured. Such conditions are not likely to be

available when capturing an unknown target, such as space debris.

Research and development of methodologies for the capture of uncooperative tar-

gets have become a significant part of research programs of space agencies around the

world. In particular, the DLR German Space Operations Center has been developing

two on-orbit servicing missions, DEutsche Orbitale Servicing Mission (DEOS) and

Orbital Life Extension Vehicle (OLEV), which focus on capturing non-supportive

and/or not specially prepared client spacecraft [9]. The creator of Canadarm, MDA

4

1.1 BACKGROUND

Figure 1.4. Space Infrastructure Servicing [5]

Cooperation is planning to launch the Space Infrastructure Servicing (SIS) in approx-

imately 2015 [6]. The SIS, equipped with a robotic arm, will be used for refueling,

performing critical maintenance and repair tasks, as well as stabilizing or towing

smaller space objects or debris. The on-orbit servicing community has not only been

concerned with the technical aspects of the problem, but also other aspects: mar-

ket potential, commercial implications, political and regulatory issues are all being

addressed by the global community [10].

A number of investigations on the capture of satellites and space debris have also

been carried out in the academic space robotics community. Due to the complexity

of capture missions in space, researchers in this field typically separate the problem

into four phases: the pre-capture, contact (impact), post-capture and compound

stabilization [11]. The illustrations of the four capture mission phases are included in

Fig. 1.5. In the pre-capture phase, the space manipulator is controlled to approach

the targeted object by maintaining a relative position between the end-effector and

the grapple fixture mechanism (if one is available) [12]. At the instant of capture, the

main issue to be considered is to minimize the impact between the space manipulator

and target, and hence the capture phase is also known as the impact phase [13,

14]. The servicer-manipulator-target becomes a serial-chain multibody system when

the capture is established. In the post-capture phase, the control strategy need to

be developed to deal with the tumbling motion as well as dynamic uncertainties

as a result of target capture. To bring the whole system to rest, the compound

5


Figure 1.5. Capture mission phases

stabilization phase has to damp out any initial angular momentum transferred to

the space manipulator system from the captured target by using the attitude control

system (ACS) devices, such as thrusters or reaction wheels [15].

A wide range of problems arise in the four aforementioned phases of the on-orbit

capture mission. The scope of the present research is limited to the post-capture

phase issues. In this phase, one critical aspect is to maintain minimum coupling

between the manipulator-target and the servicer, also referred to as base, in order to

keep its attitude unchanged. As illustrated in Fig. 1.6, this is desirable for several

reasons, in particular, to ensure that the pointing requirements for high-gain antennas

are satisfied since these antennas are not omnidirectional [16]; for efficient thermal

control of the base satellite, and to maintain the target in the field of view of the

sensors mounted on the base, such as cameras [17]. Disturbances of the base satellite’s

orientation and angular motion will require the use of corrective action from the ACS

devices and ultimately shorten the life time of the servicing spacecraft by increasing

the fuel consumption of the ACS, which is a non-renewable and expensive resource

in space. In addition, although the system’s attitude can be corrected by reaction

6

1.1 BACKGROUND

BASE

Target

RW

High Gain Antenna for Communication

On-board Camera

RW: Reaction Wheels

Thrusters(cost fuel)

Earth

Figure 1.6. Base, manipulator, target system and on-board devices

wheels powered by solar panels, the compensation they provide is limited as reaction

wheels can easily reach saturation for certain rotational disturbances. As a result of

momentum conservation, the attitude of the base can be significantly disturbed by the

tumbling motion of the target and the motion of the robotic arm during the capture

maneuver. Hence, it is necessary to design trajectories for the manipulator after

impact with the uncooperative target so that the reactions transferred to the base

are minimized. On the other hand, when capturing an unknown target, in particular,

with unknown inertial properties, the performance of existing motion planning and

control algorithms for a free-floating space manipulator will deteriorate due to the

dynamic uncertainties of the system. Therefore, the availability of target’s inertial

properties is required for the compound stabilization phase and the next motion

planning task after completing the capture phase of the mission.

At this point, it is requisite that we define more precisely the term already perused

in the above discussion to characterize the target: “uncooperative” in this dissertation

will refer to a target (e.g., satellite, debris) undergoing a tumbling motion, with

unknown angular momentum and unknown inertial properties. It is noted, however,

that the on-orbit servicing community have used the same term to describe other

7


aspects of “un-cooperativeness”, for example, lack of special grapple fixtures or visual

support devices on the target, but these aspects are not dealt with in this thesis.

To summarize the above discussion, this research is devoted to answer two spe-

cific questions: (i) How to autonomously control only the arm motion to ensure the

minimum disturbance to the base after capturing an uncooperative target with tum-

bling motion? (ii) How to identify the inertia parameters of the unknown target after

capture?

1.2 Thesis Objectives

The proposed research focuses on solving the problem of post-capture of an un-

cooperative target, such as space debris, with a free-floating space manipulator. The

principal objectives of this thesis are as follows:

1. To develop a control algorithm for space manipulator to maintain the min-

imum disturbance to the base i.e., reactionless motion, after capturing an

uncooperative target, without the use of attitude control devices. The pro-

posed control scheme is intended for use from the instant of capture till

the unknown parameters of the target are identified and/or the available

stabilization methods can be applied properly.

2. To identify the inertia properties of the captured target while executing

the above proposed control algorithm and reactionless manipulation. The

satisfaction of this objective is relevant for attaining a smooth transition

from post-capture to stabilization phase, as well as the implementation of

a model-based motion control.

3. To make the proposed control scheme practical by incorporating in it a

scheme for joint-limit avoidance. Without explicit consideration of joint

limits, the application of ARLC is likely to violate the physical joint limits

and as a result, the desired goal of reactionless motion will be severely

compromised.

8

1.3 LITERATURE REVIEW

1.3 Literature Review

According to the objectives of this thesis, the related research literature is re-

viewed in this section. First, the relevant research on kinematics and dynamics of

space manipulators is summarized. Next, current state of the art in control of space

manipulators is reviewed. Then, particular attention is focused on the literature

dealing with the target capture task by a space manipulator. The literature review is

completed by summarizing most relevant work on the problem of inertial parameter

identification.

1.3.1 Kinematics and dynamics of space manipulators. A space ma-

nipulator system is characterized by a number of unique features, distinct from the

robotic systems on Earth, since it is mounted on a freely moving base and is sub-

ject to the micro-gravity condition, as well as momentum conservation principle [18].

Based on how the base is controlled, the modeling of a space manipulator can be

categorized into two types: free-flying and free-floating. In the former, the use of an

attitude control system to control the motion of the base is considered; whereas in the

latter, the motions of the base are directly affected by the reaction of the manipulator

and target [19]. There are a number of studies that have considered how to model

the coupling dynamics between the manipulator and the base. Longman et al. [20]

proposed a kinematic and dynamic computational approach for a space manipulator

with attitude control devices, to calculate the reaction force and moment from the

motion of the manipulator to the base. From results of these calculations, the attitude

control system is commanded to compensate for the base disturbance.

The kinematic and dynamics equations of a space manipulator can be derived by

applying one of the several multibody dynamics formulations which are well defined

for a conventional fixed-base manipulator [21]. One interesting idea to simplify the

dynamics modeling methodology for a space manipulator was proposed by Vafa and

Dubowsky [22], called the virtual manipulator concept. The virtual manipulator is

constructed as an ideal kinematic chain such that one of its ends remains coincident

9


with the selected point on the real manipulator and the other end is attached to the

fixed point in inertial space. The idea behind this technique is to formulate the rules

for calculating the properties of virtual manipulator, such as link dimensions and joint

axes, and its joint movements as a function of the real manipulator joint movements.

This approach facilitates the planning and control design for space manipulators.

However, in operational space control analysis, the virtual manipulator approach is

only useful for the scenario where the orientation of the base can be directly controlled

[18, 23]. On the other hand, similarly to ground manipulator dynamics modeling

research, the dynamics model of a space manipulator is often formulated by using

the Lagrangian formulation, as presented in [18, 24–26]. With the assumption that

no external force or torque is exerted on the system, the momentum conservation

equations can be used as additional motion constraints to simplify the dynamics

equations of a space manipulator [18, 25].

1.3.2 Control of space manipulator. The development of motion control

algorithms for space manipulators has progressed based on the available concepts of

model-based motion control for fixed-base robotic arms. Due to the differences in

kinematic and dynamic characteristics, the conventional controllers available for a

ground-based manipulator cannot be applied directly to a space manipulator, if the

base of the latter is not controlled perfectly. By using reaction wheels to keep the

base attitude fixed, a conventional adaptive control was proposed in [27] for obtaining

a stable end-effector trajectory tracking. Taking into account the reaction effect of

the manipulator on the base, a well-known resolved motion rate control method was

proposed in [28] by introducing the Generalized Jacobian Matrix (GJM) which can

serve as a surrogate for a standard manipulator Jacobian in solving the kinematic

problems for a space manipulator. In fact, feedback control theory can be applied to

position control of space arm end-effector, with GJM used instead of the conventional

Jacobian [18]. In the formulation of GJM, the momentum conservation equations

are involved as the non-integrable motion constraints at the velocity level; the non-

integrability of these constraints establish the nonholonomic nature of a free-floating

10


space manipulator [29]. The implementation of nonholonomic motion planning for

trajectory control of space manipulator was presented in [30].

As mentioned above, although ground manipulator controllers can be applied to

a space manipulator, if the position and attitude of the base are perfectly controlled

during a maneuver by the attitude control systems, the development of control algo-

rithms for the free-floating base scenario is highly desirable. Dubowsky and Torres

[31] proposed the Enhanced Disturbance Map (EDM) representing graphically the

effect of the manipulator motion on the free-floating base. The EDM provides a bet-

ter understanding of the dynamic disturbance created by the arm, which is useful for

designing the minimum disturbance trajectories as a solution to the motion planning

problem. Another unique issue arising in model-based motion planning and control

of a free-floating manipulator is existence of dynamic singularities, since the relation

between the end-effector linear/angular velocity and the set of joint rates involves

the dynamic properties of the system [32]. It is noted that the dynamic singularity

configurations cannot be predicted from the kinematic configurations of the manipu-

lator, and all Jacobian-based control algorithms, such as resolved motion rate control,

will fail or produce large errors at the dynamic singularity configurations. To reduce

the effects of dynamic singularities, the damped least-squares method [33, 34] used

for computing the inverse kinematics of ground-based manipulators to find feasible

joint rates can be applied [35]. It is worth mentioning that if the base attitude is

maintained constant, the dynamic singularity reduces to the manipulator kinematic

singularity.

For space manipulator control in a free-floating scenario, the availability of redun-

dant degrees of freedom becomes important since the base and the end-effector need

to be controlled simultaneously. The redundancy of manipulator alleviates the issues

of singularity avoidance and minimum reaction motion control. To tackle the dynamic

interaction problems of free-floating robots, Reaction Null Space control (RNSC) law

was originally proposed in [36] to generate control inputs that induce minimum vi-

bration in the flexible space structure on which the manipulator system is mounted.

11


This algorithm was subsequently implemented for end-effector path tracking with

reactionless motion and vibration suppression [37]. The RNS-based reactionless ma-

nipulation was also verified with the Engineering Test Satellite VII (ETS-VII) facility

flown in space [3]. The utilization of task-priority redundancy resolution techniques

has been thoroughly investigated in ground-based manipulator motion planning lit-

erature. For example, the extra degrees of freedom are employed to alleviate the

kinematic singularity issues in [38]; and, the framework of task-priority control for

redundant robots was proposed in [39, 40] to address the joint limit and obstacle

avoidance issues. In general, these aforementioned approaches can be formulated to

control space manipulators, if the condition on degree of redundancy is satisfied. It is

noted that the degree of redundancy of a free-floating manipulator depends on which

task is to be controlled: base motion control task or end-effector motion control task.

Detailed analysis and implementation of redundancy resolution for a free-floating

space manipulator were presented in [41]. In [42], the redundancy control technique

was employed to formulate the distributed momentum control algorithm to achieve

two specific tasks: the distribution and absorbtion of the initial angular momentum

from target capture while satisfying the tracking control of the end-effector.

1.3.3 Target capture by space manipulator. Since the scope of this re-

search is the post-capture of an uncooperative target with a free-floating manipulator,

state of the art control algorithms related to the post-capture and stabilization prob-

lems are reviewed in this subsection. Researchers normally address the post-capture

issues in the ideal scenario with the assumption that the impact at the grasp is min-

imal [11], which means that the relative velocity between the end-effector and the

target’s grapple point is approximately zero. In practice, this condition is difficult to

achieve at the instant of capture. Although minimizing impact is one of the main ob-

jectives of the pre-capture mission, impact at capture is practically unavoidable and

presents an issue for the post-capture control phase. Thus, a number of investigations

have been conducted on the effects of impact on the post-capture behavior of the ma-

nipulator. Based on the angular coupling momentum equation, detailed analysis of

12


space manipulator behavior, in particular, the changes in joint rates and base angular

velocity during impact and post-impact, were presented in [36]. From these analyt-

ical results, the magnitude of impact force impulse can be minimized with proper

initial configurations of the space manipulator. In [14, 43], two types of post-impact

dynamics, soft impact and hard impact, were defined based on the relative velocity

of end-effector and target. Then, the feedback linearization controller was applied to

compensate for the differences in dynamics response depending on whether the cap-

ture is smooth or with an impact. An interesting idea was proposed in [44] on how

to use contact/impact to damp the angular momentum of an uncooperative spinning

target in order to alleviate the capture mission. In particular, a contact/push-based

control method using a cushion-type damper was presented in order to dissipate the

rotational motion of the target. However, since robot end-effector with the passive

damper has to touch a spinning target several times to reduce its angular momentum,

it is difficult to stabilize and coordinate the control of the whole robot-target system.

Reaction null-space control concept is not only useful in path planning and control

of a space manipulator in the pre-capture phase, but also can be employed as a

foundation for developing post-capture and stabilization algorithms. In [42, 45, 46],

the Distributed Momentum Control (DMC) strategy for capturing a free-floating

spinning satellite was presented. The RNS motion control was applied to facilitate

the pre-impact and post-impact control phases. Recently, the reactionless trajectory

tracking control method was derived in [47], with consideration of constraints on

the reaction wheel torques and on the angular momentum that the reaction wheels

can absorb. Moreover, a motion planner was developed in [15] based on optimal

control theory (Pontryagin’s principle), and PD quaternion feedback for base attitude

control was applied to stabilize the system under torque constraints in minimum time.

However, the methods proposed in these references require knowledge of kinematics

and dynamics of the tumbling target, in addition to those of the manipulator arm,

as well as the use of reaction wheels for controlling the base’s attitude.

13


The presence of uncertainties in the system is one of the main concerns for design-

ing controllers in the post-capture and stabilization of an unknown target. The control

algorithms to deal with the presence of dynamic parameter uncertainties caused by

the target have been well studied for ground-based manipulators, in particular, with

the implementation of adaptive control techniques [48–51]. In general, these conven-

tional adaptive control methods require that the dynamics model of manipulator can

be parameterized linearly in terms of unknown inertial parameters. The free-floating

manipulator model derived from the Lagrangian formulation presented in [24, 25, 28]

cannot be linearized in terms of the inertial parameters. Consequently, the application

of conventional adaptive control to free-floating space manipulators with dynamics

equations derived by Lagrangian formulation is not possible, since the aforementioned

dynamics equations are nonlinear in the inertial parameters [25]. This can be possible,

however, only with joint-space control if reaction wheels are available for base attitude

control [27]. To overcome this drawback, the articulated-body concept was used in

[52] to model the space manipulator system. With the linearity of unknown param-

eters in the articulated-body model, a conventional adaptive control can be applied

to the trajectory tracking problem. In addition, the linearity in terms of unknown

parameters in the momentum equation implies that the dynamics coupling equation,

which is attained from the time derivative of the momentum of manipulator arm, has

the same characteristic. Hence, by redefining the input command to the coupling dy-

namics equation, a conventional adaptive control was implemented in [35] to achieve

the positioning of the end-effector while the vibrations of the flexible-structure-base

manipulator converged to zero. The aforementioned references [35, 52], however, do

not explicitly deal with the capture of an unknown tumbling target. The authors

of [11] propose a control law to accumulate the angular momentum resulting from

capture of a tumbling target, in the presence of uncertainty in dynamic parameters of

the target. The manipulator motion is assumed to follow a pre-defined stabilization

trajectory [53] and the parameter adaptation law is based on the error in the coupling

force between the base satellite and the robot arm.

14


1.3.4 Inertia parameter identification. One of the requirements for post-

capture stabilization algorithms is that the inertial parameters of the target are ac-

curately estimated in real time, if unknown prior to capture [54]. To date, several

articles have been published addressing the inertial parameter estimation aspect of the

space-manipulator/target problem. Although the solutions proposed vary in detail,

most of them were developed based on three fundamental approaches: vision-based

[54, 55], force-based [56] and momentum-based [56–58]. The vision-based method is

very useful in motion prediction for the pre-capture phase, since it provides an es-

timation and prediction of full states as well as inertial parameters, including ratios

of moments of inertia, position of the center of mass and orientation of principal

axes of the target [54, 55]. However, the target’s mass cannot be obtained from this

method. In the force-based method, the estimation problem is formulated based on

the Newton-Euler equations of motion and all of the inertial parameters including

the mass of the target can be estimated [56]. The momentum-based method formu-

lated from the conservation of momentum principle is preferable because it requires

only velocity measurements for inertial parameter identification, these being easily

available on space manipulators. On the other hand, the two former methods require

the space manipulator to be equipped with dedicated sensors and the corresponding

processing algorithms are more susceptible to signal noise, which is further aggra-

vated in space systems [57]. The momentum-based inertial parameter identification

method is unique to space manipulators, since the key equation for identification

problem is derived from the momentum conservation equations. This procedure re-

quires a sequence of sufficiently rich motions spanning a wide range of manipulator

configurations, in order to produce large changes in the velocity of the target. In

[56–58], the momentum-based parameter identification was conducted in an off-line

mode with the motion sequences designed randomly in order to achieve robot config-

urations as different as possible. However, a control scheme for the post-capture of a

tumbling target integrated with target’s parameter identification procedure has not

been developed to date.

15


1.4 Thesis Outline

This dissertation is organized into seven Chapters. First, with the kinematic

and dynamic characteristics of free-floating space manipulators being different from

those of ground-based systems, the relevant fundamental equations are summarized

systematically in Chapter 2. To understand and point out the unique features of

space manipulator systems, we first introduce the momentum conservation equations

and their expressions with respect to two different points, the mass center of the base

and that of the whole system. Details related to the holonomic and nonholonomic

constraints of space manipulators are also discussed. Then, the dynamics equations

of space manipulators are derived by using the Lagrangian method. Since the control

concept proposed in this dissertation is developed based on the reaction null-space

control, the fundamental equations of reaction null-space control concept are reviewed

in detail. Next, the minimum kinetic energy expression of space manipulators is

derived with some relevant remarks on the post-capture control missions. Chapter

2 is closed with the description of the numerical platform using the co-simulation

of MSC.Adams and Matlab/Simulink with the two models of space manipulators, a

planar model and a 3D model of the ETS-VII system.

