Optimal Navigation Functions for Nonlinear Stochastic Systems

TL;DR

This paper introduces a novel method for designing optimal navigation functions for nonlinear stochastic systems by transforming the HJB equation into a linear PDE, enabling incorporation of optimality criteria and generalization of existing navigation approaches.

Contribution

It presents a new transformation-based methodology for creating optimal navigation functions that unify and extend previous results in stochastic nonlinear system navigation.

Findings

The HJB equation can be transformed into a linear PDE for navigation.

Optimal navigation functions relate to Laplace's equation via an exponential transform.

Analytical solutions are obtainable in simplified domains, guiding approximation schemes.

Abstract

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.