Sampling-Based Path Generation

• Sampling-Based Path Generation is a suite of algorithms that generate feasible paths in high-dimensional configuration spaces by sampling random configurations and connecting them through local planners.
• Traditional methods like PRM and RRT, along with their optimal variants PRM* and RRT*, provide foundational techniques ensuring probabilistic completeness and asymptotic optimality.
• Emerging approaches integrate learning-based strategies, bias-informed sampling, and certified methods to enhance efficiency in cluttered, dynamic, and complex planning scenarios.

Sampling-based path generation refers to a class of algorithms for path and trajectory planning in high-dimensional, continuous, or otherwise intractable configuration spaces (“C-spaces”) where explicit enumeration is infeasible. These methods operate by randomly or strategically sampling configurations in the free space, connecting them via dynamically feasible edges (local planners), and searching the resultant graph or tree for collision-free paths that satisfy geometric, kinematic, and dynamic constraints. Approaches span from classical uniform-sampling planners, through informed and learning-based methods, to advanced planners capable of exploiting structural, optimality, or task-level specifications.

1. Classical Methods: Fundamentals and Optimal Variants

Sampling-based approaches are foundational in robotic path planning, especially in the context of motion in complex or high-dimensional spaces where grid-based discretization is intractable. The two seminal methods are:

• Probabilistic RoadMap (PRM): PRM builds a multi-query roadmap by sampling random configurations and connecting neighboring nodes with local planners, such as straight lines for holonomic systems. The roadmap is then searched using standard graph search algorithms (e.g., Dijkstra or A*) for a feasible path between start and goal queries. PRM is probabilistically complete: as the number of samples increases, the probability of finding a path (if one exists) approaches one (Choudhary, 2023).
• Rapidly-exploring Random Tree (RRT): RRT incrementally builds a tree rooted at the start node by extending toward randomly sampled configurations. Each extension connects the nearest node in the tree to the sample, typically with a fixed step size. RRT efficiently explores large spaces but does not guarantee path optimality (Choudhary, 2023).

Optimality is achieved via:

• PRM* and RRT: These modify connection rules and rewiring procedures to ensure asymptotic optimality—path cost converges almost surely to the global minimum as the number of samples increases. In PRM, the connection radius shrinks as $(\log n / n)^{1/d}$ with $n$ the number of samples and $d$ the dimension (Choudhary, 2023). RRT* rewires local neighborhoods for cost reduction at each node addition.

Comparison of these algorithms is summarized below:

Algorithm	Probabilistically Complete	Asymptotically Optimal	Suited For
PRM	Yes	No	Multi-query, static
PRM*	Yes	Yes	Multi-query, static
RRT	Yes	No	Single-query, dynamic/kinodynamic
RRT*	Yes	Yes	Single-query, dynamic/kinodynamic

(Choudhary, 2023)

2. Advanced Sampling Strategies and Efficiency Improvements

Naive uniform sampling suffers from inefficiency in large, cluttered, or narrow-passage spaces, motivating numerous variants:

• Goal-biased and Front-biased Sampling: Introduced in methods such as the non-holonomic bicycle model planner, where samples are concentrated in the kinematically reachable front arc of the vehicle, sometimes with explicit goal-biased random samples to accelerate progress toward the target. Non-uniform sampling drastically reduces wasted effort in unreachable or irrelevant portions of C-space (Morsali et al., 2016).
• Locally Exploitative Sampling (LES): Mixed-mode planners combine global exploration and local exploitation, using local gradient or optimization-based perturbations to generate samples around “relevant” vertices to reduce local cost-to-come and accelerate rewiring. LES is particularly effective in high-dimensional or tightly constrained environments (Joshi et al., 2021).
• Probability Informed Trees (SPRINT): Sample-efficient heuristics prioritize promising regions, cull apparent local minima, and steer expansions away from previously detected obstacles, yielding dramatic reductions in collision checks and faster solution discovery (Rakita et al., 2021).

The acceleration gains from non-uniform, exploitative, or bias-informed strategies are particularly pronounced in environments with structure (e.g. corridors, bottlenecks, or high-dimensional kinematic constraints).

3. Learning-Based and Certified Sampling

Recent years have seen the incorporation of data-driven and learning-theoretic advances into sampling-based path generation:

• Conditional Variational Autoencoders (CVAE), LEGO: CVAEs are trained on spaces densely sampled offline to recognize bottleneck nodes (critical for near-optimal paths) and to yield diverse samples over multiple homotopy classes. This directly biases online planners towards regions required for near-optimal path existence (Kumar et al., 2019).
• GAN-Based Heuristics and Attention Models: Generative adversarial networks learn to segment spatial maps into “promising regions,” providing a probability mask for non-uniform sampling. Hybrid approaches (including spatial/channel attention mechanisms, as in CBAGAN) result in substantial reductions in planning time, node count, and improved solution rate across complex maps (Zhang et al., 2020, Sagar et al., 2023).
• Graph Neural Networks (GNNs): Used to predict guidance values on sampled roadmap graphs, reducing the number of collision checks by biasing edge evaluations and neighbor expansions toward safer or lower-cost regions (Diao et al., 2023).

