[Paper Review] Incremental Sampling-based Algorithms for Optimal Motion Planning

Sertaç Karaman, Emilio Frazzoli|arXiv (Cornell University)|May 3, 2010

Machine Learning and Algorithms4 citations

TL;DR

This paper introduces RRT∗, an incremental sampling-based motion planning algorithm that achieves asymptotic optimality—converging to the optimal path cost as the number of samples increases—by extending the Rapidly-exploring Random Tree (RRT) with a rewiring mechanism. Unlike standard RRT, which converges to sub-optimal solutions, RRT∗ maintains a tree structure while ensuring convergence to the true optimum, with computational complexity asymptotically within a constant factor of RRT.

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 have been established so far. The first contribution of this paper is a negative result: it is proven that, under mild technical conditions, the cost of the best path in the RRT converges almost surely to a non-optimal value. Second, a new algorithm is considered, called the Rapidly-exploring Random Graph (RRG), and it is shown that the cost of the best path in the RRG converges to the optimum almost surely. Third, a tree version of RRG is introduced, called the RRT$^*$ algorithm, 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.

Motivation & Objective

• To address the lack of theoretical bounds on solution quality in incremental sampling-based motion planners like RRT.
• To prove that standard RRT converges almost surely to a non-optimal solution under mild conditions.
• To develop a new algorithm, RRG, that ensures asymptotic optimality by maintaining a graph structure.
• To design a tree-based variant, RRT∗, that inherits RRG’s optimality while preserving the RRT tree structure for practical use.
• To establish that the computational complexity of RRT∗ and RRG is asymptotically within a constant factor of RRT.

Proposed method

• Propose the Rapidly-exploring Random Graph (RRG) algorithm, which incrementally builds a graph by connecting new samples to nearby vertices and rewiring edges to improve path costs.
• Introduce the RRT∗ algorithm as a tree-based version of RRG, where new vertices are connected to the nearest node in the tree, and existing edges are rewired if a lower-cost path is found.
• Use tools from random geometric graph theory to analyze the convergence properties of RRG and RRT∗, establishing almost sure convergence to the optimal path cost.
• Apply a probabilistic rewiring mechanism in RRT∗ that checks whether connecting a new sample to a parent node improves the cost-to-go of descendant nodes.
• Prove that the expected number of rewirings per iteration is bounded, ensuring that the computational overhead remains asymptotically within a constant factor of RRT.
• Validate theoretical claims through extensive simulations comparing RRT, RRG, and RRT∗ on environments with varying obstacle densities and dimensions.

Experimental results

Research questions

• RQ1 Does the standard RRT algorithm converge to an optimal solution as the number of samples increases?
• RQ2 Can a sampling-based algorithm be designed to achieve asymptotic optimality while maintaining a tree structure?
• RQ3 What is the computational complexity of an asymptotically optimal sampling-based planner compared to RRT?
• RQ4 Can the theoretical properties of random geometric graphs be leveraged to prove convergence to the optimal path cost?
• RQ5 How does the performance of RRT∗ compare to RRT and RRG in terms of path quality and runtime?

Key findings

• The standard RRT algorithm converges almost surely to a non-optimal solution under mild technical conditions, proving a fundamental limitation of RRT.
• The RRG algorithm achieves asymptotic optimality, meaning the cost of the best path in the graph converges almost surely to the optimal cost as the number of samples increases.
• The RRT∗ algorithm inherits the asymptotic optimality of RRG while maintaining a tree structure, making it suitable for real-time and online planning applications.
• The computational complexity of RRT∗ is asymptotically within a constant factor of RRT, meaning the overhead introduced by rewiring is bounded and scalable.
• Simulations show that RRT∗ achieves significantly lower path costs than RRT—e.g., 14.51 vs. 21.02 in one example—while maintaining comparable runtime ratios.
• The variance in path cost across trials decreases over time for RRT∗, indicating stable convergence to the optimal solution.