dynamics and control ofspace systems

Deift University of Technology

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Faculty of Aerospace Engineering,academic year 2001 - 2002,course AE-4-399:

Dynamics and Control of Space Systems

a collection of informal lecture notes -

Compiled by: P.Th.L.M. van WoerkomDate : 04 February 2002Office : Mekelweg2,BlockC,mom2.18Tel. : 015 - 278 2792Fax : 015-2782150E-mail address: p.vanwoerkonuwbmt.tudelft.nI

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Faculty of Aerospace Engineering,Academic year 2001 - 2002,Course AE-4-399


Part B: Control

Compiled by : P.Th.L.M. van WoerkomDate : 26 February 2002Office : Mekeiweg 2, Block C, room 2.18Tel. : 015-278 2792Fax : 015-2782150E-mail address: [email protected]

CONTENTS

INTRODUCTION

PART A

A-i SINGLE DEGREE-OF-FREEDOM ROTATION OF A RIGID BODY

A-2 SIX DEGREE-OF-FREEDOM TRANSLATION AND ROTATION OFA RIGID BODY

A-3 SPACECRAFT EQUIPPED WITH MULTIPLE ROTORS

A-4 SPACECRAFT WITH STRUCTURALLY FLEXIBLE SOLAR ARRAYS -

HYBRID, DISCRETE SYSTEM MODELLING

A-5 SPACECRAFT WITH STRUCTURALLY FLEXIBLE SOLAR ARRAYS -

HYBRID, ASSUMED MODE MODELLING

A-6 ADDENDUM: SOME SPACE SYSTEMS DISPLAYINGDYNAMIC DEFORMATION OF SYSTEM COMPONENTS

PART B

B-i CONTROL OF INSTRUMENT SCANNiNG MECHANISMS

B-2 STABILISATION AND CONTROL OF A SINGLE-SPIN SPACECRAFT

B-3 CONTROL OF FLEXIBLE SPACECRAFT: INTRODUCTION

B-4 CONTROL OF FLEXIBLE SPACECRAFT: PRACTICALCONTROL CONCEPTS

B-5 CONTROL OF VARIOUS SPACECRAFT: IMPLEMENTEDCONTROL CONCEPTS

B-6 SPACECRAFT CONTROL LAW SYNTHESIS BASED ONLYAPUNOV THEORY

B-7 ROBOTIC MANIPULATORS IN SPACE:A DYNAMICS AND CONTROL PERSPECTIVE

-0-0-0-0-0-



Part B: Control

Course note B-i:

Control of instrument scanning mechanisms

Compiled by : P.Th.L.M. van WoerkomDate : 25 February 2002Office : Mekelweg 2, Block C, room 2.18Tel. : 015-2782792Fax : 015-2782150E-mail address: [email protected]

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Part B: Control

Course note B-2:

Stabilisation and re-orientationof a single-spin spacecraft

Compiled by : P.Th.L.M. van WoerkomDate : 25 February 2002Office : Mekeiweg 2, Block C, room 2.18Tel. : 015-2782792Fax : 015-2782150E-mail address: [email protected]

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Part B: Control

Course note B-3:

Control of flexible spacecraft: introduction

Compiled by : P.Th.L.M. van WoerkomDate : 25 February 2002Office : Mekeiweg 2, Block C, room 2.18Tel. : 015-278 2792Fax : 015-2782150E-mail address: p.vanwoerkomwbmt.tudelft.nl

—2—

CONTENTS:

1. Introduction

2. Spacecraft model

3. Feedback control

4. First design: P control

5. Second design: PD control

6. Third design: adjusted PD control

7. Fourth design: add notch filter

8. Fifth design: optimal control

9. Colocated control

10. References

--,—

1. Introduction

We present an exercise in designing a controller for a verysimple flexible spacecraft. Single axis control is considered,and the spacecraft has only a single flexible mode.

First, we consider non—colocated control. The attitude of theflexible appendage is fed back to the rigid spacecraft controlsystem (reaction whheel, typically). We shall find that it isvery difficult to obtain robust, high-bandwidth control.

Then, we shall consider colocated control. Here, the attitudeof the rigid spacecraft (the “bus”) is fed back to its colocated control system. Now, we find that it is rather easy toachieve robust, high bandwidth control.

This rather detailed example was taken from the book of Franklin et al. (see the list of references).

The example illustrates the fact that controller design is ingeneral not a single shot activity, but requires repeatedsynthesis and analysis.

The reference list contains some additional references tobooks that - in the opinion of the author - are helpful ingetting to understand root locus analysis and frequency repon—se analysis.

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10. References

D’Azzo, J.J. and Houpis, C.H.Feedback Control System Analysis and Synthesis. McGraw-HillBook Co., N.Y., 1966.

DiStefano, J.J., III, Stubberud, A.R., and Williams, 1.3.Theory and Problems of Feedback and Control Systems. Schaum’sOutline Series, McGraw—Hill Book Co., N.Y., 1967.

Dorf, R.C.Modern Control Systems. Addison—Wesley Pubi. Co., Reading,Mass., 1967.

Franklin, G.F., Powell, J.D., and Emami—Naeini, A.Feedback Control of Dynamic Systems. Addision-Wesley Publ.Co., Reading, Mass., 19??

Kuo, B.C.Automatic Control Systems. Prentice Hall, Englewood Cliffs,N.J. sixth edition, 1991.

McRuer, D., Ashkenas, I., and Graham, 0.Aircraft Dynamics and Automatic Control. Princeton UniversityPress, Princeton, N.J., 1973.

For those who are interested in historical papers:

Bode, H.W.Network Analysis and Feedback Amplifier Design. Van NostrandPubi. Co.,, Princeton, N.J. 1945.(the book probably contains references to earlier papers bythe author on the subject).

Evans, W.R.“Graphical analysis of control systems”. Trans. AIEE, Vol.67,1948, p 547—551.



Part B: Control

Course note B-4:

Control of flexible spacecraft:practical control concepts

Compiled by : P.Th.LM. van WoerkomDate : 25 February 2002Office : Mekelweg 2, Block C, room 2.18Tel. : 015 - 278 2792Fax : 015-2782150E-mail address: [email protected]

—a—

CONTENTS

1. INTRODUCTION

2. TRANSFER FUNCTION DEVELOPMENT

3. BLOCK DIAGRAM REPRESENTATION

4. SOME SYSTEM PROPERTIES

5. CONTROL FOR A SPECIAL SYSTEM

6. CONTROL FOR MORE GENERAL CASES

7. APPLICATION: SOHO

8. APPLICATION: OSO-8

9. APPLICATION: SAX

10. APPLICATION: MUSES-B

11. REMARKS

12. REFERENCES

--i

1. INTRODUCTION

In Voignummer A-4 we looked at the derivation of a M

mathematical model for the dynamics of a spacecraft withstructurally flexible appendages.

The model considered there, was rather simple yet instructive.There are two identical, structurally flexible, symmetricallylocated appendages (solar arrays). It displayed angular motionof the rigid centerbody around an inertially fixed axis.Linear motion of the centerbody was assumed to be absent inthe model (but taking it into account is straightforward).Small motions were assumed, leading to linearization of thedynamics equations.

These dynaittics equations are suitable as a basis for thedesign of controllers that are to maneuver spacecraft inattitude modes such as pointing, scan, raster scan, and slew.

The resulting dynamics equations (linear, ordinarydifferential equations with constant coefficients) weredecoupled into a pair of equations displaying coupled motionof centerbody attitude and asymmetric appendage deformation,and an equation displaying symmetric appendage deformation.

The present Volgnunuuer A-5 starts with the equations for thecoupled attitude motion / asymmetric appendage motion.

In Section 2 through 4 we look at some of those properties ofthese dynamics equations that are of interest to the controldesign engineer.

In Section 2 the system transfer function is derived, relatingthe actuator torque on the centerbody to the output of asynthetic attitude sensor at some prescribed location. Thissynthetic attitude sensor measures — effectively — theinertial attitude of an element on the asymmetrically deformedappendage. It therefore measures a combination of centerbodyattitude and spatial tangent to the appendage at theprescribed location. Then, the expression for the transferfunction involving a single flexible mode is derived.

