[Paper Review] The Role of Vertex Consistency in Sampling-based Algorithms for Optimal Motion Planning

Oktay Arslan, Panagiotis Tsiotras|arXiv (Cornell University)|Apr 29, 2012

Robotic Path Planning Algorithms4 citations

TL;DR

This paper introduces RRT#, a sampling-based optimal motion planning algorithm that enforces vertex consistency in its spanning tree to accelerate convergence to the optimal path. By ensuring all vertices have minimal cost-to-come values, RRT# enables faster identification of promising regions, outperforming RRT* in convergence speed and solution quality, with three variants further improving performance through informed sampling and inconsistency reduction.

ABSTRACT

Motion planning problems have been studied by both the robotics and the controls research communities for a long time, and many algorithms have been developed for their solution. Among them, incremental sampling-based motion planning algorithms, such as the Rapidly-exploring Random Trees (RRTs), and the Probabilistic Road Maps (PRMs) have become very popular recently, owing to their implementation simplicity and their advantages in handling high-dimensional problems. Although these algorithms work very well in practice, the quality of the computed solution is often not good, i.e., the solution can be far from the optimal one. A recent variation of RRT, namely the RRT* algorithm, bypasses this drawback of the traditional RRT algorithm, by ensuring asymptotic optimality as the number of samples tends to infinity. Nonetheless, the convergence rate to the optimal solution may still be slow. This paper presents a new incremental sampling-based motion planning algorithm based on Rapidly-exploring Random Graphs (RRG), denoted RRT# (RRT "sharp") which also guarantees asymptotic optimality but, in addition, it also ensures that the constructed spanning tree of the geometric graph is consistent after each iteration. In consistent trees, the vertices which have the potential to be part of the optimal solution have the minimum cost-come-value. This implies that the best possible solution is readily computed if there are some vertices in the current graph that are already in the goal region. Numerical results compare with the RRT* algorithm.

Motivation & Objective

• To address the slow convergence of asymptotically optimal sampling-based motion planners like RRT* in high-dimensional state spaces.
• To improve solution quality and convergence speed by enforcing vertex consistency in the tree structure at every iteration.
• To enable faster identification of optimal path candidates by classifying vertices based on their potential to lie on the optimal path.
• To develop variants of RRT# that enhance convergence through targeted sampling and efficient information propagation.
• To provide a framework for integrating consistency information into incremental sampling-based planners for improved performance.

Proposed method

• The algorithm uses a Rapidly-exploring Random Graph (RRG) framework to incrementally build a geometric graph from random samples in the free configuration space.
• At each iteration, the algorithm maintains a spanning tree where vertex consistency ensures the cost-to-come of each vertex equals the optimal cost-to-come via its neighbors.
• The key procedure, ReduceInconsistency, propagates updated cost information through the tree to maintain consistency after new vertices are added.
• Three variants (RRT#_V1, V2, V3) are introduced: V1 avoids expanding into non-promising regions, V2 improves information propagation, and V3 combines both strategies.
• The algorithm uses a local planner to generate collision-free trajectories between samples and applies cost-based vertex classification to guide sampling and tree expansion.
• Vertex consistency is enforced by comparing the current cost-to-come of a vertex with the cost-to-come via its neighbors, triggering updates when a lower cost path is found.

Experimental results

Research questions

• RQ1 Can enforcing vertex consistency in the spanning tree of a sampling-based planner significantly improve convergence speed to the optimal solution?
• RQ2 How does vertex consistency enable faster identification of regions likely to contain the optimal path?
• RQ3 To what extent can informed sampling strategies based on vertex classification reduce the number of vertices needed for convergence?
• RQ4 Can the integration of consistency propagation and region pruning improve performance beyond RRT* in high-dimensional motion planning?
• RQ5 What are the quantitative gains in convergence rate and solution quality when using RRT# and its variants compared to RRT*?

Key findings

• RRT# achieves asymptotic optimality with faster convergence than RRT*, as demonstrated by numerical results in 2D and 5D environments.
• The baseline RRT# algorithm converges faster than RRT* due to vertex consistency, which allows immediate access to the best possible solution if any vertex lies in the goal region.
• The RRT#_V3 variant achieves the fastest convergence, reducing the number of vertices in the tree while maintaining high solution quality.
• Execution time of RRT# and its variants is normalized and shown to be significantly lower than RRT* over 50 trials in 2D and 100 trials in 5D spaces.
• The variance in path cost decreases more rapidly in RRT# compared to RRT*, indicating more stable and consistent convergence behavior.
• The use of vertex classification enables smarter sampling, reducing unnecessary tree expansion into unproductive regions of the state space.