In Chapter 3, a new reactionless motion control algorithm is formulated from

the momentum conservation equations for a free-floating space manipulator, without

the use of attitude control devices. In the presence of dynamics uncertainties, as

a result of capturing an unknown target, the adaptive control scheme is developed

to adaptively produce the arm motions with minimum disturbance to the base –

hereafter, it is referred to as Adaptive ReactionLess Control (ARLC) algorithm. The

co-simulation platform introduced in Chapter 2 is employed to verify the validity and

feasibility of the proposed concept. The numerical simulation results are obtained for

two basic scenarios: one where the initial (prior to capture) angular momentum of

the target is zero and the second where the target is tumbling.

The ARLC scheme proposed in Chapter 3 provides a buffer in time to segue be-

tween the post-capture and stabilization phases. In Chapter 4, the momentum-based

16

1.4 THESIS OUTLINE

parameter identification method is derived for estimating the unknown properties

of the captured target while the manipulator arm executes ARLC motions. Since

the total momentum of the system after capture is unknown, the momentum-based

identification algorithm is formulated in incremental form to eliminate the need for

knowledge of system momentum. The recursive least squares technique using QR

decomposition is employed for solving the linear estimation problem. Two control

objectives – maintaining minimum disturbance on the base and achieving conver-

gence to the inertial parameters of the unknown target – are verified through the

same models and test case scenarios as used in Chapter 3.

The manipulator arm following ARLC motions may fail to maintain minimum

disturbance on the base attitude if the joints reach their physical limits. Hence, in

Chapter 5, the ARLC is reformulated in order to implement the redundancy resolu-

tion control technique for a secondary manipulation task. In particular, the gradient

projection method is applied within the ARLC scheme to avoid the physical joint lim-

its of the manipulator. Again, the correct performance of this approach is confirmed

with the same numerical models as used in previous Chapters.

Chapter 6 presents the design concept and the initial design specifications for an

experimental test-bed which could serve as a physical validation and demonstration

platform for the algorithms developed in this thesis. The concept is based on the

air-bearing table principle to create the free-floating conditions for a planar system

configuration. The design allows for a small (3-DOF) robotic arm attached to a free-

floating base, a separate free-floating target, and the required sensing and control

hardware. Finally, Chapter 7 concludes this dissertation and suggests some directions

for future research in this area.

17

2.1 KINEMATICS OF SPACE MANIPULATOR

CHAPTER 2

Kinematics, Dynamics and Reactionless

Control of Space Manipulator

Fundamental kinematic and dynamics equations of space manipulator are system-

atically presented in this Chapter. Since the present dissertation deals with the

coupling motion between the manipulator and the base, the relation between the

end-effector/base motions and the set of joint rates and the coupling dynamics equa-

tions are discussed in detail. The system investigated in this study is comprised of a

serial manipulator mounted on a free-floating base. Since most of the space manipu-

lators in history as well as those planned for launch in the near future, for example,

the Space Infrastructure Servicing, are built with revolute joints, this study focuses

on manipulators with revolute joints only. Here, it is assumed that the system is

composed of rigid bodies and no external forces or torques are exerted. The role of

reaction wheels or other momentum exchange devices is not considered in this study.

2.1 Kinematics of Space Manipulator

Figure 2.1 shows a schematic and important parameters of a servicing system

comprised of a serial-link manipulator holding a target, and a base spacecraft. The

basic kinematic equations of a n-axis serial manipulator mounted on a free-floating

base are derived in this Section. The system is comprised of n+ 1 rigid bodies where

19

CHAPTER 2. KINEMATICS, DYNAMICS AND REACTIONLESS CONTROL OF SPACE MANIPULATOR

……..

O

Z

XY

BASE

Inertial FrameI

n

3 2

10

1

23n

Target

g0rnrnp

nna

0b

1b1a

End-effector

1k k

2k3k

nk

0r

grrp

Figure 2.1. General Model of Space Manipulator Robot

the target is assumed to be rigidized by the end-effector to the last link of the arm

when the capture is established. From Fig. 2.1, it is easy to write the position relation

as:

pr = r0 + b0 +n∑i=1

li (2.1)

where pr denotes the position vector of the end-effector from the origin of the inertial

frame; li = ai+bi with ai being the vector pointing from joint i to the mass center of

link i and bi defining the vector pointing from the mass center of link i to joint i+ 1.

There is a difference from the analogous formula for a fixed-base manipulator due to

the appearance of vectors r0 and b0 which are, respectively, the position of the mass

center of the base relative to the inertial frame at O and the vector pointing from the

mass center of the base to the first joint. Note that all vectors must be expressed in

the same inertial frame ΣI which is assumed to exist in the vicinity of the orbiting

system translating at its orbital speed for the short duration of the capture maneuver.

20

2.1 KINEMATICS OF SPACE MANIPULATOR

Differentiating both sides of equation (2.1), the end-effector linear velocity is

obtained as follows:

vr = v0 + ω0 × b0 +n∑i=1

ωi × li (2.2)

while the angular velocity of link i can be written as:

ωi = ω0 +i∑

j=1

kjφj (2.3)

Substituting (2.3) into (2.2) with the use of (2.1), we have:

vr = v0 + ω0 × (pr − r0) +n∑i=1

ki × (pr − pi) φi (2.4)

In the above, v0 and ω0 represent the linear and angular velocities of the base; pi

is the position vector of joint i; φi and ki are a joint rate and unit vector associated

with each revolute joint i, respectively. From (2.3), the angular velocity vector of the

last link can be written as:

ωn = ω0 +n∑i=1

kiφi (2.5)

From equations (2.4) and (2.5), ωn and vr can be expressed in a more compact

form as: vr

ωn

= Js

v0

ω0

+ Jmφ (2.6)

where we combine the joint rates into the joint vector and introduce two Jacobian

matrices, defined as:

φ =[φ1 φ2 ... φn

]T∈ IRn

Js =

E −p0r

0 E

∈ IR6×6, p0r = pr − r0

and

Jm =

k1 × (pr − p1) k2 × (pr − p2) · · · kn × (pr − pn)

k1 k2 · · · kn

∈ IR6×n

21


In the above, E denotes the identity matrix and (.) is a skew-symmetric matrix of

vector (.) defined as:

(.) ,

0 −(.)z (.)y

(.)z 0 −(.)x

−(.)y (.)x 0

From equation (2.6), it is apparent that the relationship for the end-effector velocity

involves two separate terms. According to [59], the coefficient matrix of the base

velocity term is defined as the Jacobian matrix for the base, whereas that of the joint

rate vector term is the Jacobian matrix for the fixed-base manipulator. In order to

derive the relationship mapping the set of joint rates to the end-effector velocities,

an additional motion constraint is needed. Under the previously stated assumptions

that there are no external forces or torques exerted on the system and no ACS devices

are used, the total momentum of the space system is conserved and this represents

the nonholonomic motion constraint which is a unique and defining characteristic of

space manipulator systems. In the next Sections, particular attention is given to the

analysis of the momentum equations because they are fundamental to developing the

control algorithms in the next Chapters.

2.2 Momentum Equations

In this Section, the momentum equations of a serial manipulator are first derived.

Next, the compact expressions for the total momentum of the system with respect

to the mass center of the base as well as the mass center of the whole system are

presented. Let P and L denote the linear and angular momenta, respectively, of the

whole manipulator and target system with respect to the inertial frame shown in Fig.

2.1. Then, by definition we have:

P =n∑i=0

miri (2.7)

L =n∑i=0

(Iiωi + ri ×miri) (2.8)

22

2.2 MOMENTUM EQUATIONS

where ri ∈ IR3 is the position vector of the mass center of link i; Ii ∈ IR3×3 is the

inertia tensor of link i w.r.t. its mass center and ωi ∈ IR3 is the angular velocity of

link i; mi is the mass of rigid body i. Again, all vectors are expressed in the inertial

frame.

By employing the forward kinematics relations for a serial space manipulator,

similarly to the derivation of equations (2.4), we have the expansion of the linear

velocity of center of mass of link i as:

ri = v0 + ω0 × (ri − r0) +i∑

j=1

kj × (ri − pj) φj (2.9)

Substituting (2.9) and (2.3) into (2.7) and (2.8), we obtain:

P = Mv0 +M rT0gω0 +n∑i=1

i∑j=1

mi [kj × (ri − pj)] φj (2.10)

L = M rgv0 +

(n∑i=0

(Ii −mirir0i)

)ω0

+n∑i=1

i∑j=1

miri [kj × (ri − pj)] + Iikj φj

(2.11)

where

M =n∑i=0

mi : Total mass of the whole system

rg =

∑ni=0 miriM

: Position of the mass center of the whole system

r0i = ri − r0; and r0g = rg − r0

We next further formulate specifically the angular momentum of the system with

the view to deriving its compact decomposition. To this end, two pertinent locations

in our system – the mass center of the whole system and the mass center of the base

satellite – can be used to decompose the total angular momentum into the angular

momentum about the aforementioned point and the linear momentum term. These

23


……..

O

Z

XY

BASE

I

Targ

et

grg0r

gL

P

0r

(A)

……..

O

Z

XY

BASE

I

Targ

et

grg0r

0LP

0r

(B)

Figure 2.2. Two decompositions of total angular momentum

decompositions are useful for the analysis and development of the manipulator control

algorithms.

Letting Lg denote the angular momentum of the system with respect to the mass

center of the whole system, as illustrated in Fig. 2.2A, the system angular momentum

L can be decomposed into:

L = Lg + rg ×P (2.12)

From (2.12), Lg can be computed with the use of (2.10) and (2.11) as follows:

Lg = L− rg ×P =

[n∑i=0

(Ii −mirir0i)−M rgrT0g

]ω0

+n∑i=1

i∑j=1

mirgi [kj × (ri − pj)] + Iikj φj

(2.13)

According to the definition of rg, we have the relation:

M rgr0g = rg

n∑i=1

mi [ri − r0] =n∑i=1

mirgr0i (2.14)

Then, (2.13) can be simplified as follows:

Lg =

[n∑i=1

(Ii +mir

Tgir0i

)+ I0

]ω0

+n∑i=1

i∑j=1

mirgi [kj × (ri − pj)] + Iikj φj

(2.15)

24

2.2 MOMENTUM EQUATIONS

Now, combining equations (2.10) and (2.11) into a compact form by making use of

(2.12) and (2.15), we obtain:P

L

=

ME M rT0g

0 Hω

v0

ω0

+

JTω

Hωφ

φ+

0

rg ×P

(2.16)

where

Hω =n∑i=1

(Ii +mir

Tgir0i

)+ I0 ∈ IR3×3

Hωφ =n∑i=1

(IiJRi +mirgiJT i) ∈ IR3×n

JT i = [k1 × (ri − p1),k2 × (ri − p2), · · · ,ki × (ri − pi),0, · · · ,0]

JRi = [k1,k2, · · · ,ki,0, · · · ,0]

JTω =n∑i=1

miJT i

The advantage of the decomposition above is that the linear velocity of the base v0 is

eliminated from the bottom part of (2.16), which is the angular momentum equation

of the system. Equation (2.16) has been used as a starting point for developing the

coupling motion control algorithms in the literature [11, 45]. This equation also plays

a key role in the derivation of the proposed control scheme presented in Chapter 3.

Similarly to the above derivation, the system angular momentum equations can

be rewritten to obtain the compact form in terms of the angular momentum of the

system w.r.t. the center of mass of the base and linear momentum, as illustrated in

Fig. 2.2B. Letting L0 be the angular momentum of the system w.r.t. the center of

mass of the base, we have:

L = L0 + r0 ×P (2.17)

Following the same procedure of the derivations of equations (2.13) and (2.14), but

now with r0 used instead of rg, we obtain the angular momentum of the system w.r.t.

25


the center of mass of the base L0 as:

L0 = L− r0 ×P =

[n∑i=1

(Ii +mir

T0ir0i

)+ I0

]ω0

+n∑i=1

i∑j=1

mir0i [kj × (ri − pj)] + Iikj φj

(2.18)

and the linear and angular momenta can be decomposed into a compact form by

using relations (2.17) and (2.18) as follows:P

L

=

ME M rT0g

M r0g HΩ

v0

ω0

+

JTω

HΩΦ

φ+

0

r0 ×P

(2.19)

where

HΩ =n∑i=1

(Ii +mir

T0ir0i

)+ I0 ∈ IR3×3

HΩΦ =n∑i=1

(IiJRi +mir0iJT i) ∈ IR3×n

The coefficient matrix of the base velocity term (first term in (2.19)) is the base

inertia matrix, while that of the joint rates is the coupling inertia matrix. Unlike the

coefficient matrix of the base velocity term resulting in (2.16), the base inertia matrix

of (2.19) is symmetric and always invertible.

Under the stated conditions, the motion of the space manipulator is subject to

the conservation of momentum, that is, the linear and angular momenta of the system

(P, L) remain constant. Hence, the momentum equations presented above impose

motion constraints on the space manipulator system. Although these constraints are

all represented in velocity form, only the linear momentum motion constraint can

be analytically integrated with respect to time. Indeed, integrating equation (2.7),

yields: ∫ t

0

Pdt =

∫ t

0

n∑i=0

miridt =n∑i=0

miri(t)−n∑i=0

miri(0)

= M (rg(t)− rg(0))

(2.20)

26

2.3 GENERALIZED JACOBIAN MATRIX

It can be seen from equation (2.20) that the mass center of the whole system does

not displace if the initial (and constant) linear momentum of the system is zero. The

linear momentum conservation constraint, which can be expressed as a configuration

or as a velocity constraint, is a holonomic constraint. On the other hand, the angular

momentum constraint cannot be expressed as a configuration constraint due to the

nonintegrability of the angular momentum of the system (see equation (2.8)). Hence,

it imposes a nonholonomic constraint on a free-floating manipulator system. This

unique feature of free-floating space manipulators can be visualized by considering

a scenario where the end-effector of the arm returns to the same initial position

and orientation in inertial space after following a closed trajectory, but the base

orientation and manipulator configuration will not be the same as the initial ones [30,

32]. As already mentioned in [20, 22], the kinematics problem of a free-floating space

manipulator is difficult to solve even for the forward kinematics because the position

and orientation of the end-effector depend on the history of manipulator maneuver.

To overcome this difficulty, the kinematics of space manipulator is usually defined

at the velocity level. Thus, the generalized Jacobian matrix was introduced in [28]

to formulate the relationship between the end-effector velocities and the manipulator

joint rates. This is a relevant concept in motion control of space manipulators because

a number of fixed-base manipulator control schemes can be adopted to their space

analogues by using this concept. The derivation of the generalized Jacobian matrix

is presented in the following section.

2.3 Generalized Jacobian Matrix

The generalized Jacobian matrix (GJM) was originally derived by Umetani and

Yoshida [28] for free-floating and free-flying space manipulators as a direct analogue

of the standard Jacobian matrix for fixed-base (ground-based) manipulators. The

formulations of GJM for two cases are presented in this Section: first, the GJM is

formulated with the additional assumption that the linear and angular momenta of

27


the system are zero; second is the general case where the system momenta can be

varied by jet thrusters, ACS devices or external forces/torques.

From the momentum equation (2.19), the motion constraint for P = L = 0 can

be written as: ME M rT0g

M r0g HΩ

v0

ω0

+

JTω

HΩΦ

φ = 0 (2.21)

Since the base inertia matrix in the above is always invertible as mentioned previously,

equation (2.21) can be solved for the base velocities with:v0

ω0

= −

ME M rT0g

M r0g HΩ

−1 JTω

HΩΦ

φ (2.22)

The above relation between the base velocities and joint rates allows us to elimi-

nate the base velocity term in the forward kinematics equation (2.6), thereby yielding

a direct relationship between the manipulator operational space (end-effector) veloc-

ities vr and ωn and the joint space rates φ:vr

ωn

=

Jm − Js

ME M rT0g

M r0g HΩ

−1 JTω

HΩΦ

φ (2.23)

and consequently revealing the Generalized Jacobian Matrix, J∗, for a space manipu-

lator:

J∗ = Jm − Js

ME M rT0g

M r0g HΩ

−1 JTω

HΩΦ

∈ IR6×n (2.24)

The GJM formula captures the fundamental invariant of a space-based system – its

momentum – thereby implicitly linking the kinematics and dynamics of the system.

It is also apparent from the above formula that the matrix J∗ reduces to the standard

Jacobian Jm for a ground-based manipulator as the mass and inertia tensor of the

base approach infinity.

Now, the GJM for a non-zero momentum scenario, where the momentum of the

system is varied through the effects of jet thrusters from ACS or external natural

forces, as for example, generated by the gradient of gravity, is derived. Proceeding

28

2.3 GENERALIZED JACOBIAN MATRIX

analogously to the derivation just presented starting with equation (2.19) to eliminate

the base velocities from equation (2.6) and making use of the definition of GJM in

(2.24), we obtain:vr

ωn

= J∗φ+ Js

ME M rT0g

M r0g HΩ

−1 P

L− r0 ×P

(2.25)

Since HΩ is always invertible, the base inertia matrix can be inverted symbolically as

follows:ME M rT0g

M r0g HΩ

−1

=

(ME−M rT0gH−1Ω M r0g

)−1 −rT0g(HΩ −M r0gr

T0g

)−1

−(HΩ −M r0gr

T0g

)−1r0g

(HΩ −M r0gr

T0g

)−1

Equations (2.23) and (2.25) describe the relation between the end-effector veloc-

ity in inertial space and a set of rates in joint space, with (2.25) also accounting for

non-zero and variable P and L. The generalized Jacobian matrix is a function of both

kinematic and dynamic parameters and depends on the robot configuration. Making

use of equations (2.23) and (2.25), the resolved motion-rate control and resolved ac-

celeration control have been employed to track the prescribed (desired) end-effector

motion with the allowance for the reaction from the arm motion onto the base. In

practice, these algorithms were successfully implemented and flown on the ETS-VII

[3]. Based on the explicit formula of GJM, it is realized that the kinematics infor-

mation alone is not sufficient for solving the inverse kinematics problem. Knowledge

of the dynamics characteristics of the whole system is necessary to formulate the

motion control algorithm. This is also the motivation for the proposed parameter

identification approach presented in Chapter 4 of this dissertation, since uncertainty

in inertial parameters of the system is introduced in the post-capture of unknown

target mission.

29


2.4 Dynamics Equations

As mentioned in Section 2.2, since the momentum equation provides direct in-

sight into the motion of the system, in particular, reactive motion between the base,

manipulator and target, the proposed control algorithm presented in Chapter 3 is not

developed from the dynamics equations. However, they of course form the dynamics

kernel of the commercial dynamics simulator, MSC.Adams, employed in this work

for testing and demonstrating the performance of the algorithms developed. There-

fore, for completeness, in this Section, the dynamics equations of a free-floating space

manipulator are formulated by employing the Lagrangian method. Starting with the

kinetic energy of the system, by definition, it can be written as:

T =1

2

n∑i=0

ωTi Iiωi +mir

Ti ri

(2.26)

By using the forward kinematics of a space manipulator, equations (2.9) and (2.3),

the kinetic energy in (2.26) can be reformulated in terms of the base velocity and

joint rates as follows:

T =1

2

[vT0 ω

T0 φ

T]

ME M rT0g JTω

M r0g HΩ HΩΦ

JTTω HTΩΦ Hφ

v0

ω0

φ

(2.27)

where

Hφ =n∑i=1

(JTRiIiJRi +miJ

TT iJT i

)∈ IRn×n

For a free-floating manipulator system without considering the attitude control

mechanisms, it is apparent that the generalized coordinates are defined to be the ac-

tive joint angles φ ∈ IRn×1 of the manipulator. Therefore, following the derivation of

the equations of motion using the Lagrangian method, it is suggested that the kinetic

energy should be expressed as a function of φ and φ. Under the assumption that

no external torques or forces are exerted on the system, the momentum conservation

30

2.4 DYNAMICS EQUATIONS

equation can be used as an additional constraint equation to eliminate the base ve-

locities from the kinetic energy function. For simplicity, we further assume that the

initial momentum of the system is zero so that the nonholonomic constraint equation

(2.21) can be applied. Hence, the kinetic energy T can be written as:

T =1

2φT

Hφ −[JTTω HT

ΩΦ

] ME M rT0g

M r0g HΩ

−1 JTω

HΩΦ

φ (2.28)

and the above expression for T can be directly applied into the Lagrangian equations.