In Section 3 two block diagram representations for the dynamicsystem are derived. One is of the closed—loop type; the otherof the open loop type. The overall transfer function of each,is identical to the one derived in Section 2.

In Section 4 the system is classified according to one of thefollowing three types:— spacecraft with appendage mode;- spacecraft with in—the—loop appendage mode;— spacecraft with in—the—loop non—minimum phase mode.

Section 5 considers control aspects for a simple two—bodysystem which represents a special case, in that it does notcontain the pair of modal zeros usually found with flexiblespacecraft.

Section 6 considers control aspects for a rigid spacecraft aswell as for spacecraft of one of the three types classified insection 4.

Sections 7 through 10 contain brief descriptions of thedevelopment of controllers for the well—known OSO—8 (launchedin 1975), and for the very recent projects SOHO (launched inDecember 1995), SAX (launched in April 1996), and MUSES—B (tobe launched in 1996).

In Section 11 there are some final remarks.

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We have had a bird’s eye view of the field of controllerdesign for flexible spacecraft.

An important point is to stress that analysis should involvethe use of various design techniques simultaneously. Forexample, the root locus will give you a quick appreciation forthe transient characteristics of the different system mode.But the frequency response curves will give you a quickappreciation of response amplitude, and of stability margins.Optimal control involving a quadratic cost function is alsouseful, as it may achieve good response were classical controlcan not easily find a solution.

Optimal control is not to be applied blindly. It should berecalled that in applications of optimal control a full statevector is needed vector and must therefore be measured orestimated. This complicates the system. Furthermore, optimalcontrol laws may be very sensitive to variations in actualsystem parameters — and by Murphy’s law that usually means atendency towards instability.On the other hand, in the design of SAX certain elementsassociated with “optimal control” have been used. And inMUSES—B the same comment holds. In other words: use “optimalcontrol” concepts carefully and wisely.

The system model contained a single mode only. The generalcase involving multiple modes can be handled similarly.

A control law that produces “good” system behavior, may excitecomponents that are not accounted for by the synthetic outputy. For example, the amplitude of the vibrational motion of theappendage tips, may be unacceptable. This amplitude can not beobtained directly from the output y, as y contains a sum ofcenterbody attitude and appendage spatial tangent. There mayeven be dynamics quantities that are not observable at all,but that may still be excited unacceptably by the controller.This should be verified! This could be done by definingadditional synthetic outputs that represent the quantitiesthat are to be monitored. In an end—to—end simulation theseadditional outputs should also be monitored. In case theydisplay unacceptable behavior, it may be necessary to redesignthe original “good” control law.

In general it is very instructive to analyzed carefully papersand reports describing controller design.Trying to repeat the analysis is of course most instructive —

but in many cases there is not enough information availableconcerning the qualitative and quantitative characteristics ofthe system. Some give different accounts, some of thentconflicting. So, some “Sherlock Holmes” analysis is required.Never mind: figure out what you can, and let your phantasy runfree (but channeled by common sense and some engineeringexperience).

References that are contained in the reference list but thathave not been cited explicitly are: Cannon et al. [1984],Gevarter [1970], Poelaert et al. [1973], Poelaert [1975],Spanos [1989). These are very nice and instructive references,that are highly recommended for background reading.

-co

12. REFERENCES

Abdel—Rahman, T. [1976].Influence of structural flexibility on a single—axis linearattitude controller. University of Toronto - Institute forAerospace Studies, UTIAS Technical Note No.198, Feb.

Bryson, A.E. [1983].“Stabilization and control of spacecraft”. Advances in theAstronautical Sciences, Vol.51: Guidance and Control 1983;Supplement to Vol.51: AAS Microfiche Series, Vol.44, papersAAS 83—087 and AAS 83—088.

Bryson, A.E., Jr. [1994).Control of Spacecraft and Aircraft. Princeton UniversityPress, Princeton, N.J.

Cannon, R.H.., Jr. and Rosenthal, D.E. [1984].“Experiments in control of flexible structures withnoncolocated sensors and actuators”. J. of Guidance, Vol.7,No.5, Sept.-Oct., p 546-553.

Dorf, R.C. [1967).Modern Control Systems. Addison-Wesley Pubi. Co., Reading,Mass.

Franklin, G.F., Powell, JD., and Emami—Naeini, A. [19..].Feedback Control of Dynamic Systems. Addison-Wesley Publ. Co.,Reading, Mass.

Gevarter, W.B. [1970].“Basic relations for control of flexible vehicles”. AIAAJournal, Vol.8, No.4, April, p 666—672.

Hughes, P.C. [1974].“Dynamics of flexible space vehicles with active attitudecontrol”. Celestial Mechanics, Vol.9, No.1, March, p 21—39.

Hughes, P.C. and Abdel-Rahman, P.M. [1979).“Stability of proportional—plus-derivative—plus—integralcontrol of flexible spacecraft”. J. of Guidance and Control,Vol.2, No.6, Nov.-Dec., p 499—503.

Hughes, P.C. and McTavish, D.J. [1989).“Dynamics modelling of viscoelastic space structures”. Proc.ESTEC Workshop on Modal Representation of Flexible Structuresby Continuum Methods, Noordwijk, June, ESA WPP-09, p 193-215.

--

Kampen, S. and Kouwen, J. [1991).“Design and development of the SAX-AOCS control software”.Proc. First ESA mt. Conf. on Spacecraft Guidance, Navigation,and Control Systems, Moordwijk, June, ESA SP-323, P 401-406.

Kwakernaak, H. and Sivan, R. [1972).Linear Optimal Control Systems. Wiley-Interscience, New York.

Poelaert, D., Liegeois, A., and Dureigne, N. [1973).“Analysis and design of an attitude control system for nonrigid satellites”. Proc. Euromech 38 Colloquium onGyrodynamics, Louvain-la-Neuve, Sept., p 215-223.

Poelaert, D.H.L. [1975).“A guideline for the analysis and synthesis of a nonrigid-spacecraft control system”. ESA/ASE Scientific and TechnicalReview, No.1, p 203-218.

Shahian, B. and Hassul, H. [1993).Control System Design using MATLAB. Prentice Hall, EnglewoodCliffs, N.J.

Spanos, J.T. £1989).“Control—structure interaction in precision pointing servoloops”. J. of Guidance, Vol.12, No.2, March-April, p 256-263.

Vigneron, F.R. [1976).“Ground-test derived and flight values of damping for aflexible spacecraft”. Proc. ESA Symp. on Dynamics and Controlof Non-Rigid Spacecraft, Frascati, May, ESA SP-l17, p 325-333.

van Woerkom, P.Th.L.M. [1994-1995].Private communications with T. Yaxaashita, NEC Corp., Kawasaki,Japan, on control algorithms for MUSES-B.

Wood, T.D. [1989].“Aspects of high accuracy attitude control systems”. Proc. ESAWorkshop on Advanced Technologies for Spacecraft AttitudeControl, Navigation, and Guidance, Noordwijk, Oct., Paper 4.1(20 pages).

Wood, T.D.[1990).“Design concepts for the SOHO spacecraft allowing forstructural flexibility”. First mt. Conf. on Dynamics ofFlexible Structures in Space, Cranfield, May, late paper (15pages).

Wood, T.D. and James, C.D. [1991).“Design aspects of the attitude control system for the SOHOspacecraft”. Proc. First ESA mt. Conf. on SpacecraftGuidance, Navigation, and Control Systems, Noordwijk, June,ESA SP—323, p 3—8.

Yamashita, T., Ogura, N., Kurii, T., and Hashimoto, T. [1995).Improved satellite attitude control using a disturbancecompensator. 46—th Congress of the International AstronauticalFederation, Oslo, Oct., paper IAF-95-A.3-03.

Yocum, J.F. and Slafer, L.I. [1978).“Control system design in the presence of severe structuraldynamics interaction”. J. of Guidance, Vol.1, No.2, p 109—116.



