Impedance Control: An Approach To Manipulation Part II Implementation

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Impedance Control: An Approachto Manipulation:Neville MoganAssociate Professor,Department of Mechanical Engineeringand Laboratory for Manufacturingand Productivity,Massachusetts Institute of Technology,Cambridge, Mass. 02139Part SI—ImplementationThis three-part paper presents an approach to the control of dynamic interactionbetween a manipulator and its environment. Part I presented the theoreticalreasoning behind impedance control. In Part II the implementation of impedancecontrol is considered. A feedback control algorithm for imposing a desired cartesian impedance on the end-point of a nonlinear manipulator is presented. Thisalgorithm completely eliminates the need to solve the "inverse kinematics problem"in robot motion control. The modulation of end-point impedance without usingfeedback control is also considered, and it is shown that apparently "redundant"actuators and degrees of freedom such vs exist in the primate musculoskeletalsystem may be used to modulate end-point impedance and may play an essentialfunctional role in the control of dynamic interaction.IntroductionMost successful applications of industrial robots to datehave been based on position control, in which the robot istreated essentially as an isolated system. However, manypractical tasks to be performed by an industrial robot or anamputee with a prosthesis fundamentally require dynamicinteraction. The work presented in this three-part paper is anattempt to define a unified approach to manipulation which issufficiently general to control manipulation under thesecircumstances.In Part I this approach was developed by starting with thereasonable postulate that no controller can make themanipulator appear to the environment as anything otherthan a physical system. An important consequence ofdynamic interaction between two physical systems such as amanipulator and its environment is that one must physicallycomplement the other: Along any degree of freedom, if one isan impedance, the other must be an admittance and viceversa.One of the difficulties of controlling manipulation stemsfrom the fact that while the bulk of existing control theoryapplies to linear systems, manipulation is a fundamentallynonlinear problem. The familiar concepts of impedance andadmittance are usually applied to linear systems and regardedas equivalent and interchangeable. As shown in Part I, for anonlinear system, the distinction between the two is fundamental.Now, for almost all manipulatory tasks the environment atleast contains inertias and kinematic constraints, physicalsystems which accept force inputs and which determine theirmotion in response and are properly described as admittances.When a manipulator is mechanically coupled to such anContributed by the Dynamic Systems and Control Division for publication inthe JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscriptreceived by the Dynamic Systems and Control Division, June 1983.environment, to ensure physical compatibility with the environmental admittance, something has to give, and themanipulator should assume the behavior of an impedance.Thus a general strategy for controlling a manipulator is tocontrol its motion (as in conventional robot control) and inaddition give it a "disturbance response" for deviations fromthat motion which has the form of an impedance. Thedynamic interaction between manipulator and environmentmay then be modulated, regulated, and controlled bychanging that impedance.This second part of the paper presents some techniques forcontrolling the impedance of a general nonlinear multiaxismanipulator.Implementation of Impedance ControlA distinction between impedance control and the moreconventional approaches to manipulator control is that thecontroller attempts to implement a dynamic relation betweenmanipulator variables such as end-point position and forcerather than just control these variables alone. This change inperspective results in a simplification of several controlproblems.Most of our work to date [3, 6, 13, 14, 16] has focused oncontrolling the impedance of a manipulator as seen at its"port of interaction" with the environment, its end effector.A substantial body of literature has been published onmethods for implementing a planned end effector cartesianpath [5, 27, 28, 32, 34, 35]. The approach is widely used in thecontrol of industrial manipulators and there is some evidenceof a comparable strategy of motion control in biologicalsystems [1, 24]. Following the lead from this prior work wehave investigated ways of presenting the environment with adynamic behavior which is simple when expressed inworkspace (e.g., Cartesian) coordinates.8 / V o l . 107, MARCH 1985Transactions of the ASMECopyright 1985 by ASMEDownloaded From: rg/ on 09/08/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

