[Paper Review] Informed Sampling for Asymptotically Optimal Path Planning (Consolidated Version)

Gammell, Jonathan D, Barfoot, Timothy D+1 more|arXiv (Cornell University)|Jun 20, 2017

Robotic Path Planning Algorithms189 citations

TL;DR

This paper proposes Informed RRT*, an asymptotically optimal path planner that directly samples the L2-informed set—prolate hyperspheroids representing states capable of improving the current solution. By focusing sampling on this compact, dimensionally adaptive set, it achieves linear convergence and outperforms existing methods by orders of magnitude in high-dimensional spaces, especially in complex planning domains like HERB’s 14-DOF manipulation tasks.

ABSTRACT

Anytime almost-surely asymptotically optimal planners, such as RRT*, incrementally find paths to every state in the search domain. This is inefficient once an initial solution is found as then only states that can provide a better solution need to be considered. Exact knowledge of these states requires solving the problem but can be approximated with heuristics. This paper formally defines these sets of states and demonstrates how they can be used to analyze arbitrary planning problems. It uses the well-known $L^2$ norm (i.e., Euclidean distance) to analyze minimum-path-length problems and shows that existing approaches decrease in effectiveness factorially (i.e., faster than exponentially) with state dimension. It presents a method to address this curse of dimensionality by directly sampling the prolate hyperspheroids (i.e., symmetric $n$-dimensional ellipses) that define the $L^2$ informed set. The importance of this direct informed sampling technique is demonstrated with Informed RRT*. This extension of RRT* has less theoretical dependence on state dimension and problem size than existing techniques and allows for linear convergence on some problems. It is shown experimentally to find better solutions faster than existing techniques on both abstract planning problems and HERB, a two-arm manipulation robot.

Motivation & Objective

• Address the inefficiency of RRT*-style planners that sample the entire state space after an initial solution is found.
• Formally define the omniscient set (states that can improve the solution) and its heuristic approximations (informed sets) using precision and recall metrics.
• Demonstrate that existing rejection-sampling techniques for informed sets suffer from factorial decay in success probability as state dimension increases.
• Propose a direct sampling method for the L2-informed set (ellipsoidal regions) to overcome the curse of dimensionality in minimum-path-length problems.
• Develop and evaluate Informed RRT*, an extension of RRT* that uses direct informed sampling and admissible pruning to achieve faster, scalable convergence.

Proposed method

• Define the omniscient set as the exact set of states that can yield a better solution, and informed sets as heuristic approximations with quantifiable precision and recall.
• Use the L2 norm (Euclidean distance) as a sharp, universally admissible heuristic for minimum-path-length problems, forming the L2-informed set.
• Introduce a direct sampling technique for prolate hyperspheroids (n-dimensional ellipses) that guarantees sampling from the L2-informed set regardless of its size.
• Integrate this sampling method into RRT* via Informed RRT*, which maintains the tree structure but restricts sampling and rewiring to the informed set.
• Apply admissible graph pruning to further reduce computational overhead by removing non-promising paths.
• Derive theoretical bounds on convergence rates using expected cost improvements and prove that Informed RRT* achieves linear convergence under optimal conditions.

Experimental results

Research questions

• RQ1 How does the probability of improving a solution in RRT*-style planners decay with increasing state dimension when using rejection sampling over informed sets?
• RQ2 Can direct sampling of the L2-informed set (prolate hyperspheroids) mitigate the factorial decay in improvement probability observed in high-dimensional problems?
• RQ3 What is the theoretical convergence rate of RRT* when sampling is restricted to the L2-informed set, and how does it compare to standard RRT*?
• RQ4 Does Informed RRT* achieve linear convergence in minimum-path-length problems, and under what conditions?
• RQ5 How does Informed RRT* perform in practice compared to RRT* and other informed sampling techniques on high-dimensional and real-world robotic platforms?

Key findings

• Existing rejection-sampling techniques for informed sets have a solution-improvement probability that decays factorially with state dimension, making them ineffective in high-dimensional spaces.
• Direct sampling of the L2-informed set (ellipsoidal regions) ensures that every sampled state is potentially beneficial, eliminating the need for rejection and enabling scalable performance.
• Informed RRT* achieves linear convergence in minimum-path-length problems, with an expected convergence rate of exactly (n−1)/(n+1), where n is the state dimension.
• Theoretical analysis shows that Informed RRT* has a better convergence rate than standard RRT* and RRT* with rejection sampling, especially in high-dimensional problems.
• Experimental results on abstract problems and the HERB robot (14-DOF) show that Informed RRT* finds better solutions faster than existing techniques, with performance gains increasing with dimensionality.
• Informed RRT* is probabilistically complete and almost-surely asymptotically optimal, and it reduces to RRT* when the informed set is large or the problem is small.