Modeling Resistance of a Four-Link Biped to Lateral Push

Transcription

1 ORIGINAL RESEARCH Modeling Resistance 99 JOURNAL OF APPLIED BIOMECHANICS, 2002, 18, by Human Kinetics Publishers, Inc. Modeling Resistance of a Four-Link Biped to Lateral Push Gilles Dietrich University of the Mediterranean Alan Mark Wing and Martine Gilles University of Birmingham Ian Nimmo-Smith MRC Applied Psychology Unit, Cambridge This paper presents a conceptual model for studying the contribution of each leg to sideways stability of a four-link biped. It was assumed that a linear feedback controller maintained balance with torque related to the deviation from a reference value of the angle made by the trunk with the vertical. Predictions for ground reaction forces produced in resisting sideways push at the pelvis, based on simulation using a simple linear controller, are presented for two special cases (using one or both legs). This simple model was then compared to experimental data in which participants were asked to resist a sideways push. It was observed that all participants employed a strategy in which one leg was used to develop the force response. With this simple model, it was possible to simulate different kinds of responses to the balance perturbation. This model could be considered the first step of a more complex model in order to include specific components related to physiological parameters. Key Words: biped model, postural control, perturbation, sideways push Introduction Many studies on balance control in humans have examined the organization of postural responses when the support base translates or rotates, resulting in a transient disturbance to balance due to inertial forces and torques (Macpherson, Horak, Dunbar, & Row, 1989; Nashner, 1976; Woollacott, von Hoston, & Rösblad, 1988). These studies reveal that postural control is organized in terms of strategies. Most G. Dietrich, Univ. of the Mediterranean, Science & Techn. Luminy 163, Case 918, Marseille Cedex 09, France; A.M. Wing and M. Gilles, Ctr for Advanced Studies in Sensory Motor Neuroscience, Univ. of Birmingham, Edgbaston Birmingham B15 2TT, UK; I. Nimmo-Smith, MRC Applied Psychology Unit, 15 Chaucer Rd, Cambridge CB2 2EF, UK. 99

2 100 Dietrich, Wing, Gilles, and Nimmo-Smith studies focus on anterior/posterior balance and describe the postural response to perturbation without differentiating the contribution of each leg (e.g., Horak, Diener, & Nashner, 1989; Nashner, 1976). This seems reasonable if the same muscles are used in each leg and if we can assume that homologous muscles have a similar response in terms of speed or efficacy. In contrast, sideways perturbation to balance will clearly activate an asymmetric response from the two legs in the sense that different muscles will be involved in each leg. The perturbation itself has an asymmetric effect by changing the position of the center of gravity within the support area, thus putting one leg at a mechanical disadvantage. We are led naturally to wonder about the respective contribution of each leg. In crowded situations, a common perturbation to standing balance is a horizontal push, possibly because of extended duration, to the upper body. Resisting such a horizontal push requires the development of shear forces at the ground by the coordinated action of the trunk and leg muscles. But few studies have examined the postural response when external forces are applied to the body (Yang, Winter, & Wells, 1990a, 1990b). Here we present an analytical approach to modeling the response to application of and release from a sustained horizontal force applied to the body at some height above the ground. In this study we consider the case in which a sustained push is delivered in the frontal plane at the level of the pelvis (Wing, Clapp, & Burgess-Limerik, 1995) and note that the response may involve differential contributions of either leg subject to initial conditions. In biomechanical models of human stance, posture is represented generally as an open-chain N-link inverted pendulum placed on a triangular foot. Such models may take the form of a simple pendulum to explain quiet standing (Winter, 1995), or a compound pendulum to explain the dynamic changes during movements (Iqbal, Hemami, & Simon, 1993; Yang et al., 1990a) or perturbations (Barin, 1989; Yang et al., 1990b). In these models it was assumed that limb segments are two-dimensional with a homogeneous mass and that they rotate about simple pin joints. Although a one-legged inverted pendulum provides a simple means of simulating backward/forward perturbations, in a sideways perturbation the individal may use the two legs in different ways. This suggest that we should consider a two-legged model (a closed chain). In this paper we provide a quantitative model of the kinematics of postural control and show how it may be fit to data from an illustrative experiment with six participants. The aim of this study was to demonstrate that it is possible to use a vary simple model to simulate postural control in the frontal plane. The first question was to investigate the behavior of this model. To answer this point we simulated two extreme cases: using one or both legs. The second point was to compare human postural responses to our model. Methods A five-segment model was used to study the postural response induced by a sideways push. All segments were assumed to be rigid and to articulate with the adjacent segments about pin joints. This model was constrained to move in the frontal plane (2D). The segments, as shown in Figure 1A, represented: (1) the HAT (head, arms, trunk); (2) the left leg; (3) the right leg; (4) the left foot; and (5) the right