The lowest-order term in any impedance is the staticrelation between output force and input displacement, astiffness. If, in common with much of the current work onrobot control, we assume actuators capable of generatingcommanded forces (or torques), Tact, sensors capable ofobserving actuator position (or angle), 6, and a purelykinematic relation (i.e., no structural elastic effects) betweenactuator position and end-point position1, X L(6), it isstraightforward to design a feedback control law to implement in actuator coordinates a desired relation betweenend-point (interface) force, Fint, and position, X. Definingthe desired equilibrium position for the end-point in theabsence of environmental forces (the virtual position) as X0, ageneral form for the desired force-position relation is:Fint ATXo-X]Compute the Jacobian, J(0):dX 3(6)d6From the principal of virtual work:Tact J'(0)FintThe required relation in actuator coordinates is:Tact J'(0)A-[Xo-L(0)](1)(2)(3)(4)No restriction of linearity has been placed on the relationK[X0 - X ] , and the displacement of X from X0 need not besmall. Note that in this equation the inverse Jacobian is notrequired.Inverting the kinematic equations of a manipulator todetermine the time-history of actuator (joint) positionsrequired to produce a desired time-history of end-pointpositions has been described as one of the most difficultproblems in robot control [28]. For some manipulators (e.g.,those with nonintersecting wrist joint axes) no explicit (closedform) algebraic solution may be possible. However, if K[X0 X] is chosen so as to make the end-point sufficientlystiff, then a controller which implements equation (4) willaccomplish Cartesian end-point position control and the needto solve the "inverse kinematics problem" has been completely eliminated. Only the forward kinematic equations for'Throughout this paper, "position" will refer to both location and orientation, and "force" will refer to both force and moment.admittanceYZ impedancemodulation by command Ac]setSf flow sourceSe effort sourceFext external forceFint interface forceX end-point positionXo commanded (virtual)positionV end-point velocityV0 commanded (virtual)velocityK[.] force/displacementrelation [ ] force/velocity relation6 actuator position or angleCO actuator velocityU-) linkage kinematic equations3(6) Jacobianthe manipulator need be computed. This is a direct consequence of the care which was taken to ensure that thedesired behavior was compatible with the fundamentalmechanics of manipulation and was expressed as an impedance.Another important term in the manipulator impedance isthe relation between force and velocity. Again, given theabove assumptions, it is straightforward to define a feedbacklaw to implement in actuator coordinates a desired relationbetween end-point force and end-point velocity such as:Fint fi[V0-V](5)From the manipulator kinematics:V J(0)a (6)The required relation in actuator coordinates is:Tact J'(0)fl[Vo - J(0)co](7)Again note that the relation B[\0 - V] need not be linear andthat inversion of the Jacobian is not required.The dynamic behavior to be imposed on the manipulatorshould be as simple as possible, but no simpler. The foregoingequations take no account of the inertial, frictional, orgravitational dynamics of the manipulator. Under somecircumstances this may be reasonable, but in many situationsthese effects cannot be neglected. To ensure dynamicfeasibility the choice of the impedance to be imposed shouldbe based on the dominant dynamic behavior of themanipulator. The choice is a tradeoff between keeping thecomplexity of the controller within manageable limits whileensuring that imposed behavior adequately reflects the realdynamic behavior of the controlled system. As a result itdepends both on the manipulator itself and on the environment in which it operates. For example, a manipulatorintended for underwater applications will operate in apredominantly viscous environment and it may be reasonableto ignore inertial effects. In contrast, a manipulator intendedfor operation in a free-fall orbit will encounter apredominantly inertial environment. For terrestrial applications (which have been the main concern of our work)both gravitational and inertial effects are important, and thedominant dynamic behavior is that of a mass driven bymotion-dependent forces, second order in displacement alongeach degree of freedom.M inertia tensor in end-pointcoordinatesm massI inertiat timeF(-) noninertial impedanceMe environmental inertiatensor1(6) inertia tensor in actuatorcoordinatesC(0,co) inertial coupling torquesK(co) velocity-dependent torquesS(6) position-dependenttorquesG(0,«) accelerative coupling termsTact actuator force or torqueTint interface torquesY(6) mobility tensor in actuatorcoordinatesW(6) mobility tensor in endpoint coordinatesJournal of Dynamic Systems, Measurement, and Controlh generalized momentum inactuator coordinatesP generalized momentum inend-point coordinatesH(.) HamiltonianT,Ti,T2 torque#1.02 absolute j oint anglePi 1P2 relative joint angle- 1 J - 2 link lengthsKe net stiffness due to elbowmusclesKs net stiffness due toshoulder musclesKt net stiffness due to twojoint musclesKx stiffness tensor in endpoint coordinatesKo stiffness tensor in jointcoordinatesEp potential energyEk kinetic energyEk* kinetic coenergyX,X2 eigenvaluesMARCH 1985, Vol. 107/9Downloaded From: rg/ on 09/08/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

