Energy-optimal control of unconstrained planar RR robot manipulators

Given the dynamic model of a planar robot manipulator with two revolute joints, the initial state, and some specifications about the final state of the system, we find the trajectory and the actuator forces that minimize the energy consumption during the motion. We solve this optimal control problem using an approach based on the Euler-Lagrange necessary condition in integral form in which time is discretized and admissible variations for each variable are approximated using a linear combination of piecewise-continuous basis functions of time. Our method consists in a numerical resolution of an unconstrained reformulation of the optimal control problem in which the dynamic equation is regarded as a constraint. Then, we use a damped Newton methodology to solve the resulting nonlinear system of difference equations. The required initial guess of the solution is obtained using a trigonometric polynomial interpolation. The use of the Euler-Lagrange necessary condition in integral form avoids the numerical intricacy caused by the presence of corner of the extremals. In this way a general method for the solution of optimal control problems is obtained in which constraints on the state can be easily introduced. The accuracy of the proposed procedure is assessed by comparing the obtained global error with different discretization sizes. Finally, the performance of the proposed method has been compared with a direct transcription method, a widely used procedure to solve optimal control problems.