Analysis of Asymptotically Optimal Sampling-based Motion Planning Algorithms for Lipschitz Continuous Dynamical Systems
This addresses a key limitation in sampling-based motion planning algorithms like RRT*, which previously only guaranteed optimality for specific systems, potentially enabling broader applications in robotics and autonomous systems, though it appears incremental by extending existing methods.
The paper tackles the problem of achieving asymptotically optimal motion planning for dynamical systems with differential constraints without requiring a steering function, demonstrating that sampling the control space directly can achieve this for any Lipschitz continuous system and providing theoretical bounds on convergence rates.
Over the last 20 years significant effort has been dedicated to the development of sampling-based motion planning algorithms such as the Rapidly-exploring Random Trees (RRT) and its asymptotically optimal version (e.g. RRT*). However, asymptotic optimality for RRT* only holds for linear and fully actuated systems or for a small number of non-linear systems (e.g. Dubin's car) for which a steering function is available. The purpose of this paper is to show that asymptotically optimal motion planning for dynamical systems with differential constraints can be achieved without the use of a steering function. We develop a novel analysis on sampling-based planning algorithms that sample the control space. This analysis demonstrated that asymptotically optimal path planning for any Lipschitz continuous dynamical system can be achieved by sampling the control space directly. We also determine theoretical bounds on the convergence rates for this class of algorithms. As the number of iterations increases, the trajectory generated by these algorithms, approaches the optimal control trajectory, with probability one. Simulation results are promising.