Hogan - Impedance Control: An Approach to Manipulation: Part 1-Theory

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Hogan - Impedance Control: An Approach to Manipulation: Part 1-Theory
ENVIRONNENT cal controller (a) of a purel possible to I sn 8. equivalent antiques for -LI 9, LIcos目 d to the END一PoNT i Cos. constraints 3 K INEMATIC COORDINATE represen ated 72 COORDINATES MT phs will be 0582 2 cose> Lysin a of physical Fig 2 (a)A planar two member linkage and(b)a bond- graph of the associated kinematic transformations, seen from the tip, this sytsem is MT i freedom (b) e physical t)is always Fig3 (a)A planar three member linkage and (b)a bond- graph of the 8 force as the output variable, defined as a function of the associated kinematic transformations. Seen from the tip this system is an effort derivative o f the input velocity variable The only difference properly described as an admittance a current) between the two representations of this linear element is that t is that no ny degree in the strictest sense differentiation is not a physically in [X,, X2 for which no point in [01, 023 exists. The latter on Its en- realizable operation as it is the limiting case of process which problem could be eliminated by suitably restricting the range it, but not requires knowledge of the future. However, it is often a of points in (XI, X2), and given a knowledge of the current perfectly reasonable operation in a model (no worse than the joint angles the angular displacement corresponding to an freedom, assumption of the existence of lumped-parameter elements) end-point displacement could be uniquely defined although physically alizable infinite po wer flow may b ces, which predicted during transients However, consider the planar linkage shown in Fig. 3(a) motion) However, the and a corresponding bond graph shown in Fig. 3(d). The he consitutive equa tion of a nonlinear dynamic kinematic transformation equations are motion) element need not be invertible. The constitutive equation for oncepts of any device which stores elastic energy is fundamentally XI=LI coS 0\+L2 cos 02+L3 cos 63 agers of written with force as the output variable, defined as a func tion of input displacement; displacement is in turn defined as X2=L1 sin 81+L2 sin 62+L, sin 03 zations of garded as Again, Joint angles uniquely define end-point position but the same the integral of input velocity The constitutive equation may the converse is not true; even given a suitably restricted set of be nonmonotonic or even discontinuous; the only restriction points in IX,X encies this is that the potential energy integral must be definable (the angles, the end-point displacement does not provide sufficient 2) and a knowledge of the current joint nonlinear coenergy integral need not be). Real physical elastic devices information to determine the joint angular displacements e the two h: exist which cannot be described in the derivative causal form In constrast, the corresponding transformation from forces t mass is ith force as the input variable and motion as the output applied at the interaction port to the resulting torques applied variable variable This inviolable causal contraint is not unique to energy to the links is always well defined urn is the秘 storing elements. The real-world phenomenon of stiction is T=-Lr sin 01 Fi+L cos 8 F2 tion typically represented by a dissipative element with a nonin T2=L2 sin 02 F1+L2 cos 02 F2 Itten o vertible relation between force and velocity, A velocity may be imposed and a resulting force is defined but the converse is T3=-Ly sin A3 FI+L3 cos ]3 F2 (7) not true In fact, exarmination of the five-port bond graph of Fi When more than one degree of freedom is considered, 3(b)will show that any combination of two efforts(forces or kinematic relations may impose a further causal constraint torques)may be impressed. Similarly, for the four-port bond Consider the planar linkage shown in Fig. 2(a). Assume that graph of Fig. 2(b)any two efforts may be impressed. The lte equa this system may interact with its environment across an in kinematic transformations X L(0 (1),(2),(3) teraction port at the tip of the linkage, a bond graph of the and (4)) are in fact part of the junction structure through linkage showing the two independent power bonds associated which the various elements in a physical system interact and put equa- &:: with this point is shown in Fig. 2 (b). The linkage equations are impose a kinematic causal constraint which is relatcd to but ate var a transformation between kinematic variables [01, 02d and distinct from the conditions