On the use of adjoint gradients for time-optimal control problems regarding a discrete control parameterization

In this paper, we discuss time-optimal control problems for dynamic systems. Such problems usually arise in robotics when a manipulation should be carried out in minimal operation time. In particular, for time-optimal control problems with a high number of control parameters, the adjoint method is probably the most efficient way to calculate the gradients of an optimization problem concerning computational efficiency. In this paper, we present an adjoint gradient approach for solving time-optimal control problems with a special focus on a discrete control parameterization. On the one hand, we provide an efficient approach for computing the direction of the steepest descent of a cost functional in which the costs and the error in the final constraints reduce within one combined iteration. On the other hand, we investigate this approach to provide an exact gradient for other optimization strategies and to evaluate necessary optimality conditions regarding the Hamiltonian function. Two examples of the time-optimal trajectory planning of a robot demonstrate an easy access to the adjoint gradients and their interpretation in the context of the optimality conditions of optimal control solutions, e.g., as computed by a direct optimization method.