Time and energy optimal control by a new way based on central difference approximation of equation of motion with application to robot control

The object of optimal control of robots is to determine the signals or torques of its actuators that will cause a motion to satisfy the constraints and at the same time minimize (or maximize) some performance criterion or functional. Optimal control of robot manipulator has a complex nature. In this paper the exact equations of motion are approximated by the central difference technique and Taylor series expansion in a new way, while the path of motion is divided into finite segments. The motion is assumed to have zero velocity at the beginning and at the end of the motion, without loss of generality. In the time optimal control, the Pontryagin principle is applied and the optimal controller is of bang bang type. the actuator torques, iscolines and switching points, can be calculated. As the maximum torques of the actuators are fixed, the time optimal control problem is an optimal control with bounded inputs. The problem of energy optimal control is reduced to minimizing a scalar function of many but finite variables with equality and inequality constraints. By applying the modified Hooke and Jeeves method, the actuator torques at any time are calculated. The algorithm described is implemented on a 2R robot manipulator, and results are presented