On the Usage of Analytically Computed Adjoint Gradients in a Direct Optimization for Time-Optimal Control Problems

This paper discusses time-optimal control problems and describes a workflow for the use of analytically computed adjoint gradients considering a discrete control parameterization. The adjoint gradients are used here to support a direct optimization method, such as Sequential Quadratic Programming (SQP), by providing analytically computed gradients and avoiding the elaborate numerical differentiation. In addition, the adjoint variables can be used to evaluate the necessary first-order optimality conditions regarding the Hamiltonian function and gives an opportunity to discuss the sensitivity of a solution with respect to the refinement of the discretization of the control. To further emphasize the advantages of adjoint gradients, there is also a discussion of the structure of analytical gradients computed by a direct differentiation method, and the difference in the dimensions compared to the adjoint approach is addressed. An example of trajectory planning for a robot shows application scenarios for the adjoint variables in a cubic spline parameterized control.