Jun Nishii

Legged insects select the optimal locomotor pattern based on the energetic cost.

Abstract. The gait transition in legged animals has attracted many researchers, and its relation to metabolic cost and mechanical work has been discussed in recent decades. We assumed that the energetic cost during locomotion is given by the sum of positive mechanical work and the heat energy loss that is proportional to the square of joint torque and examined the optimal locomotor pattern based on the energetic cost in a simple dynamical model of a hexapod by computer simulations. The obtained results well agree with characteristics in the locomotor patterns in legged animals; for example, the leg protraction time, step length, and the metabolic cost of transport are almost constant for many velocities, the leg cycling period decreases with velocity, and the energetic cost of locomotion induced by carrying loads linearly increases with mass loaded. This correspondence of the results of calculation to experimental results suggest that the heat energy loss for torque generation is proportional to the square of the torque during locomotion, and that the locomotor pattern in legged animals is highly optimized based on the energetic cost.

Trajectory planning for a leg swing during human walking

Although locomotor patterns of many legged animals have been widely studied from a viewpoint of energetic optimality, the optimality of the leg swing trajectory has not been well studied. In this paper, we examine whether the leg swing trajectory in human walking is determined based on the criterion of the minimization of the energetic cost or not. The computed optimal leg swing trajectory shows following characteristics. (1) The foot draws an arc in the first half duration of swing phase and in the second half duration it falls down with a forward swing along a slightly curved line and then is drew back for landing. (2) The maximum height of the foot increases with walking speed. (3) The speed of the foot takes its peak value in the second half duration of swing. These characteristics were well coincident with those of experimental data of human locomotion, which indicates that the leg swing trajectory would be planned so as to minimize the energetic cost

Evaluation of trajectory planning models for arm-reaching movements based on energy cost

Computational studies have suggested that many characteristics of reaching trajectories in a horizontal plane can be effectively predicted by certain models, including, the minimum end point variance model and minimum torque change model. It has also been reported that these characteristics appear to differ from those obtained by the minimum energy cost model that has been reported to explain the characteristics of locomotor patterns. Do these results imply that the human nervous system uses different strategies to resolve the redundancy problem for different tasks? In order to reexamine the optimality of reaching trajectories from a viewpoint of energy cost, we considered the corrective submovements to compensate for positional error due to signal-dependent noise in motor commands and computed the expected value of the total energy costs required to reach a target by repetition of submovements planned by each of the following models: the minimum energy cost model, minimum end point variance model, and minimum torque change model. The results revealed that when the noise is large, the total energy cost required by the minimum end point variance model and the minimum torque change model can be lower than that required by the minimum energy cost model which assumes minimizing energy cost under noise-free condition. This result indicates that the minimization of the expected value of the energy cost would be an important factor in determining the reaching trajectories.

Learning model for coupled neural oscillators

Neurophysiological experiments have shown that many motor commands in living systems are generated by coupled neural oscillators. To coordinate the oscillators and achieve a desired phase relation with desired frequency, the intrinsic frequencies of component oscillators and coupling strengths between them must be chosen appropriately. In this paper we propose learning models for coupled neural oscillators to acquire the desired intrinsic frequencies and coupling weights based on the instruction of the desired phase pattern or an evaluation function. The abilities of the learning rules were examined by computer simulations including adaptive control of the hopping height of a hopping robot. The proposed learning rule takes a simple form like a Hebbian rule. Studies on such learning models for neural oscillators will aid in the understanding of the learning mechanism of motor commands in living bodies.

A Learning Model of a Periodic Locomotor Pattern by the Central Pattern Generator

JN: Dept. of Physics, Biology, and Informatics, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 7538512, Japan. nishii@bcl.sci.yamaguchi-u.ac.jp Many basic locomotor patterns of living bodies are rhythmic, and oscillatory components of physical systems effectively contribute to the generation of the movement. The control signals for the basic locomotor patterns are generated by the central pattern generator (CPG), which is composed of collective neural oscillators, and the activity of the CPG is tightly synchronized with the movement of the physical systems. That is, appropriate locomotor patterns are realized by mutual synchronization between the physical system and the neural system. In this article a simple learning model is proposed to acquire an appropriate parameter set, the intrinsic frequency of the CPG, and the interaction between the CPG and the physical system, in order to obtain a desired locomotor pattern. The performance of the proposed learning model is confirmed by computer simulations and an adaptive control experiment of a one-dimensional hopping robot.Many basic locomotor patterns of living bodies are rhythmic, and oscillatory components of physical systems effectively contribute to the generation of the movement. The control signals for the basic locomotor patterns are generated by the central pattern generator (CPG), which is composed of collective neural oscillators, and the activity of the CPG is tightly synchronized with the movement of the physical systems. That is, appropriate locomotor patterns are realized by mutual synchronization between the physical system and the neural system. In this article a simple learning model is proposed to acquire an appropriate parameter set, the intrinsic fre quency of the CPG, and the interaction between the CPG and the physical system, in order to obtain a desired locomotor pattern. The performance of the proposed learning model is con firmed by computer simulations and an adaptive control experiment of a one-dimensional hop ping robot.

