Sampling-based Algorithms for Optimal Motion Planning (1105.1186v1)

Abstract: During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

• The paper demonstrates that traditional planners like PRM and RRT are probabilistically complete but not asymptotically optimal compared to the new PRM*, RRG, and RRT* methods.
• The paper introduces a variable connection strategy scaling as (log n/n)^(1/d) to achieve optimality while keeping computational complexity manageable.
• The paper provides a comprehensive theoretical framework and robust numerical analyses that set a new benchmark for efficient and optimal motion planning.

Sampling-based Algorithms for Optimal Motion Planning

The paper "Sampling-based Algorithms for Optimal Motion Planning" by Sertac Karaman and Emilio Frazzoli rigorously addresses theoretical and computational properties of various sampling-based motion planning algorithms, with a focus on their ability to provide optimal solutions. The authors primarily analyze existing algorithms such as Probabilistic RoadMaps (PRMs) and Rapidly-exploring Random Trees (RRTs) and introduce new algorithms PRM$^*$, RRG, and RRT$^*$ that improve upon these classical methods. The core contributions of the paper lie in providing a comprehensive theoretical framework that elucidates the probabilistic completeness and asymptotic optimality of different algorithms, complemented by robust numerical results.

Key Contributions and Findings

Probabilistic Completeness and Asymptotic Optimality

Traditional sampling-based algorithms like PRM and RRT are shown to be probabilistically complete, meaning they can find a feasible solution with high probability if one exists. However, these algorithms are not asymptotically optimal; their solutions do not converge to the optimal solution as the number of samples grows. For example, the paper proves that RRT incurs a non-negligible suboptimality gap compared to the optimal solution.

In contrast, the newly introduced algorithms PRM$^*$, RRG, and RRT$^*$ are designed to be asymptotically optimal. The authors demonstrate that these algorithms ensure the cost of the solution converges almost surely to the optimal value as the number of samples increases, provided that specific conditions on sample density and connection strategies are met.

Detailed Analysis of New Algorithms

PRM$^*$: Optimal PRM Variant

The PRM$^*$ algorithm modifies the traditional PRM by utilizing a variable connection radius that decreases as the number of samples increases, specifically scaling as $(\log n / n)^{1/d}$. This scaling ensures that the average number of connections made at each iteration is proportional to $\log n$, maintaining computational efficiency while guaranteeing asymptotic optimality. The choice of connection parameters is rigorously justified, with the threshold being dependent on the dimension of the space and the volume of the obstacle-free space.

RRG and RRT$^*$: Incremental Methods

The RRG (Rapidly-exploring Random Graph) algorithm constructs a connected roadmap incrementally and ensures that the new nodes are connected within a certain radius. Similarly, RRT$^*$ maintains a tree structure and rewires the tree to maintain minimal cost paths. Both algorithms demonstrate that ensuring connections within a radius proportional to $(\log n / n)^{1/d}$ leads to asymptotically optimal solutions. The analysis reveals the minimal additional computational complexity required for these algorithms to achieve optimality compared to their classical counterparts.

Computational Complexity

The paper provides a thorough complexity analysis. For PRM$^*$, RRG, and RRT$^*$, the computational complexity per iteration is shown to be $O(n \log n)$, which is within a constant factor of traditional PRM and RRT algorithms. Memory usage is managed efficiently since the average number of connections per node scales logarithmically with the number of samples.

Implications and Future Research Directions

Theoretical insights from this paper have far-reaching implications for the design of sampling-based planners. By establishing that optimality can be achieved with only a logarithmic increase in computations, the authors set a new standard for efficiency and efficacy in motion planning algorithms. Future research could extend these results to more complex scenarios involving differential constraints and non-Euclidean metrics, or adapt these algorithms for multi-agent coordination and decentralized planning.

Additionally, the connection between sampling-based motion planning algorithms and the theory of random geometric graphs opens up new avenues for interdisciplinary research, leveraging advances in network theory to further refine motion planning techniques.

Conclusion

Karaman and Frazzoli's work represents a significant step forward in the understanding and development of sampling-based motion planning algorithms. Their rigorous analysis provides both theoretical foundations and practical algorithms that ensure optimal path planning with manageable computational demands. As these methods are integrated into real-world robotic systems, the balance of efficiency, and optimality achieved by PRM$^*$, RRG, and RRT$^*$ will likely lead to more robust and capable autonomous systems.