Trajectory planning algorithm with path search for cooperative multiple manipulators

In this paper, we propose an algorithm to determine sub-optimal paths and trajectories for cooperative multiple manipulators which hold an object and transfer it from a given start point to a given goal point. The dynamic equations of manipulators and the object are expressed by a path parameter's which is the length along the path and does not depend on the time 't' explicitly. Using this parameter 's', we express the position of the center of gravity and the attitude of the object as p(s) and φ(s), respectively. And the joint path θ_i of the i-th manipulator is also expressed by this parameter 's' and written as θ_i(s). A quality λ(s) which connects the time 't' and the path parameter 's' is introduced. θ_i(s) and λ(s) are approximated by B-spline. Sub-optimal θ_i(s) and λ(s) are determined by considering the boundary conditions of both end points and the driving torque/force constraint of each joint of each manipulator. The proposed algorithm is applied to a two-manipulator system. Numerical results show the effectiveness of the algorithm.