Constraint and exploitation of redundant degrees of freedom during walking

What kind of leg trajectories are selected during human walking? To address this question, we have analyzed leg trajectories from two points of view: constraint and exploitation of redundant degrees of freedom. First, we computed the optimal leg swing trajectories for forward and backward walking that minimize energy cost for the condition of having some stretch of elastic components at the beginning of the leg swing and found that the optimal trajectories explain the characteristics of measured trajectories. Second, we analyzed how and when leg joints cooperate to adjust the toe position relative to the hip position during walking and found that joint coordination (i.e., joint synergy) is exploited at some control points during human walking, e.g., the toe height when it passes through its lowest position from the ground and the leg posture at the beginning of the double-support phase. These results suggest that the basic constraint in selecting a leg trajectory would be the minimization of energy cost; however, the joint trajectory is not strictly controlled over the entire trajectory and redundant degrees of freedom are exploited to adjust the foot position at some critical points that stabilizing walking.

Optimality of upper-arm reaching trajectories based on the expected value of the metabolic energy cost

When we move our body to perform a movement task, our central nervous system selects a movement trajectory from an infinite number of possible trajectories under constraints that have been acquired through evolution and learning. Minimization of the energy cost has been suggested as a potential candidate for a constraint determining locomotor parameters, such as stride frequency and stride length; however, other constraints have been proposed for a human upper-arm reaching task. In this study, we examined whether the minimum metabolic energy cost model can also explain the characteristics of the upper-arm reaching trajectories. Our results show that the optimal trajectory that minimizes the expected value of energy cost under the effect of signal-dependent noise on motor commands expresses not only the characteristics of reaching movements of typical speed but also those of slower movements. These results suggest that minimization of the energy cost would be a basic constraint not only in locomotion but also in upper-arm reaching.

Optimality of Reaching Movements Based on Energetic Cost under the Influence of Signal-Dependent Noise

As candidates of a constraint which determines human arm reaching trajectories, criteria of the minimum torque change and of the minimum end-point variance have been proposed and it has been shown that these criteria well predict the characteristics of reaching trajectories. In our previous work, we have shown that these criteria would also suppress the energy cost for reaching movements, when motor command is affected by signal-dependent noise. In this study, we computed the trajectories which minimize the energy cost under the effect of the signal-dependent noise, and compared them with those of human subjects. The optimal trajectories were in good agreement with the measured trajectories at the points that when the movement duration is short, the speed profile of the hand movement takes a bell shape, and when the duration is long, the speed profiles take a collapsed shape. These results would suggest that human brain solves the redundancy problem of trajectory planning by the constraint of minimization of the expected value of energy cost.

An analysis of leg joint synergy during backward walking

Grasso et al. (1998) proposed the hypothesis that motor commands for the backward walking is designed so as to reproduce the reversal motion of forward walking. In this study, we analyzed the leg joint synergy in backward walking by the UCM analysis and compared the results with the time reversal profile of the synergy in forward walking. Some similarities between them were observed, e.g., the body posture is controlled by utilizing joint synergy during double support phase. However, differences were also observed during swing phase, e.g., at touch down at the end of swing phase the joint synergy is utilized to adjust the foot position in backward walking, contrary in forward walking the synergy is not utilized but the variance of joint angles are suppressed. The results indicate that the backward walking is not a reversal motion of forward walking, but planned independently of forward walking.

An analysis of leg joint synergy during bipedal walking in Japanese macaques

We analyzed bipedal locomotion of Japanese macaques from the view point of leg joint synergy by the UCM (Uncontrolled manifold) analysis in order to examine how and when hip, knee and ankle joints cooperate so as to suppress the variances of the toe position relative to the hip position. Our results showed that joint synergy is exploited at some moments during walking. For instance, the variance of the vertical toe position was suppressed by joint synergy when the tip of the finger passes its lowest position from the ground. Some characteristics of the synergy pattern of macaques have been also reported in human walking, on the other hand, some differences between humans and macaques were found. For instance, high degree of joint synergy that suppresses the variance of hip height was observed around the end of stance phase in human walking, but such synergy was weak in macaques. The results suggest that different control strategies are used in bipedal walking of macaques and humans.