Pergamon J. Biomechanics, Vol. 28, No. 11, pp. 1369%1376,1995 Copyri& 0 1995 Elsevicr Science Ltd hinted in Great Britain. All rifits reserved 0021-9290/95 $9.50 + .oO

0021-9290(94)00185-J

AN INVERSE DYNAMICS MODEL FOR THE ANALYSIS, RECONSTRUCTION AND PREDICTION OF BIPEDAL WALKING

Bart Koopman, Henk .I. Grootenboer and Henk J. de Jongh University of Twente, Faculty of Mechanical Engineering, Laboratory of Biomedical Engineering, P.O. Box 217,750O AE Enschede, The Netherlands

Abstract-Walkingis a constrainedmovement which may best be observed during the double stance phase when both feet contact the floor. When analyzing a measured movement with an inverse dynamics model, a violation of these constraints will always occur due to measuring errors and deviations of the segments model from reality, leading to inconsistent results. Consistency is obtained by implementing the constraints into the model. This makes it possible to combine the inverse dynamics model with optimization techniques in order to predict walking patterns or to reconstruct non-measured rotations when only a part of the three-dimensional joint rotations is measured. In this paper the outlines of the extended inverse dynamics method are presented, the constraints which detine walking are defined and the optimization procedure is described. The model is applied to analyze a normal walking pattern of which only the hip, knee and ankle flexions/extensions are measured. This input movement is reconstructed to a kinematically and dynam- ically consistent three-dimensional movement, and the joint forces (including the ground reaction forces) and joint moments of force, needed to bring about thts movement are estimated.

INTRODUCIION are, except for the double support phase,completely determinedby the segmentaldisplacements. However, Numerous models have been developed to simulate sincethe measureddisplacements have to be differen- human walking, based on segment models in varying tiated twice with respectto time in order to obtain the complexity from three (McMahon, 1984) up to 17 accelerations,this may resultin large numericalerrors segments (Hatze, 198la).Symmetry betweenthe right whenno precautionsare taken. To avoid theseerrors, and left leg is often assumed to reduce complexity (e.g. Chao and Rim (1973)combined optimization tech- Brand et al., 1982),and the movement is often re- niqueswith the direct dynamicsmethod to calculate stricted to the sagittal plane only. the joint momentsof force in normal walking. The two ways to apply the equationsof motion to Optimization techniquesare also usedin predictive the segmentsmodel are usually referred to as the models,where both the movementand the internal ‘direct dynamicsmethod’ and the ‘inversedynamics forces have to be calculated.This approachmay find method’. In the direct dynamicsmethod, the move- a wide application in rehabilitation technology, for ments of the segmentsare calculated by integrating example,to calculatethe effect of prosthetic compo- the equationsof motion. This is only possiblewhen nentson the walking pattern or in functionalelectrical the joint momentsof force are known or assumed to stimulation.Chow and Jacobson(1971) were the first be zero. The latter is the case in ballistic walking to developa (semi)predictive modelfor walking: with (McMahon, 1984).It is alsopossible to choosefor the prescribedhip trajectories,ground reaction forcesand internal forces such values that a normal walking ankle momentsof force they predicted the hip and pattern results (e.g. Pandy and Berme, 1988) or to knee angles and moments of force by minimizing calculatethe joint momentsof force from estimations a criterion reflecting the mechanicalenergy expendi- of the muscle forces (e.g. Olney and Winter, 1985). ture. The complex model of Hatze (1981b) includes In the inverse dynamics method, the joint forces muscledynamics and is applied to predict the long and joint momentsof force are calculatedfrom a pre- jump. scribedmovement. Since the segmentalmovements, in The predictive model presentedhere is, in contrast contrast to the internal forces, can be measured,this to other existing models,based on a combinationof method is commonly appliedfor the analysisof meas- inverse dynamics and optimization techniques. There ured movements.Hereby the measuredground reac- are two reasonsfor choosing the inverse dynamics tion forces are used as an input for most of these method insteadof the direct dynamicsmethod: first, models(e.g. Brand et al., 1982).This, however,is not sincemovements can be measuredand internal forces a necessityfor a bipedal model, as was shown by and moments of force cannot, the inverse dynamics Hardt and Mann (1980).The ground reaction forces part of the model can be usedseparately for the gait analysisof measurement.Second, walking is a con- strained movement, and it is easier to implement Received in jinal form 21 March 1994. kinematic constraints in an inverse method than in

