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.

Legged locomotion requires the determination of a number of parameters such as stride period, stride length, order of leg movements, leg trajectory, etc. How are these parameters determined? It has been reported that the locomotor patterns of many legged animals exhibit common characteristics, which suggests that there exists a basic strategy for legged locomotion. In this study we derive an equation to estimate the cost of transport for legged locomotion and examine a criterion of the minimization of the transport cost as a candidate of the strategy. The obtained optimal locomotor pattern that minimizes the cost suitably represents many characteristics of the pattern observed in legged animals. This suggests that the locomotor pattern of legged animals is well optimized with regard to the energetic cost. The result also suggests that the existence of specific gait patterns and the phase transition between them could be the result due to optimization; they are induced by the change in the distribution of ground reaction forces for each leg during locomotion.

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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.

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