BONOS REPRESENT VECTOR QUANTITIESAll of the parameters in this expression are implicitly assumedto be functions of the set of control commands (c) and oftime.This set of assumptions defines a target behavior whichincludes inertial effects. The first two terms are the positionand velocity-dependent impedances of equations (1) and (5).If the environment is a simple rigid body acted on by unpredictable (or merely unpredicted) forces, its dynamicequations are:IF„1'S, 1 yI'iLEXTERNALSOURCE-. 01\'IY] S r V 0 j c )Z fc)' 'IIENVIRONMENTALADMITTANCE CONTROLLED MANIPULATORMedV/dt Fext mt(12)Fig. 1 A bond graph equivalent network representation of an impedance-controlled manipulator interacting with an environmentaladmittance. Each bond represents a vector of power flows alongmultiple degrees of freedom.and the coupled equations of motion for the complete systemof figure 1 are:When the manipulator is decoupled from its environmentthe term due to the environmental admittance disappears, andin principle the manipulator alone need exhibit no mass-likebehavior. In practice, the uncoupled manipulator still hasinertia (albeit nonlinear and configuration-dependent). Thismeans that the controlled system, both with the manipulatorcoupled to and uncoupled from its environment, can berepresented by an admittance coupled to an impedance asshown in Fig. 1.No physically realisable strategy can eliminate the inertialeffects of a manipulator but the apparent inertia seen at theend effector can be modified. The approach we have taken todeal with inertial manipulator behavior is to "mask" the truenonlinear inertial dynamics of the manipulator and imposesimpler dynamics, those of a rigid body. Most manipulatorytasks are fundamentally described in relative coordinates, thatis, in terms of displacements and rotations with respect to aworkpiece, tool or fixture whose location in the workspace isnot known in advance with certainty. As a result, taskplanning and execution will be simplified if the end-pointinertial behavior is modified to be that of a rigid body with aninertia tensor which remains invariant under translation androtation of the coordinate axes. This is achieved if:Note that in this case both the coupled and uncoupledequations for the system have the same second-order form.To implement the target behavior of equation (11), oneapproach we have used is to express the desired Cartesiancoordinate impedance in actuator coordinates (the kinematictransformations between actuator coordinates and end-pointcoordinates provides sufficient information to do this) andthen use a model of the manipulator dynamics to derive therequired controller equations. Assuming that the kinematic,inertial, gravitational, and frictional effects provide anadequate model of the manipulator dynamics as follows:m 0 0 0 0 00 m 0 0 0 00 0 m 0 0 00 0 0 /0 00 0 0 0 /00 0 0 0 0 /This is the inertia tensor of a rigid body such as a cube ofuniform density. This inertia tensor eliminates the angularvelocity product terms in the Euler equations for the motionof a rigid body [8] and ensures that if the system is at rest theapplied force and the resulting acceleration vectors arecolinear.To represent the dominant second-order behavior of themanipulator the noninertial interface forces are assumed todepend only on displacement, velocity and time:Fint F(X,V)-MdV/dt(9)If the noninertial behavior to be imposed is nodic, it may bewritten in terms of a displacement from a commanded (timevarying) position X 0 :Fint F(X0-X,Y0-\)-MdV/dt(10)Although there may be cases in which coupled nonlinearviscoelastic behavior is useful, for simplicity the position- andvelocity-dependent terms may be separated:Fint K[X0-X] B[V0-V] MdY/dt10/Vol. 107, MARCH 1985(11)(Me M)dV/dt K[X0-X] B[\0- V] FextI(8)dco/dt C(6,a ) V(co) S(d) Tact Tint(13)(14)an expression for the required actuator torque as a function ofactuator position and velocity and end-point force can bederived by straightforward substitution (see Appendix I):Tact 1(6)3 ,(6)M lK[X0-L(6)] S(6) 1(6)5',(d)M-iB[\0-3(d)u] V(o)I(ff)J 1(e)M-iFmt-Jt(9)Fmt- 1(6)3 1(6)G(6,u) C(6,w)(15)This equation expresses the required behavior to beprovided by the controller as a nonlinear impedance in actuator coordinates. It may be viewed as a nonlinear feedbacklaw relating actuator torques to observations of actuatorposition, velocity and interface force. The input (command)variables are the desired cartesian position (and velocity) andthe terms of the desired (possibly nonlinear) cartesiandynamic behavior characterized by M, /?