Neighborhood estimation in sensitivity-based update rules for real-time optimal control

In the domain of optimal control theory, an optimal control problem can be formulated with respect to a predefined task parameter set, leading to a parametric optimal control problem. In application areas such as spacecraft guidance and robotics, optimal control and trajectory planning inevitably involve dealing with motion constraints and imposing a requirement of feasibility, as well as optimality. By virtue of the Sensitivity Theorem, real-time solutions of such constrained optimal control problems can be approximated through a sensitivity-based update of a given nominal solution for nominal task parameter values. There are, however, strict requirements placed on the task parameters and on the optimization solutions to make this approximation possible, so as to place guarantees on the existence and optimality of solutions in the vicinity of the nominal solution. One of these requirements is a neighborhood of validity about the nominal parameter, the size of which is, however, not specified by the Sensitivity Theorem. In this paper, we therefore present a method by which to estimate the size of this neighborhood. We then judiciously apply the result to the whole task parameter space of interest, so as to be able to place weak guarantees on the existence and optimality of solutions throughout it. We demonstrate the efficacy of our approach on a two-link planar robot arm.