Since the system is operated under the micro-gravity condition, the system’s potential

energy is approximately zero. Hence, the system dynamics equation is derived from

the simplified Lagrange’s equations:

d

dt

(∂T

∂φ

)− ∂T

∂φ= τ (2.29)

where τ = [τ1 τ2 ... τn]T is the vector of joint torques corresponding to the generalized

coordinates φ. Substituting for T from (2.28) into (2.29), the equation of motion in

terms of φ, φ and φ is obtained as:

M(φ)φ+ C(φ, φ)φ = τ (2.30)

where the inertia matrix M is a symmetric and positive definite matrix defined ex-

plicitly in terms of the previously introduced matrices as:

M = Hφ −[JTTω HT

ΩΦ

] ME M rT0g

M r0g HΩ

−1 JTω

HΩΦ

(2.31)

and the nonlinear coriolis and centrifugal terms as:

C(φ, φ)φ = Mφ− ∂

∂φ

(1

2φTMφ

)(2.32)

Several important properties need to be noted regarding the dynamics equation

of a free-floating space manipulator as listed below:

31


• First, a relationship between matrices M and C that exists for ground-

based manipulators also holds for a space-based manipulator, in particular,

M− 2C is a skew-symmetric matrix [25].

• Second, it can be observed from equation (2.31) that the inertia matrix

M is composed of two separate terms. Analogously to the generalized Ja-

cobian matrix formula, the first term is a function of manipulator inertia,

geometry and configuration only, whereas the second term takes the cou-

pling dynamics between the base and manipulator into account. It should

be noted that the inertia matrix M for a space manipulator reduces to that

of a ground-based manipulator if the mass and the inertia tensor of the

base approach infinity.

• Finally, the most significant feature of the dynamics equation of a space ma-

nipulator is the nonlinearity of parametrization property, since the second

term in equation (2.31) cannot be expressed linearly in terms of the dynam-

ics parameters of the system [25]. This is an important consideration for

studies on post-capture of unknown target. In fact, the implementation of

nonlinear control or adaptive control methods for the post-capture problem

is preferred for ground-based manipulator control, since these approaches

have been demonstrated to offer very good performance in controlling a ma-

nipulator with unknown grasping payload [48, 60]. However, the linearity of

the parameterization is one of the prerequisite conditions for designing most

of the existing adaptive control approaches. Hence, these approaches can-

not be analogously implemented for the dynamics equations of free-floating

space manipulators derived with the Lagrangian method.

In this study, the main consideration is the disturbances of the manipulator mo-

tion on the free-floating base. Hence, it is not necessary to consider the problem at

acceleration or torque level because the investigation of motion rates and momenta

is sufficient for designing a control algorithm [28]. In the next Section, the powerful

concept of the arm manipulation producing minimum disturbance to the base, which

32

2.5 REACTION NULL-SPACE MOTION CONTROL

is the main inspiration for the present research, is presented. This concept has become

widely known as the reaction null-space motion control.

2.5 Reaction Null-Space Motion Control

Reaction null-space (RNS) motion control was introduced by Nenchev et al. [36,

61] to address the problem of minimizing coupling disturbance for a free-floating

space manipulator. RNS motion is defined as a set of joint rates producing zero

disturbance to the base. A manipulator system with non-zero degree of redundancy

(DOR) is required for the existence of RNS solutions. However, the definition of

DOR for a space manipulator is not the same as that for a conventional ground-

based manipulator, the latter defined as the difference between the degree-of-freedom

(DOF) of the arm and the dimension of the end-effector task space [40]. Since the

control task of a free-floating space manipulator does not only deal with the end-

effector motion, but also the base motion (w.r.t. the inertial frame), the DOR has

to be defined according to the dimension of the task space which poses as the main

objective for the controller to be designed. A detailed explanation of DOR in the

context of the free-floating manipulator control problem can be found in [41].

Now, the RNS concept is presented for the basic scenario where the initial linear

and angular momenta of both manipulator and target with respect to the defined

inertial frame are zero (P, L = 0), that is, neither are translating nor rotating

relative to the inertial frame. Considering the rotational motion of the base as the

main control task, we assume that the necessary condition for the existence of RNS,

which is the kinematic redundancy condition with respect to the base motion task,

holds. From (2.16), it follows that the angular momentum equation can be simplified

to:

Hωω0 + Hωφφ = 0 (2.33)

The above statement of the angular momentum of the system has formed the basis

of the RNS control algorithm [41, 62], which generates the joint motion resulting in

zero disturbance to the base under ideal conditions. Specifically, by substituting the

33


desired angular velocity of the base, ω0 = 0, into (2.33) and solving the remaining

homogeneous equation, we obtain the null-space solutions for joint rates φ:

φRNS =(E−H+

ωφHωφ

)ζ (2.34)

In the above, H+ωφ denotes the right pseudo-inverse of matrix Hωφ and ζ ∈ IRn is

an arbitrary vector with units of angular rates [rad/s]. It is apparent that the RNS

joint rates are generated by projecting an arbitrary vector ζ onto the null-space of

the inertia coupling matrix Hωφ via an orthogonal projection PRNS = E−H+ωφHωφ.

From the existence condition of RNS, the solution (2.34) exists if:

dim(RNS) = n− rank(Hωφ) > 0

where recall n is the DOF of the arm. The formulation of the RNS motion can also be

used as a benchmark to ascertain the accuracy of the numerical simulation platform

for a free-floating space manipulator as is executed in Chapter 3, in the simulation

study Section 3.3.

For the general case where the initial system angular momentum is non-zero,

although the RNS motion defined by (2.34) produces a zero net angular momentum

of the manipulator arm, there is no guarantee of zero disturbance to the base since the

momentum of the base must accumulate the non-zero total angular momentum. By

the same token, the general formula of reactionless motion for the non-zero angular

momentum scenario can be obtained as follows:

φRL = H+ωφL +

[E−H+

ωφHωφ

]ζ (2.35)

Equations (2.34) and (2.35) have been employed as fundamental equations in

many studies on trajectory planning and control of space manipulators [3, 36, 63].

Moreover, the validity of the RNS motion was experimentally proved with the ETS-

VII in orbit. In the next Chapter, this powerful concept is put to further use to

develop a control method for the post-capture problem with the presence of dynamics

uncertainties as a result of unknown target capture.

34

2.6 REMARKS ON KINETIC ENERGY OF SPACE MANIPULATOR

2.6 Remarks on Kinetic Energy of Space Manipulator

Our analysis of the space manipulator system also demonstrates that there is a

maximum amount of kinetic energy which can be dissipated by the manipulator arm

during and/or after capture of a tumbling target and this value depends on the initial

momenta of the target and the arm configuration at capture. In fact, assuming that

the space system is initially at rest, the kinetic energy of the system T is equal to the

work done by the actuators W , that is:

T = W =

∫ tf

0

τ T φdt (2.36)

where as before τ is the assembly of manipulator joint torques [Nm] and φ is the

vector of joint rates [rad/s]. From equation (2.36), it immediately follows that:

T = τ T φ (2.37)

and therefore, we can conclude that the kinetic energy of the space system reaches its

minimum value at φ = 0. From the kinetic energy of the system expression obtained

previously in equation (2.27), it then follows that the minimum kinetic energy of the

space manipulator system can be reduced to the following:

Tmin =1

2

[vT0 ω

T0

] ME M rT0g

M r0g HΩ

v0

ω0

(2.38)

Making use of the momentum conservation equations (2.19) with the initial linear

momentum P and angular momentum L, the above formula for Tmin can be expressed

as follows:

Tmin =1

2

[PT LT

0

]ME M rT0g

M r0g HΩ

−1 P

L0

(2.39)

where recall L0 can be computed from L0 = L− r0 ×P.

In the above, the matrix inverse depends on the configuration of the manipulator.

This result demonstrates that, for the given initial momentum of the system, there

is a maximum amount of kinetic energy which can be dissipated by the manipulator

35


arm. Therefore, the residual kinetic energy has to ultimately be accommodated by

the attitude control devices, such as reaction wheels, for stabilizing the whole sys-

tem.1 Simulation results verifying the minimum kinetic energy expression for a space

manipulator, equation (2.39), are presented in [64].

2.7 Numerical Co-simulation Platform

Developing an experimental facility for testing a free-floating space manipulator

on Earth is one of the biggest challenges and constraints for research scientists in

this field. Still, there are a number of test-beds that have been constructed for this

purpose: the planar air-bearing table is suitable for testing 2D scenarios; the par-

abolic flight can be used for short duration experiments; under-water facilities are

few in the world but they can provide a fully 3D neutrally-buoyant environment [65].

These methods are costly and may be rather limited, for example, the under-water

facilities are not suitable for direct testing of a space robot prototype. In Chapter

6 of this thesis, we present an initial design for a small-scale planar facility that

could be used in the future to demonstrate the algorithm developed here. For the

present, however, a numerical co-simulation platform was developed to analyze and to

demonstrate the behaviors of two space manipulators and to test the proposed control

and identification algorithms. The co-simulation platform combines the advantages of

Automatic Dynamic Analysis of Mechanical Systems (MSC.Adams) software in mod-

eling and simulating complex multibody dynamics systems, with the advantages of

Matlab/Simulink with special capabilities for control system design. It is worth men-

tioning that MSC.Adams has been used by NASA Jet Propulsion Laboratory (JPL)

as a main simulation platform for the Curiosity’s descent and landing simulation on

Mars [66].

A schematic of the co-simulation setup employed in this research is illustrated in

Fig. 2.3 where the dynamics model of space manipulators are built and computed in

MSC.Adams and the control and parameter identification algorithms are developed

1In the context of spacecraft attitude control, this is referred to as detumbling control problem[16].

36

2.7 NUMERICAL CO-SIMULATION PLATFORM

Robot Model

MSC.ADAMS

Control/IdentificationAlgorithm

Matlab/SimulinkPositions

Joint angles/rates

Base angular velocity

Control input (Torques)

Figure 2.3. Co-simulation platform using MSC.Adams and Matlab/Simulink

Base Target

Link

1 Link2

Link3

End-effector

Top view

Gripper

Target

F

F

Front view

Grapple Fixture

(A) Planar Base-3-DOF manipulator and target

Link1

Link2

Lin

k3

Link6

Lin

k4

Link5

Link7 + Target

Inertial Frame

(B) ETS-VII model

Figure 2.4. Space manipulator models built in MSC.Adams

in Simulink. The outputs from MSC.Adams including position vectors (the end-

effector, center of mass of the base and links), joint angles/rates and base angular

velocity are the inputs to the Simulink program. Then, the control outputs from

Simulink, specifically the commanded torques to each joint of the manipulator arm,

are the inputs to MSC.Adams. As previously mentioned, the dynamics kernel of the

space manipulator is executed in MSC.Adams simulator.

The investigation and verification of the proposed concepts in this dissertation are

conducted with two models of space manipulator systems created in MSC.Adams as

37


shown in Fig. 2.4. Fig. 2.4A shows a model snapshot of the planar system comprised

of the base, 3-DOF manipulator and the target. Realistic capture is considered in this

model scenario with the solid to solid impact modeling capabilities of MSC.Adams

employed to model impact and contact between the gripper and grapple fixture. At

the instant of capture, the target is grasped and held firmly with two forces F applied

to the gripper perpendicular to the gripper jaw surfaces, in the plane of the system.

Fig. 2.4B shows the model of the ETS-VII servicing satellite employed for testing the

proposed algorithms with a non-planar scenario. In this model, the target is assumed

to be attached rigidly to the end-effector at the instant the capture is established.

Thus, the last link of the 3D model in Fig. 2.4B is annotated as Link7+Target. The

detailed kinematic and dynamics parameters of the aforementioned models as well as

the specific simulation conditions will be presented in the simulation study Section

3.3 in the next Chapter.

2.8 Summary

The fundamental equations, in particular, kinematic, momentum and dynamics

equations, for developing control algorithms of a free-floating space manipulator were

presented in this Chapter. The unique features of a free-floating space manipulator

were discussed in detail including the nonholonomic motion constraints, generalized

Jacobian matrix and dynamics equations derived with the Lagrangian method. In

addition, the formula for minimum kinetic energy attainable by a space manipulator

without the use of momentum exchange devices was derived. In this Chapter, we

particularly focused on analyzing the coupling motion between the arm and the base,

and the reaction null-space control concept. These concepts and corresponding equa-

tions will be used again in the following Chapters to develop the adaptive reactionless

control and momentum-based parameter identification algorithms.

38

CHAPTER 3. ADAPTIVE REACTIONLESS CONTROL ALGORITHM

CHAPTER 3

Adaptive Reactionless Control Algorithm

The main contribution of this research is presented in this Chapter. A new reactionless

motion control algorithm is proposed to satisfy the principal objective of maintaining a

minimum disturbance to the base in the post-capture of an uncooperative target with

a space manipulator. Since the target becomes an integral part of the manipulator

after capture, the dynamics properties of the whole system are changed in a priori

known or unknown manner, depending on the available information on the target.

As already stated, the control method proposed in this study focuses on solving the

problem of capturing an unknown tumbling target.

To date, a number of control strategies have addressed the post-capture problem.

With the availability of system properties as well as knowledge of target’s motion, an

optimal control technique was designed in [15, 67] to coordinate control between the

base, manipulator and the target to bring the whole system to rest in minimum time,

where the input torque to control the attitude of the base is generated by reaction

wheels. In [42, 45, 46], a distributed momentum control algorithm was developed

based on the angular momentum equation to generate the arm motion so that the

initial angular momentum (assumed to be known) from the captured target is dis-

tributed to the arm and is absorbed by reaction wheels in the base. Considering the

dynamics uncertainties after capturing an unknown target, the adaptive control tech-

nique was proposed in [35, 52] to deal with the problem of end-effector motion control

39


while holding an unknown payload. However, none of these approaches resolved the

issues arising when capturing an unknown and tumbling target.

The unknown changes in the dynamics parameters of the system, as a result of

capturing an unknown target, must be accommodated as otherwise they may lead

to poor performance of the trajectory control and attitude stabilization systems. In

addition, the amount of unknown angular momentum of a tumbling target trans-

ferred to the space manipulator after capture can lead the reaction wheels to their

saturation state due to the limitations on the available output torque [47]. Since the

attitude control mechanisms cannot instantly absorb an arbitrary angular momen-

tum from the target in order to minimize the disturbance to the base, a task buffer

time is necessary for the stabilization phase. To address this issue, a new concept,

the adaptive reactionless manipulation to segue between target capture and system

stabilization phases of the capture mission, is proposed in this study. It is motivated

by the RNS control concept introduced in Chapter 2; however, the dynamics param-

eters as well as the total momentum of the system are not exactly known in advance.

This scheme is intended for use in the transition phase from the instant of capture

till the unknown parameters are identified and/or the available stabilization methods

can be applied properly. The proposed adaptive control scheme is referred to here as

Adaptive ReactionLess Control (ARLC) and its role as part of the complete capture

mission is illustrated in Fig. 3.1.

In Section 3.1, the new reactionless manipulation formula is developed based

on the momentum equations. Then, in Section 3.2, an adaptive control scheme is

implemented with the use of the recursive least squares algorithm for parameter

adaptation. The feasibility of the proposed concept is verified through simulations

conducted with two manipulator systems: a planar and a three-dimensioned model

presented in Section 3.3. In this study, the planar model is comprised of a free-floating

base equipped with 3-DOF manipulator to capture the target spinning about the axis

perpendicular to the plane; and, the model of the real ETS-VII system is employed

for the non-planar test cases.

40

3.1 FORMULATION OF ADAPTIVE REACTIONLESS MOTION

Approaching Target(Pre-capturing Phase)

Capturing Target(Impact Phase)

RNSC

Adaptive ReactionLess Control (ARLC)

Post-capturing Phase

System Stabilization Algorithm

Compound Stabilization Phase

Figure 3.1. ARLC as part of capture mission

3.1 Formulation of Adaptive Reactionless Motion

The important assumptions for the existence of reactionless manipulation in the

post-capture phase are first stated. Since the new formula for the reactionless mo-

tion is developed from the fundamental formula for the reaction null-space motion,

the necessary conditions for the existence of reaction null-space solutions for a free-

floating manipulator are assumed to be satisfied in the development. In this study,

the primary concern is the base attitude and not the base translation resulting from

the manipulator motion. As we pointed out in Section 1.1, in practice it is important

to avoid the attitude disturbances on the base, whereas the translational disturbance

does not pose significant undesirable side effects [15]. For the target capture mission,

since the present study focuses on the effect of the target angular momentum on the

system, and in particular on the attitude of the base, it can be assumed, without

loss of generality, that the initial linear momenta of both the target and the space

manipulator are zero, i.e., Pm = Pt = 0, where Pm and Pt are the linear momentum

of the space manipulator and target, respectively [45]. With respect to the existence

of reaction null-space motion, it is sufficient for the servicing system to be kinemati-

cally redundant with respect to the base task space. The latter, in light of the above

comments, contains attitude degrees of freedom only. It is also noted that the algo-

rithm developed here does not require knowledge of the motion of the target before

capture as it assumes that the capture itself has been completed successfully. The

impact which occurs as a result of the capture has no effect on the conservation of

41


momentum of the whole system, which in turn forms the basis of the adaptive con-

troller derived here. The target is assumed to be rigidized by the end-effector to the

last link of the manipulator when the capture is established.

Under changing inertial parameters of the arm, which result from target capture,

our objective is to develop a new procedure to incrementally compute a set of reac-

tionless joint rates in an online manner based on the current observed performance:

the joint motions of the arm and the disturbance on the base attitude. In the follow-

ing, the derivation of the regressor form based on the angular momentum equation

and the reaction null-space theory is presented.

The angular momentum of a space manipulator after capturing a tumbling target

with an initial angular momentum Lt can be restated based on the bottom part of

equation (2.16) as follows:

Hωω0 + Hωφφ = Lm + Lt (3.1)

where Lm is the angular momentum of the space manipulator before capture. The

matrices Hω and Hωφ in equation (3.1) include the inertia term of the target by

assuming that the target becomes rigidly attached to the end-effector. In the ideal

scenario where perfect knowledge of target properties is available, the reactionless

motion for the manipulator to produce zero attitude disturbance to the base can be

computed according to the generalization of equation (2.34) as:

φr = H+ωφL +

[E−H+

ωφHωφ

]ζ (3.2)

where L = Lm + Lt is the angular momentum of the whole system after capture and

φr denotes the required reactionless joint rates. Thus, in the ideal scenario, φr would

be computed from equation (3.2) and then commanded to the joint controller of the

manipulator, thereby producing reactionless motion.