[ ] and AT*].The feasibility of this approach to cartesian impedancecontrol has been investigated [6,16] by implementing thisnonlinear control law to impose cartesian end-point dynamicson a servo-controlled, planar, two-link mechanism (similar tothe nonlinear linkage in a SCARA 2 robot). A simple analysisestimating the computation required to implement thiscontroller on a six-degree-of-freedom manipulator indicatedthat the computational burden is comparable to "exact"approaches to generating forward-path manipulator commands such as the recursive LaGrangian [17] and NewtonEuler [21] methods or the configuration space method [18].If the interface forces and torques in equations (11) and (15)are eliminated and the position- and velocity-dependent termsreduced to linear diagonal forms, this implementation ofimpedance control resembles the resolved accelerationmethod [22]. However, unlike the resolved accelerationmethod, the impedance control algorithm presented above isbased on desired end-point behaviour which may be chosenrationally using approaches such as the optimisationtechnique presented in Part III. Furthermore, the impedancecontrol algorithm includes terms for coping with external"disturbances." Without the external "disturbance" terms(which have no counterpart in the resolved accelerationalgorithm) the manipulator is not capable of. controlled2Selective Compliance Assembly Robot Arm [23].Transactions of the ASMEDownloaded From: rg/ on 09/08/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

mechanical interaction with its environment. Note also thatthe above approach to defining the controller equations is notrestricted to commanded linear behavior and can be appliedequally well to achieve the more general coupled nonlinearbehavior of equation (9).It is not claimed that the above algorithm is the only way toachieve a desired end-point impedance. It is presented hereonly to demonstrate that a control law capable of modulatingthe end-point impedance of a manipulator may be formulated. The controller of equation (15) was designed by atechnique which is similar to pole-placement methods [31] inthat the desired behaviour and a model of the actualbehaviour of the manipulator were compared algebraically toderive the controller equations. In common with most approaches to manipulator control the approach is based on amodel which ignored many aspects of real manipulatorperformance, particularly the dynamics of the actuators andthe transmission system. Furthermore, like many other approaches the method assumes that the Jacobian is invertible.This technique is, of course, only one possible approach tothe design of a controller for implementing a desired cartesianimpedance, and, if one may draw from linear systems designexperience without overstretching the analogy to poleplacements methods, it is not even likely to be the best. Otherapproaches to controller design such as the model-referencedadaptive control method [9] will probably be useful.Impedance Modulation Without FeedbackModulation of end-point impedance using feedbackstrategies is not the only way to control the dynamic behaviorof a manipulator, nor is it always the best. This is particularlyevident in a biological system. One of the most distinctivefeatures of the primate neural control system is theunavoidable delay associated with neural transmission. Theshortest time for information to get from peripheral sensors(e.g., in the muscles or skin) in the human arm to the higherlevels of the central nervous system (e.g., the cortex) and backto the actuators of the arm is 70 milliseconds, and looptransmission delays of 100 to 150 milliseconds are typical [29].This problem is further exacerbated if significant computation is required (the response time to a visual stimulus issomewhere between 200 and 250 milliseconds). The effectiveness of feedback control in the presence of a delay ofthis magnitude is severely limited, particularly in dealing withtasks involving dynamic interaction. Yet primates excel atcontrolling dynamic interactions; How do they do that?One alternative to feedback which we have explored is theuse of redundancies: "excess" actuators or "excess" skeletaldegrees of freedom. From a purely kinematic standpoint theneuromuscular system is multiply redundant. For example,the kinematic chain connecting the wrist joint to the chest(clavicle, scapula, humerus, radius and ulna) has considerablymore degrees of freedom than those required to specify theposition (and orientation) of the hand in cartesian coordinates. These skeletal redundancies can serve to provide ameasure of control over the inertial component of the endpoint dynamics.In considering the apparent inertial behaviour of the endpoint it is useful to remember that an inertia is fundamentallyan admittance; flow (velocity) is determined as a response toimpressed effort (force). Dealing with kinematic redundancyis considerably simplified if the constitutive equations arewritten as a relation determining generalised velocity, a , (e.g.,the