Density Functions for Guaranteed Safety on Robotic Systems

Abstract

The recent study on density functions as the dual of value functions for optimal control gives a new method for synthesizing safe controllers. A density function describes the state distribution in the state space, and its evolution follows the Liouville Partial Differential Equation (PDE). The duality between the density function and the value function in optimal control can be utilized to solve constrained optimal control problems with a primal-dual algorithm. This paper focuses on the application of the method on robotic systems and proposes an implementation of the primal-dual algorithm that is less computationally demanding than the method used in the literature. To be specific, we use kernel density estimation to estimate the density function, which scales better than the ODE approach in the literature and only requires a simulator instead of a dynamic model. The Hamilton Jacobi Bellman (HJB) PDE is solved with the finite element method in an implicit form, which accelerates the value iteration process. We show an application of the safe control synthesis with density functions on a segway control problem demonstrated experimentally.