However, in the absence of accurate knowledge of Hωφ due to the unknown prop-

erties of the target, the manipulator motion will be governed by (3.1) with nonzero

42

3.2 RECURSIVE ADAPTATION

ω0, and the general solution for the joint rates can be written as:

φ = H+ωφ (L−Hωω0) +

[E−H+

ωφHωφ

]ζ (3.3)

In the above two equations, the null-space solutions are represented by the second

term of equations (3.2) and (3.3) where ζ ∈ IRn is an arbitrary vector. Therefore, the

set of joint rates computed from the aforementioned equations is not unique in terms

of the angular velocity of the base ω0. To satisfy the control objective, it is necessary

to choose a particular solution φr from the general solution φ in order to obtain a

desired base angular velocity. Substituting for φr from (3.2) into (3.3) yields:

φ = −H+ωφHωω0 + φr (3.4)

or more succinctly,

φr − φ = Kω0 (3.5)

where

K = H+ωφHω

The regressor form in equation (3.5) is the foundation of the proposed control

scheme. The left hand side of this equation can be viewed as the deviation from

the reactionless motion or its perturbation. It is apparent from the above that if φ

closely follows φr, then ω0 converges to zero, i.e., zero attitude disturbance to the

base is produced. Equation (3.5) forms the basis of the adaptive reactionless control

algorithm (ARLC) where the regressor formulation is coupled with the well-known

recursive least squares algorithm for parameter adaptation, as presented next.

3.2 Recursive Adaptation

Equation (3.5) can be considered as an alternative to equation (3.2) to compute

the reactionless manipulator motion, if perfect knowledge of kinematics and dynamics

of the space manipulator is available. However, for the capture mission of an unknown

43


target, the transition between the approach phase and the post-impact phase will in-

volve an unpredictable change in the inertia properties, as well as the total momentum

of the base-manipulator-target system. For a safe and smooth transition, a certain

buffer in time is needed to achieve the convergence of the target’s identification pro-

cedure or to provide the attitude control system with the additional time to correct

for the initial (pre-capture) angular momentum of the target Lt while maintaining

minimum disturbance to the base. Thus, it is necessary to setup an adaptive control

scheme to compute reactionless joint rates online, in the presence of uncertainties.

The benefit of equation (3.5) is apparent when it is utilized as a regressor form for

deriving an adaptive control algorithm. In the following, the adaptive reactionless

control scheme is developed by using the certainty equivalence adaptive control ap-

proach. In the adaptive control scheme, the recursive least squares (RLS) algorithm

[68, 69] is employed to adaptively update reactionless joint rates in online manner.

In our conception of ARLC algorithm, we presumed that it is not necessary for the

manipulator to follow any specific trajectory because the main objective of ARLC is

to ensure minimum disturbance of the base attitude right after capturing a tumbling

target. Since the target capture changes the inertial properties of the arm in a priori

unknown way, we redefine the reference joint rates φr as follows:

ˆφr − φ = Kω0 (3.6)

where φ and ω0 represent the actual joint rates and base angular velocity, respectively.

Notation . is employed to denote the unknown post-capture vector and matrices,

which incorporate the properties of the unknown target.

With the current estimates . and the exact measurements of φ and ω0, it is

natural to define the prediction error e of equation (3.6) as follows:

e(t) = ˆφr(t− 1)− φ(t)− K(t− 1)ω0(t) (3.7)

where the time index is introduced in the above equation to make clear the discrete

nature of the process, in this case characterized by two consecutive time instants,

44


or time samples in a practical control system. Now, the well-known recursive least

squares parameter adaptation algorithm [70] is used to update the time-varying ma-

trix K(t) in order to minimize the total prediction error:

J =

t∫0

e2(τ)dτ (3.8)

3.2.1 Parameter adaptation of time-varying systems. In the adaptive

control literature, although the analysis of most techniques is restricted to time-

invariant systems, their real motivation is to provide a practical control scheme for

dealing with time-varying systems [68, 71]. Indeed, the control problem for a time-

varying system can be fundamentally viewed as a combination of an on-line parameter

adaptation with an on-line control system synthesis. The primary concern for the

time-varying case is that the parameter adaptation algorithm is capable of continually

tracking the time-varying parameters.

The online parameter adaptation algorithm updating the estimated parameters

at each time sample plays a key role in designing the adaptive control algorithm. As

per [69], the general structure of a parameter adaptation algorithm can be described

as follows:New estimated

parameters

←Previous estimated

parameters

+

Adaptation

gain

Measurement

vector

Prediction

errors

From the above, we can see that the role of the parameter adaptation algorithm is to

adjust the estimated parameters based on the current measurements and the previ-

ous estimates in order to minimize the prediction error. Several types of parameter

adaptation algorithms are available in the literature such as heuristic approach, gra-

dient method and least squares minimization. The recursive least squares algorithm

is implemented for the derivation of ARLC in this study.

45


The RLS algorithm has been known to provide extremely fast initial convergence.

However, this advantage has to be weighed against the requirement for continuous

tracking of the time-varying parameters, because as the adaptation gain approaches

zero, signifying convergence, the parameter adaptation algorithm effectively comes

to a halt. In addition, to track the slowly time-varying parameters, it is necessary

to employ the forgetting factor to discount old data. However, the phenomenon of

estimator windup and bursts, when the adaptation gain becomes sufficiently large

due to poor excitation, can lead to an unstable control system [71]. Hence, the

modification of RLS with covariance resetting proposed by Goodwin and Sin [68] is

implemented to attain the initial convergence while ensuring that the gain is bounded

both away from zero and from growing excessively large.

As per [68], the RLS algorithm to compute the updates for K(t) takes the fol-

lowing form, using the current (at time t) joint rates and base angular velocity, these

in turn either measured on a real system or simulated in a simulation environment:

K(t) = K(t− 1) +[

ˆφr(t− 1)− φ(t)− K(t− 1)ω0(t)]

N(t) (3.9)

where

N(t) =ω0(t)TQ(t− 1)

λ+ ω0(t)TQ(t− 1)ω0(t)(3.10)

Q(t) =1

λ

[Q(t− 1)− Q(t− 1)ω0(t)ω0(t)TQ(t− 1)

λ+ ω0(t)TQ(t− 1)ω0(t)

](3.11)

It is noted that N ∈ IR1×3 is a row vector. The initial guess for the adaptation

gain matrix Q ∈ IR3×3 can be chosen as: Q(0) = αE for α > 1. A value of forgetting

factor λ very close to unity is desired to ensure the ARLC algorithm remains stable

during the post-capture maneuvering [72]. As already noted, since the matrix K

changes with time due to its configuration dependence and unknown properties of

the captured target, the covariance resetting algorithm is implemented to avoid the

matrix Q from becoming excessively large or approaching zero. The matrix Q is

reset to its initial value whenever the trace of Q goes beyond the preset thresholds;

46


otherwise, the ordinary least squares, i.e., equation (3.11), is employed to compute

Q(t). Letting ti be the time when resetting occurs, then, Q is reset as Q(ti−1) = Q(0)

(in the computation of adaptation gain Q(ti) as per equation (3.11)).

The RLS algorithm requires measurements of the current joint rates φ(t) and the

corresponding base angular velocity ω0(t). The manipulator reactionless joint rates

can be initialized as per (2.34) by using the known pre-capture parameters and the

ARLC algorithm is triggered at the instant of capture. Once K(t) update has been

calculated, the desired reactionless motion of the manipulator which serves as the

reference input to the manipulator joint control law is computed from equation (3.6)

as follows:ˆφr(t) = φ(t) + K(t)ω0(t) (3.12)

In this manner, the reactionless joint rates and the parameter matrix K are adaptively

updated to guarantee minimum disturbance of the base (provided the RLS algorithm

is stable), without knowledge of the inertial properties of the tumbling target. The

block diagram of the algorithm is included in Fig. 3.2 and the main steps are sum-

marized below, assuming a sampling rate of ∆t. Note that the parameter adaptation

algorithm is to be executed continually during the maneuver and this is known as the

non-vanishing adaptation [69].

Step1: Initialization (based on the known pre-capture parameters of the ma-

nipulator)

φr(0) = H+ωφLm +

[E−H+

ωφHωφ

]ζ

K(0) = H+ωφHω

t = ∆t

Step2: Measure (or simulate) φ(t) and ω0(t).

Step3: Compute K(t) from (3.9) with (3.10) and (3.11).

Step4: Using K(t), update the desired joint rates from (3.12).

Step5: t← t+ ∆t; Return to step 2 until finish.

47


Parameter Adaptation Algorithm

Space ManipulatorPlant(t)(t)(t)(t) 0r

ˆˆ ωK

tK

];[ 0 ωPD Velocity-based Controller

Inner Loop

Outer Loop

Figure 3.2. Adaptive reactionless control scheme

Two observations need to be made regarding the proposed adaptive control

scheme as follows:

(i) It is emphasized that the ultimate objective of the proposed control scheme

is to adaptively generate the desired reactionless motion for the manipulator

after capturing a tumbling target with unknown inertial properties. The

reactionless motion is computed from the estimates of the plant parameters

“as if these were the true plant parameters” [60]. This idea is well known

in the adaptive control literature as the certainty equivalence principle.

(ii) Since ARLC scheme is based solely on the observed performance of the base-

manipulator system and the total prediction-error is minimized by using the

RLS approach, the control objective of maintaining the reactionless motion,

i.e., ω0 ≈ 0 is still satisfied even though there is a mismatch between the

estimated matrix K and its true value. As illustrated in Fig. 3.2, the

proposed control scheme can be viewed as comprised of two loops: the inner

loop with the PD joint velocity-based controller and the outer loop with an

on-line parameter adaptation algorithm. The incremental perturbation of

the joint rates y = ˆφr − φ is by design driven to zero with the inner loop

controller, since the reference input to the latter is ˆφr, and the updated

reactionless motion is a solution of the on-line adaptation algorithm (RLS).

3.2.2 Stability Analysis. Following our previous discussion of the approach

to deal with a time-varying system, the stability of the proposed scheme is analyzed

48


under the assumption that the control algorithm is implemented at a sufficiently

high sampling rate, in order to ensure a relatively small change in K between two

consecutive samples. As discussed in [73], the recursive prediction-error algorithm

presented in the previous subsection can be analyzed by considering the individual

components of prediction error e, in particular, the k-th component for estimating

the k-th row of matrix K. Thus, from (3.7), we have:

ek(t) = ˆφkr(t− 1)− φk(t)− ωT0 (t)KT

k (t− 1) (3.13)

where Kk is k-th row of matrix K; k = 1, ..., n. The RLS parameter adaptation

algorithm for the k-th row Kk of matrix K has the same adaptation gain matrix

Q and regression vector ω0 as the algorithm presented in equations (3.9)-(3.11) and

therefore,

KTk (t) = KT

k (t− 1) + N(t)ek(t) (3.14)

where

N(t) =Q(t− 1)ω0(t)

λ+ ωT0 (t)Q(t− 1)ω0(t)(3.15)

Q(t) =1

λ

[Q(t− 1)− Q(t− 1)ω0(t)ωT0 (t)Q(t− 1)

λ+ ωT0 (t)Q(t− 1)ω0(t)

](3.16)

The approach used by Goodwin and Sin [68] is adopted here for proving stability

and convergence of parameter adaptation algorithm presented in (3.14)-(3.16). First,

a scalar Lyapunov function is introduced as follows:

Vk(t) = Kk(t)Q−1(t)KT

k (t) (3.17)

where Kk(t) = Kk(t) −Kkt and Kk(t − 1) ≈ Kk(t) = Kkt (based on the aforemen-

tioned assumption of small changes in K between two consecutive samples). Sub-

tracting KTkt from both sides of equation (3.14), yields:

KTk (t) = KT

k (t− 1) + N(t)[

ˆφkr(t− 1)− φk(t)− ωT0 (t)KTk (t− 1)

](3.18)

49


From the discrete form of equation (3.6), we have:

ˆφkr(t− 1)− φk(t) = ωT0 (t)KT

kt (3.19)

and substituting (3.19) into (3.18) with the use of (3.15), we obtain:

KTk (t) = KT

k (t− 1)− Q(t− 1)ω0(t)ωT0 (t)KTk (t− 1)

λ+ ωT0 (t)Q(t− 1)ω0(t)(3.20)

Further manipulating equation (3.20), it follows that:

KTk (t) = λ

1

λ

[Q(t− 1)− Q(t− 1)ω0(t)ωT0 (t)Q(t− 1)

λ+ ωT0 (t)Q(t− 1)ω0(t)

]Q−1(t− 1)KT

k (t− 1) (3.21)

or more simply with the use of equation (3.16),

KTk (t) = λQ(t)Q−1(t− 1)KT

k (t− 1) (3.22)

Considering the relation:

Vk(t)− λVk(t− 1) =[Kk(t)− Kk(t− 1)

]Q−1(t)KT

k (t) (3.23)

= −Kk(t− 1)ω0(t)ωT0 (t)Q(t− 1)

λ+ ωT0 (t)Q(t− 1)ω0(t)Q−1(t)KT

k (t) (3.24)

where we made use of equations (3.17) and (3.22), and then using the hypothesis

0 < λ ≤ 1, (3.24) can be rewritten as:

Vk(t)− Vk(t− 1) = −λKk(t− 1)ω0(t)ωT0 (t)KTk (t− 1)

λ+ ωT0 (t)Q(t− 1)ω0(t)− (1− λ)Vk(t− 1) ≤ 0 (3.25)

Inspecting (3.25) while recalling that the adaptation gain matrix Q is symmetric

and positive definite, we observe the Vk(t) must be a nonincreasing positive scalar

function, thus completing the stability proof.

50

3.3 SIMULATION STUDY

3.3 Simulation Study

In this section, the proposed approach is verified by using two servicing spacecraft-

manipulator systems: one with a three-degrees-of-freedom (3-DOF) planar manipu-

lator as illustrated in Fig. 3.3 and a second one with a 7-DOF three-dimensional ma-

nipulator model based on the real ETS-VII testing satellite. The dynamics models of

the servicer and the target, in free-floating conditions, are created in MSC.Adams and

co-simulated with Matlab/Simulink in interactive mode (at the communication inter-

val of 0.0001 sec). The detailed description of the MSC.Adams and Matlab/Simulink

co-simulation platform can be found in Section 2.7. The relevant parameters of the

base, the links and the three targets employed in this study (large, medium, small)

are summarized in Table 3.1 for the 2D model and Tables 3.2 and 3.3 for the 3D

model. Note that the target size characterizations for the 2D test case were defined

according to their mass ratios relative to the base.

The desired motion generated by ARLC is produced by driving the joints with

the low-level velocity-based controller where the joint torques are computed as per the

PD control law with constant gains for the whole motion. In particular, the ARLC

control scheme provides joint rate input commands to the manipulator presumed to

be equipped with a velocity servo controller implemented in hardware. In manipulator

control practice, it is acknowledged that the servo follows the velocity-based command

more accurately than, for example, the torque-based command due to the high-gain

hardware-velocity feedback [74]. The velocity-based control is a standard practice

both in space manipulators [11, 28, 35] and ground-based manipulator control.

3.3.1 Planar Test Case. Verification of the proposed algorithm is first pre-

sented by using the numerical simulation platform for the planar model test case. In

the following, the planar model is tested for two basic scenarios: Lt = 0 and Lt 6= 0

before capture (recall Lt is the angular momentum of the target). The first scenario

demonstrates the adaptation of ARLC in producing the reaction null-space motion

51


sradt /2.0BASE

Link1

Target

Link2

Link3o60

Y

X

Figure 3.3. Planar base-3-DOF manipulator-target model

Table 3.1. Planar 3-DOF manipulator simulation model parameters

Mass (kg) Izz (kgm2) a Dimension

Base 500 83.61 1m×1m

Link1 10 1.05 1m

Link2 10 1.05 1m

Link3 10 1.05 1m

Target 1 500 20.83 0.5m×0.5m

Target 2 250 10.41 0.5m×0.5m

Target 3 125 5.20 0.5m×0.5m

when the inertia parameters of the space manipulator are modified as a result of tar-

get capture. For the case Lt 6= 0, the simulation of the post-capture of an unknown

spinning target is realistically conducted with the incorporation of impact between

the target and manipulator. The simulation study aims to verify the capability of

the proposed scheme to minimize the disturbance to the base. The initial adapta-

tion gain matrix and forgetting factor for the RLS algorithm were chosen as Q(0) =

diag[1000, 1000, 1000] and λ = 0.99, respectively. As already noted, the reaction-

less joint rate command serves as a reference input to the closed-loop PD controller

to generate the input torques in each joint of the manipulator. For all 2D model

simulations, the actual capture/impact occurs at t ≈ 4 sec.

aThe moment of inertia values are computed by MSC.Adams based on geometry and density.

52


Table 3.2. 3D, 7-DOF manipulator simulation model parameters

Mass (kg) Ixx (kgm2) Iyy Izz

Base 1000 1200 1200 1200

Link1 35.01 1.218 0.5132 1.331

Link2 30 2.10 1.378 2.359

Link3 22.69 0.102 3.378 3.359

Link4 21.38 0.4327 2.266 1.911

Link5 16.75 0.3878 0.3963 0.07271

Link6 26.17 0.5727 0.5987 0.1288

Link7 + Target 500 50 50 50

Table 3.3. Denavit-Hartenberg parameters of 7-DOF manipulator

Link ai (m) αi (deg) bi φi (deg)

1 0 90 0.35 φ1

2 0 90 0.2 φ2 + 90

3 0.87 0 0.275 φ3

4 0.63 0 -0.275 φ4

5 0 90 -0.36 φ5 − 90

6 0.55 90 0.16 φ6 − 90

7 0.532 0 0.2 φ7

The accuracy of the simulation platform was first ascertained by examining the

conservation of system total momentum during the RNS motion without the target,

where the base is initially at rest and the manipulator was commanded with joint

motions as per (2.34) with ζ = [0.1 0.1 0.1]T for the duration of 200 sec. The reac-

tionless joint motions, the response of the base and the system linear and angular

momenta are illustrated in Fig. 3.4. It is apparent that the total momentum of the

system is conserved (to within 10−7 kgm/s for components of P and 10−4 kgm2/s for

Lz) and minimum base disturbance is produced by RNS motion. These results verify

the accuracy of the simulation platform and its suitability for testing the proposed

control concept.

53


0 50 100 150 200−0.05

0

0.05

0.1

0.15

Time [s]

Join

t rat

es [r

ad/s

]

Joint 1Joint 2Joint 3

50 100 150 200−2

0

2x 10

−4

[rad

/s]

Angular velocity of the base

50 100 150 200−5

0

5x 10

−4

[kgm

2 /s]

Total angular momentum of system

50 100 150 200−1

0

1x 10

−7

Time [s]

[kgm

/s]

Total linear momentum of system

Lx, L

y Lz

PzP

x

Py

Figure 3.4. Reaction null-space system response for model verification

54


Planar model test case with Lt = 0

After the target is captured and rigidized by the gripper relative to the end-

effector, the inertia parameters of the last link of the manipulator are changed to

the new constant values. For example, for the numerical values of the medium target

considered here, the mass of the last link is modified from 10 kg to 260 kg. To demon-

strate the effect of parameter uncertainty on the base attitude, the response of the

system is simulated by commanding RNS motion computed with the before-capture

knowledge of system parameters, again with ζ = [0.1 0.1 0.1]T . The corresponding

results are shown in Figs. 3.5 and 3.6 from which one can observe that the RNS

motion produces a significant rate disturbance (as high as 0.07 rad/s) on the base

when no adaptation to the change in the inertia of the space manipulator is imple-

mented. Simulation results of the adaptive reactionless algorithm are displayed in

Figs. 3.7, 3.8 and 3.9. In this simulation, the arm is initialized at the instant of

capture with rates φRL computed from (3.2) and the ARLC adaptively updates the

reactionless motion. It is apparent that negligible base rotational motion is achieved

with the ARLC algorithm. The small “blips” at ≈ 43 sec and ≈ 135 sec occur as a

result of the degradation in tracking of the desired ARLC joint rates with the low-

level velocity-based PD controller that occurs for the higher joint rates near these

time instants. The angular momentum profile associated with the target object, the

manipulator and the whole system while undergoing the ARLC motion are also pre-

sented in Fig. 3.9. These results illustrate the momentum exchange between the

target and manipulator resulting from the ARLC motion.

