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Deterministic Sampling-Based Motion Planning: Optimality, Complexity, and Performance
Lucas Janson, Brian Ichter, Marco Pavone
TL;DR
The paper examines whether deterministic sampling can match or improve probabilistic motion-planning guarantees while easing certification and offline computation. It analyzes low-dispersion sequences for PRM and finds deterministic asymptotic optimality, explicit convergence bounds, near-linear complexity, and generally better empirical performance. The paper’s scope includes stated assumptions and limitations in lattice-based complexity and deterministic kinodynamic planning.
Problem
The paper asks whether deterministic sampling sequences can provide theoretical guarantees and practical performance comparable to probabilistic planners while simplifying certification and enabling offline computation.
Method
The paper analyzes PRM with deterministic low-dispersion sequences and extends the results to broader algorithms, sampling strategies, and kinodynamic planning.
Results
Deterministic low-dispersion sampling yields asymptotic optimality, explicit convergence-rate bounds, near-linear PRM complexity, and no worse or sometimes substantially better empirical performance than i.i.d. sampling.
Takeaways & Limitations
The results support using non-i.i.d. low-dispersion sampling for studying and applying sampling-based motion-planning algorithms.
Takeaways & Limitations
The complexity result relies on leveraging lattice structure, and the kinodynamic analysis identifies dispersion measures as an unresolved direction.
Abstract
from arXiv · showhide
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 independently and identically distributed 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. Specifically, we first show that PRM, for a certain selection of tuning parameters and deterministic low-dispersion sampling sequences, is deterministically asymptotically optimal. 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 l2-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 logn) 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.
1. Introduction
This paper asks whether deterministic low-dispersion sampling can preserve or improve the guarantees and performance of probabilistic motion planners while simplifying certification and offline computation. It rigorously establishes deterministic asymptotic optimality, explicit convergence bounds, near-linear complexity, and generally superior empirical performance.
- Motivation: Deterministic low-dispersion sampling addresses open questions about optimal-path convergence, convergence rates, and computational and space complexity.Earlier deterministic variants were mainly restricted to convergence toward feasible paths, while this paper studies optimality and complexity.
- Theoretical results: PRM is deterministically asymptotically optimal when its samples have ℓ2-dispersion at most γ n^-1/d and its connection radius satisfies r_n ∈ ω(n^-1/d).The returned solution cost converges deterministically to the optimum as n approaches infinity.
- Theoretical results: 2D_n/(r_n − 2D_n) upper-bounds PRM’s sub-optimality factor without obstacles, explicitly relating performance to dispersion and connection radius.The obstacle-cluttered case also admits a more sophisticated bound.
- Theoretical results: An asymptotically optimal lattice-based PRM can have computational and space complexity arbitrarily close to ω(n), near the theoretical O(n) lower bound.This improves on the analogous O(n log(n)) complexity results for i.i.d. sampling.
- Extensions: The results extend, exactly or approximately, to k-nearest-neighbor and other batch algorithms, non-lattice and non-uniform sampling, and kinodynamic planning.The paper specifically discusses FMT∗ and Halton-sequence sampling among these extensions.
- Empirical results: For a fixed number of samples, deterministic low-dispersion sampling performs no worse and sometimes substantially better than i.i.d. sampling in experiments.The reported advantages include path cost, success rate, and the number of samples required for a 90% success rate, with potential runtime gains from fast nearest-neighbor indexing.
- Implications: Controlled randomness, such as randomly rotated and offset lattices, retains deterministic guarantees and deterministic-sampling benefits while addressing potential lattice axis-alignment issues.The paper identifies axis alignment along obstacle boundaries as a potential source of infeasible sample rows.
2. Background
The paper uses ℓ2-dispersion to quantify how well deterministic or random samples cover configuration space, linking lower dispersion to better path approximation. Deterministic low-dispersion sequences achieve O(n^-1/d), improving on the random-point rate by a logarithmic factor.
- ℓ2-dispersion: ℓ2-dispersion is the radius of the largest open Euclidean ball containing no sample point.It quantifies how well a point set covers a finite domain.
