DSC-Vol. 42, Advances in RoboticsASME 1992

MODELING HUMAN ARM MOVEMENTSCONSTRAINED BY ROBOTIC SYSTEMS

Karin Hollerbach' and H. Kazerooni2Department of Mechanical Engineering

University of California. BerkeleyBerkeley, California

1. INTRODUCTIONABSTRACTFundamental difierences in human arm :>ehavior

can be related to two types of movements: constrainedand unconstrained. III constrained moveme Its, thehuman arm moves in such a way that the environmentcontinuously exerts a dynamic constraint on the arm.The "environment", in this instance, is the object that isin continuous coptact with the human arm. Inunconstrained movem~nts, the arm moves in itsworkspace without contact with the environment and,therefore, without cont~ ct forces between the arm andthe environment. Exarr.ples of constrained movementscan be seen when thE' arm is moving an exercisemachine or is constrained by an orthotic device. Insystems of this nature, the human arm dynamics areintegrated with the mbchine dynamics, resulting inbehavior specific to the total system. Therefore, theperformance and stabilit,' of the system taken a.:. a wholeare both functions of not only the machine dynwnics, butalso the human arm dynamics. Instability of nlachinesthat interact with humans has been rep\.' -ted in(Kazerooni 1990). In this article, we examine thedynamics of constrained movements of the hunlan arm,where the environment dictates the position, while thehuman maintains a desired contact force.

The human arm in constrained movements, isin continuous contact with its environment. Thedefining characteristic of the human arm in suchmovements is associated with the ability of the human toimpose desired forces. Examples of con!trainedmovements can be seen in telerobotic systems, orthotic

evices, and robotic systems worn by humans\Kazerooni 1991). We describe a theoretical andexperimental analysis for constrained movements of thehuman arm.

NOMENCLATURE

(l feedback parameter representing the bandwith ofthe human central nervous system

B viscosity in GmC DC feedback gainfh the contact force imposed on the environment by

the human armfi the intended force applied by the mu~cles, as

commanded by the central nervous systerrGa operator describing muscle activation dynamicsGcns operator relating neural input to the muscles and

the arm configuration, pGi operator mapping the position constraint, p, into

the contact force, f~Gf feedback operatlr whose output affects the

descending neural signals, nGm operator represe: \ting the inertial and visco-

elastic properties of the muscles and passivetissues surrounding the jointstiffness in aminertia in Gmdescending neural input to the musclesarm configuration

In contrast to f. ur work, previous studies havefocused on posture mail:tenance, disturbance rejection,and trajectory trackinb" in the presence of disturbingsinusoidal, trapezoidal, .1nd pulse forces. (Berthoz andMetral 1970) examined the dynamic behavior of thehuman elbow in respon;e to extemal forces applied atthe forearm, while the s,lbject maintained visual controlof the foreann position. In this study, the ratio of thedisplacement to the am]: ;itude of the imposed sinusoidalforce (admittance) was Dleasured at various freCiuencies.A maximum value for the admittance was obsz:rved inthe range of 3-5 Hz with a noticeable reduction above 6

KM

np

1 Ph.D. Candidate in Bioengi'1eering Graduate Group

2Associate Professor in Mechanical Engineering DepartrlJent

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Hz. In 1950, Wilkie concluded that "rigidity" of a limb isincreased by simultan~ous contraction of antagonistmuscles; this results in i smaller joint admittance andmore effective posture maintenance. This conclusionwas confirmed by Murray and Hogan (1989), whodetermined that deliberute co-contraction of antagonistmuscles is an effective means of rejecting torquedisturbances to main...ain posture. Furthermore,Murray and Hogan deml)nstrated that co-contraction is,in fact, used for this purpose in able-bodied persons.Winters and Stark (1985) provided further evidence forthe influence of co-contraction on limb res}:l>nse toexternal loading (force disturbance rejection) if. humansingle joint motion. In this study, a lumped parameter,antagonist muscle modt;l was employed to simulate avariety of movement tri. jectories from isometric to fastballistic motions. A iligh co-contraction level wasobserved to cause a strong resistance in low frequencyforces by raising both model stiffness and viscosity. Inan antagonist muscle model of the human wrist,Agarwal et al. (1970) studied the behavior of the humanwrist in response to pulse torque disturbances. Theyreported changes in stiff~ess and viscosity by factors of10 and 70, respectively, with varying levels of co-contraction.