3 Modeling Resistance 101 Figure 1 Biological model. (A) Postural control in the frontal plane (XZ). A 5-segment body model (1 5) with 4 equivalent muscles was used to examine the effect of postural regulation with a lateral push on ground reaction forces measured at force plates (6, 7). Equivalent ankle and hip muscle torques were related to angular position ( ) and velocity. (B) Foot contact. R = ground reaction force; R x = lateral component of R; R z = vertical component of R; s = limit value of ground friction coefficient; = ground reaction force angle. foot. Segment positions, lengths, and masses were calculated using a regression equation (Chandler, Clauser, McConville, Reynolds, & Young, 1975). Four joints were used in the model: left and right hips, left and right ankles. Each joint was modeled as a pin joint with one degree of freedom without any friction. Foot contact was modeled with ground reaction forces (R x : lateral force; R R z : vertical force) calculated under each foot and friction coefficient ( = x ) subject to a limiting value ( s ). In other words, sliding occurred if the ground reaction R z force angle (relative to the vertical axis) was greater than a limit value (see Figure 1B). Posture was controlled by four equivalent muscles, one at each hip and ankle. These equivalent muscles included both passive and active components with stiffness in the range of 73 to 113 kn/m (as assumed by Pandy & Berme, 1988). Passive muscle torques (T p ) were given by a linear relation involving joint angle and joint angular velocity. T p = K p1 + K d p2 (1) dt where: T p = passive muscle torque; K p1 = elastic coefficient; = angular position; K p2 = damping coefficient; d = angular velocity. dt

4 102 Dietrich, Wing, Gilles, and Nimmo-Smith Joints and segments were moved by an active component of equivalent muscle. This active component was represented by a torque actuator, which exerted the net torque needed to maintain a postural constraint as detailed below. This biomechanical model was implemented using a mechanics simulation package (working model 3D). We assumed that a linear feedback controller maintained balance with torque (active torque of equivalent muscle, T A ) related to the deviation from a reference value of the angle made by the trunk with the vertical T ( T = 0 when vertical). T A = K a1 + K d a2 (2) dt To validate the model, we compared to extreme cases by choosing two parameter sets. Typical reaction force data for these two simulations: in one (Figures 3A, 3B) the coefficients of two legs were set equal; in the other (Figures 3C, 3D) the gain coefficients of one leg were set to zero. In this case the perturbation was maintained for 1.1 second. In the first case (Figures 3A, 3B) the relative left and right vertical forces (i.e., force minus half of body weight) curves are symmetrical about zero. Maxi- where T A = muscle active torque; K a1 = angular gain factor; = angular position; d K a2 = angular velocity gain factor; and dt = angular velocity. For each of four equivalent muscles we therefore had a set of four parameters (K p1, K p2, K a1, K a2 ): two for passive torques and two for active torques (Equations 1 and 2) yielding a total of 16 independent actuator parameters. For this study, gain parameters were chosen to minimize the perturbation during the first phase, i.e., during application of the lateral force (peak of lateral and vertical forces) by minimizing the lateral displacement. The gain values were constant during and after the application of the perturbation. Postural perturbation was induced by a lateral push. This lateral push was chosen as 4% of body weight (27.5 N) and applied horizontally to the pelvis to produce sideways sway. Simulations were performed to explore time delay parameter ( ) between angular position ( ) and torque response (T A ) and gain parameters (K p1, K p2, K a1, K a2 ). The main effect of increasing the time delay is to increase postural sway. The maximum time delay before instability was around 150 ms for all the controllers. On the basis of Gilles, Wing, and Kirker (1999), we assumed a time delay ( ) of 80 ms between active torque (T A ) and angular position: T A = f( t ). The time delay for passive torque development was assumed to be zero. This control process was implemented using Matlab and C-code in order to monitor the mechanical model and control process (Figure 2). For the simulation, we used a classical Runge-Kutta integrator. At each time step, the integrator calculated the acceleration of each part of the body through four calculation steps. In order to compare real experiment and simulation data, we computed the ground reaction forces under each model foot. To test the sensitivity of the model, we used different sets of gain parameters, for both passive and active components. It was possible to modify the gain range (relative increase or decrease: 100%) without loss of stability. Results

