Asymptotically Optimal Sampling-based Kinodynamic Planning (1407.2896v5)

Abstract: Sampling-based algorithms are viewed as practical solutions for high-dimensional motion planning. Recent progress has taken advantage of random geometric graph theory to show how asymptotic optimality can also be achieved with these methods. Achieving this desirable property for systems with dynamics requires solving a two-point boundary value problem (BVP) in the state space of the underlying dynamical system. It is difficult, however, if not impractical, to generate a BVP solver for a variety of important dynamical models of robots or physically simulated ones. Thus, an open challenge was whether it was even possible to achieve optimality guarantees when planning for systems without access to a BVP solver. This work resolves the above question and describes how to achieve asymptotic optimality for kinodynamic planning using incremental sampling-based planners by introducing a new rigorous framework. Two new methods, Stable Sparse-RRT (SST) and SST*, result from this analysis, which are asymptotically near-optimal and optimal, respectively. The techniques are shown to converge fast to high-quality paths, while they maintain only a sparse set of samples, which makes them computationally efficient. The good performance of the planners is confirmed by experimental results using dynamical systems benchmarks, as well as physically simulated robots.

• The paper introduces a Monte Carlo-based propagation strategy that replaces complex steering functions with stochastic dynamics propagation.
• The paper achieves efficiency through a sparse sampling and selective pruning method that reduces computational load while maintaining quality paths.
• The paper guarantees asymptotic optimality by converging to near-optimal solutions, as validated by benchmarks on dynamic systems like drones and robots.

Asymptotically Optimal Sampling-based Kinodynamic Planning

The paper by Yanbo Li, Zakary Littlefield, and Kostas E. Bekris presents a rigorous framework for achieving asymptotic optimality in kinodynamic motion planning without requiring a detailed two-point boundary value problem (BVP) solver. Traditional methods often struggle with high-dimensional motion planning under dynamics, especially when steering functions are hard to define. This research introduces sampling-based methodologies that claim strong theoretical guarantees and competitive empirical performance.

Problem Context

Kinodynamic planning deals with dynamically feasible paths for systems like high-velocity vehicles and balancing robots. These systems have constraints imposed by their dynamic models parting from simpler kinematic path planning. The significance of solving this lies in its application range, from aerial drones to autonomous cars, where dynamics are a critical part of feasibility.

Core Contributions

The paper tackles optimality without steering functions. Conventional sampling methods either don't assure optimal paths or depend heavily on existence and computation of steering functions. The authors propose methods based on random geometric graphs and probabilistic approaches that permit discovering near-optimal, sparsely sampled paths by a novel propagation mechanism.

1. Random Propagation Strategy: The proposed Monte Carlo-based method replaces deterministic solvers with a stochastic propagation of systems dynamics. The framework introduces probabilistic completeness and establishes the existence of near-optimal paths even without predefined BVP solutions.
2. Sparse Representation with Pruning: The paper suggests turning away from dense sampling by employing a selective pruning strategy, maintaining only necessary segments of the search space with the best improvement in cost. It practically reduces memory and computational load, making the framework scalable.
3. Asymptotic Guarantees: Two sampling-based methods, titled - and -, offer asymptotic optimality and near-optimality. The analysis confirms convergence to high-quality paths within a sparse set of samples, leading to significant computational gains.

Empirical Evaluation

The experiments used benchmarks from simple pendulum systems to complex robots like quadrotors and fixed-wing aircraft. Proven convergence rates, better than existing approaches like RRT and shooting-based methods, were demonstrated. The number of stored nodes was significantly reduced while maintaining high-quality paths, highlighting the method's efficiency in balancing computational requirements and solution quality.

Practical and Theoretical Implications

This work opens possibilities for planning under dynamics across various robotic applications without the burdensome requirement of a steering function. The methods provide a tractable approach to explore high-dimensional systems, streamline resource usage, and achieve near-optimal paths reliably. Future developments might explore integrating these methods in control architectures and explore adaptability in uncertain environments.

Speculation on Future Developments

The paper hints at extending its findings to feedback-based planning and planning under uncertainty, calling for research into leveraging the new algorithmic insights in real-world systems. The influence of witness node selection, its adaptation to different environments, and integration with diverse metrics are potential areas for further exploration, capable of advancing autonomous systems' reliability and efficiency.

This paper contributes a comprehensive and theoretically backed approach to a persistent problem in kinodynamic planning, setting a new standard in the field with implications stretching across robotics and AI. The foundational work and potential extensions promise further advances in automated mobility and manipulation.