2. DYNAMIC MODEL

We are interested in the behavior of the drm as itinteracts continuously with an environment thatimposes a position constraint on the arm, while thehuman can only attempt to regulate the contact force.This work can be consi(lered dual to the studies listedabove. The duality may be seen in the expetimentalmethods: Instead of aJ."'plying a disturbance force andmeasuring the resultallt movement, as was done inprevious studies, we apply a continuous positionconstraint and measure the resultant contact forcesbetween the source of he perturbation and the arm.Experiments of this nature will lead to an under-standing of force maintenance and position disturbancerejection. For exampl(, Crowninshield et al. (1976)studied differences between intact and injured humanknees in an effort to develop a clinically useful methodfor assessing ligamentous injury. In doing so, theycarried out a set of experiments in which they imposedposition constraints on the knee and reported tht! ratio ofthe knee force to the imposed position. Througllout thisarticle, this ratio will be referred to as the "impcdance".

We model constraIned, single joint movements ofthe human arm. As a specific example, the joint beingmodeled may be the elb..w joint, actuated by the elbowflexor (biceps) and the elbow extensor (triceps). We avoidattributing a particular class of dynamic behaviors toconstrained movements )f the human arm, as it is notclear whether the arm behaves as a force or a positioncontrol system in constrained motions. Maneuveringour hands in a stream of water from one point to anothertarget point, while struggling with the water current, isan example that shows the human arm can work as aposition control in a constrained space and cancontinuously accept a position or velocity comm&nd fromthe central nervous system. Alternatively, pushing apin into a wall is an example of a constrained movementwhere the human imposes a force on the pin, wiUloutbeing concerned with the pin position in the directionnormal to the wall; this system may be viewed as onethat accepts force commands from the central nervoussystem. Considering the above dilemma in attributing aparticular control action to constrained movements orthe human arm, we use a Norton or a Theveninequivalent concept (Senturia and Wedlock 1975) to arriveat a general substitute for the dynamic behavior of thehuman arm interacting with the environment. In Ulesame way that the choice of a Norton or Theveninequivalent does not affect the behavior of a c-'.rcuit incontact with other circuits, our choice in modE1ing thehuman arm by a Norton or a Thevenin equivaler: t has noeffect on the arm's interaction with other SYSteIr3.

Using the "force-current" analogy ~tweenelectrical and mechanical systems, a Norton equivalentis now chosen to model the human arm's dynamicbehavior as a non-ideal source of force interacting wiUlother systems. The notion of "non-ideal", as appliedhere, refers to the fact that the arm responds not only t4descending commands from the central nervous systembut also to position con:;traints imposed by interactionwith the environment. uh is that part of the contactforce that is imposed by the muscles, as commanded bythe central nervous system (Figure 1). If the arm doesnot move (i,e. the arm position, p = 0), the total contactforce, fh, is the same as uh. However, fh js also afunction of the environmental position constraint. If Ulearm moves, (i.e. the e;}vironment imposes a '>Ositionconstraint on the arm), the force imposed on theenvironment will differ from uh. The analogy can beobserved from the NortJn equivalent circuit snown inFigure l(c): The current, fh, is a function of not only thecurrent source, uh, but also the external voltage, p.Considering the above analogy, shown in Figure I, thecontact force, fh, can be lepresented by equation (1):

In this article, we arrive at a moC:el thatadequately describes a ~'ange of human elbow behaviorin constrained motions. The dynamic modelingpresented here is necE',;sary to design controllers foractive machines that interact with human arms.Section 2 is dedicated to a dynamic model of the humanarm. We use an input-output model, as the detaileddynamic behavior of the arm muscles is of less concernin this study and is i)!lplicitly accounted for in themodel. Section 3 descril-cs a set of experiments to verifythe theory. The exp'~riment employs a computercontrolled, active machine that imposes a positionconstraint on the hU)llan arm. This mat:hine isinstrumented with force sensors and encoolers formeasuring human arm forces and positions.

(1)fh = uh .Gil p

Throughout this article. Gi is referred to as the humanarm impedance and maps the environmental positionconstraint into the contzct force.

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Here we give a brief description of the internalstructure of the human arm impedance, Gi. usingFigure 2. Figure 2 consists of three elements:

1. Muscle Activation Dynamics. Ga, produces anintended muscle force, fi, in response to descendingneural commands. n.

Figure 2: The internal structure of Figure 1 (b) is presented, 'Nhere theneural feedbad mechanism, Gf, allows for regulation of the CCX\tact

fotte, fh.

2. Muscular Contraction and Passive Tissue Dynamics.Gm represents the pi operties of the muscles and thepassive tissues surrcunding the joint. and reducesthe intended force by Gm p.