Part B: Control

Course note B-5:

Control of various spacecraft:implemented control concepts

Compiled by : P.Th.L.M. van WoerkomDate : 25 February 2002Office : Mekelweg 2, Block C, room 2.18Tel, : 015 - 278 2792Fax : 015-278 2150E-mail address: [email protected]

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Part B: Control

Course note B-6:

Spacecraft control law synthesisbased on Lyapunov theory

Compiled by P.Th.L.M. van WoerkomDate : 26 February 2002Office : Mekelweg 2, Block C, room 2.18Tel. : 015-2782792Fax : 015-2782150E-mail address: [email protected]

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References cited

Junkins, J.L., Rahman, Z., and Bang, H.“Near-minimum-time maneuvers of flexible vehicles: a Liapunov control law designmethod”.In: J.L. Junkins, ed., Mechanics and Control of Large Flexible Structures, Progress inAstronautics and Aeronautics, Vol. 129, AIAA, Washington, DC, 1990, P. 565-593.

Vadali, S.R.“Feedback control of space structures: a Liapunov approach”.In: J.L. Junkins, ed., Mechanics and Control of Large Flexible Structures, Progress inAstronautics and Aeronautics, Vol. 129, AIAA, Washington, DC, 1990, p. 639-665.

Kalman, R.E. and Bertram, J.E.“Control system analysis and design via the Second method of Liapunov”.ASME Journal of Basic Engineering, Vol. 82, June 1960, p. 371-400.

-0-0-0-0-0-



Part B: Control

Course note B-7:

Robotic manipulators in space:a dynamics and control perspective

Compiled by P.Th.L.M. van WoerkomDate : 26 February 2002Office : Mekelweg 2, Block C, room 2.18Tel. 015-278 2792Fax : 015-2782150E-mail address: [email protected]

IAF-95-A.4.O1

Invited Survey paper onROBOTIC MANIPULATORS IN SPACE:A DYNAMICS AND CONTROL PERSPECTIVE

P.Th.L.M. van Woerkom

National Aerospace Laboratory N LRP.O. Box 90502, 1006 BM Amsterdam, The Netherlands

A.K. Misra

Department of Mechanical Engineering,McGill University817 Sherbrooke Street W., Montreal H3A 2K6, Canada

46th International Astronautical CongressOctober 2-6, 1995/Oslo, Norway

For permission to copy or republish, contact the International Astronautical Federation,3-5, Rue Mario-Nikis, 75015 Paris, France

Invited Survey Paper on IAF-95-A.4.0 1

P.Th.L.M. van Woerkom

ROBOTIC MANIPULATORS IN SPACE:

A DYNAMICS AND CONTROL PERSPECTIVE

by

A.K. Misra

National Aerospace Laboratory NLR,P.O. Box 90502,1006 BM Amsterdam, The Netherlands

Abstract

Large robotic manipulators are to play a key role inassembly, maintenance, and servicing of future complex in-orbit infrastructures. The first generation space manipulatoris the Canadian Remote Manipulator System (RMS), in orbitsince 1981. A second generation of large space manipulatorsis presently under development, for use on the InternationalSpace Station: the Space Station RIMS, the European Robotic Arm, and the Japanese Experiment Module RMS.

Large robotic space manipulators pose significant challenges to the dynamics and control community. This surveypaper discusses dynamics modelling issues and controlschemes for such systems.

1. IntroductionLarge robotic manipulators are to play a key role in

assembly, maintenance, and servicing of future complex in-orbit infrastructures. The first generation large space manipulator is the Shuttle-based Canadian Remote ManipulatorSystem (RMS, or Canadarm), for the first time in orbit inNovember 1981. A second generation of large space manipulators in now under development, for use on board ofthe International Space Station. These are the CanadianSpace Station RMS (or SS-RMS), the European RoboticArm (or ERA), and the Japanese Experiment Module RMS(or JEM-RMS).

Topologically, large space manipulators are articulatedmulti-body systems, carried by a free-floating base body.They have large dimensions, low specific weight, and actuators with limited torque authority. Payload mass may bevery large compared to the mass of the manipulator, and thepayload itself may display complex dynamics. Controloperations may be carried out on-line by an astronaut on-site, or entirely from the ground.

Large space manipulators pose significant challenges tothe dynamics and control community. These involve topicssuch as: accurate dynamics modelling of the multi-flexiblebody system and the system actuators and sensors, real-timesimulation, verification and validation, design of man-machine interfaces, robust yet accurate control, compactcontrol algorithms, vibration suppression, and teleoperation.

Department of Mechanical Engineering,McGill University,817 Sherbrooke Street W.,Montreal H3A 2K6, Canada

This paper discusses those topics briefly. Section 2reviews dynamics modelling issues. Section 3 reviews manipulator control concepts. Section 4 addresses the issue ofsimulation, both in software and in hardware. Section 5gives a bird’s eye view of the current large space manipulator systems under development. Conclusions are drawn insection 6.

2. Modelling of manipulator dynamicsA space manipulator is a multi-body system; for dyna

mical analysis it can be modelled as a kinematic chain ofinterconnected rigid and flexible bodies (links). This kineniatic chain can be simple or complex. It is simple if each linkis connected to, at most, two other links; otherwise it iscomplex. A simple kinematic chain is closed if all links areconnected to two other links; otherwise it is open. A serialmanipulator such as the Shuttle’s Remote Manipulator System (RMS) is therefore a simple open kinematic chain,while two cooperating serial manipulators form a simpleclosed kinematic chain. Complex kinematic chains arecomposed of more than one simple kinematic chain. Multi-armed manipulators such as the proposed Special PurposeDexterous Manipulator (SPDM), one of the manipulators inthe Mobile Servicing System (MSS) of the InternationalSpace Station, has a tree-structure which is a complex kinematic chain. Another type of manipulator, parallel mariipulators, do not have any near-term application in space, butmay be useful in planetary exploration; these comprise ofseveral kinematic loops.

From the kinematic and dynamic modelling point ofview, serial and multi-armed manipulators can be consideredto be topologically similar. On the other hand, cooperatingserial manipulators, parallel manipulators and multi-armedmanipulators in a coordinated activity (Fig. 1) have kinematicloops and require a different approach.

The couplings between adjacent manipulator links provide relative motion between the links. The most commoncouplings found in manipulators are: (a) the simple hinge orturning joint, usually called the revolute joint; and (b) the

Copyright © by the International Astronautical Federation.All rights reserved.

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sliding or rectilinear joint, usually called the prismatic joint.While a revolute joint provides relative rotational motionbetween adjacent links, the prismatic joint provides relativetranslational motion. In industrial robots joints and links canbe considered as rigid. However, space manipulators usuallycontain one or more long, slender links, whose structuralflexibility must be considered. Similarly, slippage occurs inthe presence of joint friction, and restraint occurs when themotor shaft that drives the joint exhibits deformation. Thegearbox stiffness must be modelled. Thus both joint as wellas link flexibility can be important for space manipulators.For the RMS, joint stiffness is more important than linkflexibility, at least in its operational range. On the otherhand, for the SS-RMS (Space Station Remote ManipulatorSystem) both joint and link flexibility are important.

In the following, various methods for modelling manipulator dynamics are stated briefly. This is followed by discussion of several modelling issues of current interest. Finally,the dynamical model and simulation results of the SSRMSare given in Section 5.4 as an example to focus attention onsome salient features of manipulator dynamics.

2.1 Equanons of motionConsiderable efforts have been directed in the last three

decades toward the formulation of dynamical equations ofmulti-body systems. The initial investigations were related tospacecraft systems involving interconnected rigid bodies;Hooker and Margulies in 1965 were probably the first tostudy such space systems. Soon after this, in the l970s,effects of structural flexibility were taken into account andresearch work on multi-body dynamics with applications tomechanisms as well as robotics was undertaken. At present,there is a vast body of literature on multi-body dynamicsincluding several books [1, 13, 23, 27]. There are alsoseveral commercial generic codes such as DISCOS [4],AUTOLEV (26], and TREETOPS [28], which have beenused extensively. Since the literature is vast, an exhaustivereview is difficult and is probably unnecessary. Only a briefreview is presented here, along with the discussion of somecurrent issues.