- ℓ2-dispersion: Dispersion bounds how closely paths through sampled points can approximate an optimal path.An optimal path may pass through an empty ball whose radius is the dispersion.
- Random sampling: Random independent samples have ℓ2-dispersion O((log(n)/n)^(1/d)) with probability 1.The corresponding ℓ∞-dispersion bound implies this ℓ2 result.
- Deterministic sequences: Deterministic low-dispersion sequences achieve ℓ2-dispersion O(n^-1/d), improving over i.i.d. sampling by a factor of log(n)^(1/d).The paper calls sequences with this order of dispersion low-dispersion sequences.
- Deterministic sequences: The analysis focuses on Sukharev and Halton sequences, with triangular lattices also considered in two dimensions.Other deterministic sequences with the same dispersion order are noted.
3. Problem Statement
The paper formulates optimal motion planning in a configuration space with obstacles, an initial state, and a goal region, using arc length as the cost. It analyzes a generic PRM framework that permits deterministic or random samples and varied connection radii.
- Planning problem: The configuration space is X=[0,1]^d, with closed obstacles, obstacle-free space Xfree, initial state xinit, and goal region Xgoal.The goal region lies within Xfree.
- Clearance assumptions: The goal region is assumed regular, meaning each boundary point is associated with an interior ball of fixed radius ξ.Regularity is a technical condition used in the theoretical results.
- Planning problem: An optimal path minimizes arc length among all feasible paths, with failure reported if no feasible path exists.Feasibility requires collision avoidance, the initial state, and reaching the goal region.
- Clearance assumptions: A path has strong δ-clearance when it remains entirely at least distance δ from obstacles.δ-robust feasibility permits a sequence of strongly δn-clear paths with δn→0.
- Generic PRM: gPRM samples n points, connects nearby collision-free pairs within radius rn, and returns a shortest graph path to the goal.SampleFree may generate deterministic or random sequences, and rn is left unspecified for analysis.
- Generic PRM: The gPRM framework deliberately leaves the sampling rule and connection radius open, encompassing established PRM variants.The algorithm constructs an undirected graph and reports failure when no graph path exists.
4. Theoretical Results
The theoretical results establish deterministic asymptotic optimality for gPRM on low-dispersion sequences, give finite-sample convergence bounds in terms of dispersion, and show near-linear complexity for lattice sampling. They also connect these bounds to performance certification for paths with prescribed clearance.
- Asymptotic optimality: gPRM is deterministically asymptotically optimal on low-dispersion sequences when rn∈ω(n^-1/d).This improves on prior asymptotic-optimality requirements of at least Ω((log(n)/n)^(1/d)).
- Convergence rate: The returned path cost is bounded using the sample dispersion Dn and connection radius rn, with explicit finite-sample control.The analysis compares the returned path with the best path having strong δ-clearance.
- Convergence rate: In obstacle-free space, gPRM suboptimality is at most 2Dn/(rn−2Dn) for rn>2Dn.This follows because the shortest strongly δ-clear path has cost equal to the straight-line distance.
- Certification: For feasible δ0-clear problems, the analysis can certify a returned path within factor 1/(ψ−1) of the best δ0-clear path.Given δ0 and ψ, the required sample count and computation time can be determined.
- Complexity: A cubic-lattice gPRM can be asymptotically optimal with computational and space complexity ω(n), below the O(n log(n)) complexity of existing asymptotically optimal algorithms.The lattice structure supports efficient neighborhood processing and shortest-path computation.
- Complexity: Theoretical complexity depends on lattice structure because shortest-path algorithms on general graphs with n vertices require Ω(n log(n)).The lattice is leveraged to improve the final graph-processing step.
5. Extensions
The paper extends its deterministic convergence analysis to alternative batch-processing algorithms, non-uniform sampling, and kinodynamic planning. These extensions preserve guarantees under stated sampling or system assumptions while exposing implementation and modeling limitations.
- Alternative batch-processing algorithms: Theoretical convergence results extend to alternative implementations and other batch-processing algorithms, including Lazy-PRM and FMT∗.Lazy-PRM returns the same path as gPRM with the same radius, while FMT∗ requires a modified proof using suitable sequences of balls.