1369 1370 B. Koopmanet al.

a direct method. One kinematic constraint is, for segments(Fig. 2): the upper legs,lower legs,feet, pelvis example, that the feet must be exactly on the floor and the head, arms and trunk (HAT) segment.The during the stance phase. However, the distinction number of segmentsis a compromisebetween the betweeninverse and direct dynamicsis not essential wish to avoid unnecessarycomplexity and large com- sinceboth methodsshould eventually yield the same putation times, on the one hand, and to simulatethe results. movement adequately on the other hand. The seg- Dynamic constraintsare appliedto achievethat the mentsare connectedin the joints; the point of contact segmentsmodel is in balancefor the completewalking betweenthe foot and the floor is modeledas if it were cycle,which may bestbe explainedin the frontal plane a joint aswell. In this view, the floor isa segmentwith (Fig. 1). With an input consistingof the sagittal hip, zero velocity and infinite mass,which makesit pos- knee and ankle rotations only, the foot will move sible to calculate the ground reaction forces in the straight under the hip joint. Sincethe ground reaction sameway as the joint forces. A referenceframe is forces can only apply to the foot and are in general attached to the floor, with the x-axis pointing in the not in the line of action of the static and dynamic walking direction, the y-axis pointing upward and the forces acting on the total center of massof the seg- z-axis perpendicular to the xy-plane in the lateral mentsmodel, an imbalancemoment M is necessaryto direction (Fig. 2). keep the body upright [Fig. l(a)]. This imbalanceis In each segmenta local frame is definedaccording reducedby estimatingthe leg adduction in an optim- to the method usedby Brand et al. (1982),which is ization schemeand thus meeting the dynamic con- basedon the location of someanatomical landmarks. straints [Fig. l(b)]. It is extendedwith definitionsfor the segmentsof the The application of kinematic and dynamic con- feet and the HAT segment.A detailed description of straints eventually results in a movement of the the construction of theseframes, as well as a list of all segmentsmodel which is consistent with physical segmental parameters, has been given elsewhere demands:all deviations due to measuringerrors and (Koopman, 1989). approximationsin the segmentsmodel are corrected The mass,the position of the center of massin the for. As an example,the model is appliedto a normal local frame and the moment of inertia tensor are gait pattern which is basically two-dimensional:only determinedwith the regressionequations of Chandler the hip, knee and ankle flexions/extensionsare meas- et al. (1975).These relations depend on the local ured. The non-measuredrotations are estimatedwith dimensionsof the segmentsand the total body weight, the constraints,and the resulting three-dimensional and are in some respectsadapted to the segments movementis analyzed. model usedin this study: the properties for the foot- segmentare corrected for the influenceof the shoe, METHODS and a distinction is madefor the HAT and the pelvis.

The segmentsmodel In the segmentsmodel the human body is modeled as a coupledsystem of right bodiescontaining eight

Fig. 2. The segmentsmodel for the body at rest,with seg- ments: pelvis (I), R-L femur (2-31, R-L tibia (4-5), R-L foot Fig. 1. Imbalanceof the segmentsmodel. The imbalance (6-7) and HAT (8);with joints: R-L hip (l-2), R-Lknee (3-4), momentM canbe viewed as resulting from an eccentricity of R-L ankle (5-6), R-L foot-floor (7-8) and HAT-pelvis(9); the groundreaction force FR with the body weightF, (a) and the definition .of the. reference. frame: forward axis (x), andis correctedwith a hip adduction(b). verticalaxis (y) andlateral axis (2). An inverse dynamics model of bipedal walking 1371

