Incremental Sampling-based Algorithms for Optimal Motion Planning (1005.0416v1)

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.

• The paper demonstrates that standard RRT converges almost surely to a sub-optimal solution, highlighting its limitations in achieving optimal motion planning.
• It proves that RRG converges to the optimal path as the number of samples increases, ensuring asymptotic optimality under practical conditions.
• The study introduces RRT* as an efficient variant that combines tree structure with optimality, maintaining similar computational overhead to RRT.

Incremental Sampling-Based Algorithms for Optimal Motion Planning

The paper by Sertac Karaman and Emilio Frazzoli investigates the theoretical underpinnings and practical implications of various incremental sampling-based motion planning algorithms, specifically Rapidly-exploring Random Trees (RRTs), Rapidly-exploring Random Graphs (RRGs), and a novel variant called RRT$^*$.

Overview and Key Contributions

Incremental sampling-based motion planning algorithms have become pivotal in addressing the complex navigation problems faced by modern robots. These algorithms, such as RRTs, PRMs, and other related methodologies, are celebrated for their probabilistic completeness and excellent performance in high-dimensional state spaces. The principal thrust of this paper is to bridge the gap in theoretical guarantees related to the optimality of these algorithms.

The paper makes several notable contributions:

1. Negative Result for RRT: The authors demonstrate that under mild technical assumptions, the RRT algorithm asymptotically converges to a sub-optimal solution almost surely. This reveals a critical limitation in RRT’s applicability towards optimal motion planning.
2. Asymptotic Optimality of RRG: They prove that a new algorithm, RRG, converges to an optimal solution almost surely, thereby addressing the optimality gap inherent in RRTs.
3. Introduction of RRT$^*$ Algorithm: The RRT$^*$ is introduced as a variant of RRG that retains the simplicity and tree structure of RRT while ensuring asymptotic optimality. RRT$^*$ rewires the tree to discover lower-cost paths incrementally.
4. Computational Complexity Analysis: The paper shows that the asymptotic computational complexity of RRG and RRT$^*$ remains within a constant factor of RRT.

Theoretical Insights and Results

Feasibility and Probabilistic Completeness:

• The probabilistic completeness of RRTs and RRGs is established, and the authors prove that the probability of finding a feasible path exponentially approaches one, given an attraction sequence, which is an intrinsic property of environments with good visibility.

Negative Result on RRT Optimality:

• The authors rigorously prove that RRT almost surely converges to a non-optimal cost under the assumptions that the environment's visibility properties are good, the sampling is absolutely continuous, and the cost function is monotonic. This result significantly impacts the deployment of RRT in scenarios where optimality is desired.

Asymptotic Optimality of RRG:

• By leveraging results from random geometric graph theory and constructing novel sequences of paths and sets, the authors show that RRG samples can almost surely achieve optimality as the number of samples approaches infinity. They establish that RRG has low computational overhead while maintaining this optimality property.

Computational Complexity:

• The paper details the computational analysis, proving that both RRG and RRT$^*$ algorithms incur an overhead of $O(\log n)$ calls to the collision-checking primitive compared to RRT, translating this into computational efficiency bounded within a constant factor of RRT.

Practical Implications

The introduction and analysis of RRG and RRT$^*$ have practical consequences for robotics. In applications where the cost and quality of paths are critical, RRT$^*$ and RRG offer robust alternatives to RRT. These algorithms promise not only to find feasible paths but to continually improve the path's cost, converging to the optimal solution over time. This makes them particularly valuable in fields like autonomous driving, aerospace, and robotic surgery, where optimal trajectories are crucial.

Future Directions

The promising results of RRT$^*$ and RRG spur several future research directions:

1. Extension to Differential Constraints: Adapting RRT$^*$ to environments with differential constraints would enhance its applicability to more complex robotic systems with non-holonomic properties.
2. Advanced Specifications: Exploring motion planning under more stringent temporal or logic-based constraints, such as Linear Temporal Logic (LTL), could expand the notion of optimality under different specifications.
3. Online and Dynamic Environments: Further research could focus on real-time implementations and adaptations of these algorithms to handle dynamic and unpredictable changes in the environment seamlessly.
4. Interdisciplinary Applications: Beyond robotics, these algorithms can have implications in solving PDEs, optimization in logistics, and operations research where finding near-optimal solutions efficiently is critical.

Conclusion

This paper significantly advances understanding in the field of motion planning by formally establishing the sub-optimality of RRT and proposing computationally feasible, asymptotically optimal alternatives in RRG and RRT$^*$. These contributions are not only theoretical milestones but also provide practical tools that can be directly applied to real-world robotic systems, ensuring paths that are not just feasible but optimal in terms of cost.