imposed by zero- and one interaction port variables (X, X2I te equa- XI=LI coS 81+L2 cos B2 As an aside, it is the fact that in bond graphs functional relations are (1)represented at graph nodes which makes the equivalence of transformers, X2=L1 sin 81+L2 sin 62 (2)gyrators and junctions clear !n contrast, in linear graphs[2-5]or Mason(signal For every point in (B1, @2) there is a correspanding point in transformers and gyrators masquerade as elements, and the equivalence is not L XI, X23 but the transformation is, in general, not uniquely graphs over invertible and there exists a two-dimensional infinity of points graphing physical dynamic systems. Paynter has pointed out some other more mportant reasons [211 ASME e Journal of Dynamic Systems, Measurement, and control MARCH 1985, VoL. 107/3 Impedance Control, Force Control, and Compliance If the environment as an admittance, then the manipulator Se:Ft) must always impress a force on the environment. It might then be concluded that what is required in general is the ENVIRONMENTAL CONTROLLED control of a vector of interaction forces. because the con ADMIT TANCE I MANIPUL atOR trolled manipulator corresponds to some equivalent physical ystem, it may be represented by a network of physical system elements such as a bond graph. An equivalent physical net- i work representing pure force control along a single degree of ENV ADM freedom is shown in Fig. 4(a). In this graph the forcc com mands from the high-level supervisor to the low-level con Fig. 5 A troller are represented by an effort source, an ideal element including b which may impose any time-history of force an the rest of th system independent of its motion o-1s,:v() If it is assumed that at a minimum the manipulator should giving the be capable of stably positioning a simple mass it can be seen y bodies an that this networ k is an incomplete description of the necessary described controller action: Stable positioning requires at a minimum a easonabl static rclation between force and position; some spring-like inertial ENV IRONMENTAL CONTROLLED element must be included in the equivalent physical network ADMITTANCE MANIPULATOR The controller must specify a vector quantity such as the admittance desired position, but it must als tity which is This distin ly different: a relationship, an impedance, whic important has properties similar to those of a second-rank, twice- generality Fig.4 Bond graph equivalent network representations of (a)pure covariant tensor, it operates on a contravariant vector of Generali force control and (b)impedance control vector of interface forces. In fact, linearized components of Is the s the impedance such as the stiffness and the viscosity are Wr by the equ junctions [20]. Any one bond may be causally indifferent but second-rank twice covariant tensors multi-axis its causality is constrained relative to the others The simplest equivalent physical network representing of the con The point of this discussion is that the distinction between impedance control is shown in Fig 4(b). The position com- flow soure admittance and impedance is fundamental: Real physical manded by the high-level supervisor is represented by a flo\ impedance systems exist which can be described in one form and not the source,an ideal element which may impose any time history y The separ other. A spring with a nonmonotonic constitutive equation of velocity on the rest of the system. To prevent causal componen can only be described as an impedance; seen from an in- conflict between this element and the environmental ad- b properties teraction port at its tip, the behavior of a kinematically mittance(which must experience an impressed effort)a zero- N: nonlinear constrained system such as the linkage of Fig.3 can only be junction is interposed between the two. The impedance assumptio described as an admittance coupled to this zero-junction represents the relation between s Figure The most important consequence of dynamic interaction force and motion commanded by the supervisor and includes impedance both the static force/ displacement relation plus the possible invariance between two physical systems is that one must physically dynamic terms required to ensure controlled dynamic change ir complement the other: Along any degree of freedom, if one is an impedance, the other must be an admittance and vice behavior Consider versa. NoW, for almost all manipulatory tasks the en The problems of controlling the mechanical interaction i The nodic vironment at least contains inertias and/or kinematic con between a manipulator and its environment