The equations of motion of a flexible multi-body systemcan be obtained by using the Newton-Euler method, theEuler-Lagrange procedure, or the Lagrangian form ofD’Alembert’s principle on which Kane’s method is based.The first method is based on the force and moment equilibrium of component bodies and subsequent elimination ofinter-body forces. The latter two methods are based onwork/energy principles and consider the system as a whole.The final outcome of either of these methods is a matrixdifferential equation of the form

H 4 + v(q,q,r) = U (1)

where q is the vector of joint angles (revolute joints), jointtranslational displacements (prismatic joints) and elasticgeneralized coordinates. Furthermore, H is the generalizedmass matrix, v is a nonlinear function of generalized co-ordinates and velocities arising due to inertia effects such asCoriolis and centrifugal acceleration and of elastic forces,while u is a vector of external generalized forces includingactuator torques and forces.

The various methods cited above as applied to a spacemanipulator have been discussed in [35]. It is often thoughtthat the energy based methods are more efficient than theNewton-Euler method; however, it is possible to make thelatter method almost equally efficient. The Lagrange’smethod involves many partial differentiations and yieldslengthy expressions and terms which evetitually cancel.Kane’s method does not suffer from this drawback. To avoidlengthy expressions in the Lagrangian procedure, it is advantageous to derive the equations of motion for componentbodies and subsequently eliminate the non-working constraint forces utilizing the natural orthogonal complement ofthe velocity constraint matrix [8].

As mentioned earlier, a space manipulator is a simple orcomplex kinematic chain of bodies. This allows computationally efficient recursive formulations in which the components are dealt with sequentially, taking advantage of thekinematic and dynamic equations for the previous body.Typically, kinematic relations are formulated in a forwardpass from body 1 to body N (the last body), which are usedin deriving dynamical equations in a backward pass frombody N to body 1. This can make the algorithm to be oforder-N, instead of a possible order-N3. The first systematicrecursive formulation was possibly carried out by Vereshchagin in 1974, for a rigid manipulator. The first extensionto flexible manipulators was probably carried out by d’Eleuterio in 1987. Many other recursive formulations exist inliterature which are too numerous to list here; see e.g. [2,35].

An important factor affecting the dynamical behaviourand control of space manipulators is whether the base is mobile or fixed. As far as the dynamical formulation and control concepts are concerned this poses no special difficulty.

2.2 Issues in dynamical modellingSome of the current issues in modelling space manipu

lator dynamics are: (a) consideration of constraints andkinematic loops; (b) incorporation of geometric nonlinearityeffects; and (c) modelling of impact, contact and inelasticeffects. A brief discussion of items (b) and (c) now follows.A discussion of item (a) is beyond the scope of this paper;see [1] for a discussion of constrained motion.

2.2.1 Geometric nonlinearity effects

For quite some time, researchers have recognized thatthe stiffness of an elastic beam, whose base is undergoingfast rotation around an axis normal to its centre line (e.g.flexible appendages of a spinning spacecraft), increases witha rate proportional to the square of the angular velocity ofrotation. This phenomenon is called geometric stiffening orsometimes dynamic stiffening. Thus a flexible link of aspace manipulator will experience geometric stiffening if itrotates rapidly.

One of the first investigators to consider this problemwas Likins in 1974, who probably coined the words geometric stiffening. Vigneron [31] studied the effect of dynamic stiffening in multi-body dynamics while consideringstability of a spinning satellite with a crossed-dipole confignration.

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Since then a large number of works on flexible spacecraft

systems has included this effect in dynamical models; how

ever, workers on space manipulators have started to incorpo

rate this effect into their dynamical models only recently,

probably because the operational speeds so far have been

small enough to ignore this effect.Even though the phenomenon has been known for over

two decades, there is no universally accepted approach tocapture this effect in multi-body dynamics modelling [34].

Note that conventional linear modelling leads to erroneousresults, that is, the beams become softer rather than stifferwith increasing spin rate. There are basically three differentapproaches that can correctly account for the so-called geometric stiffening effects (in reality, the correction terms canalso modify the mass matrix H in Eq.(l) in addition to thestiffness terms in v).

The first method is to consider a nonlinear strain-displacement field and use a strain energy expression in whichthird or higher order terms in elastic displacements areretained; e.g. [11, 14]. The second approach is to consider a

pseudo-potential field caused by the stresses generated in theelastic body due to the inertia forces and torques; e.g. [2,32]. The third method is to consider nonlinear kinematicsand avoid ‘premature linearization’, i.e. carry out linearization only after all partial derivatives in the formulation havebeen calculated; e.g. [15, 18, 25]. Consideration of axialshortening effect to account for geometric stiffening [17,3 1]is rather equivalent to the third approach. The differencebetween the first and the third approaches is that the formeraffects only the stiffness matrix since it is derived from thestrain energy expression, while the latter can affect bothstiffness and mass matrices. The second and the third aremore reliable. It should be pointed out here that althoughlarge base rotational rates lead to stiffening effect, basetranslational acceleration may lead to stiffening or softeningdepending on its direction; hence the expression ‘geometricstiffening” is not always appropriate.

2.2.2 Post-impact dynamicsSpace manipulators will increasingly be used to retrieve

malfunctioning satellites or for rendezvous between twospacecraft systems. Ideally these operations should besmooth; but in practice, there is always a velocity differential between the end-effector and the grapple point leadingto an impact. In Space Station construction and operation aswell there can be soft collisions with obstacles and impactwith environment even though operational control schemeswill aim to avoid them. An impact causes elastic oscillationsand joint rotations, and in the case of retrieval of largesatellites possible attitude drift of the carrier spacecraft aswell. However, study of post-impact dynamics has not yetreceived sufficient attention.

The impact dynamics during robotic operations may bemodelled using a graphical method and assuming that the

robots are rigid (33]. Also, the extended generalized inertiatensor and the virtual mass concept can be used to formulatethe dynamics of the system of free-floating links impacting apoint mass in space [39]. The links were modelled as rigidallowing the rigid body impact equations for the formulation

of a frictionless collision of a fully elastic or a plastic case.This yielded information related to what is the best approachpose. Another study considered the capture of a disabledspinning satellite by a spacecraft-mounted manipulator, andsimulated the post-impact dynamics [8]; it was assumed thatthe target was successfully captured, i.e., the impact wasfully plastic. The work was extended subsequently [9] toimpact scenarios from fully plastic to fully elastic, definedby two parameters - energy loss parameter and frictionparameter - which had significant effect on the post-impactdynamics.

In all the above investigations, it is assumed that theimpact duration is small. The outcome of this assumption isthat the generalized co-ordinates of the spacecraft-manipulator system remain virtually unchanged and only generalizedvelocities change; thus the details of the dynamics duringimpact need not be modelled. Although this is a reasonableapproximation, to obtain accurate results one must considerthe details of deformation at the contact surface using theories of continuum mechanics, of which Hertz’s contactmodel is a special case.

3. Manipulator controlThe primary objective for controller design is to be able

to let the six-dimensional end-effector ‘pose” (position andorientation) track a desired trajectory. There may also besecondary objectives, such as: collision avoidance, actuatortorque Limitation, and minimization of induced structural excitation.

Whereas the detailed development of operational controllaws is a truly challenging task, beyond the scope of thispaper, we shall outline various conceptual controller designsfor different levels of system complexity.

3.1 Rigid manipulator on fixed baseThe simplest case is that of a rigid manipulator which is

attached to an inertially fixed base. For this elementary casethe dynamics equations read:

+114 = Uq (2)

where q is the vector of joint angles, u the vector of actuator torques acting in the joints, H he generalized massmatrix, and v the nonlinear coupling vector.

