Incremental Sampling-based Algorithms for Optimal Motion Planning

Abstract

Abstract-During the last decade, incremental sampling-based motion planning algorithms, such as the Rapidly-exploring Random Trees (RRTs), have been shown to work well in practice and to possess theoretical guarantees such as probabilistic completeness. However, no theoretical bounds on the quality of the solution obtained by these algorithms, e.g., in terms of a given cost function, have been established so far. The purpose of this paper is to fill this gap, by designing efficient incremental samplingbased algorithms with provable optimality properties. The first contribution of this paper is a negative result: it is proven that, under mild technical conditions, the cost of the best path returned by RRT converges almost surely to a non-optimal value, as the number of samples increases. Second, a new algorithm is considered, called the Rapidly-exploring Random Graph (RRG), and it is shown that the cost of the best path returned by RRG converges to the optimum almost surely. Third, a tree version of RRG is introduced, called RRT * , which preserves the asymptotic optimality of RRG while maintaining a tree structure like RRT. The analysis of the new algorithms hinges on novel connections between sampling-based motion planning algorithms and the theory of random geometric graphs. In terms of computational complexity, it is shown that the number of simple operations required by both the RRG and RRT * algorithms is asymptotically within a constant factor of that required by RRT.

Minimum Constraint Displacement Motion Planning

“…, q m that are close to q and such that the path through the milestone to q would be J max -reachable, assuming all constraints are feasible at q. As demonstrated by Karaman and Frazzoli [9], the choice m = (1 + 1/d)e log |V | ensures that the optimal path in the roadmap asymptotically approaches the true optimum as more samples are drawn. Extend-Toward(q i , q, δ, J max ) extends the roadmap with a new edge from q i to a configuration q in the direction of q.…”

Incremental Sampling-Based Algorithms for a Class of Pursuit-Evasion Games

“…This algorithm is closely related to the RRT * algorithm recently introduced in [35], which will be discussed first. RRT * is an incremental sampling-based motion planning algorithm with the asymptotic optimality property, i.e., almost-sure convergence to optimal trajectories, which the RRT algorithm lacks [35]. In fact, it is precisely this property of the RRT * that allows us to cope with the game introduced in the previous section.…”

“…(This particular scaling rate is cho-sen since it ensures both computational efficiency and asymptotic optimality of the RRT * algorithm [35,39].) Finally, we define Near α (G, z, n) := V ∩ Reach α (z, l(n)).…”

“…In particular, the Rapidly-exploring Random Tree (RRT) algorithm proposed in [33] has been shown to be very effective in practice, and was demonstrated on various platforms in major robotics events (see, e.g., [34]). Very recently, optimality properties of incremental sampling-based planning algorithms were analyzed and it was shown that, under mild technical assumptions, the RRT algorithm converges to a non-optimal solution with probability one [35]. In [35], the authors have proposed a new algorithm, called RRT * , which converges to an optimal solution almost surely, while incurring essentially the same computational cost when compared to the RRT.…”

“…Very recently, optimality properties of incremental sampling-based planning algorithms were analyzed and it was shown that, under mild technical assumptions, the RRT algorithm converges to a non-optimal solution with probability one [35]. In [35], the authors have proposed a new algorithm, called RRT * , which converges to an optimal solution almost surely, while incurring essentially the same computational cost when compared to the RRT. The RRT * algorithm can be viewed as an anytime algorithm for the optimal motion planning problem.…”