- k-nearest-neighbor implementations: k-nearest gPRM replaces a fixed connection radius with an adaptive neighbor count and retains deterministic asymptotic optimality when kn is selected to approximate the rn-neighborhood.The construction uses kn = (1+ϵ)nζdrd under the stated assumptions.
- Complexity across sampling schemes: The improved shortest-path implementation does not generally apply to non-lattice sequences, k-nearest-neighbor graphs, or algorithms that do not separate graph construction from shortest-path computation.The other computational advantages of Theorem 3 remain except for the shortest-path speedup in these cases.
- Non-uniform sampling: Deterministic non-uniform strategies remain covered when their ℓ2-dispersion is suitably bounded, including mixtures of low-dispersion and biased samples.A construction using n/2 low-dispersion points yields dispersion bounded by 21/dγn−1/d, and the analysis depends on dispersion rather than a specific strategy.
- Kinodynamic planning: For linear affine dynamics, gDPRM with deterministic low-dispersion sampling is asymptotically optimal under the assumptions of the corresponding kinodynamic theorem.The analysis uses controllability indices, while Euclidean-ball approximations can introduce an exponential factor and may inadequately represent differential constraints.
6. Numerical Experiments
The experiments compare deterministic low-dispersion sequences, especially lattices and Halton sampling, with i.i.d. sampling across geometric, rigid-body, kinematic-chain, and kinodynamic planning problems. Across these tests, deterministic sequences generally achieve better solution cost, success rate, or sample efficiency.
- Experimental setup: The simulations compare lattices and Halton sequences against i.i.d. sampling across point-robot, rigid-body, kinematic-chain, and kinodynamic problems.The implementations include MATLAB, OMPL, and Julia experiments, while results are compared by sample count rather than runtime.
- Experimental setup: The connection radius is selected as rn = γPRM (log(n)/n)1/d, a choice stated to ensure asymptotic optimality for both i.i.d. and deterministic low-dispersion sequences.The experiments use γPRM = 2.2 for most simulations.
- Sampling strategies: Lattices can suffer from axis-alignment effects, where an obstacle may invalidate an entire row of samples; rotating the lattice is used as a remedy without changing its l2-dispersion.The paper also notes that SE(2) and SE(3) examples lie technically outside the stated theory because their cost is not configuration-space arc length.
- Sampling strategies: For rigid-body and kinodynamic settings, samples are spread spatially or by velocity to improve coverage, with alternating 180-degree rotations used to reduce unfavorable lattice structure.The rigid-body extension spreads orientations around their original translational points, while the double-integrator extension offsets velocity samples proportionally to velocity.
- Results: Deterministic low-dispersion sequences find solutions with fewer or equal samples and lower or equal cost than i.i.d. sampling in the reported comparisons.Randomly rotated lattices perform as well as or better than random sampling, indicating that low dispersion need not require deterministic orientation.
7. Conclusions
The paper concludes that low-dispersion sampling, particularly deterministic sampling, benefits optimal path planning through deterministic guarantees, reduced computational complexity per sample count, and stronger empirical performance. It also identifies extensions to kinodynamic planning, other algorithms, alternative sequences, and certification as future work.
- Conclusions: Low-dispersion sampling strategies, particularly deterministic ones, provide substantial benefits for optimal path planning with sampling-based algorithms.The stated benefits include deterministic performance guarantees, reduced computational complexity per given number of samples, and superior practical performance.
- Future work: Future work includes deterministic kinodynamic planning with tailored sampling notions, dispersion measures, and more general dynamical models.The paper also proposes extending the results to other sampling-based algorithms, especially anytime methods such as RRT and RRT*.
- Future work: The authors plan to study alternative low-dispersion strategies, particularly incremental sequences for anytime algorithms, and non-uniform sampling that incorporates prior problem knowledge.They also identify fast nearest-neighbor indexing and precomputed data structures as practical directions, especially for kinodynamic planning.
- Future work: Further work will examine convergence relative to the true optimal cost and use the results to study certification for sampling-based planners in drone and self-driving-car trajectory planning.These directions connect the theoretical convergence analysis to safety-critical applications.