Siulating the effects of fatigue and pathologies in gait T Koura 1, A Nagano, H Leung 3 and Y Shinagawa 4 1,3 Departent of Coputer Engineering and Inforation Technology, City University of Hong Kong, 83 Tat Chee Ave, Kowloon, HONG KONG Coputational Division, RIKEN, -1 Hirosawa Wako, Saitaa, JAPAN 4 Departent of Electrical and Coputer Engineering, University of Illinois at Urbana-Chapaign, 45 N. Mathews Avenue, Urbana, Illinois 6181, USA 1 taku@ieee.org, a-nagano@riken.jp, 3 howard@cityu.edu.hk, 4 sinagawa@uiuc.edu 1 www.it.cityu.edu.hk/~itaku, www.it.cityu.edu.hk/~howard, 4 netfiles.uiuc.edu/sinagawa/www/ ABSTRACT In this study, we propose a new ethod to siulate the effect of fatigue and pathologies in huan gait otion. The ethod is based on Angular Moentu inducing inverted Pendulu Mode (AMPM), which is the enhanced version of 3D linear inverted pendulu ode that is used in robotics to generate biped locootion. By iporting the gait otion captured using a otion-capture device, the value of AMPM-paraeters that define the trajectory of the center of ass and the angular oentu are calculated. By iniizing an objective function that takes into account the fatigue and disabilities of uscles, the original otion is converted to a new otion. Since the nuber of paraeters to define the otion is sall in our ethod, the optiization process converges uch ore quickly than in previous ethods. 1. INTRODUCTION Coputer siulations based on usculoskeletal odels are often used to analyze the role of specific uscles during otions. Many researchers have analyzed the contribution of soe specific uscles to the perforance of certain otions by using optiization ethods based on forward dynaics. For exaple, Pandy and Zajac (1991) analyzed the role of biarticular uscles in axial juping, and they concluded that biarticular uscles contribute to juping perforance by redistributing segental energy within the usculoskeletal syste but do not contribute to the energy of juping. Neptune et al. (1) analyzed the role of the plantarflexor uscles during gait, and they calculated the degree to which these uscles contribute to propelling the trunk in the forward direction. Huan gait otions have also been siulated by cobining forward dynaics and optiization (Yaaguchi and Zajac, 199; Anderson and Pandy, 1; Neptune et al., 1). Although these researches have succeeded in siulating realistic noral gait, there are several shortcoings with these ethods that ake it difficult to apply the for the diagnosis of patients due to the following reasons: 1. First of all, researchers ust provide a good initial guess of the activation data to prevent the optiization process fro getting caught into soe local inia. In ost of the cases, this is done through trial-anderror by researchers. This is quite tie consuing, and the researcher needs special experience to deterine the good set of uscle activations to generate realistic gait otion. In addition, since this process is done copletely anually, the otion can becoe subjective to the researcher.. Next, since all the tie sequence data for the uscles ust be deterined through the optiization, costly coputation is required until the result is converged to the optial value. Neptune (1999) reported that ore than thousands of iterations are needed before the optiized value is obtained. 3. Third, although balance-keeping is one of the ost iportant factors for gait otion, there was no explicit way to describe the balanced otion in these researches. As a result, the search space for the paraeter is uch larger since it also includes those paraeter values that lead to unbalanced otion. 111
In robotics, biped locootion of huanoid robots is one of the ost exciting topics these days. Many researchers have developed huanoid robots that are capable of perforing biped gait otion (Kajita et al., 199, 1). A ethod coonly used to generate gait otion is the 3D Linear Inverted Pendulu Mode (3DLIPM) (Kajita et al., 199, 1). The great advantage of the 3DLIPM approach is that the trajectory of the center of ass (COM) can be written in an explicit for. The 3DLIPM approach is a hierarchical approach, in which the abstract otion such as the trajectory of COM is deterined first, and the details of the otion such as the kineatic data are calculated later. The advantage of the hierarchical approach is that the controller does not have to waste tie deterining low level control signals such as the torque exerted at the joints or the uscle activation data. With the hierarchical approach, it is possible to siplify coplex odels of huanoid robots that have too any degrees of freedo (DOF) to be controlled directly. In addition, as the otion of the COM is explicitly controlled, the balance of the huanoid robot is assured in the feedforward stage. However, no angular oentu has been generated by the 3DLIPM since it assues that the COM is a ass point and that the ground force vector always passes through the COM of the syste (Figure 1(a)). Figure 1. The standard 3DLIPM (a) and AMPM (b). The 3DLIPM restricted the ground force to pass through COM while the AMPM allows the horizontal coponent of the ground force to have linear relationship with the position of the COM. It is known that angular oentu is consistently