velocities of the manipulator joints) as a function ofgeneralized momentum, h:o Y(6)h(16)Y{6) is the inverse of the more commonly used inertia tensor,and to help distinguish the two, the term "mobility" isJournal of Dynamic Systems, Measurement, and Control(a)3(b)(c)(d)Fig. 2 A schematic representation of the influence of kinematicredundancies on the mobility (inverse effective mass) of the end-pointof a planar linkage. The ellipsoid of gyration associated with themobility tensor is shown in (a). The eigenvalues of the mobility tensorare inversely proportional to the effective mass in the direction of thecorresponding eigenvectors and the square root of their ratio determines the ratio of the major and minor axes of the ellipsoid, which arecolinear with the eigenvectors. For a planar, three-member linkage withlinks of uniform density and cross section and lengths in the ratio 1: 2:3 the effect on the ellipsoid of gyration of changing the linkage configuration for a fixed position of the end-point is shown in (b), (c), and d .suggested. The elements of the mobility tensor in general willdepend on the manipulator configuration.At any given configuration, the kinematic transformationsbetween joint angles and end-point coordinates define notonly the relations between generalized displacements, flowsand efforts in the two coordinate frames, (see equations (2),(3), and (6)) they also define the relation between thegeneralised momenta in joint coordinates, h, and end-pointcoordinates, p, through the Jacobian (see Appendix II):h J'(0)p(17)Consequently, the mobility tensor in end-point coordinatesW{6) is related to the mobility in joint coordinates Y{&) asfollows:(18)W(0) 3(0)Y{0)3 {8)(19)The physical meaning of the end-point mobility tensor is thatif the system is at rest (zero velocity) then a force vectorapplied to the end-point causes an acceleration vector (notnecessarily co-linear with the applied force) which is obtainedby premultiplying the force vector by the mobility tensor (seeAppendix II).Note that the Jacobian in the above equation need not besquare, and that the end-point mobility is configurationdependent. As a result, redundant degrees of freedom can beused to modulate the end-point mobility. Consider thesimplified three-link model of the primate upper extremity(arm, forearm and hand, each considered to be rigid bodies,linked by simple pin-joints) moving in a plane as shown inFig. 2. For simplicity, assume the links are rods of uniformdensity with lengths in the ratio of 1: 2: 3.Any real linkage such as the skeleton is a generalised kineticenergy storage system. Kinetic energy is always a quadraticform in momentum:Ek lAWY(6)\\(20)Thus the locus of deviations of the generalised momentumfrom zero for which the kinetic energy is constant is anellipsoid, the "ellipsoid of gyration" [33]. It graphicallyMARCH 1985, Vol. 107/11Downloaded From: rg/ on 09/08/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Table 1 Variation of apparent end-point mass with linkageconfigurationDistal linkorientation(degrees)90135180Effective massX\ -direction(kgm)0.3220.5681.824Effective massX2 -direction(kgm)1.8230.5680.323Link Lengths: 1, 2, 3 meters; Linear density: I kgm/mrepresents the directional properties of the mobility tensor.The eigenvalues of the symmetric mobility tensor define thesize and shape and the eigenvectors the orientation of theellipsoid of gyration (see Appendix II). An ellipsoid ofgyration can be associated with the mobility tensor in anycoordinate frame, e.g., end-point coordinates (see Fig. 2(a)).Figures 2(b) through 2(d) show the profound effect on theellipsoid of gyration of changes in arm configuration whilekeeping the position of the end-point fixed. The inertialresistance to a force applied radially inward toward theshoulder (vertically downward in the figure) changes byalmost a factor of six as the hand rotates through ninetydegrees (see Table 1). In the configuration of Fig. 2(d) theapplied force has to accelerate all three links; in that of Fig.2(b) it primarily has to accelerate the distal link. Clearly,kinematic redundancies in a linkage provide a vehicle forchanging the way the end-point will react to externaldisturbances without recourse to feedback strategies.As an aside, an alternative representation of inertialbehavior is via the ellipsoid of inertia [33]. Asada [4] hassuggested its use as a tool for designing robot mechanisms.However, the ellipsoid of gyration is the more fundamentalrepresentation; it is readily obtained even when the Jacobianof the linkage is noninvertible. Also, while the matrix Y(8)may never have zero eigenvalues, (assuming real links withnonzero mass) the matrix W(d) may, because of thekinematics of the linkage. If the inertial behavior of the tip iswritten in the conventional (impedance) form:p M(0)V(21)there exist locations in the workspace for which the eigenvalues of the tensor M(6) become infinite. Thus the end-pointinertia tensor can not be defined for some linkage