Monolithic and Staggered Solution Strategies for Constrained Mechanical Systems in Optimal Control Problems

Abstract

This paper deals with the optimal control of constrained mechanical systems, with potential additional kinematic constraints at the final time. Correspondingly, the equations of motion of the underlying mechanical system assume the form of differential-algebraic equations with end constraints. The proposed discretisation of the optimality conditions yields a scheme which is capable of preserving control angular momentum maps resulting from the rotational symmetry of the underlying optimal control problem. The numerical solution of the discretised system is first tested with two solution strategies: Monolithic and staggered approaches, and then also solved with hybrid approaches, which combine salient features of each individual strategy. The monolithic strategy solves all the optimality conditions for all time steps as a single system of non-linear equations and relies on a Newton-Raphson scheme, which guarantees quadratic rates of convergence in the vicinity of the optimal solution trajectory. The staggered strategy is based on the Forward-Backward Sweep Method (FBSM), where state and adjoint equations are solved separately, and the control equations provide an update of the control variables, which we here achieve with also a Newton-Raphson scheme. The proposed hybrid strategies combine the advantages of a conventional gradient-based FBSM with the individual Newton-based solution procedures once the solution is close to the optimal trajectory. The strategies are developed and compared through three representative numerical examples, which show that all schemes yield very similar solutions. However, the hybrid approaches become more advantageous in the computation time when the time-step decreases or the size of the problem increases.