A major development is the introduction of Conformalized Non-uniform Sampling (CP-RRT*), which leverages a prediction set constructed via conformal prediction around initial heuristic paths. The empirical guarantee is that, with user-specified confidence $1-\alpha$, the optimal trajectory lies within the “certified” sampling regions, offering probabilistic correctness on top of speedup (Natraj et al., 6 Nov 2025).

4. Specialized Sampling-Based Techniques: Temporal Logic, Informative Planning, and Path Constraints

Sampling-based frameworks can accommodate complex objective and constraint classes:

• Temporal Logic Path Planning: The extension to temporal logic using Büchi automata allows specification of tasks such as “visit these regions infinitely often, avoid these obstacles always.” An incremental, sparse roadmap is built in the continuous C-space, and its synchronous product with a Büchi automaton is maintained. Probabilistic completeness is preserved, but focus is on logical satisfaction rather than path length (Vasile et al., 2013).
• Information-Theoretic Path Planning (TIGRIS): Sampling distributions are biased using precomputed proxies of expected information gain over discretized spatial regions, and edge-wise information collection is modeled explicitly, enabling efficient informative planning under sensor/kinodynamic constraints (Moon et al., 2022).
• Passage-Traversing Optimal Path Planning (PTOPP): Path optimality is generalized from path length to functions of passages traversed. These methods: (i) detect passages via proximity graphs and Gabriel conditions, (ii) decompose free space into passage-bounded cells, and (iii) define costs based on passage widths or other traversed-region properties. PTOPP enables explicit accessible free-space objectives and highly configurable constraints with only modest computational overhead (Huang et al., 30 Jun 2025).

DDPS (Discrete Diffusion Posterior Sampling) demonstrates that diffusion models tailored to discrete valid-path representations (e.g., layered graph PALMs) can sample only valid, constraint-satisfying paths and offer posterior-guided sampling for preference over edges or path features, which has not previously been achieved by continuous-sampling approaches (Luan et al., 29 Apr 2025).

5. Performance Guarantees, Complexity, and Optimality

Sampling-based methods possess strong theoretical guarantees under suitable conditions:

• Probabilistic Completeness: As the number of random samples increases to infinity, the probability of finding a feasible path (if one exists) approaches one (Choudhary, 2023, Huang et al., 30 Jun 2025).
• Asymptotic Optimality: PRM*, RRT*, and extensions such as BIGIT* guarantee that with enough samples, the minimum-cost path found converges almost surely to the true optimum under compatible cost functions (Wang et al., 7 Dec 2024, Choudhary, 2023, Huang et al., 30 Jun 2025).
• Sample Complexity for Path-Integral and Stochastic Optimization: Path-integral control, a sampling-based stochastic optimal control method, exhibits sample complexity dependent on the expected (exponential) cost of sampled trajectories; explicit bounds are available for general nonlinear and linear time-varying systems with quadratic and indicator costs (Yoon et al., 2022).

Space and time complexity are heavily influenced by nearest-neighbor data structures and the number of collision checks; learning-based and heuristic-informed sampling can reduce these costs by orders of magnitude in practice (Joshi et al., 2021, Rakita et al., 2021, Diao et al., 2023).

6. Application Domains and Algorithmic Extensions

Sampling-based path generation underpins diverse application domains:

• Robotic Manipulation and Mobile Navigation: Classic, informed, and learning-based planners are standard in high-DOF arm, UAV, and vehicle navigation tasks (Choudhary, 2023, Flores-Aquino et al., 3 Jul 2025).
• Dynamic and Replanning Scenarios: Online, continuously updating sampling-based planners maintain solution validity and safety in the presence of dynamic obstacles and environmental changes, as in OpenMORE for ROS (Tonola et al., 2023).
• Library-based Planning and Experience Transfer: Reusing libraries of previously-computed paths for similar object-environment instances, as in RRT-LIB, accelerates planning in repeat environments or for similar task/geometry classes (e.g., bin picking, assembly) (Minařík et al., 14 Oct 2025).
• Structural and Informative Objectives: Advanced frameworks encompass requirements beyond geometric feasibility, such as maximizing information gain, satisfying temporal logic, or optimizing for passage traversability (Moon et al., 2022, Vasile et al., 2013, Huang et al., 30 Jun 2025).

Hybrid approaches employing skeletonization via deep networks (SkelUnet), variational inference (SV-PRM), and sampling-informed constraints (CP-RRT*) are now reaching practical maturity, providing both theoretical guarantees and strong empirical gains (Flores-Aquino et al., 3 Jul 2025, Lambert et al., 2021, Natraj et al., 6 Nov 2025).

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In conclusion, sampling-based path generation encompasses a spectrum from foundational, probabilistically complete graph and tree-building schemes to domain-specialized, optimization-guided, and learning-augmented methods with optimality and certified guarantees. The field continues to evolve through integration of experience, environmental priors, deep learning, uncertainty quantification, and task-level logic, supporting performance across increasingly complex and high-dimensional planning scenarios.