Planar model test case with Lt 6= 0

In this scenario, the implementation of the ARLC when the space manipulator

grasps a spinning unknown target of small, medium and large mass, with respect

to the mass of the base, are conducted. It is assumed that prior to target capture,

the space manipulator has negligible linear/angular momentum and the target is

spinning about the axis perpendicular to the plane of the system with a constant

55


0 20 40 60 80 100−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time [s]

Join

t rat

es [r

ad/s

]


Figure 3.5. Joint rates, Lt = 0, without ARLC

0 20 40 60 80 100−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

Figure 3.6. Base angular rate, Lt = 0, without ARLC

56


0 50 100 150 200−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Join

t rat

es [r

ad/s

]


Figure 3.7. Joint rates, Lt = 0, with ARLC

0 50 100 150 200−1

0

1

2

3

4

5

6

7x 10

−3

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

Figure 3.8. Base angular velocity, Lt = 0, with ARLC

57


0 50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

The

rel

atio

nshi

p be

twee

n L

m a

nd L

t [kgm

2 /s]

Lt

Lm

Lm

+Lt

Figure 3.9. The relationship between Lm and Lt, Lt = 0, with ARLC

angular velocity ωt = 0.2 rad/s, as shown in Fig. 3.3. The target is grasped firmly

by the gripper to the end-effector at the grasping point after the capture.

1) Medium target, mt = 250 kg

The principle objective of the proposed control scheme is confirmed through the

simulation results of ARLC joint rates and the response of the base (angular velocity

and attitude) shown in Figs. 3.10 and 3.11, respectively. The impact between the

manipulator and the target is reflected in the abrupt changes in the joint rate and base

angular velocity profiles at t = 4 sec. However, these perturbations are very quickly

damped out with the ARLC controller. From these results, it is observed that the

ARLC algorithm is able to maintain the base attitude nearly unchanged with the

online adaptation of the joint rates, while the manipulator holds the unknown target.

The reason for the higher values in the base angular velocity at around 30 sec and 85

sec is again the degradation in the joint tracking of the PD controller that occurs for

the higher joint rates. Indeed, the error profiles of the low-level velocity-based PD

controller shown in Fig. 3.12 match the base angular velocity profile in Fig. 3.11.

58


0 50 100 150 200−0.2

−0.1

0

0.1

0.2

0.3

Time [s]

Join

t rat

es [r

ad/s

]


Figure 3.10. Joint rates, Lt 6= 0, mt = 250 kg

This suggests that with a better tracking controller, the base angular motion could

be reduced even further.

In Fig. 3.13, the angular momentum distribution between the target and the arm

is illustrated as well as conservation of the total angular momentum of the system,

Lm + Lt at the value equal to the target’s pre-capture angular momentum. The

joint acceleration profiles of ARLC motion are also presented in Fig. 3.14 showing

that they are quite low and feasible. To visualize the motion of the arm during the

adaptation, five snapshots of the system are shown in Fig. 3.15, where the fixed frame

is also drawn to illustrate the uncompensated translational motion of the base.

2) Large and small targets, mt = 500 kg and mt = 125 kg

The capability of ARLC in dealing with a wide range of target masses is demon-

strated through two sets of simulations in capturing a large (500 kg) and a small (125

kg) target, relative to the base. The targets’ properties are shown in Table 3.1 and in

both cases, the targets are spinning at angular velocity ωt = 0.2 rad/s before capture,

59


0 50 100 150 200−8

−6

−4

−2

0

2

4

6x 10

−5

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

0 50 100 150 200−2

−1.5

−1

−0.5

0

0.5

1x 10

−3

Time [s]

Bas

e at

titud

e [r

ad]

Figure 3.11. Base response: angular velocity and attitude, Lt 6= 0, mt = 250 kg.

0 50 100 150 200−1

−0.5

0

0.5

1 x 10−3

Time [s]

Join

t rat

es c

ontro

l err

or [r

ad/s

]


Cause of spikes in the base angular velocity

Figure 3.12. ARLC joint rates error, Lt 6= 0, mt = 250 kg.

60


0 50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time [s]

The

rel

atio

nshi

p be

twee

n L

m a

nd L

t [kgm

2 /s]

Lt

Lm

Lm

+Lt

Figure 3.13. The relationship between Lm and Lt, Lt 6= 0, mt = 250 kg

0 50 100 150 200−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time [s]

Join

t acc

eler

atio

ns [r

ad/s2 ]


Figure 3.14. Joint accelerations, Lt 6= 0, mt = 250 kg

61


X

Y

X

Y

X

Y

X

Y

Y

X

Figure 3.15. Five configurations (snapshots) of the arm after capturing aspinning target, Lt 6= 0, mt = 250 kg

62


0 50 100 150 200−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time [s]

Join

t rat

es [r

ad/s

]


(A)

0 50 100 150 200−1.5

−1

−0.5

0

0.5

1x 10

−3

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

(B)

Figure 3.16. ARLC with Lt 6= 0, mt = 500 kg

0 50 100 150 200−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Join

t rat

es [r

ad/s

]


(A)

0 50 100 150 200−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1x 10

−4

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

(B)

Figure 3.17. ARLC with Lt 6= 0, mt = 125 kg

thus, resulting in Ltz of 4.16 and 1.04 kgm2/s for the large and small target, respec-

tively. With the similar simulation setup as the previous test case, the reactionless

joint rates and base responses are presented in Fig. 3.16 for 500 kg target and Fig.

3.17 for 125 kg target. The above angular rate disturbances remain on the scale of

10−3 rad/s and 10−4 rad/s for the 500 kg and 125 kg targets, respectively. These

results confirm the capability of ARLC scheme to perform well for a wide range of

target masses.

63


0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1x 10

−3

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

ωx

ωy

ωz

Figure 3.18. Base angular velocity, mt = 500 kg, 3D case.

3.3.2 3D Test Case. The implementation of the ARLC algorithm for a

general 3D model is presented in this section. The proposed control algorithm is tested

by simulating the capture of a tumbling target with the 7-DOF manipulator based on

the model of ETS-VII system. For the purposes of this simulation, it is assumed that

the target is attached rigidly to the last link immediately after capture, that is, the

impact is not modeled. As a result of capture, the mass of the last link is modified

from 18.07 kg to 500 kg and the corresponding properties are changed accordingly, as

per information in Table 3.2. To emulate the tumbling motion, an external impulsive

torque of 15Nm is applied to the end-effector about each of the three axes for the

duration of 0.2 seconds. This results in the initial angular disturbance to the base

about the three axes, as shown in Fig. 3.18. The reactionless joint rates are shown in

Figs. 3.19 and 3.20 for all seven joints. As can be seen from Fig. 3.18, the impulsive

angular momentum causes a spike in the base response at ≈ 1 sec, after which time

the ARLC maintains the minimum disturbance to the base (on the order of 10−4

rad/s) by generating the reactionless motion of the ETS-VII arm.

64


0 20 40 60 80 100−0.06

−0.04

−0.02

0

0.02

0.04

Time [s]

Join

t rat

es [r

ad/s

]


Figure 3.19. Joint rates φ1, φ2, φ3, mt = 500 kg, 3D case.

0 20 40 60 80 100

−0.1

−0.05

0

0.05

0.1

Time [s]

Join

t rat

es [r

ad/s

]

Joint 4Joint 5Joint 6Joint 7

Figure 3.20. Joint rates, φ4 – φ7, mt = 500 kg, 3D case.

65


3.4 Summary

A new formulation as well as a control scheme to adaptively produce reactionless

manipulation for a space manipulator during and after capture of an unknown tum-

bling target are presented in this chapter. Similarly to the reaction null-space control,

the proposed adaptive reactionless control is derived from the conservation of angu-

lar momentum for the system and the adaptation of the desired manipulator motion

is computed by using the recursive least squares procedure and the use of certainty

equivalence concept. The feasibility of the approach was verified with the numerical

simulations of a base-3-DOF manipulator-target planar model and the ETS-VII 3D

model, both co-simulated in MSC.Adams and Matlab/Simulink. It was shown that

the adaptive reactionless control algorithm is effective in generating the required joint

rates which produce minimum reaction to the base after capturing an unknown tar-

get. Moreover, although the parameter matrix K does not converge to its true value

with the correct dynamics parameters of the system, the algorithm still produces the

reactionless motion of the arm. The benefits of ARLC in the post-capture mission

will be demonstrated in the next chapter with the identification of the target’s inertial

properties while executing ARLC motion.

66

CHAPTER 4. ADAPTIVE REACTIONLESS CONTROL FOR PARAMETER IDENTIFICATION

CHAPTER 4

Adaptive Reactionless Control for

Parameter Identification

One of the critical issues for control of space manipulator after capturing a target

satellite or space debris is the uncertainty, or in the extreme case, complete lack

of information on the inertial properties of the target. In the case of satellite cap-

ture, the inertial parameters may be available from the design specifications; however,

even then, they can be different in orbit as a result of uncertain fuel consumption,

mechanical malfunctions, or damage accrued from years of operation in orbit. In

this Chapter, the adaptive reactionless control scheme presented in Chapter 3 for the

post-capture of an unknown target is integrated with the target’s parameters iden-

tification procedure. In literature, the on-orbit stabilization algorithms addressing

the post-capture issues require accurate models of kinematics and dynamics of the

space-manipulator-target system. Moreover, the available control methods for space

manipulators which have been successfully demonstrated on systems in orbit, such

as resolved motion rate control using generalized Jacobian matrix [28] or reactionless

manipulation control [3], also require precise knowledge of inertial parameters of the

system. Indeed, the formulations of reaction null-space motion and generalized Ja-

cobian matrix for space manipulator require not only geometric parameters but also

inertial parameters. Hence, the unknown inertial parameters which are the mass,

67


center of mass and moment of inertia of the payload need to be identified accurately

to achieve such controllers after the manipulator grasps an unknown object.

The problem of dynamics parameter identification for a ground-based manipula-

tor holding a payload were addressed in the literature by using estimation procedures

based on the Newton-Euler formulation of the system dynamics equations [75]. How-

ever, the estimation procedure based on the relationship between acceleration and

torque/force in each joint is difficult to implement for the space manipulators due to

the difficulty and accuracy of joint acceleration and torque measurements. Hence, the

parameter identification procedure derived from the momentum conservation equa-

tions is preferable for space manipulator systems since this method requires only

velocity data measurements. It is known as the momentum-based estimation method

and is unique to space manipulator systems. In Chapter 3, the ARLC scheme is

proposed as a means to transition between post-capture and compound stabilization

phases, thus providing a buffer in time during which the base is undisturbed. Now,

this capability of ARLC is exploited to implement the momentum-based parameter

identification algorithm [56, 58] while the reactionless motion of the arm is generated

by ARLC. The principal differences of the proposed procedure from the previous

related studies can be stated as follows:

(i) Two objectives are satisfied conjointly: minimization of disturbance to the

base and estimation of inertial parameters after capturing an unknown tum-

bling target (without using the attitude control devices).

(ii) The parameter identification task is performed online while the arm is

executing ARLC motion.

(iii) The regression form of the parameter identification problem is formulated

for the case of the tumbling target with an unknown initial angular mo-

mentum.

The implementation of ARLC and momentum-based parameter identification in the

capture mission is illustrated in Fig. 4.1.

68

4.1 MOMENTUM-BASED PARAMETER IDENTIFICATION

Approaching Target(Pre-capturing Phase)

Capturing Target(Impact Phase)

RNSC

Adaptive ReactionLess Control (ARLC)

Post-capturing Phase

System Stabilization Algorithm

Compound Stabilization PhaseJoint Rates and Base

Velocity Measurements

Momentum-BasedParameter Identification

Figure 4.1. Incorporation of ARLC and momentum-based parameter iden-tification into the capture mission

4.1 Momentum-based Parameter Identification

The momentum-based parameter identification for a space manipulator is formu-

lated in this Section. In the following derivation, it is assumed that shortly after the

manipulator grasps the target, the target is rigidized relative to the end-effector, and

hence the inertial parameters of the last link of the arm are changed after capture.

Under this assumption, the unknown parameters to be identified are: mn, nan, and

nIn, which denote the mass, the position of center of mass w.r.t. Σn frame and the

inertia tensor of the last link, respectively. The kinematic and dynamics parameters

of the manipulator, except for the last link with the unknown target, are assumed to

be known a priori and it is also assumed that the joint angles and rates are available

from standard joint encoder measurements. The momentum-based identification to

estimate the inertial parameters of the last link of space manipulator after target

capture, without the knowledge of the target’s initial angular momentum, is now

derived.

The momentum-based method is developed based on the momentum conservation

equations. To identify the inertial properties of the last link, the conservation of

momentum equations can be decomposed into the measured terms, involving joint

69


rates and base velocity, and the estimated quantities, which depend on the inertial

parameters of the last link. Then, the momentum equations are cast into a regressor

form in order to apply the least-squares technique to estimate the unknown quantities.

From their definitions, the linear and angular momentum equations, (2.7) and

(2.8), are thus rewritten by separating the terms with the known and measured quan-

tities from those involving the unknown parameters of the last link. For clarity, the

latter are surrounded by square brackets as follows:

P =n−1∑i=0

miri + [mnrn] (4.1)

L =n−1∑i=0

(Iiωi + ri ×miri) + [Inωn + rn ×mnrn] (4.2)

Realizing that the position of the center of mass of the last link, the vector nan, is

embedded in rn, rn is decomposed as:

rn = pn + IRnnan (4.3)

where IRn is the rotation matrix from frame Σn to ΣI . Furthermore, rn is itself

embedded in the derivative rn which can be expanded by using forward kinematics

of the system as:

rn = v0 + ω0 × (rn − r0) +n∑j=1

(kj × (rn − pj)) φj (4.4)

Introducing u =∑n−1

i=0 miri for compactness of presentation to denote the known

quantities in Eq. (4.1) and with P = 0, (4.1) becomes:

mnrn = −u (4.5)

70

4.1 MOMENTUM-BASED PARAMETER IDENTIFICATION

Substituting for rn from (4.4) with rn replaced as per (4.3) and separating the terms

involving the parameters to be identified on the right-hand-side gives:

−

(v0 + ω0 × (pn − r0) +

n∑j=1

(kj × (pn − pj)) φj

)

= u

[1

mn

]+ (ω0

IRn +n∑j=1

φjkjIRn) [nan]

(4.6)

where again the square brackets were placed around the terms involving the param-

eters to be identified. Note that ω0 and kj notation is used in Eq. (4.6) to allow

a matrix product computation where the corresponding terms multiply a rotation

matrix resulting from substitution of Eq. (4.3).

Proceeding similarly for the angular momentum equation, after replacing rn and

rn in the last term of (4.2), we obtain:

L =n−1∑i=0

(Iiωi + ri ×miri) + u× pn +[Inωn + uIRn

nan]

(4.7)

and again introducing the known quantities q as:

q =n−1∑i=0

(Iiωi + ri ×miri) + u× pn (4.8)

Separating out the known terms, the angular momentum equation (4.2) is rewritten

as:

L− q = uIRnnan + IRn

nInnωn (4.9)

However, the angular momentum of the whole system after the target capture is

equal to L and is unknown. Since the momentum is conserved for the free-floating

space manipulator maneuvering with ARLC motion, the angular momentum L can

be eliminated by replacing (4.9) with its incremental form. Thus, the increment of

the total angular momentum L(t), as the system evolves from configuration at time

tk to another configuration at the instant tk+1, is zero. This results in:

−∆q = ∆(uIRn)nan + ∆Ωnin (4.10)

71


where the symbol ∆(·) now denotes the increment of the (·) between time tk and

tk+1. The matrix Ωn and the vector in are defined as follows:

Ωn = IRn

nωnx

nωnynωnz 0 0 0

0 nωnx 0 nωnynωnz 0

0 0 nωnx 0 nωnynωnz

in = [I11, I12, I13, I22, I23, I33]T

The left-hand-side of (4.10) contains known quantities only, while the right-hand-

side contains the unknown parameters nan and in, the latter stringing the moments

and products of inertia in a column vector. Combining equations (4.6) and (4.10)

into matrix form and factoring out the vector of unknown parameters, the standard

linear equation for the estimation problem is obtained as follows:

−

v0 + ω0 × (pn − r0) +∑n

j=1 (kj × (pn − pj)) φj

∆q

=

u ω0IRn +

∑nj=1 kjφj

IRn 0

0 ∆(uIRn) ∆Ωn

1/mn

nan

in

(4.11)

Equation (4.11) represents an underdetermined system of equations for the ten

inertial parameters of the last link after capture; it, therefore, requires more than one

set of measurements for a solution. It can be solved with any of the existing least

squares techniques for linear estimation problems and in this study, solution of (4.11)

is obtained recursively by using the conventional QR decomposition recursive least

squares algorithm. The estimation is performed online with the set of measurements

generated during the ARLC motions.

4.2 QR Decomposition Recursive Least Squares

The QR decomposition recursive least squares (QRD-RLS) is a numerically reli-

able method which uses orthogonal transformation of the coefficient matrix for solving

72

4.2 QR DECOMPOSITION RECURSIVE LEAST SQUARES

the linear least squares problem [76]. It has been found to produce more accurate

and smoother results compared to the standard RLS procedure [77]. In this Section,

the implementation of QRD-RLS method to recursively estimate the unknown quan-

tities in equation (4.11) is presented. The momentum-based parameter identification

problem is formulated as the least squares solution to a system of linear equations as

follows:

y = Ax (4.12)

where

y = −

v0 + ω0 × (pn − r0) +∑n

j=1 (kj × (pn − pj)) φj

∆q

∈ IR6×1

A =

u ω0IRn +

∑nj=1 kjφj

IRn 0

0 ∆(uIRn) ∆Ωn

∈ IR6×10; and x =

1/mn

nan

in

∈ IR10×1

Assuming that the measurements are available at each time step, up to and

including time step k − 1 we have the measurement equations of the form:

y(k−1) = A(k−1)x (4.13)

where

y(k−1) =

y1

...

yk−1

, A(k−1) =

A1

...

Ak−1

In the above, yi and Ai are the known vector/matrix quantities obtained at time step

i. It is noted that the vector/matrix ∆(·) in y and A can be computed as follows:

∆(·)i = (·)i − (·)l (4.14)

where (·)l represents the known quantities measured at time step l < i and (·)l can

represent a fixed value at a certain configuration during the identification procedure.