Assume tiat the pose of the end-effector is representedby xe. Then, x0 is a nonlinear function of q,

x = f(q)

For simplicity we shall assume that q, like x, is a six-dimensional vector. Upon differentiation one has

a). -

xg=_qJq, xg=Jq+J

(3)

(4)

where 3 is known as the Jacobian. Solving Eq.(2) for q andsubstituting the result into Eq.(4) gives

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(8)b =fb(P’f)

Let the error between actual pose Xe and desired pose Xd bedenoted by e, and let the controller be designed such thatthis error will display dynamic behavior that is described bythe second order system with constant gains K and

+KE +Ke 0, C -X (6)

The control required to produce the required error dynamicsthen reads

ug=HqqJL[d_J4_KyE_KpC1+Vq (7)

Hence, assuming the system parameters to be known, aswell as the joint angles and their rates, one can use Eq.(7) tocompute the torque vector required to track Xd(t) with theprescribed error dynamics. This control is recognized as acontrol law of the ‘operational space” type.

Different control laws can of course be derived. Forexample, a control law may be derived from a Lyapunovfunction defined as the sum of the system Hamiltonian (herethe kinetic energy) and a penalty term which is quadratic inthe joint angle error q

-

The most common control law, perhaps, is based on inverse kinematics; i.e., one transforms xd(t) to q(t), and thenapplies local Pt) control in each joint such that each localjoint angle q. tracks its desired value 1(t).

In actual space robot manipulators, such as the RMSand the upcoming SS-RMS, control can be executed invarious modes, such as: control of a single joint with prescribed q,(t); and control with prescribed xd(t) and transformed to their corresponding desired joint parameters q(t).This is basically the third control law mentioned above [22].

3.2 Rigid manipulator on moving baseSpace manipulators are attached to free-flying space

craft, which represent moving bases. Specifically, the dynamics of the manipulator will affect the dynamics of thebase, and vice versa. This may be particularly importantwhen the manipulator is holding a heavy payload. Thisincludes the case in which the manipulator is used to assistin docking the Shuttle to a space station. For this next levelof complexity we shall indicate how the operational spacecontrol law derived above, can be modified to handle thisnew case. For simplicity a rigid moving base is assumed.

In addition to the degrees-of-freedom for the manipulator described by the joint angle vector q, there are nowalso the six degrees-of-freedom for the moving base. Let thevector f3 represent the true base coordinates (position andattitude), and let b represent the vector of derivatives ofquasi-coordinates (that is, b consists of the linear velocity ofthe base reference frame, and the angular velocity of thatframe). Hence,

The system dynamics then read

[Hqq Hqblf1 [Vq[Uq1II ,_l[H Hj[ j [Vbl Hal

(9)

where the new vector u represents the external forces andtorques applied to the base body by its AOCS system.

In order to reduce the present problem to the one ofsection 3.1, one eliminates b from Eq.(9). This gives a resultof the form

Hq,’ c7 + Vq = Uq (10)

The matrix Hqq is a modified generalized mass matrix, incorporating the matrices Hqb = HbT and the matrix Hbb.The vector Vq is a modified coupling vector, which nowcontains also the weighted vector and base load ub. Butnote that the structure of the equation is the same as forEq.(2).

If operations are prescribed in a reference frame fixed tothe base body, the pose will depend on q only. Hence, theapproach in section 3.1 applies.

On the other hand, operations may be prescribed out inan inertial reference frame. One example could be catchingof a free-flying payload, or the release of a payload withcertain injection conditions expressed with respect to inertialspace. In that case, one has the relation

Xe “fe(’P) (11)

To proceed with this case, one takes the first and secondtime derivatives of Eq.(I 1). The second derivative of Xcontains the first derivative of b and the second derivative ofq on the right hand side. Eliminate b in favor of q, and theneliminate q using Eq.(l0). This gives an expression of theform

Xe = + A uq (12)

similar to Eq.(5). From this point, one can proceed developing the control law as if the base were rigid (Section 3.1).For example, the operational space control law Eq.(7) can beused, provided the quantities on the right hand side are reinterpreted as indicated in the present section.

3.3 Flexible manipulator on fixed baseFor large space manipulators the structural flexibility in

the links and in the gear boxes may contribute appreciablyto the motion of the end-effector. Much attention has therefore been given to this case. Consequently we can sketchapproaches to control law design only in very broad terms.

= [3 - J Hqq1 vq I • J H’ Uq (5)

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Denote the generalized coordinates associated with thestructural deformations in links and gearboxes by the vectorq of dimension nr Assuming again a fixed base, the dynamics equations read

[Hqq H1fq ÷[Vq1 lB 1[H Hffj[j

[v,j[BjUq

where in general the mathx B is not zero.The first step is to introduce the transformation

[]=cTT] [], [:][] [Jwhere the matrix I denotes the 6x6 unit matrix. The dynarnics equation (14) then becomes, after pre-multiplicationwith TT,

1 r 1[H11 H12][Z11 [V I+1 lI=j[H21 Hj[j 1V21 oJ”q

Note that this equation has the same mathematical structureas Eq.(9), and can therefore be dealt with in the same man-ncr as shown in section 3.2. That is, an operational spacecontrol law of the type of Eq.(7) can again be synthesized,as well as any other control law referred to in sections 3.1and 3.2. However, see also the cautionary remarks later inthis section.

A more common type of controller for the case of aflexible manipulator is the perturbation controller. Here, onesplits the motion in a nominal motion produced by a nominal, feedforward control command uff. and a correctingfeedback command u. The nominal motion is usually thatof a rigid manipulator, with the nominal control uff obtainedfrom inverse dynamics analysis. Application of this technique therefore involves some amount of pre-planning. Thedeviations form the nominal motion are to be suppressed bythe feedback control uth. In order to compute u (on-line),one must have available a mathematical model for the dynamics of the perturbed motion. Moreover, due to the factthat the number of generalized coordinates has risen from 6(in section 3.1) to 6+nr the six-dimensional pose vector Xemust be augmented with the deformations vector q in orderthat the Jacobian associated with this augmented system benon-singular. After some analysis there results the linearsystem with time-varying coefficients in standard form [6):

f =A,i1 6Xe=C11

where 6Xe is the difference between actual pose and nominalpose, and where the coefficents are pre-computed along thenominal trajectory (and are therefore known). Given thissystem, an appropriate feedback control law for uth must benow be synthesized.

Even though the system looks familiar, synthesizing aneffective control law may not be so easy. This is in largepart due to the fact that the basic dynamics model Eq.(13)may not be known accurately, and may moreover be toocomplex to handle. Furthermore, obtaining the coefficientsin Eq.(16) for a variety of maneuvers and a variety of pay-

(13) loads, and processing these in an onboard computer, maypose a rather heavy burden to onboard memory and computational speed. Even so, one may approximate the modelby a frozen’ approximation over a given time interval, anddevelop a control law with coefficients to be adjusted to thelocal values of the frozen parameters.

An approach that seems at present more practical forspace manipulator operations, involves the pre-planning of

(14) the control uq such, that the perturbation around the nominaltrajectory due to the excitation of structural dynamics, remains within bounds. That is, the control is designed suchthat the manipulator nearly behaves like a rigid one. Oftenthese control laws are formulated in parametric form, suchas polynomials in time or sinusoidal functions of time, withadjustable parameters. Convenient is the description of thedesired trajectory x,(t) in terms of accelerating and dcccl

(15) crating sections connected by a constant speed section. Atthe initial and final points of the nominal trajectory, and atthe points connecting the three sections, one imposes therequirement that the acceleration or even the jerk be zero.Only this pre-planned control is implemented; i.e. u = u.At the end of the completed nominal maneuver one waitstill the stuctural dynamics have subsided sufficiently, beforecommanding the next maneuver. This next maneuver may beof the closed loop type, even perhaps a simple PD control,dc-signed to correct residual errors (biases) only.

3.4 TeleoperationAt present, Shuttle-based manipulator operations are car

ried out by an astronaut inside of the Shuttle, viewing themanipulator through windows and through video monitors.In future operations, the astronaut may be inside of thespace station, with no direct view of the manipulator. Or,even more demanding, the manipulator is operated from theground, again with no direct view, but hampered additionally by appreciable time-delays in the loop. These new developments call for a renewed appreciation of the allocationof tasks between operator control and onboard control,between onboard local controllers and onboard hierarchicalcontrollers, and between operator autonomy and autonomyof groundbased automatic control systems (in the latter case,we are talking about the scala of possible activities rangingfrom full manual control all the way to purely supervisorycontrol). Entering into detail is beyond the scope of thepresent paper. The reader is advised to get a first appraisalof the field from papers such as [3, 5, 19, 29].

