Sampling-based algorithms for optimal motion planning

Abstract

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM * and RRT * , which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

A Survey on Aerial Swarm Robotics

“…Trajectories generated independently of the task assignment algorithm can be thought of in the same light as traditional optimal motion planning or boundary value problems. Popular randomized algorithms, such as PRM [81], RRT [82], and RRT* [83], may not be effective for obtaining optimal and safe flight of multiple 6-DOF aerial robots; not only can they not effectively handle 6-DOF nonlinear dynamics, but they also use a finite set of primitives predicated on asymptotic optimality without using higher-fidelity dynamic models, which could preclude a large set of otherwise flyable trajectories in a high-dimensional space. The rapid advancement in computing capacity combined with algorithmic improvements has enabled the development of tools that are capable of solving constrained optimization problems in real-time, which can better provide explicit or approximate solutions to an optimal control problem of the form…”

Convex Hull Asymptotic Shape Evolution

“…Let J = (J i, j ) be the linear part of (5). In particular, if g (π/3) > 0, the regular triangles are stable.…”

“…The lengths of RRT paths and their lack of optimality was studied in two recent works. Karaman and Frazzoli prove that RRTs are not asymptotically optimal and propose RRT*, which is a variant that converges to optimal path lengths [5]. Nechushtan, Raveh and Halperin developed an automaton-based approach to analyzing cases that lead to poor path quality in RRTs [12].…”

Robust Online Motion Planning with Regions of Finite Time Invariance

“…Planning algorithms like the Rapidly-exploring Randomized Tree (RRT) [12], RRT [11], and related trajectory library approaches [13] [5] can handle large state space dimensions and complex differential constraints, and have been successfully demonstrated on a wide variety of hardware platforms [22] [21]. However, a significant failing is their inability to explicitly reason about uncertainty and feedback.…”