An Incremental Sampling and Segmentation-Based Approach for Motion Planning Infeasibility

Technical Summary: An Incremental Sampling and Segmentation-Based Approach for Motion Planning Infeasibility

Problem Statement
Motion planning is a fundamental challenge in robotics, requiring the computation of a collision-free path between a start configuration ($q_s$) and a goal configuration ($q_g$). While sampling-based planners (e.g., RRT, PRM) are efficient at finding feasible paths, they are only probabilistically complete; they cannot definitively prove that no path exists (infeasibility) within a finite time, often requiring arbitrary timeouts. Conversely, resolution-complete planners based on cell decomposition can prove infeasibility but become computationally prohibitive in high-dimensional configuration spaces (C-spaces). This paper addresses the under-explored problem of efficiently certifying motion planning infeasibility without constructing the entire C-space obstacle region.

Methodology
The proposed approach operates on a discretized C-space represented as a bitmap ($CB$), where each cell corresponds to a specific configuration. The core insight is that proving infeasibility does not require mapping the entire obstacle region; it only requires sampling a sufficient subset of the obstacle region to isolate $q_s$ and $q_g$ into disconnected free-space components.

The algorithm proceeds iteratively:

1. Initialization: The entire C-space bitmap is initialized as free space (value 1).
2. Incremental Sampling: In each iteration, the algorithm samples configurations from the C-space. It specifically targets the obstacle region ($C$-obstacle) by:
   • Randomly sampling indices from a set of unvisited cells.
   • Checking for collisions using a standard collision checker.
   • SpeedUp Heuristic: If a sampled configuration $q$ collides, the algorithm identifies the specific robot link(s) causing the collision. It then infers that all configurations sharing the same coordinates for the colliding link(s) (regardless of the values of other links) are also in collision. This allows the algorithm to mark a large volume of the bitmap as obstacles without explicit collision checks, significantly reducing computational overhead.
   • Neighbor Sampling: It also checks $d$ neighboring configurations to leverage local continuity in the C-space.
3. Segmentation: The updated bitmap is segmented into connected components of free space using a Union-Find algorithm.
4. Connectivity Check: The algorithm verifies if $q_s$ and $q_g$ belong to the same connected component. If they are in different components, infeasibility is certified, and the process terminates. If they remain connected, the loop continues with new samples.

Resolution and Topology
The paper emphasizes the importance of selecting an appropriate discretization resolution ($\delta$). A resolution that is too coarse may fail to capture narrow passages or incorrectly merge distinct free regions, leading to false feasibility claims. The authors propose a sampling-based procedure (Algorithm 4) to estimate a theoretical minimum resolution $\delta^*$ that ensures the discrete bitmap preserves the topological structure (connectivity) of the continuous C-space.

Key Contributions

• Algorithmic Framework: A simple, easy-to-implement algorithm that incrementally constructs the C-space bitmap only until infeasibility is proven, rather than exhaustively mapping the entire space.
• Optimization via SpeedUp: A heuristic that exploits the kinematic structure of the robot to classify large subsets of the C-space as obstacles based on partial collision information, avoiding redundant collision checks.
• Scalability: The method is demonstrated to be effective for robots with up to 5 degrees of freedom (DOF) and is extended to 6-DOF and 7-DOF manipulators by leveraging quotient-space topology to reduce the problem dimensionality.
• Parallel Feasibility: The approach can be run in parallel with a standard motion planner; if a path exists, the planner finds it, while the proposed method provides an infeasibility certificate if no path exists.

Experimental Results
Experiments were conducted on six scenarios involving 2D and 3D workspaces with robots ranging from 3-DOF to 5-DOF, as well as 6-DOF and 7-DOF arms.

• Performance: The method successfully certified infeasibility for 4-DOF and 5-DOF scenarios in seconds to minutes, depending on resolution and obstacle density.
• Comparison with Search: The segmentation-based approach was found to be 2 to 4 orders of magnitude faster than a traditional A* search-based infeasibility check on the same discretized spaces.
• Parameter Sensitivity: The study analyzed the impact of sample count ($n_s$) and neighbor count ($d$), identifying $n_s=100$ and $d=5$ as practical trade-offs for the tested scenarios.
• High-Dimensional Scaling: By reducing the 6-DOF and 7-DOF problems to 3-DOF subspaces (based on the specific link causing the blockage), infeasibility was proven in under 50 seconds, demonstrating the potential for handling higher dimensions.

Significance and Claims
The paper claims to provide a practical solution for certifying motion planning infeasibility, a critical capability for Task and Motion Planning (TAMP), manipulation in clutter, and Navigation Among Movable Obstacles (NAMO). Unlike previous methods that rely on learning manifolds or computing complex separating polytopes (often limited to 4-5 DOF), this approach uses a straightforward sampling and segmentation strategy. The authors assert that their method offers a computationally efficient alternative to exhaustive grid search and a definitive alternative to probabilistic planners that cannot guarantee infeasibility. The work highlights that for many high-dimensional problems, infeasibility can often be established in lower-dimensional subspaces, offering a pathway to scalable certification without the full cost of high-dimensional exploration.