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 when no 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 movement and the internal ‘direct dynamics method’ and the ‘inverse dynamics 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 calculate the effect of prosthetic compo- the equationsof motion. This is only possiblewhen nentson the walking pattern or in functional electrical 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 combination of 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 instead of 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 analysis of 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 dynamics is 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 is a 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 defined according 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 extended with definitions for 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 applied to 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 influence of 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 coupled systemof 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-L knee(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.