Indirect optimal trajectory planning of robotic manipulators with the homotopy continuation technique

Indirect solution for the optimal trajectory planning of the robotic manipulators is studied from the practical point of view even for robots with a high degree of freedom. Derivation of the optimality conditions for these mechanical systems is symbolically cumbersome. It is convenient to release some unnecessary symbolical steps and to replace by some numerical shortcuts. On the other hand, generally, the Pontryagin maximum principle gives the optimal trajectory between initial and final states. But since state space computation of the robot manipulators needs consideration of the direct dynamics, thus it makes a practically serious problem chiefly caused by necessity of the inertia matrix inversion. Symbolically, finding inverse of the inertia matrix is time and memory consuming, since it cannot be found by eliminating methods. This paper extends the traditional optimality condition extraction to a case that no symbolical inversion of inertia matrix is needed. Also the optimality conditions incorporated with holonomic and non-holonomic constraints is considered using the projection method. Moreover, to overcome the convergence problem, simple homotopy continuation is employed. Consideration of the acceleration in the cost function is also becomes clear. Simulation indicates that the indirect solution with homotopy continuation is highly accurate and it can be intended for the mechanical systems. Additional codes are appended to the article for convenience.