[t is assumed that the axes of the local frames of the With known joint rotations, all segmental move- segments are in the principal directions, so that ments are defined relative to an arbitrary reference the principal moments of inertia as determined by point and with an unknown reference orientation. Chandler et al. can be used. These are constructed by applying the kinematic All joints are modeled as potential ball-hinges with constraints. three independent rotational degrees of freedom. -The reference orientation (as a function of time) However, the nature of the joint (ball-hinge, line- is deduced from the demands that at the time of heel hinge) is determined by the input joint rotations of the contact, both feet touch the floor and during a part of inverse dynamics model and the constraints: when the single stance phase (foot flat) the rotation of the the joint angular velocity vector has a fixed direction, foot relative to the floor is zero. the joint will effectively act as a line-hinge. The posi- -The displacement of the reference point is cal- tions of the joints in the local frames of the segments culated from the constraint that during the stance may depend on the joint rotation and thus vary impli- phase no slipping between foot and floor may occur. citly with time, so joint translations are allowed but Other kinematic constraints check for the kinematic only as functions of the joint rotations. consistency of the movement and, if necessary, correct An example of this is the modeling of the roll-over for these: of the foot segment: the point of contact between foot -During the double support phase both feet must and floor shifts from heel to toe during stance. It is remain on the floor and no slipping between foot and assumed that the instantaneous point of contact can floor may occur. Due to measuring errors, deviations be defined as the point where the resultant ground of the segments model or in this case simply the fact reaction forces apply to the foot and that the location that not all joint rotations are measured, this con- of this point in the local frame of the foot depends on straint will always be violated. The necessary correc- the rotation of the foot relative to the floor only. The tions are preferably performed on rotations which are set of all contact points defines the effective shape of not measured, in this case on the pelvic segmental the (deformed) foot. From a defined foot shape it is rotations around the forward and vertical axes. possible to calculate the location of the contact point -During the swing phase the foot must remain in foot coordinates as well as in floor coordinates as above the floor and the legs may not collide. Also a function of the foot rotation. The foot shape that is here-to corrections are performed on the not-meas- used in this study is based on measurements and ured rotations (i.e. pelvic rotations). is shown in Fig. 3. Not allowing for independent joint -The calculated velocity is corrected for the meas- translations effectively reduces the number of inde- ured velocity by assuming a sinusoidal pelvic rotation pendent degrees of freedom. around the vertical axis.

Construction of the movement Kinetics The movement of a segment is the combination of The ground reaction forces are unequivocally de- translation and orientation of the local frame relative termined by the movementsof the segments,e.g. the to the reference frame. The orientation is uniquely acceleration of the total center of mass,with one defined by a rotation tensor, its components are cal- exception: the force component that acts on the culated using Euler parameters to assure ortho- straightline betweenthe two contact points of the feet gonality (Koopman, 1989). Rotations are presented in with the floor. It is assumedthat this counteracting clinical angles. force is zero. Instead of defining four sets of force

