Time-Optimal Control of Dynamic Systems Regarding Final Constraints

Within the framework of this article, we pursue a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g., if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been generally considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this work is the development of an iterative, gradient-based solution strategy, which can be applied even to complex multibody systems. The so-called adjoint method is a promising way to compute the direction of the steepest descent, i.e., the variation of a control signal causing the largest local decrease of the cost functional. The proposed approach will be more robust than solving the underlying boundary value problem, as the cost functional will be minimized iteratively while approaching the final conditions. Moreover, so-called influence differential equations are formulated to relate the changes of the controls and of the final conditions. In order to meet the end conditions, we introduce a descent direction that, on the one hand, approaches the optimum of the constrained cost functional and, on the other hand, reduces the error in the prescribed final conditions.