The numerically reliable approach using orthogonal transformations is implemented

73


to obtain the least-squares estimate xk−1 of x. Supposing at time step k− 1, we have

obtained the QR decomposition of A(k−1) as:

A(k−1) = Qk−1Rk−1 (4.15)

where Qk−1 is orthogonal and Rk−1 is nonsingular upper triangular. From (4.13) and

(4.15), the estimate xk−1 of x can then be obtained by solving the upper triangular

equation:

Rk−1xk−1 = QTk−1y

(k−1) (4.16)

The benefits of employing the QR decomposition are two-fold: it allows for a numer-

ically reliable solution for the parameter estimates as per equation (4.16), and it also

provides an efficient algorithm to recursively compute the QR decomposition at the

next time step without repeating the whole procedure from the beginning. At time

step k, the manipulator evolved to a new configuration with the new measurement

equation, and hence the total measurement equation is written as:

y(k) = A(k)x (4.17)

where

y(k) =

y(k−1)

yk

, A(k) =

A(k−1)

Ak

At this point, we can obtain xk by integrating the latest measurements in the estimate

but without repeating the full algorithm. This is possible by computing the QR

decomposition of A(k) based on the already obtained QR decomposition of A(k−1).

In particular, the QR decomposition of A(k) is computed by appending the new

measurements Ak to the upper triangular Rk−1, which is stored from the previous

time step, and annihilating Ak by a sequence of Givens rotations. As a result, the

new QR components of A(k), Qk and Rk, are obtained as follows:

QTk

Rk−1

Ak

=

Rk

0

(4.18)

74

4.3 PERSISTENT EXCITATION OF ARLC MOTIONS

Analogously to (4.16), the least squares estimate xk of x is also obtained by solving

the upper triangular equations.

The implementation of the parameter identification algorithm for the planar

model can be simplified to efficiently obtain the estimates and avoid numerical prob-

lems. Since the translational motion takes place in the x-y plane and all rotational

motions are about the z-axis, the number of unknown inertial parameters reduces to

4 and the vectors and matrices in the linear equation (4.12) are redefined as:

y = −

[v0 + ω0 × (pn − r0) +∑n

j=1 (kj × (pn − pj)) φj

]xy

∆qz

∈ IR3×1

A =

u2×1

[ω0

IRn +∑n

j=1 kjφjIRn

]2×2

02×1

0[∆(uIRn)

]1×2

∆ωnz

∈ IR3×4; and x =

1/mn

nanxnany

I33

∈ IR4×1

The above simplified formulas are employed for the planar test cases presented in the

simulation study Section of this Chapter.

4.3 Persistent Excitation of ARLC Motions

In the previous Section, the identification problem for the inertial parameters

of the target was formulated. As is well known, convergence of the identification

algorithm to accurate parameter values can only be guaranteed if the input motion of

the arm is sufficiently or persistently exciting [78]. The issue of persistent excitation of

input trajectories for a robotic manipulator has been tackled extensively in the context

of robot dynamics identification [78–80]. In the dynamics identification of ground-

based manipulator problem, the objective is to determine the inertial parameters of

every link of the arm as well as the payload, and hence the problem addressed here

can be viewed as a special case of this more general problem. The complexity of

finding the persistently exciting trajectories for the manipulator to allow complete

75


identification of all inertial parameters of the robot links has been acknowledged

[78, 81]. This same issue for inertial parameter estimation of the captured object

with a space manipulator is discussed in [56] and [58] where the trajectories are

chosen intuitively.

A commonly used measure of persistent excitation is the condition number of the

normalized global regression matrix [79], which is formed by assembling the regression

matrices at the individual time steps over the whole maneuver. Thus, in our case, the

global regression matrix is A(k) defined in Section 4.2. The condition number measure

is employed in Section 4.4 in order to demonstrate that the trajectories generated with

the ARLC scheme are indeed persistently exciting.


In this Section, the verification of the ARLC with momentum-based parameter

identification is presented by using two servicing spacecraft, the planar and 3D mod-

els, presented in Section 3.3. Our primary objective here is identification of inertial

parameters of the target tested for the scenario of capturing an unknown tumbling

target with initial angular momentum Lt 6= 0. Recall that the objective of minimiz-

ing the disturbance to the base with ARLC was already verified with the simulation

results presented in Section 3.3. Now, the momentum-based parameter identification

method is executed in parallel with ARLC scheme after the capture is established.

4.4.1 Planar Test Case. The same simulation setup for testing the planar

base-3-DOF manipulator capturing a spinning unknown target with Lt 6= 0 is em-

ployed here for the implementation of momentum-based identification algorithm. The

target is realistically captured by the manipulator with the incorporation of impact.

After the impact, the target is assumed to be firmly held by the gripper mechanism

as illustrated in Fig. 2.4A. The momentum-based identification algorithm is derived

based on the assumption that the target is rigidized to the end-effector immediately

after capture. However, in practice, this does not happen as it takes a certain time af-

ter impact to rigidize the connection between the gripper and grapple fixture. In our

76


simulations, where the simple gripper mechanism and the grapple fixture are modeled

on the end-effector and the target respectively (see Fig. 2.4A), small sliding motion

still occurs between the end-effector and the target after the capture is established.

This will delay the convergence of the identification algorithm. With the same con-

ditions as stated in subsection 3.3.1 for the scenario Lt 6= 0, the momentum-based

identification simulation results for three different targets with small, medium and

large mass, and the inertial properties indicated in Table 3.1, are presented in the

following.

1) Medium target, mt = 250 kg

The implementation of the ARLC for inertial parameter identification when the

space manipulator grasps a spinning unknown target of intermediate weight (relative

to the base), mt = 250 kg, is presented. The motions generated by ARLC to minimize

the disturbance on the base attitude (see results in Fig. 3.10 and 3.11) provide the

measurements required for solving the parameter identification problem stated in

equation (4.11). In particular, with the ARLC trajectory, the condition number of

the normalized global regression matrix for this case is 18.5 which is very low, thus,

promising a good quality of estimation. The corresponding estimation results are

displayed in Fig. 4.2 where the dot-dashed lines indicate the real values. It is noted

that these values correspond to the properties of the last link including the target. It

can be seen that approximately 110 seconds after capture, the parameter estimates

for mn, nan, and I33, converge to their real values.

2) Large and small targets, mt = 500 kg and mt = 125 kg

To demonstrate the capability of momentum-based estimation for a range of

targets, estimation results for a large target (mt = 500 kg) and a small target (mt =

125 kg) are presented in Figs. 4.3, 4.4 and 4.5. These estimation results were obtained

with the ARLC motion shown in Figs. 3.16 and 3.17. Similarly to the capability of

ARLC in keeping the attitude of the base unchanged for different size targets, it is

apparent that the momentum-based identification scheme is also able to handle a wide

range of target masses. It is noted that convergence of the estimates is somewhat

77


0 50 100 150 200−100

−50

0

50

100

150

200

250

300

Time [s]

Est

imat

ion

of th

e m

ass

[kg]

0 50 100 150 200−1

0

1

2

nanx

0 50 100 150 200−1

−0.5

0

0.5

Time [s]Est

imat

ion

of th

e po

sitio

n of

the

mas

s ce

nter

[m]

nany

0 50 100 150 200−5

0

5

10

15

20

25

Time [s]

Est

imat

ion

of I 33

[kgm

2 ]

Figure 4.2. Estimation of medium target, mt = 250 kg

78


0 50 100 150 200−100

0

100

200

300

400

500

Time [s]

Est

imat

ion

of th

e m

ass

[kg]

Target 3

Target 1

Figure 4.3. Estimation of target mass, mt = 500 kg (Target 1) and mt =125 kg (Target 3)

faster for the larger target, which is also corroborated by the condition number of the

regression matrix: 13.36 and 20.05 for targets 1 and 3, respectively.

The simulation results presented in this section demonstrate the important role

of the proposed ARLC scheme in the post-capture mission. Indeed, the availability of

inertial parameters for the whole base-manipulator-target system facilitates the post-

capture stabilization task, since precise knowledge of space manipulator dynamics is

necessary for many model-based control algorithms, such as trajectory control [28],

optimal control [15] or distributed momentum control [45].

4.4.2 3D Test Case. In this subsection, the proposed parameter identifica-

tion scheme is verified with a general 3D model with the 7-DOF manipulator based

on the model of ETS-VII system. Under the same assumptions and simulation setup

as stated in Section 3.3.2, the momentum-based parameter identification algorithm

begins execution immediately after the capture impulse is applied to the end-effector.

It is noted that the target is attached rigidly to the last link at the same instant, that

79


0 50 100 150 200−10

−5

0

5

10

15

20

25

30

35

Time [s]

Est

imat

ion

of I 33

[kgm

2 ]

Target 1

Target 3

Figure 4.4. Estimation of I33, mt = 500 kg and mt = 125 kg

0 50 100 150 200−1

0

1

Est

imat

ion

of th

e po

sitio

n of

the

mas

s ce

nter

[m]

nanx

0 50 100 150 200−1

−0.5

0

0.5

Time [s]

nany

Target 1 (500kg)Target 3 (125kg)

Target 1

Target 1

Target 3

Target 3

Figure 4.5. Estimation of position of the center of mass, mt = 500 kg andmt = 125 kg

80


0 20 40 60 80 1000

100

200

300

400

500

600

700

Time [s]

Est

imat

ion

of th

e m

ass

[kg]

Figure 4.6. Estimation of target mass, mt = 500 kg, 3D case

is, the impact is not modeled. The target properties are incorporated in the last link

and this results in the change of the inertial parameters of the last link as indicated

in Table 3.2 (Link7 + Target). The estimation results for the mass, position of the

center of mass and diagonal components of the inertia tensor (products of inertia

of the target are set to zero for this test case) are presented in Figs. 4.6, 4.7 and

4.8, respectively. These results confirm that the objectives of the proposed control

scheme are also achieved for the non-planar manipulator model. The parameters con-

verge to their real values after ≈ 50 sec. Because the 3D simulation test case does

not model the actual capture and the associated impact between the target and the

end-effector, the rigidization of the target to the last link occurs instantaneously and

the convergence rate of the estimation procedure is faster than what is observed in

the 2D test case. In the planar simulation examples, modelling of the actual impact

initially incurs a relative motion between the target and the last link which results in

a slower convergence of parameter estimation.

81


0 20 40 60 80 1000

0.5

nanx

0 20 40 60 80 100−0.4−0.2

00.2

Est

imat

ion

of th

e po

sitio

n of

cen

ter

of m

ass

[m]

nany

0 20 40 60 80 100−0.2

0

0.2

0.4

Time [s]

nanz

Figure 4.7. Estimation of position of the center of mass, mt = 500 kg, 3D case

0 20 40 60 80 1000

204060

Ixx

0 20 40 60 80 1000

204060

Est

imat

ion

of in

ertia

tens

or [

kgm

2 ]

Iyy

0 20 40 60 80 1000

204060

Time [s]

Izz

Figure 4.8. Estimation of moment of inertia, mt = 500 kg, 3D case

82

4.5 SUMMARY

4.5 Summary

The results obtained in this Chapter show the relevance and utility of the pro-

posed adaptive reactionless control scheme to the post-capture stabilization. The

ARLC is effective in generating arm joint rates which produce minimum disturbance

to the base, thus eliminating the need for operation of the attitude control system,

while providing a certain buffer in time to allow for target’s properties estimation.

This is supported with the numerical results presented in this Chapter showing the

quality of the inertial parameter estimates obtained. With the combination of ARLC

and momentum-based parameter identification approach, two relevant problems of

post-capture control of space manipulator are addressed simultaneously. However,

the reactionless manipulation under ARLC as presented in Chapter 3 will inevitably

result in the arm reaching and exceeding practical joint limits after a certain time. In

the next Chapter, the ARLC will be reformulated in order to avoid the joint limit con-

straints of the arm by using the redundancy resolution control scheme for redundant

manipulators.

83

CHAPTER 5. ADAPTIVE REACTIONLESS CONTROL WITH JOINT-LIMIT AVOIDANCE

CHAPTER 5

Adaptive Reactionless Control with

Joint-Limit Avoidance

In this Chapter, the previously developed adaptive reactionless control algorithm is

extended to minimize the disturbance transferred to the base with consideration of

joint limit constraints. Avoiding the joint limits is necessary for a practical imple-

mentation of the proposed ARLC scheme on space manipulators, as otherwise the

space manipulator may reach physical joint limits when executing the ARLC motion

after capturing an unknown tumbling target. Moreover, if the joint limit constraints

are not explicitly taken into account and the ARLC motion violates these constraints,

a significant base disturbance may result.

In literature, local optimization methods are widely used to address the manipu-

lator control problem with more than one manipulation task [82]. These approaches

require the manipulator arm to be kinematically redundant, i.e., the manipulator has

to be designed with non-zero degree-of-redundancy. As previously mentioned, the

free-floating space manipulators are designed to be kinematically redundant for flex-

ibility and versatility. For instance, the ETS-VII and Dextre arms, the latter known

as the Special Purpose Dextrous Manipulator of the ISS servicing system, are both

redundant manipulators. In this Chapter, the redundancy of space manipulators is

85


exploited to address the joint-limit avoidance issue. In particular, redundancy res-

olution through local optimization using the gradient projection method [40, 83] is

formulated within the ARLC scheme with the primary task to maintain minimum

angular disturbance to the base, without the use of attitude control devices, and the

secondary task to avoid the physical joint limits. In the following, the ARLC is refor-

mulated in a suitable form to apply a redundancy resolution technique for joint-limit

avoidance. Then, the implementation of the gradient projection method within the

ARLC scheme is presented. To verify the validity and feasibility of the proposed al-

gorithm, the same simulation platform as presented in previous Chapters is employed

to obtain the numerical results for a planar base-manipulator-target model as well as

the three-dimensional model of ETS-VII servicing satellite.

5.1 Formulation of ARLC for Joint-Limit Avoidance

In this Section, the ARLC scheme is reformulated in order to incorporate a local

optimization technique to satisfy a secondary task. In particular, two specific tasks

are tackled with the proposed control scheme. The primary task is to update the

reactionless motion of the manipulator in an online manner, under the conditions of

changing inertial parameters of the arm resulting from target capture. The secondary

task is to ensure that the joint limit constraints are satisfied while executing the ARLC

motion.

Under the same conditions as stated in the derivation of ARLC (see Section 3.1),

the angular momentum of a space manipulator after capturing a tumbling target with

an initial angular momentum Lt can be rewritten as follows:

Hωω0 + Hωφφ = Lm + Lt = L (5.1)

The reactionless motion for the manipulator to produce zero attitude disturbance to

the base can be computed from equation (5.1) as:

φr = H+ωφL +

[E−H+

ωφHωφ

]ζ1 (5.2)

86

5.1 FORMULATION OF ARLC FOR JOINT-LIMIT AVOIDANCE

where ζ1 ∈ IRn is an arbitrary vector. However, in the presence of disturbances on the

base, the manipulator motion will be governed by (5.1) with nonzero ω0, the general

solution of which for the joint rates can be written as:

φ = H+ωφ (L−Hωω0) +

[E−H+

ωφHωφ

]ζ2 (5.3)

with ζ2 ∈ IRn an arbitrary vector. The difference between the general and the

reactionless joint rates can be obtained by subtracting (5.3) from (5.2) to yield:

φr − φ = H+ωφHωω0 +

[E−H+

ωφHωφ

]ζ (5.4)

where ζ = ζ1 − ζ2 ∈ IRn is an arbitrary vector; the matrix[E−H+

ωφHωφ

]is a

projection operator, which projects an arbitrary vector ζ ∈ IRn onto the null-space

of Hωφ. The motion generated by the second term on the right hand side of equation

(5.4) does not cause any rotational motion of the base. Since the matrix Hω is always

invertible and assuming Hωφ has a full row rank, (5.4) can be rewritten as follows:

φr − φ =[H−1ω Hωφ

]+ω0 +

E−

[H−1ω Hωφ

]+ [H−1ω Hωφ

]ζ (5.5)

or more succinctly,

φr − φ = Kω0 +[E−KK+

]ζ (5.6)

where the matrix K is the same as defined in Chapter 3, repeated here for quick

reference:

K = H+ωφHω

Equation (5.6) is now in the standard redundancy resolution form where the

secondary task can be implemented by defining ζ ∈ IRn as a gradient of a function

to be optimized [40, 84]. Hence, this equation is the foundation of the ARLC control

scheme with joint-limit avoidance. The left hand side of this equation can be viewed

as the deviation from the reactionless motion, the primary manipulation task, or its

perturbation. In this study, the conventional gradient projection method originally

87


proposed by Liegeois [83] is implemented for avoiding the joint limits, and will be

presented in the next Section.

Since the target capture changes the inertia properties of the arm in a priori

unknown way, we redefine the reference joint rates φr in equation (5.6) as follows:

ˆφr − φ = Kω0 +

[E− KK+

]ζ (5.7)

where recall φ and ω0 represent the actual joint rates and base angular velocity. As

before, the notation . is employed to denote the unknown post-capture vector and

matrices, which incorporate the properties of the target. Similarly to the procedure

presented in Chapter 3, the well-known recursive least squares parameter adaptation

algorithm [70] is used to update the time-varying matrix K(t). However, with the

view to implementing the gradient projection approach for joint-limit avoidance, the

prediction error e is now defined as:

e(t) = ˆφr(t− 1)− φ(t)− K(t− 1)ω0(t)−

[E− K(t− 1)K+(t− 1)

]ζ(t) (5.8)

As per [69], the RLS algorithm to compute the updates for K(t) takes the fol-

lowing form, using the current (measured) joint rates and base angular velocity:

K(t) = K(t− 1) +

ˆφr(t− 1)− φ(t)− K(t− 1)ω0(t)

−[E− K(t− 1)K(t− 1)+

]ζ(t)

N(t) (5.9)

where

N(t) =ω0(t)TQ(t− 1)

λ+ ω0(t)TQ(t− 1)ω0(t)(5.10)

Q(t) =1

λ

[Q(t− 1)− Q(t− 1)ω0(t)ω0(t)TQ(t− 1)

λ+ ω0(t)TQ(t− 1)ω0(t)

](5.11)

In the above, the initial guess values as well as the implementation of covariance

resetting algorithm to estimate the time-varying matrix K(t) are provided in detail

in Chapter 3 (see Section 3.2). Then, the reactionless joint rate update is adaptively

88

5.2 ARLC WITH JOINT-LIMIT AVOIDANCE

computed based on the current estimate K(t) as:

ˆφr(t) = φ(t) + K(t)ω0(t) +

[E− K(t)K(t)+

]ζ(t) (5.12)

The above signal represents the commanded/desired input to the joint control law.

It was previously noted that the development of the ARLC scheme in Chapter

3 is based on the assumption that the space manipulator system is kinematically re-

dundant with respect to the base task space (attitude degrees of freedom). However,

with the reactionless joint rates calculated from (5.12) with an arbitrary ζ, some joints

may be driven to their physical limits thus reducing the degree of redundancy in the

whole system. Therefore, the ARLC motion may fail to maintain the base attitude

unchanged, unless there is an explicit joint-limit avoidance criterion incorporated in

the control. From a practical standpoint, the implementation of the proposed adap-

tive reactionless control scheme on a space manipulator requires that the problem of

joint-limit avoidance be addressed. In literature, a motion planning algorithm for a

free-flying manipulator considering the joint limit constraints was presented in [53].