E -O.OO Q 5 .g s 8 j. -0.10 -

-0.10 0.00 0.10 0.20

x-coordinate [ml Fig.3. Lateralview of the foot shapein localfoot coordinateswith the anklejoint at theorigin x = y = 0. 1372 B. Koopman et al. equations for each phase of the walking cycle (e.g. tinuous and at each point of time of the exact history Hardt and Mann, 1980), a shift function f is intro- and future of the movementthis is known. duced which defines the shift of body weight from one The number of Fourier coefficientsis chosensuch foot to the other during the double support phase; that the differencebetween the Fourier seriesrepre- fequals one during single support and zero during the sentation and the actual, measuredand averagedro- swing phase. The set of vector equations to solve the tation is very small comparedto the measuringand joint forces and ground reaction forces becomes averagingdeviations. In practice, a representationup to the 10th waveform,resulting in a total of 21 Fourier Nij coefficients,is for most walking movementssufficient. C Fij = mi(%i - g) Insteadof usinga minimal energy criterion for the j=l objective function, such as proposedby Chow and FR =f ; m,(& -g), Jacobson (1971) and also used by Vaughan et al. i=l (1982a,b),a slightly modified criterion, U, is mini- mized: FL = (1 -f) ; mi(%i -g), (1) i=l where Fij is the force vector in joint j acting on wherek denotesthe coordinate number,Tis the stride segment i, mi is the mass,%i is the accelerationvector time, Mjk is the calculatedjoint momentof force and of the center of mass,g is the accelerationof gravity M ,,,*=,jk is(depending on the signof Mik) the maximal vector, Nij is the number of joints at segmenti, N, is positive or negative isometricjoint momentof force, the number of segmentsand FR and FL are the right obtained from the literature (Koopman, 1989) or and left floor reaction force vectors.The equationsfor basedon the resultsof musclemodels (Thunnissen, the left and right floor reaction forcesare equivalent. 1993).ajk and fl are optimization parameters,with ajk Likewise,the equationsfor the joint momentsof force equalto one or zero (when no joint rotation in direc- are tion k is allowed)and #I has a chosenvalue of 2. This Nfi criterion showsa large resemblancewith the fatigue C (Mil + rij x FiJ = !!!fd, criteria usedby Crowninshieldand Brand (1981)and j=l Dul et al. (1984a,b) to determine the load sharing Ma =f Mci betweensynergistic muscles. In criterion (3) the load sharingbetween the joints is basedon the properties of the joints. Therefore, this criterion will also be referred to as a fatigue criterion. (2) Kinematic constraintsare usedto construct a con- sistent movement for each iteration of the optimi- where Mij is the moment of force vector in joint zation problem. Other kinematic constraints are j working on segmenti, Ji is the inertia tensor,Oi is the implementedfor the optimization only and definethe rotation velocity vector, and rij is the vector pointing range in which the solution has to be found. from the center of massto joint j. MG is the total -The rangesof motion in eachjoint defineallow- ground reaction momentof force vector which must able joint angles. be zero except for the vertical componentthat repre- -Allowable rangesfor the Fourier coefficientsof sentsthe friction betweenfeet and floor. The horizon- the rotations restrict the maximal angular acceler- tal componentsof MO representthe imbalanceof the ations. segmentsmodel that is dealt with in the dynamic A violation of theseconstraints resultsin an increase constraints. of a penalty function that is multiplied with the objec- tive function U. Optimization With the dynamic constraints,the imbalancemo- In general,the optimization problemcan be stated ment of force Ma is minimized. This is implemented as follows. Find, in subsequentiterations, estimates as a dimensionlesstime integral of the componentsof for the rotations that are not measuredby minimizing Ma. weightedwith momentsof force which represent an objective function. The measuredjoint rotations an allowableeccentricity betweenthe calculated and have fixed values,while the non-measuredrotations prescribedcontact points on the surfaceof the foot. are initially zero. The solutionshould not violate any This time integralis addedto the objectivefunction as constraints. a penalty function. As a result, the solution of the The joint and segmentalrotations are represented optimization is both kinematically and dynamically by finite Fourier series.One of the reasonsfor this consistent. representationis that the Fourier coefficientsprovide The optimization is performed with standard a relatively smallset of optimization variables,which numerical rout@ utilizing a quasi-Newton algo- are necessaryto avoid large computation times.Fur- rithm (Numeric&Algorithms Group, 1984),which is thermore,the movementis implicitly cyclic and con- modified for this specificproblem. An inverse dynamics model of bipedal walking 1373