4. SimulationSimulations play a key role in the development and

operation of manipulators. They are carried out on one ormore generic software simulators, usually one being an‘engineering’ simulator and another one an “operations and

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training’ simulator.The Engineering Simulator is a generic software simu

lator for hardware analysis and design, controller analysisand design, system level software tests, pezformance determination, and sensitivity investigation. The simulatorshould accept changing requirements and changing specifications throughout the life of the manipulator. It must therefore be generic.- Initially one will develop a highly accurate non-real-timesimulator which accounts for all dynamics phenomena beingpossibly of relevance and allows investigation of techniquesfor computational speed enhancement. Data from a finiteelement package are transferred through a special interface.The simulator provides simulated sensor data. Visualizationand animation are key tools for interpretation of the systembehavior.- Concurrently with the development of this non-real-timesimulator, a real-time version of the simulator is developed.This simulator is used for tests involving hardware-in-the-loop, and also for the study of interaction between operatorand system (man-machine interaction). In the latter application it gives information on the suitability of the operatordisplays and controls, it allows study of operator workload,and it allows fine-tuning of the control system parameters.In order to achieve real-time performance, it may be necessary to trade off accuracy versus computational speed.

The Operations and Training Simulator is a genericsoftware simulator designed to assist in mission planning,failure analysis, and training of the operator. This simulatoris largely based on the real-time Engineering Simulator, butenhanced by applicable detailed payload models, specificflight control systems, applicable operator control station,instructor station, and special image generation system forrealistic simulation of the manipulator environment.

To achieve sufficiently fast computational speed for thedynamics kernels in the simulators, is a considerable challenge. It involves for example the selection of structure ofdynamics algorithm (e.g. Order-N versus Order-N3, andserial versus parallel computation), of computer hardware, oflevel of complexity of actuator models and structural deformation field models, the possible neglect of base body dynamics, the desired detail of interaction dynamics during grappling, and the selection of numerical integration routine.References of interest include [7, 10, 16, 20, 36].

In all cases, verification as well as validation of thedynamics kernels in the simulators is a key requirement.“Verification” refers to the comparison of output from thedynamics kernel in the software simulator under consideration with the output of the equivalent dynamics kernel in a“truth” or “master” software simulator. “Validation” refers tothe comparison of the output from the software simulatorunder consideration with actual hardware behavior. The latter comparison clearly represents a level above that of “verification”. (Note that the terms validation and verification aresometimes found intermixed in literature.)

Verification starts in the first instance with comparisonwith the output of established generic dynamics simulatorsoftware such as DADS, DCAP, DISCOS, TREETOPS, orsimilar packages. Then, comparison is made with the dy

nanucs output of “truth’ or “master’ dynamics simulatorswhich have already been validated with respect to the actualmanipulator hardware. An example of the latter is the ASADsimulator, which served as “truth’ simulator for the ShuttleRMS, and which was used for the verification of the SIMFAC real-time simulator.

Validation may be carried out first on individual hardware component level, and then on system level. In turn, itwill be carried out first in the laboratory, and then in space.Validation on the system level on the ground is carried out,to the extent possible, in flat-floor facilities [21]. Here,dynamics and operations can be investigated in a singlehorizontal plane. However, these facilities are also used forfunctional testing of the entire manipulator before launch. Inthe case of the Shuttle RiMS a large flat floor is used. Herethe RMS is cradled on a test rig with air bearing supports,with RMS and rig floating over the floor. For the (H)ERA aflat floor with an ingenious support system is used, involving one or two servo-controlled trolleys supporting themanipulator and following it during its motion.

5. External space manipulatorsIn space operations, robotic manipulators already play

key roles, and their versatility and degree of involvementwill only increase. Spearheading has been the CanadianRemote Manipulator System (RMS) on the US Space Shuttle, in operation since 1981. Its main use has been to releasespacecraft, to retrieve spacecraft, and to assist astronauts inthe repair of spacecraft. The repair of the Hubble SpaceTelescope is a recent and most spectaculair example. It hasindeed added a new dimension to space operations.

A variety of new manipulator systems is now underdevelopment, with applications such as: International SpaceStation (Fig.2) internal and external operations; servicingoperations between unmanned spacecraft; and planetaryoperations with a planetary rover as moving base. In thecontext of this presentation we shall restrict ourselves to abrief description of the large external manipulator found onthe Shuttle, and to the large external manipulators to befound on the future International Space Station. Their analysis, design, and operation provide powerful challenges andstimuli for applied dynamics and control specialists workingin the field of multi-body systems.

5.1 RMSThe only space manipulator that is currently operational

is the Shuttle Remote Manipulator System (RMS). TheShuttle RMS in the reference configuration is shown inFig.3. It has six degrees of freedom: yaw and pitch at theshoulder, elbow pitch and all three rotations pitch, roll andyaw at the wrist. The manipulator is therefore a six link-, sixjoint- system. Each of the middle two links is approximately7 meters long with significant structural flexibility. Therevolute joints have fairly large flexibility as well. We referto the appropriate literature cited earlier for a further discussion of the RMS.

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5.2 ERAThe European Robotic Arm (ERA) is being developed

under the guidance of ESA, for assembly, maintenance, andservicing of the Russian part (MW-2) of the InternationalSpace Station. It is a symmetric arm of about 10 m long,consisting of two long, flexible links, and a wrist assemblywith end-effector at each far end of the arm (Fig.4). Oneend-effector is attached to the trolley moving over the Russian segment, and serves as shoulder joint assembly. Theother (identical) end-effector is free, or attached to a payload. The symmetrical design allows relocation of the ERAfrom one attachment point to the next. The latter featuregives access to all areas of the Russian Segment. The massof ERA is about 330 kg. It will be capable of handling payloads of up to 12 000 kg mass. The arm has in total sevenrotational joints, each with an electric motor, brake, gearbox,encoder, and tacho. The ERA Control Computer is attached

near the elbow joint. Various cameras are also attached, nearthe elbow and near the end-effectors.

Control of the arm is commanded by the dedicated ERAControl Computer, and the Man-Machine Interface panelheld by an astronaut in EVA (i.e., involved outside of theSpace Station). Man-Machine operations conducted by anastronaut inside of the Space Station are also foreseen.

Aspects of modelling and simulation of the systemdynamics has been described in [30, 36]. Structural modelling includes bending and torsion of the large links, actuatordynamics, and gearbox characteristics.

The basic control modes are: the free space mode,involving maneuvering over large angles; the approachmode, involving vision control, taking place when ERAapproaches a target; and the compliant motion mode, whenERA makes contact with its environment, while a payloadmay or may not be attached to the end-effector. The controlmodes are may be automatic, or may involve the astronautas human operator. In all cases the controllers in each of thejoints are provided with angular setpoint evolving in time,and the local feedback control component is basically of theProportional-Integrating-Differential (PH)) type:- The free space control makes use of pre-planning (on theground) of the path in task space, leading to feedforwardcontrol between way points, with pre-shaped control sectionsto reduce structural excitation and to facilitate astronautintervention. The astronaut is able to modify the motion atailtimes;- The approach control mode uses feedback control, usingposition errors derived from the vision sensor data;- The compliant motion control mode makes use of a force-torque sensor built into the end-effector. The control algorithm is to ensure that the actual force-torque data are closeto the prescribed ones (impedance control).

5.3 JEM-RMSThe Japanese contribution to the International Space

Station is the Japanese Experiment Module (JEM). The configuration is shown in Fig.5, and its location on the SpaceStation is shown in Fig.2.