However, the optimization problem with inequality constraints was formulated based

on perfect knowledge of the robot system, unlike the situation treated here. Joint lim-

its violation is also an issue with the conventional adaptive control techniques [35, 52]

where the adaptive scheme is not formulated for utilizing the redundancy resolution

control. According to the reformulated ARLC joint reference motion equation (5.12),

the implementation of a redundancy resolution technique to minimize a desired per-

formance index is feasible [51]. In the next Section, the gradient projection method

(GPM) is employed within the ARLC scheme for utilizing the redundancy of the arm

to avoid its physical joint limits.

5.2 ARLC with Joint-Limit Avoidance

5.2.1 Redundancy resolution. In literature on kinematic control of re-

dundant manipulators, redundancy has been widely utilized to avoid joint limits,

89


singularities, and obstacles, while tracking the desired primary task such as end-

effector trajectory. There are two fundamental approaches to a redundancy utilization

problem: local optimization and global optimization [82]. Since the latter requires

knowledge of the complete trajectory information in advance and is computationally

costly, it is not well suited for tasks requiring on-line real-time implementation. On

the other hand, although the local optimization method does not guarantee global

optimality, its computation cost is inexpensive and suitable for most real-time appli-

cations [40]. In this Section, the gradient projection method [83] is utilized to satisfy

the joint limit constraints while simultaneously maintaining the minimum reaction

disturbance transmitted to the base after capturing an unknown tumbling target.

The gradient projection method has been widely used in literature because a

variety of performance criteria can be easily incorporated into the control scheme

with this approach [82]. The performance criterion for avoiding joint limits is defined

as a smooth potential function of joint angles and their limits, p(φ), which is to be

optimized. According to [40], the “joint limit” potential function can be defined as:

p(φ) =n∑i=1

1

(φ2imax − φ2

i )(5.13)

where φimax > 0 is the mechanical limit of the ith joint, i.e., the corresponding joint

limit constraint is |φi| ≤ φimax. As per equation (5.13), p(φ) is a scalar function

dependent on the manipulator configuration; it represents a penalty function on the

joint limits by returning high weights for the joints approaching their limits and going

to infinity at their bounds. Then, to satisfy the secondary task, the arbitrary vector

ζ ∈ IRn in equation (5.6) can be defined as the gradient of the joint limit function at

φ as follows:

ζ = −k ∂p∂φ

= −k

2φ1

(φ21max−φ21)2

2φ2(φ22max−φ22)2

...

2φn(φ2nmax−φ2n)2

(5.14)

90

5.2 ARLC WITH JOINT-LIMIT AVOIDANCE

where k is a user-defined positive constant. From the above, the reactionless motion

necessary to optimize the performance criterion can be obtained by projecting the

gradient of the performance criterion onto the null-space of matrix Hωφ.

5.2.2 Integration with ARLC scheme. In the context of adaptive re-

actionless control scheme where the matrix K is recursively updated based on the

observed performance, the concept of using the null-space of the approximate matrix

to minimize a performance index as mentioned in [51] is implemented in the form of

(5.7).

The proposed ARLC scheme with joint-limit avoidance requires measurements

of the current joint rates φ(t) and the corresponding base angular velocity ω0(t).

Also, the current joint angles φ(t) are needed for computing the potential function

gradient in (5.14). Since the main task of the proposed control scheme is to generate

the reactionless maneuver for the manipulator without using the momentum exchange

devices, it is not necessary for the manipulator to follow a specific desired trajectory.

Hence, the manipulator reactionless joint rates can be initialized as per (2.34) by

using the known pre-capture parameters and the ARLC algorithm is triggered at

the instant of capture. The schematic of the complete algorithm with joint-limit

avoidance is included in Fig. 5.1 and the main steps are summarized below:

Step1: Initialization (based on pre-capture parameters of the manipulator)

φr(0) = H+ωφLm +

[E−H+

ωφHωφ

]ζ

t = ∆t

Step2: Measure (or simulate) φ(t) and ω0(t).

Step3: Compute ζ from (5.14).

Step4: Compute K(t) from (5.9) with (5.10) and (5.11).

Step5: Using K(t), update the desired motion from (5.12).

Step6: t← t+ ∆t;

Return to step 2 until finish

91


Parameter Adaptation Algorithm

Space ManipulatorPlant )(ˆˆ

ˆˆ0r

t(t)(t)

(t)(t)(t)(t)

ζKKE

ωK

tK

];;[ 0 ωPD Velocity-based Controller

Inner Loop

Outer LoopT

pkt

)(ζ

Figure 5.1. Adaptive reactionless control scheme with joint-limit avoidance

Table 5.1. Joint limit constraints for 7-DOF and 3-DOF manipulators

Joint # 1 2 3 4 5 6 7φimax [deg] 3-DOF 90 140 140φimax [deg] 7-DOF 180 90 90 120 120 90 120


Before presenting the verification of the ARLC with joint-limit avoidance, we first

demonstrate the violation of joint limit constraints while executing the ARLC without

joint-limit avoidance. Similarly to the planar test case presented in Section 3.3.1,

simulations are conducted with the initial angular momentum of the target Lt 6= 0

and consideration of the impact between the capture mechanism and target. In this

scenario, the space manipulator grasps a spinning unknown target of intermediate

weight, mt = 250 kg. All simulation parameters are the same as those used in

Chapter 3 and the joint limits for the 3-DOF manipulator are stated in Table 5.1.

The value of constant k was set to 0.0005 for planar test simulations and 0.001 for

3D model test case for ARLC with joint-limit avoidance. Results for joint angle and

base time histories response for ARLC without joint limits are shown in Fig. 5.2 from

which we observe that the ARLC motion violates the joint limits while maintaining

minimum disturbance to the base.

92


0 50 100 150 200−200

−100

0

100

200

300

400

Time [s]

Join

t ang

les

[deg

]

Joint 1

Joint 2

Joint 3

Joint 1 Limit

Joint 2 and 3 Limit

0 50 100 150 200−8

−6

−4

−2

0

2

4

6x 10

−5

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

Figure 5.2. ARLC motion without joint-limit avoidance, Lt 6= 0, mt = 250 kg

5.3.1 2D Test Case.

5.3.1.1 Medium target, mt = 250 kg. The ARLC with joint-limit avoidance is

applied to the same test case scenario as described above. The primary task of the

proposed control scheme is verified through the simulated response of the base and

joint rates shown in Fig. 5.3. The impact between the manipulator and the target

is reflected in the abrupt changes in the motion profiles at t = 4 sec. Figure 5.4A

shows the joint angles of ARLC with joint-limit avoidance to demonstrate that the

secondary task of the proposed control scheme is completely satisfied. The potential

function p is presented in Fig. 5.4B. Similarly to the simulation results presented in

[83] demonstrating the use of task-priority for joint-limit avoidance, it is seen that

the potential function is forced to decrease whenever the joints approach their limits

(different joints at different times). In Fig. 5.5, the angular momentum distribution

between the target and the arm is illustrated, as well as conservation of the total

angular momentum of the system, Lm + Lt, at the value equal to the target’s pre-

capture angular momentum. From these results we observe that the ARLC algorithm

is able to maintain the base attitude nearly unchanged and avoid the joint limits

by updating the reactionless joint rates according to the base disturbance and the

arm configuration, after capturing the unknown spinning target. However, we also

observe that the enforcement of joint limit constraints does compromise somewhat

93


0 50 100 150 200−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time [s]

Join

t rat

es [r

ad/s

]


0 50 100 150 200−5

0

5x 10

−4

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

Figure 5.3. ARLC motion with joint-limit avoidance, Lt 6= 0, mt = 250 kg

the primary task. In fact, comparing the results in Figs. 3.10 and 3.11 with the

results in Fig. 5.3, it is apparent that the performance of the ARLC with joint limits

is worse as evidenced by the larger in magnitude spikes in the base response (O(10−5)

vs. O(10−4)), especially when the joints approach their limits. By comparing the

joint rates profiles, we can see that the spikes observed in the base response while

executing ARLC with joint-limit avoidance result from high variation of the joint

rates commanded.

5.3.1.2 Large and small targets, mt = 500 kg and mt = 125 kg. The proposed

control scheme is also tested with Target1 (500 kg) and Target3 (125 kg) to verify its

capability in dealing with the wide range of target masses. The joint rates, response

of the base and joint angles of the ARLC motion with joint-limit avoidance after

capturing Target1 and Target3 are shown in Figs. 5.6 and 5.7, respectively. These

results demonstrate that the two specific tasks of the proposed adaptive control are

also achieved with different target masses. Note that the same constant coefficient k

is used for computing ζ from (5.14) in the three planar test cases. Again, the com-

promise between the base disturbance and violating the joint limits can be observed

in the test cases with Target1 (comparing base angular velocity response in Fig. 5.6

vs. Fig. 3.16) and Target3 (comparing base angular velocity response in Fig. 5.7 vs.

Fig. 3.17).

94


0 50 100 150 200−150

−100

−50

0

50

100

150

Time [s]

Join

t ang

les

[deg

]

Joint 1Joint 2Joint 3Joint 2 and 3 Limit

Joint 1 Limit

(A)

0 50 100 150 2000.8

1

1.2

1.4

1.6

1.8

2

Time [s]

Pot

entia

l fun

ctio

n [r

ad−2 ]

(B)

Figure 5.4. Joint angles and potential function for 2D case, Lt 6= 0, mt = 250kg

95


0 20 40 60 80 100 120 140 160 180 200

−2

−1

0

1

2

3

4

5

Time [s]

The

rel

atio

nshi

p be

twee

n L

m a

nd L

t [kgm

2 /s]

Lt

Lm

Lt+L

m

Figure 5.5. Momentum exchange between Lt and Lm, mt = 250 kg

5.3.2 3D Test Case. In this subsection, the proposed control algorithm

is tested for the general 3D model by simulating the capture of a tumbling target

with the 7-DOF manipulator (see Table 3.2). The joint limit constraints of the

manipulator are summarized in Table 5.1.1 As in the previous simulations, we assume

that the target is attached rigidly to the last link after capture and its tumbling

motion is emulated with an external impulse torque of 15 Nm applied to the end-

effector about the three axes for the duration of 0.2 seconds. As can be seen from

Fig. 5.8, this creates an initial angular disturbance on the base. This disturbance,

however, is successfully reduced with the ARLC motion presented in Fig. 5.9. As well,

satisfaction of the joint limit constraints is illustrated in Fig. 5.10. The compromise

in satisfying two manipulation tasks is not obvious in the 3D test case: comparison

of Fig. 5.8 and Fig. 3.18 does not show significant differences in the base response.

This is because the joints do not approach their limits closely, and not at the same

time in this test case, unlike was the case for the planar tests.

1Since information on the real joint limit constraints of ETS-VII is not available in public domain,these constraints are made up for the purpose of this study.

96


0 50 100 150 200

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time [s]

Join

t rat

es [r

ad/s

]


0 50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

0 50 100 150 200−150

−100

−50

0

50

100

150

Time [s]

Join

t ang

les

[deg

]

Joint 1

Joint 2

Joint 3

Joint 1 Limit Joint 2 and 3 Limit


97


0 50 100 150 200−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time [s]

Join

t rat

es [r

ad/s

]


0 50 100 150 200−3

−2

−1

0

1

2

3

4x 10

−4

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

0 50 100 150 200−150

−100

−50

0

50

100

150

Time [s]

Join

t ang

les

[deg

]

Joint 1

Joint 2

Joint 3

Joint 1 Limit Joint 2 and 3 Limit


98

5.4 SUMMARY

0 20 40 60 80 100−8

−6

−4

−2

0

2

4

6x 10

−4

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

ωx

ωy

ωz

Figure 5.8. Base angular velocity, 3D test case, Lt 6= 0, mt = 500kg

5.4 Summary

The proposed adaptive reactionless control algorithm is reformulated in this

Chapter for implementing kinematic redundancy resolution through a local optimiza-

tion method. Specifically, the gradient projection method is employed to avoid the

physical joint limits while executing the reactionless motion in the post-capture sce-

nario. Simulation results show that two specific tasks, minimizing disturbance on

the base and avoiding joint limits, are satisfied. Although the standard gradient pro-

jection method has been implemented within the ARLC scheme to avoid the joint

limits, these results also demonstrate the possibility of utilizing a redundancy reso-

lution technique within the ARLC to satisfy a general secondary manipulation task.

This feature is significant in controlling a free-floating space manipulator especially

for the post-capture and stabilization missions.

99


0 20 40 60 80 100−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]

Join

t rat

es [r

ad/s

]


0 20 40 60 80 100−0.1

−0.05

0

0.05

0.1

Time [s]

Join

t rat

es [r

ad/s

]


Figure 5.9. Joint rates, 3D test case, mt = 500 kg

100

5.4 SUMMARY

0 20 40 60 80 100

−3

−2

−1

0

1

2

3

Time [s]

Join

t ang

les

[rad

]

Joint 1Joint 2Joint 3Joint 1 limit

Joint 2 and 3 limit

0 20 40 60 80 100

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Join

t ang

les

[rad

]


Joint 4,5 and 7 limit

Joint 6 limit

Figure 5.10. Joint angles, 3D test case, mt = 500 kg

101

CHAPTER 6. ANALYSIS AND PRELIMINARY DESIGN OF EXPERIMENTAL TEST-BED

CHAPTER 6

Analysis and Preliminary Design of

Experimental Test-bed

In this Chapter, the development of a free-floating test-bed for testing autonomous

capture algorithms is presented. Truthful emulation of free-floating conditions of

space here on Earth is one of the main challenges for space systems researchers and

engineers around the world. To date, a number of methods to simulate the zero-

gravity condition in space have been proposed [65], including the air-bearing planar

systems, suspension systems, parabolic flight and under water environment. All these

methods have associated advantages and disadvantages depending on the specific

testing purposes. As well, for testing the pre-capture phase of the capture mission,

some researchers have used two robotic arms, where one arm moves a satellite mockup

according to its orbital dynamics and the other is used to simulate the approach and

capture of the mockup satellite [85]. This test-bed works under the assumption that

the base attitude is controlled perfectly; as well, the coupling motion of the arm and

the base can not be simulated. Validation of autonomous capture algorithms for a

free-floating manipulator with consideration of disturbances on the base generated

by the arm motion and capture of a tumbling target is difficult, in particular for a

3D scenario. In addition to gravity compensation, the test-bed must be designed to

allow the arm and the base to move freely in any direction. Moreover, the movement

103


of the base, arm and the disturbance of the arm motion onto the base have to be

measurable and trackable.

A well-known approach to laboratory testing of planar motion of space robotic

systems in micro-gravity is by using the air-bearing table to support the equipment

to be tested; this approach has been used to validate hardware and control algorithms

for space manipulator systems for several decades. In the academic community, the

first air-bearing table equipped with a manipulator arm was developed at the Stanford

Aerospace Robotics Laboratory (ARL) for research on semi-autonomous construction

in space [86]. Similarly to the Stanford ARL test-bed, researchers at the Tokyo In-

stitute of Technology built the Experimental Free-Floating Robot Satellite Simulator

(EFFORTS) [87]. This facility was used to verify a number of control algorithms such

as the resolved rate motion control using GJM and payload inertial parameter identi-

fication algorithms [56]. With the same principle, the dual-arm manipulator test-bed

with significant link and joint flexibility developed at the University of Victoria was

used for many years to study dynamics and control of flexible and macro-micro ma-

nipulators [88]. The aforementioned test-beds were designed with the granite or glass

surface to support the manipulator/satellite equipment, the particular surfaces cho-

sen in order to minimize frictional effects. The base and manipulator are floated on

the surface by a cushion of air supplied at particular points on the main components

of the system.

In the following, we present the analysis and preliminary design of a free-floating

space manipulator test-bed that could be used to validate the algorithms developed

in this dissertation, as well as other methodologies relevant to on-orbit servicing. The

particular design was motivated by the existing facility, hardware, as well as technical

and scientific expertise available at the Canadian Space Agency, combined with the

knowledge base of researchers in the Aerospace Mechatronics Laboratory at McGill

University. We anticipate that the proposed test-bed will be constructed and housed

at CSA and ultimately used to evaluate control techniques for the capture mission of

a spinning target with a free-floating manipulator mounted on a chaser craft.

104

6.1 EXPERIMENTAL APPARATUS

Air Table

I/O card

Wireless Communication

Target satellite

Base

Pressure Control System

Magnetic Contact

Compressor

Antenna for wireless communication

Global Sensing System

Figure 6.1. Experimental apparatus

6.1 Experimental Apparatus

Figure 6.1 shows a schematic of the proposed experimental apparatus including

the supporting table, the base body with a 3-DOF arm, the target satellite, computer

control interface and sensing system. Since the objective of the designed test-bed is

to simulate the free-floating condition of a space system, a frictionless table surface

is required to minimize the external forces in a horizontal plane. In this design, the

air-table with pinholes through which high pressure air can flow, similarly to the air

hockey table concept, will be employed for creating a near frictionless surface to simu-

late the free-floating condition in space. In the following sections, we first present the

analysis and design of the base-3-DOF manipulator chaser system and target. Then,

the air-bearing table design is detailed based on the selected manipulator model. Fi-

nally, a discussion of the sensing system options and control interface is presented.

105


Granite Table

Free-flyingSpacecraft

Wireless Communication Attenna

Air Source Connection

25 cm

Figure 6.2. Free-flying test-bed at Canadian Space Agency

6.2 Base-3-DOF Manipulator-Target

The dimensions of the base and the arm were determined by scaling the test-

ing servicing satellite system, the ETS-VII. In that system, the mass scaling factor

between the arm and the base is:

Total mass of links

Mass of the base=

340

1000= 0.34 (6.1)

Using the above scale as a guideline, three base-manipulator models were defined as

summarized in Table 6.1, corresponding to the mass scaling factors of 0.1, 0.2 and 0.4,

approximately. The dimensioning of the facility was done by using the dimensions

of the existing free-flying satellite test-bed at CSA (see Fig. 6.2) [89] as a guideline,

while keeping the overall size of the facility reasonable. In terms of dimensions, all

three models have the same dimensions of the base and links, as shown in Fig. 6.3

and stated in Table 6.1. Model1 has the heaviest base, Model2 and Model3 have the

identical lighter base and Model3 has the heaviest links. Note that the three links of

the manipulator are assumed to be identical in dimensions and inertia, and these are

listed in Table 6.1 for a single link for each model.

Since one of the behaviors we wish to demonstrate with the experimental facility

is the disturbance created by the motion of the arm/target onto the base, the three

106

6.2 BASE-3-DOF MANIPULATOR-TARGET

Table 6.1. Base-3-DOF manipulator testing models

Mass (kg) Izz (kgm2) Dimension (cm)

Base Model1 31.25 0.32 25× 25× 25

Link Model1 1.14 1.23 10−4 25

Base Model2 15.6 0.16 25× 25× 25

Link Model2 1.14 1.23 10−4 25

Base Model3 15.6 0.16 25× 25× 25

Link Model3 2.04 1.29 10−2 25

BASELink1 TargetLink2 Link3

Y

X

25 cm

25 cm

Magnetic Contact

Figure 6.3. Top view of the base-manipulator-target system

models just described were simulated in order to determine how visible and measur-

able the reaction from the arm to the base is, based on the resulting motion of the

base. For this purpose, a joint rate PD controller is implemented and the response

of the models is simulated with the desired joint rate inputs switching between the

two setpoints, −0.1 and 0.1 rad/s, as shown in Fig. 6.4A. The base angular veloc-

ity response and four configurations (snapshots) of the base-manipulator system are

shown in Figs 6.4B and 6.5, respectively, for Model1. These results illustrate that the

disturbances generated by the arm motions onto the base are visible and measurable.