3. Neural Feedback. Gf is a feedback operator regulatingthe force imposed by human arm on theenvironment. The following is the bri,f discussion of the three features

listed above:

Muscle Activation D~amics.Ga represents muscle activation dynamics and

maps the descending neural control signals into theintended muscle force, fi (Zahalak 1990). We rely on thismodel, since this type of model for muscle activationdynamics has been used with some success in otherwork (Zajac 1989). D'~coupled operators3 for muscleactivation dynamics arE: generally used with Hill-typemuscle models (Winters 1990). We are interested,however, in the overall behavior of the arm and wish tosuppress the details of muscle behavior. Th\JS, Ga isthe only operator we use that explicitly describe:; musclebehavior. A first order time lag h~s been suggested in(Zajac 1989) to represent Ga.

f",~~,>'i'ti.;.

II~.t~j,!

it'ii

Muscular Contraction arid Passive Tissue DmamicsGm is an impedance that reduces the intended

force, resulting in the total contact force, fh. Gmimplicitly takes into acl'ount both the internal muscledynamics, such as the f,'rce-velocity and length-tensionrelationships (Rack and Westbury 1984, Wilkie 1950, Hill1970), and the changing effective stiffness of the arm thatresults from varying the co-contraction of theantagonistic muscles; <'}m also includes the dynamicbehavior of the passive tissues surrounding the joint.Equation (2) describes Gm.

(a)

----fh=U h- Gip Gm = M 82 + B 8 + K (2)

where M represents the mass of the human arm, and Kand B define the visco-~lastic behavior of the musclesand passive tissues.

(b) (c)

Figutt 1 : (a) In constrained movements, the contact fon:e, f}\, is afunction of not only the commmds, n, from the central nervous system but

also of the imposed position Cllnstraint, p. (c) ttpresents the Nortonequivalent circuit of the dynamics shown in (b). The cun-ent, f}\, is a

function of not only the cun-ent source, n, but also of the imposed voltagedrop, p.

Note that in th~ human arm, the descendingneural control signals have two functions: (1) causingthe arm to move, and (2) altering the arm impedance,Gm. Here, we model the first of these functions by n in

3 Use of a decoupled operator for the activation dynamics impliesthat muscular activation is assumed to be independent of themuscular contraction process.

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3. EXPERIMENTAL Ml:THODS AND RESULTS

In this section, we provide preliminaryexperimental evidence in support of the model that wehave presented.

In order to relate the experimental evidence to themodel. we note that the measured force between therobot handle and the human hand is represented by thecontact force. th. in the model.

Figure 2. The second function, altering the armimpedance, is not explicitly shown but is accounted forin Gm. See Humphrey and Reed (1983) f,)r moreinformation on neurophysiological evidence for theexistence of two separate cortical systems, one fordetermining the commanded limb trajectory, tIle otherfor specifying the level of co-contraction.

Neural FeedbackThe interaction force, th, exerted by the arm on its

environment is used as a feedback signal to modulatethe descending neural signals. This feedback is effectiveonly at relatively low frequencies because of the limitedbandwidth of the central nervous system. Since thehuman hand is able to very accurately apply a desiredinteraction force at very low frequencies, we deduce thatGf is very large at low frequencies. (The experimentalresults, discussed in Sec{ion 3, will clarify this.) On theother hand, since the human cannot apply a desiredinteraction force at high frequencies, Gf must be verysmall in the high frequency range. Assuming Gf as alinear transfer function, equation (3) is an appropriatechoice

Several experinlents have been conducted tomeasure the human arm impedance represented in themodel by OJ. In the experiments, the subject grabbed therobot shown in Figure 3. The robot was commanded tooscillate via small-amplitude sinusoids of knownfrequency in the sagittal plane (parallel to the plane ofsymmetry in the human body) along the y directionshown in Figure 3.

At each oscillation frequency of the robot, thehuman operator attempted to follow the robot so that zerocontact force was maintained between his hand and therobot. Two experiments were conducted at two differentco-contraction levels. In the first experiment, thesubject maintained a relatively loose grip on the robothandle. The contact force. £'h. and position, p, weremeasured at each oscilla lion frequency. The ratio of fhto P. at each frequency, represents Gi and is shown inFigure 4. In the second t!xperiment, the subject grabbedthe handle with a relatively tight grip. Figure 5 showsthe results of the second experiment.

cGr =~ (3)

where a. is a small number representing the bandwith ofthe central nervous systE"m, and C is a feedback gain. Inaddition, one may introduce a pure time delay operatorin order to account for nl:ural conduction delay.