The JEM Remote Manipulator System (JEM-RMS) ispart of the JEM. It has a total mass of about 1 000 kg. Itconsists of two connected arms:- The main arm is a 10 m long manipulator attached to thepressurized module. It is used for large payload handling,like JEM assembly and exchange of experiments on the exposed facility. The arm consists of two links each of about 4m length, and a wrist assembly with end-effector. It containssix rotational degrees-of-freedom. The end-effector is similarto the one for the Shuttle RMS, The main arm can carry apayload mass of 7 000 kg;- The small fine arm is a 2 m long daughter manipulatorattached to the end-efector of the main arm. Is is used fordexterous tasks such as the replacement of small units(ORU’s) on the exposed facility. It contains also six rotational degrees-of-freedom. A special end-effector has beendeveloped, including force-torque sensor and a gripper withthree ‘fingers for capture and release of a payloads. Thesmall fine arm can carry a payload mass of up to 300 kg inthe absence of force feedback.

Some aspects of modelling and simulation of the systemdynamics have been presented in [12]. Modelling includesbase body motion and bending and torsion of the two largelinks, with up to seven modes per link. The joint modelsinclude actuator dynamics, gearbox characteristics, andbrake.

Control of the main arm is primarily preprogrammed,but it can also be controlled by the astronaut (using the twohand-controllers). The small fine arm is primarily controlledby the astronaut. Control concepts for the main arm includeend-effector control in task space and single joint control.The controllers in the joints are provided with angular set-points evolving with time. The local feedback component inthe joint is basically a Proportional-Differential (PD) controller with band pass filters. Control concepts for the smallfine arm include end-effector control in task space, singlejoint control, and active compliance control. See also [37,38].

5.4 SS..RMSThe Space Station Remote Manipulator System (SS

RMS), shown in Fig.6, is an integral part of the MobileServicing System (MSS), to be supplied by Canada as itscontribution to the International Space Station. The SSRMSis a large manipulator system, about 17 meters long whenstretched. It has a mass of about 1500 kg, and has sevenmodular rotary joints. The two middle links have significantstructural flexibility as 40 the joints. The manipulator architecture is symmetrical about the elbow joint and has alatching end effector at each end. The manipulator base ismobile and the symmetry enables either end to act as thebase of the manipulator. The two end effectors have interface mechanisms to transfer power, data and video.

Several dynamical models have been developed byanalysts for various purposes: some include a large numberof details, while others are simplified. Even a basic, simplified model must include the effects of joint flexibility andlink flexibility. Furthermore, the rigid body and elasticmotions considered must be three dimensional. A simplified

8

model is described below to give a flavour of dynamicsmodelling of space manipulators. The equations of motionare obtained using the flexible multi-body code FLXSTM[24). The code uses Kane’s method and the algebra is car

ried out using the symbolic manipulation software MAPLE.

The output from the MAPLE code is a FORTRAN codewhich is used as a subroutine of the integrator. As men

tioned earlier, the model developed for the simulation takes

into account the joint flexibility and link flexibility Thegeneralized co-ordinates 0., 1 = l,2,...,7 represent the joint

angles. The joint flexibility can be modelled by introducinganother set of seven angles. Here, the motor angles , i =

1,2 7 were defined as the additional generalized co-ordi

nates. The joint and motor angles, 0 and ‘, are relatedthrough

= fli 0+

, I = 1,2 7

where n. and 6. are respectively the gear ratio of the gearbox

and the elastic rotational deflection of the gearbox as seenfrom the input side. If the joint flexibility is neglected, then

6. vanishes.To model the joint flexibility, one needs to know J and

K3, the motor inertia and stiffness. The latter can be calculated from the torquelflexible rotation relation. The torque Tand joint elastic rotation 6 are related by

Ta = sign(8)... , for (18)

T8 = sign(8) [KJ—A)÷T,], for 81 i. (19)

Here, is the backlash half-angle, T is the torque intercept,while K3 is the linearized joint stiffness. In developing thisjoint flexibility model (Fig.7) it was assumed that:* the gearbox has negligible inertia; it is included in themotor inertia J* with T representing the input torque to the gearbox, theoutput torque would be = n T5 which is the net torquedelivered to the (i÷l)th link (or the i-th joint torque);* only the gearbox has stiffness; the motor stiffness isincluded in the gearbox stiffness.

The long links of the manipulator are modelled asEuler-Bernoulli beams. Only the bending of the links, bothin-plane and out-of-plane, are considered, while their torsionis neglected. The bending deflections in the two directionsare discretized by using the assumed modes method:

= E1 t) Ix1)

w1 = E T’./t) 4/x?

for i = 4 and 5; and are generalized co-ordinates associated with the in-plane and out-of-plane deformations of thei-th link, while .(x.) are a set of admissible functions.Fixed-based cantilever modes are selected here as these

admissible functions.The simulations of controlled maneuvers using a PD

computed torque control basically lead to the conclusion thatboth joint and link flexibility are equally important, and thatthey can not be neglected.

6. ConclusionsThe paper presents a survey of dynamics and control

issues of current interest in the development of large spacemanipulator systems.

Such manipulator systems display considerable structuraldeformation in their long links and in their gearboxes. Evenwhen restricting oneself to the dynamics of such relativelysimple mechanical models, the development of mathematicalformulation of the system dynamics and the development of

(17) the software simulator displaying sufficient computationalspeed represent daunting tasks.

The development of automatic control as well of manualcontrol of such large manipulators is an equally challengingtask. On the one hand one would like to use detailed mathematical system models for high performance control; on theother hand one may not have sufficiently accurate models atone’s disposition, or the onboard computing facilities do notallow the incorporation of such necessary large models.

The International Space Station project constitutes adriving force in the field of multi-body dynamics and control. The development of the second generation large spacemanipulators forces the advancement of the state-of-the-artin multi-body dynamics and control. As has already beenwitnessed in the past, there will also be an ongoing spin-offinto many terrestrial fields of application.

7. References1. Amirouche, F.M.L. Computational Methods in Multi-

body Dynamics. Prentice Hall, NJ, 1992.2. Banerjee, AK. ‘Recursive formulation with geometric

stiffness in multibody elastodynamics’. AAS/AIAAAstrodynaniics Specialist Conf., Halifax, Nova Scotia,Aug. 1995, paper AAS-95-398.

3. Bassett, D.A., Wojcik, Z.A., et al. Ground based control of robots aboard Space Station. 44th IAF Congress, Graz, Oct. 1993, paper IAF-93-T.4.5 15.

4. Bodley, C. S., Devers, A. D., et al. A Digital ComputerProgram for the Dynamic Interaction Simulation ofControls and Structures (DISCOS). NASA TP-1219,Vol.1 and II, May 1978.

5. Bos, 3.F.T., Stassen, H.G., et al. “Aiding the operator in

(20)the manual control of a space manipulator”. ControlEngineering Practice, Vol.3, No.2, Feb. 1995, p 223-

(21) 230.6. Choi, B.-O. and Krishnamurthy, K. “Unconstrained and

constrained motion control of a planar two-link structurally flexible robotic manipulator”. 3. of RoboticSystems, Vol.11, No.6, Sept. 1994, p 557-571.

7. Cyril, X., Huculak, M., et al. “Validation of a real-timespace robotics simulator”. Proc. Conf. on Modellingand Simulation 1994, Barcelona, June 1994, p 573-577.

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8. Cyril, X., Jaar, G. J., et al. Post-impact dynamics of aspacecraft-mounted manipulator. 44th IAF Congress,Graz, Oct. 1993, paper IAF-93-027.

9. Cyril, X., Kim, S.W., et al. Dynamics and control oftwo flexible multi-body systems attempting docking!berthing. 45th IAF Congress, Jenisalem, Oct. 1994,paper IAF-94-A.3.023.

10. Cyril, X., Kucük, N., et al. “An engineering and training simulation facility for space robotics. Paper,presented at the SCS Simulation MultiConf., Orlando,April 1992 (11 pages).

11. Hanagud, S. and Sarkar, S. “ProbLem of the dynamicsof a cantilever beam attached to a moving base”. J. ofGuidance, Control, and Dynamics, Vol.12, No.3, May-June 1989, p 438-441.

12. Hashimoto, H., Matsueda, T., et al. ‘Simulation fordeveloping iBM Remote Manipulator”. Proc. Symp. onArtificial Intelligence, Robotics, and Automation inSpace (i-SAIRAS), Toulouse, Sept. 1992, p 343-352.