generated around the COM when huans walk. The aount of angular oentu gets even larger as the otion involves larger bending of the thorax, which can be often observed in pathological gait. It is therefore difficult to odel huan gait otion by using 3DLIPM. To represent trajectory of COM and angular oentu of huan gait otion in an explicit for, we have extended the 3DLIPM odel so that angular oentu can be induced by the ground reaction force (Kudoh and Koura, 3). This odel is called Angular Moentu inducing inverted Pendulu Model (AMPM). In this study, we propose a new hierarchical approach to calculate huan gait otion using AMPM. By cobining AMPM with the usculoskeletal odel, it is even possible to calculate pathological gait. The effects of deactivating the gluteus edialis uscles were siulated and copared with the otion of patients with siilar disabilities. The trajectory of the COM and angular oentu is first calculated by using AMPM. Next, the trajectories of the generalized coordinates including the position of the pelvis and the joint angles of the whole body are deterined using inverse kineatics. After deterining the uscles of the body to be deactivated, an objective function based on the tie series data of the force exerted by this uscle is fored. By searching for an optial set of AMPM paraeters that iniizes the objective function defining the gait otion, the pathological gait otion is calculated. The ethod proposed in this study has the following advantages copared with previous ethods: Since the gait otion is described by the AMPM paraeters, there is no need to specify all the input paraeters of the uscles. As a result, the coputational cost for the optiization is uch less than previous ethods. As the balance of the huan body odel is explicitly kept by using the AMPM odel, the optiizer only needs to search for the optial set of paraeters in ters of uscle force, and hence does not need to go through a large nuber of trials to generate the balanced otion. 11
. METHODS In this section, the ethod to siulate the effects of pathologies and fatigue during huan gait is explained. First we review the Angular Moentu inducing inverted Pendulu Model (AMPM) (Kudoh and Koura, 3) and also explain further enhanceents so that it can handle iported huan otion data. Next, the details of the usculoskeletal odel we used in this research are explained. Finally, the ethod to siulate the effects of pathologies and fatigue by optiization is explained..1 Angular Moentu inducing inverted Pendulu Model The AMPM enhances the 3DLIPM in a way the ground force vector is calculated to be not only parallel to the vector connecting the ZMP and the COM; its horizontal eleent can be linearly correlated to the ZMP- COM vector (Figure 1(b)). As a result, rotational oent will be generated by the ground force. In the previous studies, we assued the ground force vector to be linearly correlated to the position of the COM: & x = Ax + B (1) As we assue the single support phase of huan gait follows this rule, the paraeters A and B can be calculated using the position and acceleration of the COM at the lift-off and heel-strike during gait. However, when actually we substitute data into this equation, the trajectory of the acceleration does not atch well with those by huans. As shown in Figure, oscillatory gap is observed between the acceleration of the COM and that generated by AMPM. Figure. Acceleration of the COM during gait (dashed line) and that by AMPM (bold line). In order to decrease the gap between the trajectory by the AMPM and the trajectory of the COM during gait, we propose to add trigonoetric functions in the right side of Equation (1) as follows: & x = Ax + B + C cos ωt + D sinωt () The paraeters C, D, ω are decided in a way that trajectory of the differential equation atches well with the trajectory of the COM. Equation () has the explicit solution as follows: x = B + C 1e A At + C e At cos ω t + D sin ω t A + ω where C 1, C are constant values which can be calculated fro the terinal conditions. In this study, we ake the following two assuptions through observation of huan otion: The position of the ZMP is linearly correlated to the position of the COM. Therefore, the position of the ZMP can be defined by (zp, ) = (cx + d, ), where (c, d) are constant paraeters. (3) 113
The height of the COM can be expressed by the following equation: t z g = H + zh sin T where H is a constant value, T is a half cycle of gait otion, z h is the aplitude of the otion, and z g is the height of the COM. Then, the ground force vector can be written as At At ω F = x && x = AC1e + ACe + (cosωt + Dsinωt) A + ω 1 1 Fy = g sin T T where is the ass of the syste. The rotational oent r around the y-axis can be calculated by t r = (( 1 c) x + d) Fy ( H + zh sin ) Fx T Using this oent, it is also possible to calculate the angular oentu generated by the rotational oentu in an explicit for, although we cannot list it here due to its length. This for was calculated using the software Matheatica. AMPM is used to represent the otion of the COM/ZMP/angular oentu in the frontal and sagittal plane. Then, the generalized coordinates of the body are then calculated using inverse kineatics (Kudoh and Koura, 3).. Musculoskeletal