Similar conclusions were reached from simulations of Model2 and Model3. Hence, all

three models are satisfactory for demonstrating the coupling effects between the base

and the arm.

6.2.1 Base design. The base with all the necessary on-board devices includ-

ing cold air tank for thruster control, possibly a reaction wheel for attitude control,

107


0 10 20 30 40 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Join

t rat

es [r

ad/s

]


(A)

0 10 20 30 40 50

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]

Bas

e an

gula

r ve

loci

ty [r

ad/s

]

(B)

Figure 6.4. Joint rates profile and base response for Model1

Y

X

Y

X

Y

X

Y

Xt = 0st = 5s

t = 30s t = 45s

Figure 6.5. Four configurations of Model1 during simulated motion

motor driver boards, Inertial Measurement Unit (IMU) for attitude measurement,

and either the DAQ card or a motion control board is schematically illustrated in

Fig. 6.6. The base satellite is expected to be of similar size and design as the free-

flying spacecraft of the existing facility at CSA, shown in Fig. 6.2. Based on the

information collected to date, we anticipate the weight of the base to be in the range

of 10–20 kg, in line with Model2 and Model3 sizing. It is noted that installation of

108


25 cm

Motor Driver Boards

DAQ on-board card

Air tank for thruster control

Reaction Wheel

Weight: 10 - 20 kg

IMU Sensor

DAQ card is available at CSA; alternatively DAQ cards from Quanser Inc. can be considered for this application.

Model: IMU400CC-100 (Crossbow Technology, Inc.)

These would be procured with the motors purchased for the test-bed

One wheel rotating about the Z-axis (optional)

Cold-gas thrusters for base satellite attitude control (optional)

Figure 6.6. Satellite base with on-board devices

a cold air tank and a reaction wheel are not necessary for testing the control algo-

rithms put forward in this thesis for a free-floating space manipulator, but they will

be required for experiments on complete stabilization of the system.

6.2.2 Arm design. A single link of the arm, the driving motor and their inter-

connection are illustrated in Fig. 6.7. Engineers at CSA have previously constructed

a 3-DOF arm for a different project and it was decided that similar construction of

the arm would be used for this test-bed (with the same and/or similar motors and

possibly reusing the links). Based on the 3-DOF arm designed at CSA, no coupler

is required to make the motor/link interconnection. The motor stator is attached

directly to the previous (aft) link and the motor shaft is connected to the following

(fore) link. Since the joint has to be supported by the air-table, a circular pad of

8–10 cm in diameter will need to be installed at each joint to support its weight. In

addition, the magnetic contact is installed at the end-effector of the arm in order to

attach to the grapple fixture on the target.

109


Table 6.2. Selection of motor for manipulator joints

Model number RH-8D-6006-TE100AL

(Harmonic Drive, LLC)

Max. Continuous Torque 1.5 Nm

Max. Output Speed 100 rpm

Weight 0.39 kg

including tachometer (optional) and encoder

Encoder resolution 1000 P/rev

An important consideration is the maximum torques required to produce the

maneuvers that we wish to demonstrate with the facility. This information is required

for sizing and choosing the motors to be used on the arm. Based on the simulations

conducted with the three base-manipulator design models, it is apparent that the

torque requirements for the ARLC algorithm tend to be higher than for reasonably

slow, arbitrary motion of the arm, as for example the motions shown in Fig. 6.4A.

In particular, the joint rates and torque profiles obtained with ARLC incorporating

joint-limit avoidance algorithm for Model3 are presented in Fig. 6.8. The target in

this test case has the same mass and moment of inertia as the base (see Table 6.1) and

is spinning with the initial angular velocity of 0.1 rad/s. It is attached rigidly to the

end-effector of manipulator at 1 second so that impact is not modeled for simplicity.

According to these results, the peak torques observed under a range of conditions

for the 3 joints are below 0.15 Nm. Although the peak torques will be different for

the 3 joints, it would be more practical to use the same motors and links for the

designed arm. For this test-bed, we suggest that the joints are driven by harmonic

drive actuators from Harmonic Drive LLC, with motor specifications stated in Table

6.2. With the combination of precision harmonic drive gear and DC servo motors, the

selected RH actuator provides the precise velocity-based control required for testing

the ARLC algorithms. The joint angles are measured by encoders mounted on the

motors. Including tachometers in the motors is optional and is not necessary for the

current design.

110


8-10 cm

20-25 cm

Link

Link 2Link 1

Pad of 8-10 cm in contact with the air table

Motor

Motor

Motor mount

Figure 6.7. Link and motor-link interconnection

0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

Time [s]

Join

t rat

es [r

ad/s

]


(A)

0 10 20 30 40 50−0.15

−0.1

−0.05

0

0.05

0.1

Time [s]

Tor

que

Con

trol

Inpu

ts [N

m]


(B)

Figure 6.8. Joint rates and torque profiles for ARLC with joint-limit avoid-ance (Model3)

6.2.3 Target Design. The target is expected to have the same dimensions as

the base, and would be equipped with magnetic contact mechanism to attach rigidly

to the end-effector when the capture happens (see Fig. 6.3). The mass and moment

of inertia of the target must be adjustable by the user in order to allow testing of the

111


developed algorithms for different conditions. To generate a spinning motion for the

target, the user can manually apply a torque to the target of a specific magnitude

and direction to ensure that the magnetic contact mechanism attaches the target to

the end-effector. It is not necessary to have an IMU installed on the target because

its motion can be tracked by the global sensing system described in Section 6.4, for

the purposes of capture by manipulator.

6.3 Air-bearing Table

In our design of supporting planar surface, the key difference from the common

air-table designs employed for testing space manipulators on the ground is that the

cushion of air is generated from the table surface itself. One advantage of this ap-

proach is that it eliminates the need of an on-board air-tank, which is limited in vol-

ume, and, compared to the designs with an off-board air supply, the present method

does not suffer from potential disturbances on the base-arm resulting from the pipe

connections to the air source. Several special issues and considerations, however, as

related to the construction and operation of the air-table need to be considered, as

listed below.

Weight of system to be supported by air-table: Different loads were tested with

the free-floating satellite on the granite table available at CSA (Fig. 6.2) and the

results showed that the mockup satellite can maintain the free-floating conditions

with a very high load at the supplied pressure. As indicated in [90], the supplied

pressure of 100 psi can support payloads up to several thousand pounds (order of 104

N) with a flow rate of few cubic feet per minute (order of 10−2 m3/min).

Table sizing and construction: Determining the appropriate table size is very

important because of construction costs and available space constraints. For the

envisioned facility, assuming that the target has the same dimensions as the base, the

length of the whole system in a full straight out configuration is 125 cm. Since the base

is intended to move freely in translation and rotation, it is necessary to consider the

workspace required for possible testing maneuvers. In this preliminary design, the size

112

6.3 AIR-BEARING TABLE

and the aspect ratio of the table are determined by inspecting the system’s workspace

for a series of simulations conducted with MSC.Adams for arbitrary motions of the

arm as well as for motions resulting from application of ARLC algorithm. To this end,

by using MSC.Adams in the post-processing mode, animations of arbitrary motions

of the arm and those obtained with the ARLC algorithm are setup to execute within

a wire-frame around the workspace of the system, as shown in Fig. 6.9. From these

results, the minimum frame for all simulations that have been conducted to date

with the three models, without violating the frame boundaries, was determined to

be 125×125 cm2. Allowing for a margin of error and keeping in mind other possible

future experiments with the facility, we suggest that the designed table be larger than

the minimum frame by approximately 40–50% on each side, giving a tentative size

of 175×200 cm2. Additional length along the lengthwise dimension of the base-arm-

target is suggested in order to have additional workspace to maneuver the target.

It is noted that since we expect to conduct most experiments with the initial linear

momentum of the system at zero, the mass center of the whole system will remain

nearly stationary (assuming minor frictional disturbance). Hence, a square table is

preferred.

Two options were considered for the construction of the air table of the afore-

mentioned size: (i) using a single piece (175×200 cm2) or (ii) by joining together

several smaller pin-hole pieces. Personnel at CSA have previously constructed and

tested a 30×30 cm2 air-table piece and thus using such pieces as a “tiles” for the big

table (second option) is viable. This would allow for a modular table construction

and hence easy resizing of the table if desired, but also reduce the possibility of large

pressure fluctuations which may occur for a single-piece construction, resulting in un-

even air-pad. On the other hand, it is necessary to ascertain that joining two pieces

in the “tile” approach will not compromise the air-pad quality as the load moves

over the joint between the two tiles. Another relevant issue for table construction is

the separation of the pin-holes. Preliminary testing conducted at CSA with 30×30

cm2 air-table piece showed that 2.5 cm is a reasonable separation distance of the

113


Base

Target

Link1

Link

2

Link3

Wire-frame

25 cm

25 cm25 cm

Figure 6.9. Animation snapshot for dimensioning air-table, 125×125 cm2

wire-frame test case

pin-holes; based on that geometry, we suggest the use of 8–10 cm in diameter pads

for supporting manipulator joints as shown in Fig. 6.7.

Initial stabilization: This issue relates to how much movement of the base-

manipulator system is likely to occur at the beginning of the experiment when the

air-table is turned on. This is a concern since one of the goals of the experiment

is to demonstrate the feasibility of reactionless motion of the arm, which, however

may be compromised if there is significant motion over the table when the arm is not

actuated. At this time, it is difficult to predict whether this effect will pose a serious

problem. That said, it was also suggested that the motion due to floatation can be

considered as an external disturbance on the system and in fact could allow us to

demonstrate the robustness of ARLC scheme.

6.4 Sensing System

The global sensing system located above the test-bed, as illustrated in Fig. 6.1,

can be used to measure the position and attitude of the base and target. The over-

head system complements the on-board sensors which are directly used by the ARLC

114

6.5 COMPUTER AND CONTROL INTERFACE

controller: it provides measurements for the validation of the methodology. In addi-

tion, it can be used to track the motion of the target in order to enable its capture

by the grapple fixture (magnetic contact mechanism).

Two options for the global sensing system have been considered in this preliminary

design: a CCD camera system currently available at CSA and the stereo Bumblebee

camera. The CCD camera at CSA is a highly accurate measuring system that was

acquired and used by CSA for the active flatness control of membrane antenna pro-

totype project. In order to use this system for the present facility, some involvement

from the supplier would be required to update and modify the software interface to

the camera system. Also, the previously used update frequency of the camera is 1Hz

which is too low for the present application. The second option would involve the use

of two Bumblebee cameras: one installed over the vicinity of the base and the other

over the target.

For the purpose of testing the coupling motion control and the identification

algorithms developed in this thesis for a capture mission, it is not necessary to install

a force sensor at the end-effector of manipulator to measure the contact forces and

moments. The instant of capture can be detected either by using a switch arrangement

to detect the impact and possibly with the overhead camera as well. As already noted,

the feedback signals required for the ARLC algorithm are obtained from the IMU and

encoders for the angular velocity of the base and joint angles, respectively.

6.5 Computer and Control Interface

In this design, wireless communication between the DAQ card on board the satel-

lite base and the ground station is a viable option and already exists on the facility

at CSA. Although issues might arise with the reliability of wireless communication,

avoiding wired connection is beneficial in terms of eliminating undesirable distur-

bances on the free-floating system. Another option is to use a dedicated motion

control board mounted on the satellite base, which would allow all control computa-

tions to be done on board the system.

115


We expect that standard motion control method is applied and velocity control

mode is employed to control the actuators of the manipulator. The control code

will be developed and executed within the RT-Lab software from Opal-RT, already

available on the CSA’s facility.

6.6 Summary

The analysis and preliminary design of experimental test-bed for testing au-

tonomous capture missions in a plane is presented in this Chapter, taking the ex-

isting facility at CSA as a starting point. The air hockey table concept is proposed

here for emulating the free-floating conditions in space. The base satellite and the

arm are designed with sensors for closed-loop control on board the system, similar

to the real application. Two options for the global sensing system are put forward

for validation of the proposed methodologies. This design is recommended for testing

and validation of control and system identification algorithms for on-orbit servicing

missions.

116

7.1 SUMMARY AND CONCLUSIONS

CHAPTER 7

Conclusions and Recommendations for

Future Research

7.1 Summary and Conclusions

The main goal of this research is to develop a control scheme for the problem

of post-capture of an unknown tumbling target by using a free-floating space ma-

nipulator. In the course of on-orbit servicing and space debris removal missions, a

priori information on the tumbling motion and inertial properties of the target to be

captured are normally unavailable. These uncertainties deteriorate the control per-

formance in the post-capture and stabilization phases, in particular, they may cause

disturbances on the base attitude which is a critical issue for free-floating space ma-

nipulator control. These considerations motivated the present research and led to the

formulation of a new adaptive reactionless control algorithm to adaptively compute

reactionless motion commands to the arm, thereby, not requiring the use of attitude

control devices to maintain the attitude of the base. The relevance of adaptive reac-

tionless motion to facilitating the stabilization phase is demonstrated by coordinating

the proposed control scheme with the momentum-based parameter identification al-

gorithm to obtain the inertial properties of the target. The combination of parameter

identification and adaptive reactionless control scheme allow for a time buffer as a

117

CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

segue from the instant of capture (after impact) to the final control phases in the

capture mission.

We began this dissertation by analyzing the kinematics and dynamics of a free-

floating space manipulator. The system under investigation is subject to the con-

servation of momentum, following from the assumption that no external forces nor

torques are exerted on it. To obtain a thorough understanding of the coupling motion

between the arm and the base, which is our main concern in this research, the mo-

mentum conservation equations as well as the reaction null-space control are analyzed

in detail. These are the key ingredients in the development of adaptive reactionless

control and momentum-based parameter identification algorithms. In addition, the

momentum conservation and reaction null-space motion are used to ascertain the ac-

curacy of the co-simulation platform employed in this study. From the analysis of

energy balance of the free-floating space manipulator, we also demonstrate that the

kinetic energy that can be dissipated by the manipulator arm is limited and this value

depends on the initial momentum of the target and the arm configuration.

Based on the momentum conservation equations and inspired by the reaction

null-space control, a new control algorithm to adaptively produce the reactionless

manipulation in the presence of uncertainty is presented in Chapter 3. The proposed

Adaptive ReactionLess Control scheme plays an important role in generating the joint

motions in the post-capture phase, as it serves as an intermediate step between the

instant of capture and the final stabilization phase. Based on extensive numerical

simulations both for a planar and three-dimensional scenario, for a range of target

sizes, we conclude that the proposed ARLC scheme is viable and produces the desired

reactionless motion after capturing a target with unknown inertial properties and

unknown angular momentum. The results obtained in this study show that the

proposed algorithm is effective in generating arm joint rates which produce minimum

disturbance to the base, thus eliminating the need for using the attitude control

system.

118

7.1 SUMMARY AND CONCLUSIONS

In Chapter 4, the online momentum-based parameter identification method is de-

rived to identify the inertial properties of the target without knowledge of the system

momentum, while the arm is executing adaptive reactionless motion. The relevance

and utility of this proposed scheme to the post-capture stabilization are supported

by the quality of the inertial parameter estimates obtained for a wide range of target

sizes relative to the base. The combination of ARLC and identification procedure ad-

dress two critical aspects of the post-capture control mission: maintaining minimum

disturbance on the base created by the manipulator-target motion and identifying

the target’s properties.

It has been argued in this thesis that the use of ARLC scheme can be beneficial for

the post-capture and stabilization operations since it provides a buffer in time after the

instant of capture. However, the duration of the time buffer may be very short because

the ARLC algorithm developed in Chapter 3 will fail to meet its objective if any of the

manipulator joints reach their limits during the execution of the algorithm. This issue

is addressed in Chapter 5 by reformulating the ARLC scheme to perform with the joint

limit constraints. To this end, the redundancy resolution control scheme is formulated

within ARLC in order to implement a secondary manipulation task. Although the

secondary task validated in the present thesis is joint-limit avoidance, it is possible

to satisfy other secondary tasks that are relevant to the present application, many of

which have been previously demonstrated in the literature on redundancy resolution

control for redundant ground-based manipulators. There is a compromise on the

performance of ARLC with joint-limit avoidance, although the base disturbances

observed in our simulations still remain small. We also note that it is not necessary

to have a higher degree of redundancy beyond what is already required for ARLC to

satisfy a secondary task. This is the main idea behind the task-priority resolution

techniques where the control scheme performs the primary task preferentially and the

secondary task is satisfied with the remaining degrees of freedom.

119

CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

The ARLC scheme derived in this dissertation is conceptually simple as it em-

ploys the certainty equivalence principle. Moreover, the desired reactionless manip-

ulation as well as the momentum-based parameter identification are computed with

the recursive least squares procedure, which is computationally very efficient. This is

particularly important for real-time implementation of the proposed control scheme

for space applications.

Finally, the analysis and preliminary design of a base-3-DOF-manipulator-target

system to be operated on an air-bearing table to simulate the free-floating condition

in two dimensions is presented in Chapter 6. This preliminary design can eventually

serve as a basis for the construction of an experimental facility recommended for

evaluating control algorithms that have been and will be developed for autonomous

capture missions.

7.2 Recommendations for Future Research

Several recommendations for future research based on the results presented in

this dissertation can be made as follows:

• A coordinated post-capture methodology based on the ARLC algorithm

presented here and momentum exchange mechanism control can be devel-

oped to stabilize the whole system and complete the capture mission. A

number of detumbling motion control and momentum distribution algo-

rithms from literature [15, 45, 67] can be implemented with the availability

of target inertial properties and the system’s total momentum, as obtained

with the ARLC control/identification scheme.

• The formulation of adaptive reactionless control algorithm derived in this

study has not taken into account the issue of joint rate measurements.

Hence, performance of ARLC should be investigated in the presence of noise

and uncertainty in joint rate estimation. Modifications to ARLC in order

to make it more robust to noise and uncertainty in joint rate estimation is

recommended for future research.

120

7.2 RECOMMENDATIONS FOR FUTURE RESEARCH

• The effect of joint rate constraints was not investigated in this thesis, but

these are likely to limit the possibilities for reactionless motion when very

large tumbling targets are to be captured. Similarly, momentum exchange

device requirements and limitations as far as final stabilization of the system

must be studied and defined.

• Self-collision avoidance for a manipulator is a critical aspect for operations

in space, especially after capturing a large-size unknown target. Although a

number of studies have addressed this problem for redundant ground-based

manipulators, solving this problem for space manipulators is recommended

for future research.

• Another concern when designing motion control for a free-floating space

manipulator is the dynamic singularity issue [32]. In fact, due to a lack of

knowledge of dynamics properties, the singularity hypersurfaces in the ma-

nipulator joint space of the free-floating space manipulator after capturing

an unknown target is unpredictable. Hence, dynamic singularity avoidance

should be investigated as the next phase of this study by exploiting the

benefits of the ARLC/identification scheme proposed in this dissertation.

• In Chapter 6, a preliminary design for an experimental test-bed for au-

tonomous capture missions was laid out. However, construction of this

facility has not taken place to date due to budgetary constraints at CSA.

Constructing the experimental test-bed and validating the proposed control

algorithms are strongly recommended for future work.

121

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