have been ad- .maintaine straints, physical systems which accept force inputs and which dressed by many researchers. The inadequacies of con interactIo determine their own motion in response. However, as ventional position control are widely recognized and the ss: written ir alternatives are typically referred to as "force control, g than an a described above, while a constrained inertial object can always be pushed on, it cannot always be moved; These 'compliance, ' compliant motion control"'or< fine motion control"'[12, 13, 15, 19, 22 30 As discussed abi onent systems are properly described as admittances. Seen from the force control is also inadequate; however, the term is applied manipulator, the world is an admittance position When a manipulator is mechanically coupled to its en loosely to control strategies using force feedback in com Examples bination with other feedback variables such as positio S finite wot ironment, to ensure physical compatibility with the en- and/or velocity. The concept of tuning stiffness, damping compone vironmental admittance, the manipulator should assume the and other aspects of the dynamic behavior of a manipulator be havior of an impedance. Because the mechanical in teraction with the environment will change with different has been explored by several researchers [18,19, 24, 301, and clude bo tasks, or even in the course of a single task-the manipulator the two possible causal forms of manipulator behavior were :. controller may be coupled to the environment in one phase and discussed by Nevins and Whitney [16]. However, they argued decoupled from it that when the manipulator was in contact with the en- positions another-the behavior of the manipulator should be adaptable, Thus the controller should vironment the appropriate strategy was to " command a a control a be capable of modulating the impedance of the manipulator position or velocity and look at feedback forces"and this a these pos as appropriate for a particular phase of a task approach was used in their subsequent work [30] and that of paper is v Thus a general strategy for controlling a manipulator is to many other researchers [12, 13, 19]. This is equivalent to equiva control its motion(as in conventional robot control) and in E: complet ddition give it a" disturbance response for deviations from In keeping with standard bond graph practice, the imposition of either aa. engineers that motion which has the form of an impedance The position or a velacity is represented by a flow source. The assumption is that the w under ste dynamic interaction between manipulator and environment position is uniquely defined by the integral of the velocity. Either the velocity is 绕1sual)s nay then be modulated, regulated, and controlled by known for the infinite past, or an inicial position and the subsequcnt time underlyir changing that impedance, and hence the approach described history of velocity are known [203 A zero-junction means that all systems connected to it experience th in this paper has been terned impedance control"[1, 6-11 A onc- effort whereas their flows sum to zero flow where 4/VoL. 107. MARCH 1985 Transactions of the ASme Journal liance the separation of unilateral power transmission effects from nipulator Y110∠srvo(t bilateral dynamic interaction effects For any general physical It Ingl system the equivalent source term seen at an interaction port eral is the is defined as that required to ensure zero power flow across the con- the port. The differential equation relating port variables t physical under conditions of zero net power flow is the impedance or NON-NODiC NODIC cal system admittance, Note that nonlincarity docs not enter into thcsc IMPEDANCE MPEDANCE ysical net definitions. Unfortunately, the junction structure(common ENVIRONMENTAL CONTROLLED MANIPULATCR effort or common flow) and concomitant superposition DMITTANCE orce co properties of the norton and Thevenin equivalent networks is Flg.5 A bond graph equivalent network representation of the only guaranteed for linear systems. This means that in a al element minimum necessary structure of an impedance-controlled machine nonlinear system the separation of effects is possible, but including both nodlc(Zo)and non nodic(Zn)impedance rest of the reassembling the pieces is not necessarily eas The superposition properties may be preserved by assuming or should giving the manipulator the behavior of an admittance, em- that the structure of the manipulator controller is such that it an be seen bodies an implicit assumption that the environment can be is always capable of determining an equilibrium position of necessary described as an impedance, and the approach might an unconstrained inertial object. If the system is not at mInimum a reasonably be termed "admittance control. 