Measured, constructed, reconstructed and predicted ankle ad/abduction, leg joints endo/exorotations)are rotations not optimized and keep their initial zero values. For each leg three electrical goniometersare used to measurethe hip, kneeand ankle flexions/extensions RESULTS as functions of time. The goniometersare connected to eachother in a light-weight external frame which is The componentsof the imbalancemoment MG in attached to the leg, with the axes of rotation posi- the forward (x-)direction and lateral (z-)direction tioned in the assumedjoint axesof rotation. To allow reach maximal values of about 120 and 80 N m re- for variations in the directions and positionsof the spectively,which cannot be neglectedin comparison joint axes of rotation, the external frame is not rigid to the joint momentsof force (i.e. the ankle plantar but consistsof rodsand tubesthat canslide relative to flexion moment has a maximum at about 120N m). each other. Relatively smalljoint rotations are sufficientto reduce The time parametersof the walking movement, the imbalanceand fulfil the dynamic constraints,as is which are determinedwith the points of time of heel shown in Fig. 5. The reconstructedhip adduction is contact (HC) and toe-off (TO), are measuredwith suchthat the foot is placed straight under the middle electricalfoot contact switches.Under eachfoot, two of the pelvis during the stance phase. During the switchesare placedin parallel, oneat the heeland one swingphase, the leg tendsto swingstraight under the joint in front of the ball of the foot. hip joint [Fig. 5(a)]. The reconstructedHAT rotation The output voltages are measuredon a personal showsthat the segmentsmodel hasto lean forward in computer with a samplingrate of 200 Hz. The subject order to keep balance[(Fig. 5(b)]. The dynamiccom- of the experimentis askedto walk a distanceof about ponent of the forward rotation, representedby the 25 m at a comfortable speed;the measurementstake amplitude,is essentialin the balancingprocess. Small place during 10 s in the middlepart of the walkway. angular fluctuations result in large angular acceler- Here the averagevelocity is also measured. ations due to the relatively large momentof inertia of A normal walking pattern of a healthy subject the HAT segment.The reconstructedrotations agree (male,24 yr, 83 kg, 1.83m) is measuredas an example well with measurements,reported in the literature (e.g. to be analyzed. Of this subject, someanatomical di- Inman et al., 1981;Kadaba et al., 1990),at leastwhere mensionswere measuredto define the segments the tendenciesare concerned.The rotations’ offsets model. The data of subsequentcycles are averaged may differ more which is mainly causedby the differ- following standardprocedures (Koopman, 1989),as ent definitionsof the offsetsin the local experimental well as the data for left and right leg, for which setups. symmetry is assumed.The averagejoint rotations The pelvic rotations around the forward (x-)axis with their standarddeviations are shown in Fig. 4. and vertical (y)axis [Figs. 6(a) and (b)] are determined From the non-measuredrotations, the pelvic rota- by the kinematic constraints:the x-rotation is neces- tions are constructed with the kinematic constraints. sary to keep both feet on the floor during the double The hip adduction and the HAT rotation around the support phaseand the y-rotation to increasethe step lateral axis are reconstructed mostly from the dy- length and meet the prescribed average velocity. namic constraints. Minimization of the fatigue cri- There is a good agreementwith measuredrotations, terion II predictsthe HAT rotations around the verti- found in the literature (e.g.Inman et al., 1981;Kadaba cal and forward axes.The other rotations (knee and et at., 1990,Vink and Karssemeijer,1988): the general

a Y 0 E = -30 % 0 20 40 60 60 100 8 -60 E -40 0 20 40 60 60 100

time [K of cycle]

a 0 20 40 60 60 100

time 1% of cycle] Fig.4. Measuredaverage joint rotations:hip flexion(a), knee extension (b) and ankle dorsiflexion (c) with standarddeviations during one cycle. The supportphase is from O-60%, the swingphase from 60to lOO%, and the doublesupport phases from 0 to 10%and from 50 to 60%. 1374 B. Koopman et al.

0 20 40 60 60 100 0 20 40 60 60 100

time [% of cycle] time [% of cycle] Fig. 5. Hip adduction (a) and HAT segment forward (z-) rotation (b) reconstructed with dynamic constraints.

0 20 40 60 60 100 0 20 40 60 60 100

time 1% of cycle] time [SC of oycle] Fig. 6. Rotations around the forward (x)-axis (a) and vertical (y-)axis (b): pelvic rotations constructed with kinematic constraints (solid lines) and predicted HAT rotations (dashed lines). measured pattern for the x-rotation is a zero rotation component also shows a good agreement, although at or just after heel contact and a maximum rotation the assumption that no counteracting force between of about 5” just after toe-off. both feet exists during the double support phase re- The predicted HAT rotations are shown in the sults in an underestimation. same figure. The rotation around the forward (x-)axis Figure 8 shows the resulting moments of force at is more or less contrary to the pelvic x-rotation the major joints. The hip abduction moment of force [Fig. 6(a)]. By rotating this way the HAT is moved reflects the vertical ground reaction force, which is not towards the direction of the supporting leg, thus surprising since the foot is placed under the body adding to the mechanisms that keep the body in during the support phase. The largest hip flexion/ balance during walking. The rotation of the HAT extension moments occur during the double support segment around the vertical (y-)axis is also inverse to phase: when both feet are on the floor, the body is in the pelvic y-rotation [Fig. 6(b)]. This can also be its most stable position to control the rotation of the observed in normal walking (e.g. Inman et al., 1981) HAT segment where most of the body mass is located. where the arms are swung contrary to the legs. The The knee extension moment controls the knee flexion effect of this inverse rotation is that the variation in during double support and with the ankle plantar the vertical angular momentum of the total body is tlexion moment some of the energy, needed for the minimal. push-off, is generated. The estimated ground reaction forces for this move- ment of the segments model are compared in Fig. 7 with measured data from Winter (1987). The esti- DISCUSSION mated vertial force component shows the character- istic pattern of weight acceptance and push-off. The The estimated rotations resulting from the con- agreement is reasonably good when the trends are struction, reconstruction and prediction agree well compared, although the amplitude of the measured with expectations based on the mechanics of walking. vertical accelerations is somewhat larger. This can Moreover, there is a good agreement with measured partly be explained with the effect of smoothing func- data, reported in the literature, at least when the tions, which were necessary when determining the tendencies are considered. A sensitivity analysis segmental accelerations numerically, and partly with showed that the fatigue criterion [equation (3)] only inevitable modeling deviations. The forward force had a small effect on the constructed and reconstruc- An inverse dynamics model of bipedal walking 1375