configurations. On the other hand the worst the eigenvalues ofW(6) will do is go to zero, which is easier to deal with computationally. Again, a reminder of the fact that the differencebetween impedance and admittance is fundamental.Impedance Modulation Using Actuator RedundanciesIt is also possible to modulate the position- and velocitydependent components of end-point impedance withoutfeedback by exploiting the intrinsic properties of the actuators, and again apparent redundancies are useful.Although a muscle is by no means thermodynamicallyconservative, it exhibits a static relation between force andlength (for any given fixed level of neural input) similar tothat of a mechanical spring, i.e., one which permits thedefinition of a potential function analogous to elastic energy.3Muscle force also exhibits a dependence on velocity similar toa mechanical damper. It has been shown that the mechanicalimpedance of a single muscle may be modulated by neuralcommands both in the presence and in the absence of neuralfeedback [7, 11, 12, 25, 26]. Simultaneously activating two ormore muscles which oppose each other across a joint is onestrategy which permits impedance to be modulated independent of joint torque [15, 20]. (This is what happens, forCuriously, the force/length behaviour of most muscles is such that the coenergy integral is not defined and thus no compliance form is definable [29]:Muscles are impedances, not admittances.12/Vol. 107, MARCH 1985(b)WORKSPACEFig. 3 A schematic representation of the influence of the polyarticularmuscles of the primate upper extremity on the range of end-pointstiffnesses which may be achieved without recourse to feedbackstrategies by simultaneous activation of opposing muscles. Theellipsoid associated with the symmetric differential stiffness tensor isshown in (a). The eigenvalues of the stiffness tensor are proportional tothe stiffness in the direction of the corresponding eigenvectors and thesquare root of their ratio determines the ratio of the major and minoraxes of the ellipsoid, which are colinear with the eigenvectors.Assuming the upper extremity may be modelled as a two-memberlinkage with equal link lengths, without biarticuiar muscles, anecessary condition to achieve an end-point stiffness with equaleigenvalues (hence a circular ellipsoid) is only satisfied at the point pon the workspace boundary as shown in (b). With biarticuiar musclesacting at equal moment arms about each joint an end-point stiffnesswith equal eigenvalues and a circular ellipsoid may be achievedthroughout the region R shown in (c).example, when one tenses the muscles of the arm withoutmoving; the impedance of the limb increases.)There are also considerably more skeletal muscles thanjoints, even beyond the antagonist pairing required to permitunidirectional muscle force to produce bidirectional jointtorques. For example, the torque flexing the elbow joint (oneof the simpler joints in the primate upper extremity) isgenerated by brachialis, brachioradialis, biceps capitus brevis,and biceps capitus longus. Does this complexity serve anypurpose? If the control of end-point impedance of the limbwithout feedback is considered it will seen that these apparentactuator redundancies may have a functional role to play [13].Consider the simplified two-link model of the primateupper limb (forearm and hand treated as a single rigid body,pin-jointed to the upper arm) moving in a horizontal plane asshown in Fig. 3. In the absence of feedback, the staticcomponent of the total end-point impedance will solely be dueto the spring-like properties of the individual muscles. Foreach muscle, a potential function may be defined, and thecombined effect of multiple muscles is to define a totalpotential function (which could be determined by adding thepotential functions of the individual muscles). The totalpotential at any point is invariant under coordinate transformations and the total potential function may be expressedin any coordinate system by direct substitution.Now, for simplicity, assume that the relations betweenmuscle force and length and muscle length and joint rotationresult in a linear torque/angle relation for each muscle. Firstconsider the monoarticular (single-joint) muscles whichgenerate torques about only a single joint: their combinedTransactions of the ASMEDownloaded From: rg/ on 09/08/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

effect is to define a diagonal stiffness tensor in relativeangular coordinates:"7VKs0 P\ KeP2 0.T2(22)Each of the terms Ks and Ke may vary. For example, thestiffness about the human el

To ensure dynamic feasibility the choice of the impedance to be imposed should be based on the dominant dynamic behavior of the manipulator. The choice is a tradeoff between keeping the complexity of the controller within manageable limits while ensuring that imposed behavior adequately reflects the real dynamic behavior of the controlled system.

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