Model To siulate the otion by pathological patients, it is necessary to prepare a physiological odel of the huan body. The usculoskeletal odel developed by Delp (199) was used in this study. This data includes the attachent sites of 43 uscles on each leg and physiological paraeters such as the tendon slack length, optial fiber length, pennation angles, and axiu exertable force. Muscles are attached only to the legs, and no uscles are put on the upper half of the body. The upper half of the body is coposed of the thorax, head, upper ars, lower ars, and hands. However, only the thorax (3 degrees of freedo (DOF)) and the upper ars (3 DOF each) are allowed to ove aong these segents. The lower half of the body is coposed of the pelvis, and the feur, tibia, patella, talus, calcaneous, and toes in each leg. The joints of each leg are coposed of a 3-DOF gibal joint (hip joint) and 1-DOF joint (knee, ankle, calcaneous, and etatarsal joint). Therefore, the total degrees of freedo of the body, including the 6DOF of the pelvis in the global coordinate syste, is twenty nine. The usculotendon odel of Hill (1938) is used, and paraeter values were derived according to Delp (199). The usculotendon odel is coposed of three eleents: a contractile eleent (CE, representing all the uscle fibers), a parallel elastic eleent (PEE, representing all connective tissues around the uscles fibers), and a series elastic eleent (SEE, representing all series elasticity, including tendons). At each tie step, the usculotendon length was deterined fro the posture (i.e., as a function of joint angles). Thereafter, it is possible to calculate the axia and inia that bound the uscle force at each tie step: F in F t) F (4) ax ( in ax where F (t) is the usculotendon force of the -th uscle, F and F are the iniu and axiu force developed by this uscle, which are deterined by their force-length-velocity properties: F F in ax = f ( F, l = f ( F, l where function f ( F, l, v, a ) is the force-length-velocity surface assued in the usculoskeletal odel (Zajac, 1989), F is the axiu force constant calculated fro the average cross-sectional area of the uscle, l is the length and v is the velocity of the -th uscle being contracted, and last paraeter ( and, v, v,),1) 114
1) in these equations is the activation level of the uscle a that deterines the aount of force exerted by the contractile eleent ( a 1)..3 Calculating the effects of fatigue and pathologies Torque τ j (t) developed at joint j is theoretically generated as follows by the uscles crossing the joint: τ j ( t ) = F ( t) r, j, j =,..., ndof (5) where r,j is the oent ar of uscle about the j-th joint axis, and n dof is the nuber of degrees of freedo whose torque are assued here to be generated by the uscles. They include the flexion/extension, adduction/abduction, and rotation at the hip, flexion/extension at the knee, and plantarflexion/dorsiflexion at the ankle, and therefore, by taking both legs into account, n dof = 1. The uscle forces at each tie step can be calculated by iniizing the uscle stress (Crowninshield and Brand, 1981): n ( ) = F ti J (6) F where n is the total nuber of uscles (n = 43 = 86). J was iniized using quadratic prograing, which is a ethod to iniize a quadratic for while satisfying linear equality and inequality constraints. In suary, the uscle forces can be calculated by iniizing Equation (6) while using Equations (5) and (4) as constraints. To describe the process to calculate the pathological gait, a new function is defined here that suarize all the processes explained previously, including the AMPM, inverse kineatics, inverse dynaics and static optiization: F f ( p, t)( =,..., n 1) (7) = where p is the vector coposed of paraeters defining the gait otion by the AMPM, and F is the force by uscle calculated by iniizing Equation (6). To siulate the effect of weakening uscle, the following criterion is iniized until it is saller than the threshold value: T J = [ f( p, t) + α ( p, t) ]dt (8) where T is one cycle of the gait otion, and α(p, t) is a penalty function based on the external torque that has to be applied to the body to assist the usculoskeletal odel accoplish the otion when there is no solution found for the static optiization proble at tie t (Koura et al, ; Koura and Shinagawa, 1). The penalty function helps to avoid the otion to converge to one that is not feasible by the usculoskeletal odel. This optiization is done by sequential quadratic prograing. 3. EXPERIMENTAL DATA ANALYSIS To exaine the validity of the ethod proposed in this paper, we have generated noral and pathological huan gait otion using our ethod, and copared the kineatical and dynaical data with those by huans. First of all, the noral gait otion that was captured using a VICON otion capture syste was iported, and the AMPM paraeters were calculated. Next, by iniizing an objective function based on the gluteus edialis that has the for of Equation (8), the effects of weakening the gluteus edialis during gait were siulated. As the optiization proceeds, features known as lateral trunk bending appears in the otion. The trunk swings fro one side to the other, producing a gait pattern known as waddling. During the double support phase, the trunk is generally upright, but as soon as the single support phase begins, the trunk leans over the support leg, returning to the upright attitude again at the beginning of the next double support phase. 115