2 As described equilibrium, assume the set of commands (which may in spring-like above, because of the nature of kinematically constrained general vary with time) are frozen''at their current in- networ k inertial objects, the environment is properly dcscribed as an stantaneous values and impose steady-state conditions. The ch as the admittance and the manipulator should be an impedance. manipulator behavior (assumed to be nodic) is then y which is This distinction is not merely one of terminology, but has characterized by a static relation between force and position nce. which important consequences, as discussed further below. First, the (modulated by the command set nk. twice. generality of impedance control is considered F=S(X cE vector of v Generalized Equivalent Networks By assumption the manipulator is interacting with an un- covariant onents of Is the simple single-axis impedance controller represented constrained inertial object, thus at equilibrium in steady state cosity are by the equivalent network of Fig 4(b)applicable to a general the interface force is zero. Now assume that zero interface multi-axis manipulator? That network depicts the separation force defines an unique equilibrium position. That is, the class present of the controller action into two distinct components, one(the o f impedances considered is restricted so that if the gradient flow source)representing the control of motion, the other (the of the static force/position relation is nonzero, zero force tion com- y: impedance) representing the control of dynamic interaction e history %w:: The separation of the controller action into a(vector)motion always defines an equivalent equilibrium positio mmand set defines an unique position. As a result the co y a flow nt causal component and a impedance component (which has the Xo=Xo: ( c ental ad- properties of a tensor) can be achieved for a general class of This is the position with respect to which the input t)a zero- nonlinear controlled manipulators but some further assumptions about the controller structure are required displacements to the nodic impedance are measured. It may be thought of as the position toward which the manipulator is n between Figure 4(b) represents only the nodic component of the impedance seen at the interaction port. Nodicity refers to the heading at any point in time. The actual position of the 明qsod invariance of the constitutive equation of an element under a manipulator end-point may, of course, be different and as the dynamic change in the reference value (origin) of its argument commands may change with time, the manipulator need never Consider again the static relation between force and position reach the position Xo. Consequently, this position need not be The nodic component of this relation is the part which may be restricted to lie within the wor kspace of the manipulator, It is nteraction a convenient fiction and is a summary statement of one been ad of con- s:. maintained invariant under a change in the coordinates of the consequence of the commands. To keep this distinction cleax interaction port, i. e, when the manipulator moves. It may be Xo is referred to as a"virtual position"'and its time history and the written in terms of a displacement of the end-point rather xo()is referred to as a virtual trajectory. control than an absolute position of the end-point, a general relation By defining the virtual trajectory the behavior of the ne notion between force and position may include non-nodic com- ove, pure ponents, relations which may only be written in terms of the controlled manipulator has been decomposed into a vector of is applied position of the end point in some fixed reference frame port variables which may be commanded and a relation In com- Examples of the latter include the constraints imposed by the between port variables, an impedance, which may also be position finite workspace of a nonmobile manipulator. The non- nodi lic commanded The value of this exercise is that by definition components should be coupled to a one-junction'shared by the two components may now be reassembled in the simple damping ,. i nipulator the manipulator and the environmental admittance. To in- manner represented by a zero-junction. The superposition E PuE ' TOE clude both of these components the minimum necessary properties o properties of the Norton equivalent network have