In comparing the data in the two figures, we observethat the human arm iI!lpedance at low co-cor tractionlevels (Figure 4) is significantly smaller than it is athigh co-contraction levels (Figure 5) in the low fI'aquencyrange. These results are hardly surprising, g.ven theevidence discussed above that both stiffness and viscosityare roughly proportional to mean muscle tenbion andthat co-contraction has an especially strong influence onboth parameters at low frequencies (Winters and Stark1985).

Simplifying the block diagram shown in Figure 2results in the transfer functions shown in Figure 1.

fi = Gcns n + Gi p (4)

where

(5)

At high frequenciE.., the forearm is moved withoutthe benefit of the active feedback loop, Gf. In the highfrequency region, we observe that the impedancebehaves like a purely iDt'rtial load in both plots. Thesedata agree with previouo; results (Lehman and Calhoun1990, Wieneke et al. 1971\), in which passive wrist motionwas shown to be second order and dominated by themoment of inertia.

GmGi = 1 + GfGa (6)

Gcns and Gi. represent. respectively. the effect of thecentral nervous syst~m commands and of theinteraction force. fh. These operators are shown inFigure 1 (b). Note that Figure l(b) mathematicallyrepresents the concepts of Figure 2: The force imposedby the human arm on the environment is the result ofboth the central nervo'ls system commands and theenvironmental position (onstraint.

From the curve fitted to the experimental datadepicted in Figures 4 and 5, we obtain the th.,oreticalimpedance calculation:

A comparison of our modeling approach with thatof Wieneke and Denier van der Gon (1974) leads to theduality described in Section 1. In their modelingapproach, an interactilffi force was modeled as theimposed disturbance, w:1ile the resultant position wasmodeled as the feedback variable. In contrast, in ourmodeling approach, the interaction force is modeled asthe feedback variable, \It hile the position is modeled asthe imposed disturbance

Gi = 0.143 s 2+1 s+ 2.51 lbf/ft(for Figure 4) (7)

Gi = 0.277 s 2 + 2.38 s + 12..1

(8)Ib£fft.(for Figure 5)

Cross over frequencies in experimental Gi of 3-6rad/sec were observed. ['he cross over frequency is the

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lQfJyimum frequency at which the subject was able toa ltely control the constrained movement, for agi~ _oj co-contraction level. The cross over frequencycreates the distinction between the low and the highfrequency regions of operation and is itself defined by thetime delays associated with finite neural conductio~velocities.

120

100

~80

~~60p'V

«IIn the experiments described here, the humanelbow joint was made to operate solely within the mid-portion of its full range of motion. Thus, we avoidedsignificant non-lineariti~s associated with joint torquescaused by passive tissues around the joint. Lehman andCalhoun (1990) determined that passive elastic torquesin the human wrist are small and relatively constantwithin 40 degrees in either direction from the middle ofthe wrist's range of mot.ion, and rise rapidly fit eitherextreme of the full range of motion. Similar resultswere observed by Hayes and Hatze (1977) in the humanelbow joint. Moreover, the passive viscous torque wasfound to vary by a factor of approximately 5, with anobvious minimum at a point within the mid-range of thejoint.

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II' ~

10-1 100 101mq ~1ICy (r8dlnc)

loZ 101

Figun 4: The experimental and theoretical plot of the impedance, Gj,of the human ann at the elbow joint; the operator is maintaining a loose

grip on the robot (low co-contraction levels).

i'"~P'G

Figure 5: The experimental and theoretical plot of the impedance, Gj,of the human arm at the elbow joint; the operator is maintaining a

tight grip on the robot (high co-contraction levels).

DISCUSSION AND CONCLUSION4.

The experimental results discussed here representpreliminary evidence in favor of the model's predictionsof arm impedance at varying co-contraction levels andfrequencies, when the position of the arm is constrainedby the arm's environmelt.

At very high frequencies, the neural feedback loopis opened, as neural conuuction velocities are too slow toallow the body to employ corrective measures based onafferent neural information. The high frequencyimpedance between the applied position constraint andthe contact force repre~ents a measure of the model'sperformance without the aid of voluntary feedbackinformation. The experimental data regarding the armimpedance support our model's predictions: Since thehuman arm cannot keep up with the high frequencYmotion of the robot, the arm's mass dominates itsdynamic behavior. The contact force is equal to theproduct of the robot acceleration and the human arminertia, by a direct application of Newton's Second Law,and a second order tran:;fer function is expected for theimpedance at high freQ"lencies. Consequently, a large

human arm

Figure 3: The experimenul set-up: the robot is commanded to movesinusoidally, while the huma:t operator grasps the robot handle and? pis to maintain a zero ctlntact force between his hand and the

handle.

