A study on optimal motion of a biped locomotion machine

In this paper we propose a calculation method for the optimal trajectory of a biped locomotion machine which is based on inverse kinematics and inverse dynamics. First, the trajectory of the waist is expressed by a Fourier series, where the bases are selected appropriately so that the periodic boundary conditions are strictly satisfied. A biped locomotion machine establishes optimal walking by using kicking forces to the ground at the moment of switching legs. In order to include the effecs of the kicking forces, additional terms that indicate the impulsive forces at the moment of switching legs are included in the formulation. Then the angles of each joint are determined by inverse kinematics, and using inverse dynamics, the input torques of each joint are expressed in terms of Fourier coefficients. By defining the performance index as a quadratic form of the input torques, the motion planning problem is formulated as an optimization problem of the trajectory of the waist, whose paramaters are Fourier coefficients of the trajactory of the waist. Using the successive quadratic programming (SQP) method, the optimal trajectory of a biped locomotion machine is obtained. This work was presented, in part, at the Third International Symposium on Artificial Life and Robotics, Oita, Japan, January 19–21, 1998.