5 Modeling Resistance 103 Figure 2 Control loop model, implemented using working model 3D. Controlled object represents biomechanical model of body. At each time step the angular position and angular velocity of each joint (state variables) were computed. Torque generated at joint j at time t was a function of: the state variables at time t of joint j for the passive part of torque; and the state variables of the same joint j at time t- for the active part. Actual torque = Tp + Ta. This control loop was implemented in Matlab and C code. Feedback delay and interface between controlled object and controller was implemented in C. mum or minimum vertical force appears 304 ms after the lateral push. For the lateral forces, the time curves are the same for the same set of muscle parameters in both left and right sides. Peak lateral forces (22.7 N) occur slightly after peak vertical force (314 ms). By modifying muscle parameters, i.e., increasing the stiffness on the left side (Figures 3C, 3D), the curve shapes are modified and the shear forces are greater on the left side (56 N) compared to the right side ( 9 N) and the left force peak occurs 30 ms after the right force peak. The previous section demonstrates that it was possible to simulate a wide variety of conditions. The next step is now to compare experimental data to simulation in order to explore the model s ability to fit real data. We will first describe the setup and then present a typical recording for one participant. Participants were instructed to stand upright and resist a sustained lateral push directed to the left side. Perturbations to balance were delivered by forceserved linear motors (Linear Drives, Ltd., Bristol, UK). The motors were connected to the participant via a 6-axis load cell (ATI Technologies, Inc., Aper, NC) attached to a semirigid belt worn around the pelvis just below the iliac crest. Position of the pelvis in anteroposterior and lateral was measured by linear motors. Application of force begain 0.3 s after an auditory signal, took 8 ms to change level, and was sustained for approximately 3 s. During the push, the participants had to resist the applied force to keep their initial posture and minimize sway (defined as displacement of the pelvis). Between trials the participants were encouraged to keep the pelvis in a consistent position in the transverse plane. Between blocks of trials they were allowed to lift either foot for relaxation, but if they did this they had to return it to the same position as shown by markings on the platform. This perturbation corresponded to approximately 4% of body weight. Six individuals participated (2 M, 4 F), ranging

6 104 Dietrich, Wing, Gilles, and Nimmo-Smith A) Figure 3 Computer simulation data. Vertical F z (A and C) and lateral F x (B and D) forces calculated for two sets of parameters in a lateral left push and release. A and B show ground reaction forces (left: LF x and LF z ; right: RF x and RF z ) calculated for the same active force parameter (K a1 and K a2 = 300) in both legs; the curves were symmetrical for relative vertical forces and were the same for lateral forces. (continued) B) in age from 21 to 29 years. All were healthy, with no known neurological or musculoskeletal disorders. Two 6-axis force plates (Bertec) were used to sample ground reaction forces. We focus our attention on the lateral (F x ) and the vertical (F z ) ground reaction forces. All the traces were synchronized by using the onset of the applied force. Latency of any event was calculated as the duration between the time of force onset and the event. Typical results for one participant (S1) are shown in Figure 4. All participants showed similar functions, as may be inferred from the similar latencies and peak forces in Table 1. Here we describe the form of the force-time functions for S1. The first major modification of the ground reaction force we observed after the push was an increase of the vertical force under the left foot. This occurred ± seconds after the lateral push was initiated. Almost simultaneously (0.106 ± s) we

7 Modeling Resistance 105 C) Figure 3 (Cont.) In C and D, ground reaction forces were calculated with different active muscle parameters for right leg (K a1 and K a2 = 0) and left leg (K a1 and K a2 = 300); with no contribution of the right leg, vertical forces were still symmetrical and the main difference was in lateral force curves (D). D) observed a release of the vertical force under the right foot. There was no statistical difference between the absolute peak value of the amplitude of the variation of the left and right vertical forces (left: 221 ± 48.1; right: 214 ± 44.0 N; see Table 1). These peaks occurred at approximately the same time (0.410 ± s for the left foot; ± s for the right foot; Table 1). After attaining peak value, left and right vertical forces returned to a plateau. This plateau can be considered as a new stable state resulting from the reaction against the sustained applied force. When this force was released, the left and right vertical forces resumed their initial values. Onset of the lateral forces was delayed compared to onset of the modification of the vertical forces. Lateral shear forces onsets were s (± 0.010) for the left foot (Figure 4A) and s (± 0.014) for the right foot (Figure 4B). So, as with the vertical forces, the onsets of the lateral forces were nearly simultaneous.

