Informed RRT*: Optimal Sampling-based Path Planning Focused via Direct Sampling of an Admissible Ellipsoidal Heuristic (1404.2334v3)

Abstract: Rapidly-exploring random trees (RRTs) are popular in motion planning because they find solutions efficiently to single-query problems. Optimal RRTs (RRT*s) extend RRTs to the problem of finding the optimal solution, but in doing so asymptotically find the optimal path from the initial state to every state in the planning domain. This behaviour is not only inefficient but also inconsistent with their single-query nature. For problems seeking to minimize path length, the subset of states that can improve a solution can be described by a prolate hyperspheroid. We show that unless this subset is sampled directly, the probability of improving a solution becomes arbitrarily small in large worlds or high state dimensions. In this paper, we present an exact method to focus the search by directly sampling this subset. The advantages of the presented sampling technique are demonstrated with a new algorithm, Informed RRT*. This method retains the same probabilistic guarantees on completeness and optimality as RRT* while improving the convergence rate and final solution quality. We present the algorithm as a simple modification to RRT* that could be further extended by more advanced path-planning algorithms. We show experimentally that it outperforms RRT* in rate of convergence, final solution cost, and ability to find difficult passages while demonstrating less dependence on the state dimension and range of the planning problem.

• The paper introduces a novel ellipsoidal informed subset that confines the search to states capable of enhancing path optimality.
• It details a direct sampling method using geometric transformations from the unit n-ball to focus the exploration within the informed subset.
• Empirical results show that Informed RRT* outperforms standard RRT* with faster convergence and lower cost solutions in complex, high-dimensional spaces.

Informed RRT*: Optimal Sampling-based Path Planning Focused via Direct Sampling of an Admissible Ellipsoidal Heuristic

The paper "Informed RRT*: Optimal Sampling-based Path Planning Focused via Direct Sampling of an Admissible Ellipsoidal Heuristic" by Jonathan D. Gammell, Siddhartha S. Srinivasa, and Timothy D. Barfoot presents a significant contribution to the field of motion planning, extending the capabilities of optimal sampling-based planners such as Rapidly-exploring Random Trees (RRT*).

Summary

This research advances the RRT* framework by introducing a novel sampling methodology that enhances the efficiency and convergence properties of the algorithm. The authors propose the Informed RRT* (Informed RRT*) algorithm, which leverages direct sampling from an ellipsoidal subset of the state space, defined by admissible heuristics, to improve the solution quality and convergence rate.

Key Contributions

1. Ellipsoidal Informed Subset: The primary theoretical insight presented in the paper is that for path length minimization problems in $\mathbb{R}^n$, the states that can improve a given solution are confined within a prolate hyperspheroid. This subset is termed the ellipsoidal informed subset and reduces the search space for optimization.
2. Direct Sampling Methodology: The paper describes an exact method for sampling directly from the ellipsoidal informed subset, ensuring that all samples contribute to potentially improving the current solution. This direct sampling method involves transforming uniform samples from the unit $n$-ball using a series of geometric transformations, effectively focusing the search within a relevant subset of the planning domain.
3. Algorithm Design: The Informed RRT* algorithm begins with a conventional RRT* search until an initial feasible path is found. Subsequent samples are drawn exclusively from the ellipsoidal informed subset, significantly improving the probability of finding better paths while retaining the probabilistic guarantees of completeness and optimality inherent in RRT*.

Experimental Results

The experimental evaluation demonstrates that Informed RRT* outperforms standard RRT* in several key aspects:

• Convergence Rate:

Informed RRT* consistently finds paths that converge more rapidly towards the optimal solution compared to RRT*. This improvement is quantified across various problem sizes and complexities, including challenging high-dimensional configurations.

• Solution Quality:

Informed RRT* achieves lower cost solutions within a shorter computational time frame, evidenced by various simulated scenarios and random world simulations. This is particularly notable in configurations with narrow passages and other complex environments where traditional RRT* methods tend to struggle.

• Dimensionality and Range Resistance:

The dependency of the algorithm's performance on state dimension and problem range is reduced due to targeted sampling, making Informed RRT* more robust and scalable in high-dimensional spaces.

Practical and Theoretical Implications

The implications of this research span both practical and theoretical domains of robotics and motion planning:

• Practical Use:

By efficiently focusing the sampling process, Informed RRT* reduces computational load, making real-time applications more feasible. This is particularly relevant for robotics tasks that require rapid re-planning and adaptation in dynamic environments.

• Theoretical Impact:

The introduction of a method to directly sample from an informed subset provides a new vantage point for understanding and developing sampling-based algorithms. It also opens avenues for integrating additional heuristic-informed sampling techniques in motion planning.

Future Directions

Future work could explore the extension of this informed sampling framework to other optimization criteria beyond path length minimization, such as energy efficiency or safety margins. Additionally, combining this direct sampling approach with other optimization techniques, like path-smoothing, could further enhance the utility and performance of the algorithm.

In conclusion, Informed RRT* represents a sophisticated and methodologically rigorous advancement in motion planning, offering a practical enhancement to the widely-used RRT* framework while maintaining its theoretical robustness. This algorithm has the potential to significantly influence the development and deployment of autonomous robotic systems in complex, high-dimensional environments.