The trajectory of the gait otion before and after the optiization is shown in Figure 3 (a) and (b) as well. The objective function converges to the optial value within one hundred iterations. Figure 3. The trajectory of the AMPM-generated otion before (a) and after (b) optiization. Next, in order to siulate the effect of fatigue, the axiu aount of force exertable by the uscles that are used during gait were gradually decreased. As a result, the strides of the gait gradually decreased as shown in Figure 4. Figure 4: Siulating the effect of fatigue during gait. The strides before the optiization (a) is shortened after reducing the upper bound of the uscle force that is used during gait (b). 4. DISCUSSIONS A ethod to calculate pathological gait by cobining the usculoskeletal syste and the inverted pendulu ode was proposed in this research. The otion is planned by the AMPM, and it is evaluated by using the usculoskeletal odel. As the nuber of the paraeters to deterine the otion is less than previous ethods, the optiization process converges ore quickly than previous ethods. In our experient, by iniizing the aount of force exerted by the gluteus edialis during the gait otion, the waddling gait otion was autoatically induced. This is a natural phenoenon, as it is known that weak abductor uscles cause waddling. Waddling is a well-known abnorality of gait otion which is caused not only by weak abductor uscles, but also by congenital dislocation of hip joints and pain in the joints. Waddling reduces the torque and the bone-on-bone contact force at the hip of the support leg during the single support phase. The uscle force history of the usculoskeletal odel perforing noral gait and waddling gait calculated using static optiization are shown in Figure 5. 116
Figure 5: The gluteus edialis (GMED) force calculated by static optiization ethod using huan otion by a healthy subject (bold line) and by a patient with congenital dislocation (dashed line). Copared with dynaic optiization ethods that have recently been used in bioechanics to siulate huan gait and estiate uscle force, the ethod proposed in this study has the following advantages. 1. Our ethod to calculate huan gait otion requires only a sall nuber of paraeters that define the otion of the AMPM. For exaple, if the usculoskeletal syste is coposed of 86 uscles, and during the gait cycle each uscle is allowed to change its input value 1 ties, the total nuber of paraeters that define the whole otion becoes 8,6. This eans at least this nuber of iterations are needed for calculating the derivative of the criteria of optiization. Although techniques such as using the sae input signals for uscles with siilar roles (Pandy et al., 199), or controlling uscles by Bang-Bang ethods to reduce the search space (Yaaguchi and Zajac, 199), are used, the nuber of paraeters still reains very large. On the other hand, when using the proposed ethod, the nuber of paraeters is less than ten. Since the nuber of the paraeters is sall, the optiization converges uch ore quickly than dynaic optiization ethods. Coparing the aount of coputation needed, our ethod requires less than one tenth of the forward dynaics approach.. Since the balance of the otion is assured by the algorith of the otion generation, the optiizer does not have to spend further effort in keeping the balance through the gait cycle. This is one of the ost iportant contributions of the paper, because when using optiization ethods based on forward dynaics to siulate bipedal gait, ost of the iterations are done for unbalanced otions tubling down onto the ground. If the syste can avoid evaluating such otions, the search for the optial otion can be done uch ore efficiently. 3. It is not necessary to search anually the initial guess of the uscle-activation paraeters that decide the otion, as the paraeters that define the gait based on the AMPM are intuitive kineatic paraeters such as the position and velocity of the COM. As explained in the Introduction, when using optiization ethods based on forward dynaics, a good initial guess of the activation data ust be provided in order to prevent the optiization process fro getting caught into soe local inia. This is quite tie consuing, and the researcher needs special experience to deterine the good set of uscle activations to generate realistic gait otion. In addition, as such process is done copletely anually, the otion can becoe subjective to the researcher. 5. SUMMARY AND FUTURE WORK In this study, we have proposed a new ethod to siulate the gait otion when uscles are deactivated. The ethod is based on the AMPM which is an enhanced odel of the 3DLIPM that is often used in robotics to generate gait otion. Copared with previous ethods, the proposed ethod is siple, and the coputational cost is uch saller. As further work, we are planning to.t the AMPM odel into huan otion data, and siulate the effects of physiological disabilities and also the process of pathology for rehabilitation use. 117
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