Quantitative examinations for multi joint arm trajectory planning—using a robust calculation algorithm of the minimum commanded torque change trajectory

Abstract

In previous research, criteria based on optimal theories were examined to explain trajectory features in time and space in multi joint arm movement. Four criteria have been proposed. They were the minimum hand jerk criterion (by which a trajectory is planned in an extrinsic-kinematic space), the minimum angle jerk criterion (which is planned in an intrinsic-kinematic space), the minimum torque change criterion (where control objects are joint links; it is planned in an intrinsic-dynamic-mechanical space), and the minimum commanded torque change criterion (which is planned in an intrinsic space considering the arm and muscle dynamics). Which of these is proper as a criterion for trajectory planning in the central nervous system has been investigated by comparing predicted trajectories based on these criteria with previously measured trajectories. Optimal trajectories based on the two former criteria can be calculated analytically. In contrast, optimal trajectories based on the minimum commanded torque change criterion are difficult to be calculated, even with numerical methods. In some cases, they can be computed by a Newton-like method or a steepest descent method combined with a penalty method. However, for a realistic physical parameter range, the former becomes unstable quite often and the latter is unreliable about the optimality of the obtained solution. In this paper, we propose a new method to stably calculate optimal trajectories based on the minimum commanded torque change criterion. The method can obtain trajectories satisfying Euler–Poisson equations with a sufficiently high accuracy. In the method, a joint angle trajectory, which satisfies the boundary conditions strictly, is expressed by using orthogonal polynomials. The coefficients of the orthogonal polynomials are estimated by using a linear iterative calculation so as to satisfy the Euler–Poisson equations with a sufficiently high accuracy. In numerical experiments, we show that the optimal solution can be computed in a wide work space and can also be obtained in a short time compared with the previous methods. Finally, we perform supplementary examinations of the experiments by Nakano, Imamizu, Osu, Uno, Gomi, Yoshioka et al. (1999). Estimation of dynamic joint torques and trajectory formation from surface electromyography signals using a neural network model. Biological Cybernetics, 73, 291–300. Their experiments showed that the measured trajectory is the closest to the minimum commanded torque change trajectory by statistical examination of many point-to-point trajectories over a wide range in a horizontal and sagittal work space. We recalculated the minimum commanded torque change trajectory using the proposed method, and performed the same examinations as previous investigations. As a result, it could be reconfirmed that the measured trajectory is closest to the minimum commanded torque change trajectory previously reported.