Abstract
Probabilistic sampling-based algorithms, such as the probabilistic roadmap (PRM) and the rapidly-exploring random tree (RRT) algorithms, represent one of the most successful approaches to robotic motion planning, due to their strong theoretical properties (in terms of probabilistic completeness or even asymptotic optimality) and remarkable practical performance. Such algorithms are probabilistic in that they compute a path by connecting independent and identically distributed (i.i.d.) random points in the configuration space. Their randomization aspect, however, makes several tasks challenging, including certification for safety-critical applications and use of offline computation to improve real-time execution. Hence, an important open question is whether similar (or better) theoretical guarantees and practical performance could be obtained by considering deterministic, as opposed to random sampling sequences. The objective of this paper is to provide a rigorous answer to this question. The focus is on the PRM algorithm—our results, however, generalize to other batch-processing algorithms such as \(\text {FMT}^*\). Specifically, we first show that PRM, for a certain selection of tuning parameters and deterministic low-dispersion sampling sequences, is deterministically asymptotically optimal, i.e., it returns a path whose cost converges deterministically to the optimal one as the number of points goes to infinity. Second, we characterize the convergence rate, and we find that the factor of sub-optimality can be very explicitly upper-bounded in terms of the \(\ell _2\)-dispersion of the sampling sequence and the connection radius of PRM. Third, we show that an asymptotically optimal version of PRM exists with computational and space complexity arbitrarily close to O(n) (the theoretical lower bound), where n is the number of points in the sequence. This is in stark contrast to the \(O(n\, \log n)\) complexity results for existing asymptotically-optimal probabilistic planners. Finally, through numerical experiments, we show that planning with deterministic low-dispersion sampling generally provides superior performance in terms of path cost and success rate.
This work was supported by NASA under the Space Technology Research Grants Program, Grant NNX12AQ43G. Lucas Janson was partially supported by NIH training grant T32GM096982. Brian Ichter was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.
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Notes
- 1.
For \(f, g:{\mathbb {N}}\rightarrow {\mathbb {R}}\), we say \(f\in O(g)\) if there exists \(n_0\in {\mathbb {N}}\) and \(k\in {\mathbb {R}}_{>0}\) such that \(|f(n)|\le k\, |g(n)|\) for all \(n\ge n_0\). We say \(f\in \varOmega (g)\) if there exists \(n_0\in {\mathbb {N}}\) and \(k\in {\mathbb {R}}_{>0}\) such that \(|f(n)|\ge k\, |g(n)|\) for all \(n\ge n_0\). Finally, we say \(f\in \omega (g)\) if \(\lim _{n\rightarrow \infty } \, f(n)/g(n) = \infty \).
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Janson, L., Ichter, B., Pavone, M. (2018). Deterministic Sampling-Based Motion Planning: Optimality, Complexity, and Performance. In: Bicchi, A., Burgard, W. (eds) Robotics Research. Springer Proceedings in Advanced Robotics, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-60916-4_29
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