Optimal Navigation Functions for Nonlinear Stochastic Systems
This work addresses navigation system design for nonlinear stochastic systems, offering a generalized framework that connects existing methods, but it appears incremental as it builds on prior results without claiming broad new applications.
The paper tackles the problem of designing navigation functions for nonlinear stochastic systems by transforming the Hamilton-Jacobi-Bellman equation into a linear partial differential equation, showing that optimal navigation functions relate to Laplace's equation through an exponential transform and providing analytical solutions in simplified domains.
This paper presents a new methodology to craft navigation functions for nonlinear systems with stochastic uncertainty. The method relies on the transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear partial differential equation. This approach allows for optimality criteria to be incorporated into the navigation function, and generalizes several existing results in navigation functions. It is shown that the HJB and that existing navigation functions in the literature sit on ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. In particular, it is shown that under certain criteria the optimal navigation function is related to Laplace's equation, previously used in the literature, through an exponential transform. Further, analytical solutions to the HJB are available in simplified domains, yielding guidance towards optimality for approximation schemes. Examples are used to illustrate the role that noise, and optimality can potentially play in navigation system design.