been .vior were controller structure is as shown in Fig. 5. However, in most retained without restriction to linear systems The behavior of the manipulator may now be written as ey argued practical situations the primary concern is to be able to specify positions of the wor kpiece in the workspace and to be able to follows(assuming a state-determined system mmand a control aspects of the behavoir of the workpiece at any of Vo=Vo: cI Virtual Source and this these positions. Accordingly, the immediate concern of this paper is with the nodic component of the impedance f=0 Junction Equations 11) valent to Equivalent net works of the Norton form (Fig. 4(b))or the dz/dt=Zs(z, f): Ic] (12) complementary Thevenin Form are familiar to systems Nodic impedanc engineers but they are normally applied only to linear systems f either a on is that the under steady-state conditions [25]. with nonlinear systems(as F=Z0(23f):{c (13) e velocity is usual)some difficulties are encountered The basic concept As before, following standard bond graph convention the quent time- underlying both Thevenin and Norton equivalent networks is imposition of a virtual position or a virtual trajectory has ce the same A one-junction means that all systems connected to it experience the same nlow whereas their efforts sum to zero or, if the equilibrium point is unstable, away from which it is heading he AsME Journal of Dynamic Systems, Measurement, and Control MARCH 1985,Vol. 107/5 BONDS REPRESENT VECTOR QUANTi TIES else is exl ien the ce 0k22 ntcraciior decompose edance controlled Teraction EN∨| RONMENTAL CONTROLLED and ADMITTANCE MANIPUL ATOR reassembly structure Fig.6 A bond graph equivalent network representating a multiaxis manipulator with controlled nodic impedance interacting with an dmittance-type environment the bond graph for the manlpulator is a a linearity w generalized Norton equivalent notwork. Part II nd applie been represented by a flow source. Writing the environmental al Fig. 7 A bond graph equivalent network representation of the admittance in general form superposition of muliple impedances coupled to an admittance. Each eN Acknowl component of the total impedance is represented by a generalized s dy/dt=Ys(y, F) (14) Norton equivalent network. Non-nodic impedances may be included in :Portion this system by setting the corresponding virtual flow source to zero. admittance NSF Gr v=Yo(y) When the manipulator is decoupled from its environment Mg NIHR 9 s in the dynamic equations due to the environ 6i:The wE The two sets of equations may be combined to write the as Polaron complete system cquations in standard (integrable) form dmittance disappear and in principle the manipulator alone The jol need exhibit no inertial behavior. In practice the uncoupled The ra dz/dt= Zs[z, ( Vo: c)-Yo(y)] [cI (16) manipulator still has inertia (albeit nonlinear and con- figuration-dependent). Because of the inevitable inertial Americ dy/dt=Ys[y, Zo(z, Vo: (c-Yo(y)]]: (c) (17)dynamics of the isolated manipulator the superposition of The T impedances holds even when the manipulator is uncoupled This sur F=Zo(z,[Vo: c-Yo(y)D: (c) (18)from its environment as there is always an admittance to sum awere per V=Yo(y) nd impede M: laborato This simple observation has many important consequences, Acoustics The purpose of the foregoing discussion was to demon- some of which will be pursued in the subsequent part s of this and the L strate that a broad and useful class of nonlinear manipulator paper. One which is immediately apparent is that different behaviors may be represented by a simple equivalent network, controller actions aimed at simultaneously satisfying different Referenc The only assumptions made were that the manipulator is task requirements may be superimposed. Each task com- sufficiently controllable to be able to determine an ponent may be represented by a generalized Norton equivalent waimplementit equilibrium position of an unconstrained inertial object such network, but referred to a different node (or virtual position manufa Book, w component is such that if its gradient is nonzero then zero manipulator behavior may be included in this equivalent 52 Brady force defines an unique position-not a restrictive set of network by associating it with a flow source identically equal Mr (Eds ) 1983 assumptions. Thus a general class of manipulation problems to zero and thus the assumption of sodicity made earlier is not CuitO have the same basic structure as Fig. 4(b). The behavior of a restrictive Wx:austria! Ma multiaxis impedance-controlled manipulator interacting with