The resultsindicate that moving the trunk is an important balancingmechanism during walking. Due to the large massand momentof inertia of the trunk (HAT segment),only small rotations and displace- mentswill sufficeto reducethe effect of deviations in the walking pattern. The displacementsof the segmentsrelative to the floor and the pelvic rotations are,due to the kinematic constraints, partly determined by the foot shape.

-0.50 ' ' ' ' ' ' ' ' A realistic foot shape will improve the results, as 0 20 40 60 60 100 numericalexperiments with different foot shapes(e.g. a stilt foot with a singlecontact point and an ellipsoid time 1% of cycle] foot shape)have shown.Likewise, the foot shapethat Fig. 7. Estimatedforward and verticalground reaction for- is usedaffects the estimatedground reaction forces. ces(solid lines), 1) = 1.4ms-‘, normalizedwith the body The proposedmethods may find various applica- weight,compared with measureddata from Winter(1987) tions in gait analysissystems: firstly, it is possibleto (dashedlines), u = 1.39m s-l, ensembleaverage 19 subjects. perform a full three-dimensionalanalysis, even when no force-plate data is present and only a limited number of joint rotations are measured.Although it ted rotations. However,the agreementbetween expec- will be difficult to assessthe quantitative accuracy of tations and the predicted rotations (the HAT rota- the estimatedforces and torques, it may well be used tions around the forward and vertical axes)does not for a qualitative comparison.Secondly, in a setting prove the validity of the fatigue criterion, it merely with a three-dimensionalmeasuring system, the con- showsthat this type of criterion yields sensibleresults straints provide a valuablecheck for the accuracy of for this type of movement. the measuringsystem and the segmentsmodel. Espe- Relatively smalljoint angleswith an amplitudeof cially the imbalancemoments of force and the differ- about 5” have a large impact on the dynamicequilib- ence between the estimatedand measuredground rium of the segmentsmodel, as is shown with the reaction forces could be minimized to improve the reconstruction of the hip adduction and the HAT accuracy. rotation with the dynamicconstraints. The imbalance Finally, the model can be usedto predict walking momentof force can be viewed asan overall measure movementsin situationswhere measurementsare not for the accuracy of the measureddata and segments feasible,for example, in the design processof new model.A good estimateof the joint momentsof force prostheticand orthotic componentsor in studying the is only guaranteedwhen this imbalancemoment is effect of a variation of parameters.In this case the small; the measuringsystem should be such that the parametersdescribing the measuredmovement may small anglesare still measuredaccurately. Unfortu- be optimized as well, while the propertiesof the new nately, in most reported gait analysis studies,this designare incorporatedin the segmentsmodel. Again, dynamic consistencyhas not beenchecked. it will bedifficult to assessthe quantitative accuracyof

120 120 a b P 5 60 c .g J 8 0' -.

s ,_~-60 c

-120 - -I 0 20 40 60 SO 100 0 20 40 60 60 100

0 20 40 60 .60 100; 0 20 40 60 60 100

time [% of cycle] time I% of cycle]

