Finding Control Synthesis for Kinematic Shortest Paths

TL;DR

This paper analyzes the properties of shortest path control synthesis for kinematic rigid body systems, providing necessary conditions, verification methods, and a procedure applicable to 2D and 3D systems, verified on a 2D Dubins vehicle.

Contribution

It introduces a novel procedure to find shortest kinematic paths and control synthesis boundaries using gradients, applicable to both 2D and 3D rigid body systems.

Findings

Shortest paths have similar adjoint functions for nearby configurations.

Gradients of control constraints can verify control synthesis regions.

Procedure verified on a 2D Dubins vehicle system.

Abstract

This work presents the analysis of the properties of the shortest path control synthesis for the rigid body system. The systems we focus on in this work have only kinematic constraints. However, even for seemingly simple systems and constraints, the shortest paths for generic rigid body systems were only found recently, especially for 3D systems. Based on the Pontraygon's Maximum Principle (MPM) and Lagrange equations, we present the necessary conditions for optimal switches, which form the control synthesis boundaries. We formally show that the shortest path for nearby configurations will have similar adjoint functions and parameters, i.e., Lagrange multipliers. We further show that the gradients of the necessary condition equation can be used to verify whether a configuration is inside a control synthesis region or on the boundary. We present a procedure to find the shortest kinematic paths and control synthesis, using the gradients of the control constraints. Given the shortest path and the corresponding control sequences, the optimal control sequence for nearby configurations can be derived if and only if they belong to the same control synthesis region. The proposed procedure can work for both 2D and 3D rigid body systems. We use a 2D Dubins vehicle system to verify the correctness of the proposed approach. More verifications and experiments will be presented in the extensions of this work.