Industrial an admittance-type environment may be represented by the Summary 4 Drak Automatic This paper has presented a unified approach to equivalent network. Not only does this graph provide a manipulation termed"impedance control. "Because by its 5 Evart! compact representation of manipulation, the parallel with the nature manipulation requires mechanical interaction between s Eds, neuro 6 He standard Norton equivalent network is quite complete: The systems, the focus of the approach is on the characterization Manipulato superposition properties of the Norton equivalent network and control of interaction To understand interaction con- 氵 San Fra have been preserved cepts drawn from bond graph network analysis of dynamic systems are useful, particularly the concept of causality. By Superposition of Impedances assuming that no control algorithm may make a physical 3 The most interesting consequence of the assumptions system behave like anything other than a physical system the underlying impedance control is that if the dynamic behavior network concepts of bond graphs may be applied to describe of the manipulator is dissected into a set of component im the way the controller may modify the behavior of the pedances, these may be reassembled by simple addition even manipulator. Several simple but fundamental observations when the behavior of any or all of the components is may then be made: Command and control of a vector such as nonlinear. This is a direct consequence of the assumption that position or force is not enough to control dynamic interaction the cnvironment is an admittance. That admittance sums the between systems; the controller must also command and forces applied to it and determines its motion in response as control a relation between port variables. In the most com represented by the one-junction of Fig, 5. The admittance also mon case in which the environment is an admittance (e.g ,a acts to sum any impedances coupled to it. All of the systems mass, possibly kinematically constrained)that relation should connected to the one-junction associated with the admittance be an impedance, a function, possibly nonlinear, dynamic,oT experience the same input velocity; the total force they apply even discontinuous, specifying the force produced in response to the admittance is simply the sum of their individual force to a motion imposed by the environment, Even more im responses to the motion of the environmental admittance. portant, if the environment is an admittance, the total im Linearily of the impedances is not a consideration pedance coupled to it (due to the manipulator or anything 6/Vol. 107. MARCH 1985 Transactions of the AsmE journa else)is expressible as a sum of component impedances, even Hogan, N ,Impedance Control of a robotic manipulator, ""Presented when the components are nonlinear at the winter Annual Meeting of the American Socicty of Mechanicat Under a set of reasonable and unrestrictive assumptions the Engineers, Washington, D.C,1981 interaction port behavior of the manipulator may be Manipulators, "Proceedings of the Conference on CAD/CAM in Mechanical decomposed into a vector motion component and an Engineering, MIT, Mar. 1982 pedance component with some of the characteristics of a 9 Hogan, N Mechanical Impedance Contral in Assistive Devices and Manipulato: s, pp. 361-371 in: Robot Motion: Planning and Control, Eds second-rank, twice-covariant tensor. The vector component Brady, M, Hollerbach, J, Johnson, T, Lozano-Perez, T and Mason, M may be expressed as a virtual trajectory towards which the MIT Press, Cambridge, 1983 i: controlled manipulator dynamics are trying to drive the in- 10 Hogan, N. " Adaptive Control of Mechanical lmpcdancc by Coactivation teraction port. Its significance is that it permits the motion of Antagonist Muscles, ''in: IEEE Transactions on Automatic Control vol and impedance components of the manipulator behavior te obe^C;20,No。.8, gus1984,p:68-690 擦 reassembled Dy superposition as depicted by the junction Nonlinear Manipulator, "pp 121-128 in: Robotics Research and Advanced Ap structure of a generalized Norton equivalent network. Note. plications, ed. Book, W.J., american Society of Mechanical Engineers New that no restrictive assumptions of small displacements or York, 1982 Part Il and Ill of this papcr will discuss the implementation 13 Ishida, T,"Force Control in Coordination of Manipuh linearity were required 2 Inoue, H,,"Computer-Controlled Bilateral Manipulator, Bi etin ofthe Japan Society of Mechanical Engineers, Vol. 14, ND.69,1971 199-207 Fine mo and application of