8 106 Dietrich, Wing, Gilles, and Nimmo-Smith A) B) Figure 4 Experimental data (A and B) from a single trial, S1, vs. simulation data (C and D). Ground reaction force in mediolateral direction F x and vertical direction F z were recorded under left and right foot. Before the push, lateral forces were equal under both feet. (continued) The maximal increase of the lateral force was under the left foot 44.7 N (± 6.04), and we observed very small changes under the right foot, 3.11 N (± 3.52) (see Table 1). In summary, with a lateral push to the left of 4% body weight, displacement of the pelvis is arrested after 320 ms. Vertical forces at each foot change in opposite directions with very little weight remaining on the right at maximum displacement. Lateral force is principally exerted by the left leg. Vertical F z and lateral F x forces were determined from the simulation using a set of parameters chosen to produce an approximate fit to the experimental data (Figures 4C, 4D). Again vertical force curves were symmetrical about zero but lateral force curves show a different contribution for the two legs. The linear correlation coefficients were calculated between computer simulation and the ensemble average of the participant for the lateral push and release. The correlation was high for the lateral forces (r = 0.987) and vertical forces (r = 0.880).

9 Modeling Resistance 107 C) Figure 4 (Cont.) At 300 ms a lateral force of 4% body weight was applied at hip level for several seconds. This participant in this trial used one foot to develop the force response. After a few seconds the applied force was released. Vertical forces at each foot returned to zero in a symmetrical manner. D) Discussion As we stated in the Introduction, it was possible to use a very simple model to simulate postural control in the frontal plane. In the present study, with the two possible extremes for resisting a lateral push with the adjacent or opposite leg contributing shear force resistance, illustrative data from a single participant clearly show an intermediate response. One possible parameterization of the model to fit the data was demonstrated. As in Barin s study (1989), we used a simple feedback loop to control the postural stability of the model (PD controller). Such linear controllers (PD and PID controllers and spring-like model) are often used as a model of postural control (Hogan, 1985; Iqbal et al., 1993; Lacquaniti & Soechting, 1986) and they can generate complex behaviors previously explained by nonlinear approaches (Peterka, 1998). In our study we have used this simple controller for each joint. We chose linear feedback under the assumption that the postural sway is restricted to ±10. We have also used a full decoupling feedback process. This kind of controller was

10 108 Dietrich, Wing, Gilles, and Nimmo-Smith Table 1 Peak Changes in Force Values and Mean Latency (±SD) for all 6 Participants Left vertical Right vertical Left lateral Right lateral Left vertical force (N) force (N) force (N) force (N) peak latency(s) S ± ± ± ± ±0.067 S ± ± ± ± ±0.068 S ± ± ± ± ±0.027 S ± ± ± ± ±0.072 S ± ± ± ± ±0.071 S ± ± ± ± ±0.050 Note: Given are mean change values of the peak of ground reaction force during the push (±SD) and mean latency for each participant under left and right foot relative to perturbation baseline. There were no peak forces for the right lateral side (see Figure 4). used in a similar modeling study (Iqbal et al., 1993) for small-range voluntary movements (body sway). In our present study we could demonstrate that we can use more or less the same kind of controller to produce a postural reaction to a sustained perturbation. However, it is apparent that the analysis is limited in a number of respects. First, variability is not treated, yet it is present in the data. This could be modeled by a modification of the time delay ( ) and gains (K p1, K p2, K a1, K a2 ) by muscle which suggests an important direction for future research. For example, ground reaction forces (lateral and vertical) show oscillations during and after the perturbation. This seems to indicate that, during these two phases, the system was under-damped. The gain factors were chosen to minimize the perturbation during the first phase, i.e., during application of the lateral force. To keep this same approach (linear controller), one solution is to not use a constant value for the gain all the time but rather an adaptive gain or an adaptive controller (see below). A second limitation comes from the fact that the movement is constrained to the frontal plane. In practice the response to perturbation involves very little anterior-posterior contribution; however, this may change as a function of forward lean, bias to one side or the other. Thus we are currently extending the approach to 3D. The third limitation is related to the control mechanism itself. By using a PD or PID controller (linear controller), we could only use this model for small feedback delay values. If the time delay between feedback information and action increased, we would observe instability. In order to respond to this limitation, it is possible to use nonlinear approaches (Slotine & Li, 1991) or constrained systems (Hemani & Wyman, 1979). In further analysis we could use a third approach by adding to this simple model an adaptive controller related to an internal model of postural control. Such models are already used for goal-directed arm movements (see Wolpert, Mial, & Kawato, 1998, for a review). One limitation of this model deserves comment. The parameters used are difficult to interpret in physiological terms. In this study the gain components may

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