Sample Complexity of Probabilistic Roadmaps via ε-nets

Tsao, Matthew; Solovey, Kiril; Pavone, Marco

doi:10.1109/icra40945.2020.9196917

Cited by 12 publications

(14 citation statements)

References 30 publications

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“…A key component in our result is a new sampling scheme that we develop, which we call the staggered grid, for finite-sample motion planning for individual robots. This sampling scheme requires significantly less samples than our previous work [32], to achieve near-optimality for the single-robot case.…”

Section: Related Workmentioning

confidence: 96%

“…We provide a formal definition of the Probabilistic Roadmap (PRM) method [17], which constructs a discrete graph that captures the connectivity of C f via sampling. PRM plays a critical role in various sampling-based planners (see, e.g., [32,33]). PRM is also instrumental to our result both on single-robot motion-planning in this section and on multirobot motion-planning in Section III.…”

Section: B Probabilistic Roadmaps and Sample Setsmentioning

confidence: 99%

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Near-Optimal Multi-Robot Motion Planning with Finite Sampling

Dayan

Solovey

Pavone

et al. 2020

Preprint

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An underlying structure in several sampling-based methods for continuous multi-robot motion planning (MRMP) is the tensor roadmap (TR), which emerges from combining multiple PRM graphs constructed for the individual robots via a tensor product. We study the conditions under which the TR encodes a near-optimal solution for MRMP-satisfying these conditions implies near optimality for a variety of popular planners, including dRRT*, and the discrete methods M* and CBS when applied to the continuous domain. We develop the first finite-sample analysis of this kind, which specifies the number of samples, their deterministic distribution, and magnitude of the connection radii that should be used by each individual PRM graph, to guarantee near-optimality using the TR. This significantly improves upon a previous asymptotic analysis, wherein the number of samples tends to infinity, and supports guaranteed high-quality solutions in practice, within bounded running time. To achieve our new result, we first develop a sampling scheme, which we call the staggered grid, for finite-sample motion planning for individual robots, which requires significantly less samples than previous work. We then extend it to the much more involved MRMP setting which requires to account for interactions among multiple robots. Finally, we report on a few experiments that serve as a verification of our theoretical findings and raise interesting questions for further investigation.

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Section: Related Workmentioning

confidence: 96%

Section: B Probabilistic Roadmaps and Sample Setsmentioning

confidence: 99%

Near-Optimal Multi-Robot Motion Planning with Finite Sampling

Dayan

Solovey

Pavone

et al. 2020

Preprint

Self Cite

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“…One possible reason for existing approaches being so computationally expensive is their insistence on solving any feasible problem instance-including those that require the robot to get arbitrarily close to obstacles in order to reach its target. It has been already observed in sampling-based planning (see, e.g., Tsao et al (2020)) that the difficulty of solving a motion-planning problem increases as its clearance decreases. The clearance of a problem δ ≥ 0 denotes the minimal distance of the robot from the obstacles that is necessary to be achieved in order to find a solution.…”

Section: Future Direction For Researchmentioning

confidence: 98%

Complexity of Planning

Solovey

2020

Preprint

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Computational complexity, time complexity. DefinitionsComplexity analysis of complete algorithms for robot motion planning. OverviewThis chapter is devoted to the study of complexity of complete (or exact) algorithms for robot motion planning. The term "complete" indicates that an approach is guaranteed to find the correct solution (a motion path or trajectory in our setting), or to report that none exists otherwise (in case that for instance, no feasible path exists). Complexity theory is a fundamental tool in computer science for analyzing the performance of algorithms, in terms of the amount of resources they require. (While complexity can express different quantities such as space and communication effort, our focus in this chapter is on time complexity.) Moreover, complexity theory helps to identify "hard" problems which require excessive amount of computation time to solve. In the context of motion planning, complexity theory can come in handy in various ways, some of which are illustrated here.Kiril Solovey (kirilsol.github.io)

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“…Sampling-based planners can also use deterministic sampling strategies (e.g., Branicky et al 2001;LaValle et al 2004), which can result in deterministic optimality guarantees (Janson et al 2018). The finite-time properties of asymptotically optimal planners are analyzed by Dobson and Bekris (2013), Janson et al (2018), andTsao et al (2020).…”

Section: Problem Definitionmentioning

confidence: 99%

“…The r-disc strategy can result in better performance than the k-nearest version but the computation of the r-disc radius must be adjusted to the properties of the state space (Kleinbort et al 2016(Kleinbort et al , 2020b. Faster-decreasing radii are presented in Janson et al (2015Janson et al ( , 2018, Solovey and Kleinbort (2020), and Tsao et al (2020), but are not used in AIT* and EIT* for direct comparison to existing algorithms as they are presented in the literature. The forward search tree is shown with black lines ( ) and the reverse search tree with gray lines ( ).…”

Section: 21mentioning

confidence: 99%

AIT* and EIT*: Asymmetric bidirectional sampling-based path planning

Strub¹,

Gammell²

2021

Preprint

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Optimal path planning is the problem of finding a valid sequence of states between a start and goal that optimizes an objective. Informed path planning algorithms order their search with problem-specific knowledge expressed as heuristics and can be orders of magnitude more efficient than uninformed algorithms. Heuristics are most effective when they are both accurate and computationally inexpensive to evaluate, but these are often conflicting characteristics. This makes the selection of appropriate heuristics difficult for many problems. This paper presents two almost-surely asymptotically optimal sampling-based path planning algorithms to address this challenge, Adaptively Informed Trees (AIT*) and Effort Informed Trees (EIT*). These algorithms use an asymmetric bidirectional search in which both searches continuously inform each other. This allows AIT* and EIT* to improve planning performance by simultaneously calculating and exploiting increasingly accurate, problem-specific heuristics. The benefits of AIT* and EIT* relative to other sampling-based algorithms are demonstrated on twelve problems in abstract, robotic, and biomedical domains optimizing path length and obstacle clearance. The experiments show that AIT* and EIT* outperform other algorithms on problems optimizing obstacle clearance, where a priori cost heuristics are often ineffective, and still perform well on problems minimizing path length, where such heuristics are often effective.

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Sample Complexity of Probabilistic Roadmaps via ε-nets

Cited by 12 publications

References 30 publications

Near-Optimal Multi-Robot Motion Planning with Finite Sampling

Near-Optimal Multi-Robot Motion Planning with Finite Sampling

Complexity of Planning

AIT* and EIT*: Asymmetric bidirectional sampling-based path planning

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