UC Santa Barbara

Presented here is research focused on numerical advancements in Hamilton-Jacobi (HJ) theory as they provide a fundamental tool to address many problems in autonomous robotics such as optimal trajectory planning, safety critical reactive control, pursuit-evasion, and the optimal gathering of information of an unknown environment. HJ equations have had limitations in the past for computing usable solutions due to poor scaling with respect to system dimension. This was because a spatial grid had to be constructed densely in each dimension. Two general frameworks are proposed to overcome this limitation. First, methods based on trajectory optimization are presented and in particular, those based on generalizations of the Hopf formula are developed. This class of methods leverage the fact that for many real-world problems, only pointwise solutions are necessary, and this enables these optimization-based approached to be implemented on embedded hardware for real-time operation. Second, decomposition methods are developed. This class of methods leverage certain problem structures that allow smaller-dimensional subproblems to be formed. The solutions of these subproblems can then be aggregated to compute the global solution of the original HJ equation.