Comparing cost functions for the optimal control of robotic manipulators using Pontryagin's Maximum Principle

In the framework of Pontryagin's Maximum Principle, the choice of the Hamiltonian has an impact on the resulting controls. In this framework, the cost function composes the Hamiltonian which leads to a desired control and motion behavior when appropriately chosen. In order to optimally control motion of robotic manipulators we compare the numerical impact of two cost functions on the resulting controls and positions of the optimal trajectory. The selected cost functions focus on the controls so that minimum effort is achieved during motion. We compared a cost function that is typically found in the literature with one that involves the robot mass tensor components. We applied these to the optimal control simulation of two robotic manipulators. Numerical trials showed that the mass tensor acts as a stabilizing factor that leads to lower motion amplitudes, increased numerical stability and reduced CPU computing times. Our proposed cost function may therefore be beneficial for the optimal path planning and control of robotic manipulators.