Fig. 8. Calculatedhip abduction(a), hip extension(b), kneeextension (c) and ankle plantar flexion (d) moments of force. 1376 9. Koopmanet al. the predictions,but the resultswill yield a qualitative Chao,E. Y. and Rim, K. (1973) Application of optimization impressionof the changesthat will take place. principles in determining the applied moments in human leg joints during gait. J. Biomechanics 6, 497-510. The choice for the inverse method instead of the Chow C. K. and Jacobson, D. H. (1971) An optimal pro- direct method in the predictive model is somewhat gramming study of human gait. Math. Biosci. IO, 103-l 10. arbitrary. However,the inversemethod offersthe op- Crowninshield. R. D. and Brand. R. A. (1981) A nhvsiolo- portunity of an easyimplementation of the kinematic gically based criterion of muscle force prediction in loco- constraints.With the use of the constraintsand with motion. J. Biomechanics 14, 793-801. Dul, J., Townsend, M. A., Shiavi, R. and Johnson, G. E. the representation of the movement with finite (1984a, b) Muscular synergism I-II. J. Biomechanics 17, Fourier series,all solutionsof the optimization prob- 663-673, 675-684. lem will automatically be valid and cyclical walking Hardt, D. E. and Mann, R. W. (1980) A five body-three movements.In fact, the constraints define what is dimensional dynamic analysis of walking. J. Biomechanics 15, 741-745. walking. Hatze, H. (1981a) Quantitative analysis, synthesis and op- The main disadvantageof the inverse dynamics timization of human motion. Hum. Mvmt Sci. 3, 5-25. methodin comparisonto the direct dynamicsmethod Hatze, H. (1981b) A comprehensive model for human motion is often consideredto be the amplification of noiseby simulation and its application to the take-off phase of the numerically differentiating twice. To avoid smooth- long jump. J. Biomechanics 14, 135-142. Inman,V. T., Ralston, H. J. and Todd, F. N. (1981) Human ing, sometimesthe direct dynamicsmethod is chosen Walkina. William & Wilkins. Baltimore/London. to analysea measuredmovement, which is then com- Kadaba, M. P., Ramakrishnan, H. K. ad Wootten, M. E. bined with optimization techniques(e.g. Chao and (1990) Measurement of lower extremity kinematics during Rim, 1973).However, sincethis is essentiallya curve- level walking. J. orthop. Res. 8, 383-392. fitting techniqueand the information containedin the Koopman, H. F. J. M. (1989) The three-dimensional analysis and prediction of human walking. Ph.D. thesis, University system(the segmentsmodel, the measurementsand of Twente, Enschede, The Netherlands. the equationsof motion) is not changed,this method McMahon, T. A. (1984) Muscles, Rej7exes, and Locomotion. should yield exactly the sameresults as the inverse Princeton University Press, Princeton, NJ. dynamics method. When this is not so, the curve- Numerical Algorithms Group (1984) NAG Fortron Library Manual, Mark 11, Vol. 3. EO4-Minimizing and maximiz- fitting is not exact, in which casethe direct dynamics ing a function. Oxford/Downers Grove. method isjust a difficult way to implicitly smooth the Olney, S. J. and Winter, D. A. (1985) Predictions of knee and input movement. When the right precautions are ankle moments of force in walking from EMG and taken in an inversedynamics method, the amplifica- kinematic data. J. Biomechanics 18, 9-20. tion of noisecan be controlled to reasonablelimits. Pandy, M. G. and Berme, N. (1988) A numerical method for simulating the dynamics of human walking. J. Bio- mechanics 21(12), 1043-1051. Thunnissen, J. (1993) Muscle force prediction during human REFERENCES gait. Ph.D. thesis, University of Twente, Enschede, The Netherlands. Brand,R. A., Crowninshield,R. D., Wittstock,C. E., Peder- Vaughan, C. L., Hay, 1. G. and Andrews, J. G. (1982a, b) sen,D. R., Clark, C. R. and van Krieken, F. M. (1982) Closed loop problems in biomechanics I-II. J. Bio- A modelof lowerextremity muscular anatomy. J. biomech. mechanics 15, 197-200,201-210. Engng 104,304-310. Vink, P. and Karssemeijer, N. (1988) Low back muscle Chandler,R. F., Clauser,C. E., McConville,J. T., Reynolds, activity and pelvic rotation during walking. Anat. Em- H. M. andYoung, J. W. (1975)Investigation of the inertial bryol. 178,455-460. propertiesof the humanbody. Report DOT HS-801430. Winter, D. A. (1987)The Biomechanics and Motor Control National Technical Information Service, Springfield and Human Gait. University of Waterloo Press, Waterloo, Virginia. Ontario, Canada.