13. Huston, R.L. Mukibody Dynamics. Butterworth-Heinemann, Boston, 1990.

14. Ider, S.K. and Amirouche, F.M.L. “Influence of geomethc nonlinearities in the dynamics of flexible treelikestructures”. J. of Guidance, Control, and Dynamics,Vol.12, No.6, Nov.-Dec. 1989, p 830-837.

15. Kane, T.R., Ryan, R.R. et al. “Dynamics of a cantileverbeam attached to a moving base”. 3. of Guidance,Control, and Dynamics, Vol.10, No.2, March-April1987, p 139-151.

16. McCullough, J.R., Sharpe, A., et al. “The role of thereal-time simulation facility, SIMFAC, in the design,development, and peformance verification of the ShuttleRemote Manipulator System (SRMS) with man-in-the-loop”. Proc. 11th Space Simulation Conf., Houston,Sept. 1980, NASA CP 2150, p 94-112.

17. Modi, V.J. and Mali, H.W. “On the nonlinear dynamicsof a space platform based mobile flexible manipulator”Acta Astronaunca, Vol.32, No.6, June 1994, p 419-439.

18. Padilla, C.E. and von Flotow, A.H. “Nonlinear strain-displacement relations and flexible multibody dynamics”. 3. of Guidance, Control, and Dynamics,Vol.15, No.1, Jan.-Feb. 1992, p 128-136.

19. Pelletier, G. The Space Station Remote ManipulatorSystem, human computer interface consideration. 42nd[AF Congress, Montreal, Oct. 1991, paper IAF-91-075.

20. Prjns, J.J.M., Dieleman, P., et al. “Flexible space-basedrobot modelling and real-time simulation”. Proc.AGARD Symp. on Space Vehicle Flight Mechanics,Luxembourg, Nov. 1989, chapter 31.

21. Pronk, Z. and van Woerkom, P.Th.L.M. Hat-floorfacilities in support of configurable space structuresdevelopment. 46th lAP Congress, Oslo, Oct. 1995,paper IAF-95-I.2.02.

22. Ravindran, R. and Doetsch, K.H. “Design aspects of theShuttle Remote Manipulator control”. Proc. AIAAGuidance and Control Conf. Aug. 1982, p 456-465,paper AIAA-82-158l.

23. Roberson, R.E. and Schwertassek, R. Dynamics ofMultibody Systems. Springer Verlag, Berlin, 1988.

24. Sadigh, 3. and Misra, A.K. “Symbolic multi-bodyformulation for space manipulators”. Proc. IMACS/SICE mt. Symp. on Robotics, Mechatronics and Manufacturing Systems, Kobe, Japan, Sept. 1992, p 1393-1398.

25. Sadigh, M.J. and Misra, A.K. “More on the so-calleddynamic stiffening effect”. 3. of the AstronauticalSciences, Vol. 43, No. 2, April-June 1995.

26. Schaechter, D.B. and Levinson, D.A. “Interactive computerized symbolic dynamics for the dynamicist”. J. ofthe Astronautical Sciences, Vol. 36, No. 4, Oct.-Dec.1988, p 365-388.

27. Shabana, A.A. Dynamics of Multibody Systems. WileyPubi., NY, 1989.

28. Singh, R.P., van der Voort, R. J., et a!. “Dynamics offlexible bodies in tree topology - a computer orientedapproach”. J. of Guidance, Control, and Dynamics,Vol.8, No.5, Sept.-Oct. 1985, p 584-590.

29. Stieber, ME. and Trudel, C.P. “Advanced controlsystem features of the Space Station Remote Manipulator System”. Preprints, 12th IFAC Symp. onAutomatic Control in Space - Aerospace Control ‘92,Ottobrunn, Sept. 1992, p 3 19-326.

30. van Swieten, A.C.M. and Schoonejans, P. “Verificationand performance of the ERA Simulation Facility”. Proc.Third Workshop on Simulators for European SpaceProgrammes, Noordwijk, Nov. 1994, ESA WPP-084,p 453-462.

31. Vigneron, P.R., “Stability of a freely spinning satelliteof crossed-dipole configuration”. Canadian Aeronauticsand Space Institute Trans., Vol. 3, No. 1, March 1970,p 8-19.

32. WaLlrapp, 0. and Schwertassek, R. “Representation ofgeomethc stiffening in multibody system simulation”.mt. 3. for Numerical Methods in Engineering, Vol.32,No.8, Dec. 1991, p 1833-1850.

33. Wang, Y. and Mason, M. “Modeling impact dynamicsfor robotic operations”. Proc. 1987 IEEE hat. Conf. onRobotics and Automation, Raleigh, NC, 1987, Vol. 2,p 686-695.

34. van Woerkom, P.Th.L.M. “Modified dynamics modelling for maneuvering flexible space manipulators”. 3. ofSound and Vibration, Vol.179, No.5, Feb. 1995, p 777-792.

35. van Woerkom, P.Th.L.M. and de Boer, A. “Development and validation of a linear recursive Order-N algorithm for the simulation of flexible space manipulatordynamics”. 43rd lAP Congress, Washington, DC, Aug.-Sept. 1992, paper IAF-92-030.

36. van Woerkom, P.Th.L.M., de Boer, A., et al. Developing algorithms for efficient simulation of flexiblespace manipulator operations. 45th IAF Congress,Jerusalem, Oct. 1994, paper IAF-94-A.4.032.

I,

37. Yamawaki, K., Kuraoka, K., et al. Design and evaluation of man-in-the-loop control system of JapaneseExperimental Module Remote Manipulator System.40th IAF Congress, Malaga, Oct. 1989, paperIAF 89 090

38. Yarnawaki, K., Kuraoka, K., et al. ‘Remote Manipulator System of the Japanese Experiment Module.Proc Sixteenth Tnt Symp on Space Technology andScience, Sapporo, 1988, Vol.11, p 1825-1829.

39. Yoshida, K., Kurazume, R., et al. “Modeling of collision dynamics for space free-floating links with extended generalized inertia tensor. Proc. 1992 IEEEConf. on Robotics and Automation, Nice, France, May1992.

10

SSRMS

SPOM

MT

s

Fig. 1 Robotic manipulators with kinematic loops

Science Power PlatformPhoto-Voltaic Arrays Port PV Arrays

Russian Modules- ,,

Thermal Control II -Z’RTh1-Th>-.Systems Radiators

IntegratedTruss

Assemb

Canadian RoboticsrcModules & Expenments

ESA ModuleU.S. Lab ModuleHab Module

Starboard PV Arrays -“

Fig. 2 Configuration of the International Space Station

1.1

Wrist Pitch Wrist CCTVJoint & Light

Wrist Yaw End EffectorLower Arm

JointElbow CCTV Boom& Panflilt Unit

MPM-Upper ARM

UpperArm-,Boom

WristJettison Subsystem

Roll JointMRL- Wrist

MPM-WflStMRLLowerArm

MPM - Lower ArmElbow Pitch Joint

MRL - Upper Arm—. Shoulder Brace

Shoulder Pitch Joint MPM = Manipulator Positioning MechanismShoulder MRL = Manipulator Retention LatchYaw Joint

Orbiter Note RMS Jettison Interface is at BaseLongeron of MPM on Longerori

Fig. 3 Configuration of the Shuttle RMS

ORUs

Fig. 4 ERA attached to trolley sliding slong the RussianSegment MIR-2

JEMRMS

0

Fig. S The JEM-RMS on the Japanese Experiment Module

12

Latching End Effector

Wrist Roll Joint

Camera

Wrist BoomWnst Pitch

Elbow Pitch Joint Joint

Shou:Jo’nt

Shoulder Pitch Joint

— Camera

Shoulder Roll Joint

Latching EndEffector

Fig. 6 Configuration of the Space Station RMS

+1

Ts Gear box (n)

Motor (Jm)

tm1 ÷Ts(n-1)

Fig. 7 Sketch ofmodelforjoin:fiexibility

dynamics and control ofspace systems

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