impedance control tions, Fift International Conference on Artificia! Intelligence, 1977 tion of the 14 karnopp, D, and Rosenberg, R, System Dynamics: A Unified Ap tance. Each Acknowledgments proach, wiley, New York, 1975 generalized 5 Mason, M.T., Compliance and Force Control for Computer-Controlled included in Portions of the work reported in this paper were supported Manipulators, "IEEE Transactions on Systents, Man, and Cybernetics,Vol a to zero SMC-11, No 6, 1981 NSF Grant No. PFR 7917348 16 Nevins, J. L, and Whitney, D. E, "The Force Vector Assembler Con ept, "Proceedings of the Ist. CISM IFToMM Symposium on Theory and Prac- Evironment NihR Grant No. G00 820 0048, Department of Education ice of Robots and manipulators, Vol. [L, 1974, pp. 273-288 ronmental The whitaker health Sciences Fund 17 Nevins, J. L, and Whitney, D.E., "Computer-Controlled Assembly, Polaroid Corporation Scientific American, Vol. 238, No. 2, Feb. 1978, pp. 62-74 ator alone 18 Paul, F.W., Gettys, T.K., and Thomas, J. D, <Definning of Iron uncoupled 3: The John and Fannie hertz Foundation Castings Using a Robotic Positioned Chipper, ""in: Roborics Researcht and Ad The Ralph E Cross Fund vanced ApplicationS, Ed. Book, W. J, American Society of Mechanical le inertial American Can Company Engineers, New York 1982 The tRw Foundation Faculty Fellowship 19 Paul,R. P.C., and Shimano, B, "Compliance and Control, Pro- osition of This support is gratefully acknowledged. Portions of the work ceedings of the Joint Auromatic Control Conference, 1976, pp. 694-699 uncoupled 10 Paynter, H. M., Analysis and Design o/ Engineering Systems, MIT P. nce to sum Laboratory for Biomechanics and Human Rehabilitation, the 2I Paynter H M. " Systcm Graphing Concepts, Instruments and Control equences, Acoustics, Vibrations and Machine Dynamics Laboratory 2 Raibert, M.H., and Craig J. J,"Hybrid Position/Force Control of arts of this and the Laboratory for Manufacturing and Productivity Manipulators, ASME JOURNAL OF DYNAMIC SYSTENS, MEASUREMENT AND different CONTROL, Vol 102, June 1981, pp. 126-133 different References 23 Rosenberg, R C, and Karnopp D. C, Introducton to Physical System Dynamics, McGraw-Hill, New York, 1983 ask com 24 Salisbury, J. K, " Active Stiffness Control of a Manipulator in Cartesian quivalent 1 Andrews, I R, and Hogan, N, "Impedance Control as a Framework for Coordinates, ")IEEE Conference on Decision and Control, New Mexico, 1980. position I mplementing Obstacle Avoidance in a Manipulator, "pp. 243-251 in: Control 25 Shearer, J L, Murphy, A. T, and Richardson H. H, Introduction to lent of the )f Manufacturing Processes and Robotic Systems, Eds. hardt, D, E,, and System Dynamics, Addison Wesley, Mass, 1967 Book, W.I., American Society of Mechanical Engineers, New York, 1983 26 Sheridan, T. B, and Ferrell, W.R., Man-Machine Systems: Information quivalent Brady, M, Hollerbach, I, Johnson, T,, Lozano-Perez, T, and Mason, Control, and Decision Models of Human Performance, MrT Press, Mass, ally equal Sg M,(Eds ) Robol Molion: Planning and Control, MIT Press, Cambridge, 1974 lier is not ::/3 Cutkosky, M. R. and Wright, P K. "Position Sensing Wrist s for It 27 Takahaski, Y. Rabins, M. J, and Auslander, D. M.. Control and Dynamic Systems, Addison-Wesley, Mass. 1970 astral Manipulators, "Proceedings af the 12th International Symposium on 28 Tanner,W.R,,(Ed )Industrial Robots, Volume 2: Applications, Society Industrial Robots, Paris, france, June 1982 of Manufacturing Engineers, Mich, 1981 4 Drake, S,"Using Compliance in Lieu of Sensory Feedback for 29 Turing, A. M.,Computing Machinery and Intelligence, "Mind, vol oach to Automatic Assembly, "Charles Stark Draper Laboratory Report T-657, Sept. LIX, No. 236.1950 197 ase by its 5 Evaris, E, V, Bizzi, E, Burke, R E, DeL ong, M, and Thach, w.t., tions, " ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT AND CoNTROL between Eds, Neuro sciences Research Progran Bulletin, Vol.9, No.1, 1971 June1971,pp.91-97 erization 6 Hogan, N,"Mechanical Impedance Control in Assistive Devices and 31 Whitney, D. E, and Nevins, J. L,, "What is Remote Center Compliance tion con- Manipulators, ""Proceedings of the Join( Automatic Controls Conference, Vol. and What Can It Do?, "Proceedings, Ninth /SIR, Washington, D.C,,Mar E, San Francisco, Aug. 1980 1979. dynamic sality. By physical ystem the describe r of th ervations such a teraction andand ost com- e(e.g,,a n should tamic, or response ore im- otal im- anything e ASME Journal of Dynamic Systems, Measurement, and Control MARCH1985,o|.107|7

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