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impedance, resulting in large contact forces, is expectedat high frequencies.

Murray, W. R. and Hogan, N. (1989) Experimentalobservations on the maintenance of elbow posture in thepresence of disturbances. Issues in The Modeling andControl of Biomechanical Systems (Edited by J. L. Stein,J. A. Ashton-Miller, M. G. Pandy), pp.19-28. ASME.

Rack, P. M. and Westbury, D. R. (1984) Elasticproperties of the cat soleus tendon and their functionalimportance. J. Physiol. 347, 479-495.

Senturia, S. D. and Wedlock, B. D. (1975) ElectronicCircuits and Applications, pp. 51-58. John Wiley & Sons,Inc.

At very low frequencies, the neural feedback loopis closed, and the arm impedance approximates aconstant value that depends upon experimentalparameters. In this region, we expect much lessagreement between experimental results and modelpredictions than we expect in the high frequency region,because the value of the impedance depends uponfeedback and muscle parameters as well as muscularco-contraction levels. The effect of these parameters ismore difficult to quantify than that of the arm inertia,the significant parameter at high frequencies. Thepredicted impedance of the arm, in the low frequencyregion, can only be given as a bounded range ofimpedances. At low enough frequencies, thehuman can follow the imposed motion comfortably, andhe can always establish approximately constant contactforces between his hand and the environment, resultingin a constant impedance when contact forces are small.Although the value of this impedance increases withincreasing muscular co-contraction, it remainsessentially constant throughout the low frequencyregion, once a given level of co-contraction has beenestablished.

S. REFERENCES

Agarwal, G. C., Berman, B. M., and Stark, L. (1970)Studies in postural cor.trol systems. Part I. Torquedisturbance input. IEEE Trans. Systems ScienceCybernetics 6,116-121.

Berthoz, J. A. and Metral, S. (1970) Behavior of amuscular group subjected to a sinusoidal andtrapezoidal variation in force. J. Appl. Physiol. 29, 378-384.

Wieneke, G. H. and Denier van der Gon, J. J. (1974)Variations in the output impedance of the human motorsystem. Kybernetik 15, 159-178.

Wilkie, D. R. (1950) The relation between force andvelocity in human muscle. J. Physiol. KIlO, 248-280.

Winters, J. M. and Stark, L. (1985) Analysis offundamental human movement patterns through theuse on in-depth antagonistic muscle models. IEEETrans. on Biomed. Engr. BME32, 10, 826-839.

Winters, J. M. (1990) Hill-based muscle models: asystems engineering perspective. Multiple MuscleSystems: Biomechanic.: and Movement Organization(Edited by J. M. Winters and S. L-Y. Woo), pp. 69-93.Springer Verlag, NY.

Zahalak, G. I. (1990) Modeling muscle mechanics (andenergetics). Multiple kluscle Systems: Biomechanicsand Movement Organiz.'1tion (Edited by J. M. Wintersand S. L-Y. Woo), pp. 1':23. Springer Verlag, NY.

Zajac, F. (1989) Muscle Ilnd tendon: properties, models,scaling and applicatiol} to biomechanics and motorcontrol. CRC Grit. Rev. i."t Biomed. Engr. 17,359-415.

Hayes, K C. and Hatze, H. (1977) Passive vis(o-elasticproperties of the structures spanning the hum&fi elbowjoint. Eur. J. Appl. Physiol. 37, 265-274.

Hill, A. V. (1970) First 3nd last experiments in musclemechanics, pp. 23-41, 12'i-129. Cambridge Univ. Press.

Humphrey, D. R. and Reed, D. J. (1983) Separatecortical systems for the control of joint movement andjoint stiffness: reciprocLI activation and coactivation ofantagonist muscles. Adu. Neurology 39,347-372.

Kazerooni, H. (1990) Httman-Robot Interaction via theTransfer of Power and Information Signals. IEEETrans. on Systems and Cybernetics, Vol. 20, No.2, 450-463.

Kazerooni, H., Mahoney, S. (1991) Dynamics anc Controlof Robotic Systems Worn By Humans, ASME Jc ;trnal ofDynamic Systems, Measurement, and Control. Vol. 113,No.3, September 379-387

Lehman, S. L. and Calh.)UD, B. M. (1990) An identifiedmodel for human wrist movements. Exper. Brain Res.81,199-208.

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