Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Antony Thomas, Fulvio Mastrogiovanni, Marco Baglietto

TL;DR

This work targets the problem of certifying the non-existence of a collision-free path in robotic motion planning by focusing on infeasibility proof rather than path finding. It introduces an incremental, discretized C-space approach that builds a configuration-space bitmap $\mathcal{CB}$ by sampling from the C-obstacle, segments the partially constructed space into connected free regions, and checks whether the start and goal lie in the same region. The method is resolution-complete: with sufficient discretization, it will certify infeasibility or feasibility; it demonstrates feasibility up to 5-DOF through extensive experiments and analyzes parameter choices ($ns$, $d$) and their impact on runtime. The contributions offer a practical, implementation-friendly tool for infeasibility certification that complements probabilistic planners and can scale via reduced-dimensional proofs and potential accelerations in collision checking and parallel computation.

Abstract

We present a simple and easy-to-implement algorithm to detect plan infeasibility in kinematic motion planning. Our method involves approximating the robot's configuration space to a discrete space, where each degree of freedom has a finite set of values. The obstacle region separates the free configuration space into different connected regions. For a path to exist between the start and goal configurations, they must lie in the same connected region of the free space. Thus, to ascertain plan infeasibility, we merely need to sample adequate points from the obstacle region that isolate start and goal. Accordingly, we progressively construct the configuration space by sampling from the discretized space and updating the bitmap cells representing obstacle regions. Subsequently, we partition this partially built configuration space to identify different connected components within it and assess the connectivity of the start and goal cells. We illustrate this methodology on five different scenarios with configuration spaces having up to 5 degree-of-freedom (DOF).

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

TL;DR

This work targets the problem of certifying the non-existence of a collision-free path in robotic motion planning by focusing on infeasibility proof rather than path finding. It introduces an incremental, discretized C-space approach that builds a configuration-space bitmap by sampling from the C-obstacle, segments the partially constructed space into connected free regions, and checks whether the start and goal lie in the same region. The method is resolution-complete: with sufficient discretization, it will certify infeasibility or feasibility; it demonstrates feasibility up to 5-DOF through extensive experiments and analyzes parameter choices (, ) and their impact on runtime. The contributions offer a practical, implementation-friendly tool for infeasibility certification that complements probabilistic planners and can scale via reduced-dimensional proofs and potential accelerations in collision checking and parallel computation.

Abstract

We present a simple and easy-to-implement algorithm to detect plan infeasibility in kinematic motion planning. Our method involves approximating the robot's configuration space to a discrete space, where each degree of freedom has a finite set of values. The obstacle region separates the free configuration space into different connected regions. For a path to exist between the start and goal configurations, they must lie in the same connected region of the free space. Thus, to ascertain plan infeasibility, we merely need to sample adequate points from the obstacle region that isolate start and goal. Accordingly, we progressively construct the configuration space by sampling from the discretized space and updating the bitmap cells representing obstacle regions. Subsequently, we partition this partially built configuration space to identify different connected components within it and assess the connectivity of the start and goal cells. We illustrate this methodology on five different scenarios with configuration spaces having up to 5 degree-of-freedom (DOF).
Paper Structure (18 sections, 1 theorem, 2 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 18 sections, 1 theorem, 2 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Approach
  5. Sampling C-obstacle
  6. $\mathcal{CB}$ Segmentation
  7. Analysis of $\mathcal{CB}$ Resolution
  8. Resolution-completeness
  9. Experimental Analysis and Discussion
  10. 3-DOF scenario
  11. 4-DOF experiments
  12. 5-DOF scenario
  13. Choosing $ns$ and $d$
  14. $\mathcal{CB}$ Resolution
  15. Obstacle Volume
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\mathcal{C}$ be the configuration space corresponding to $\mathcal{R}$ in $\mathcal{W}$. Given $q_s$ and $q_g$ configurations in $\mathcal{C}$ such that motion planning is infeasible, then for any discretized configuration space $\mathcal{CB}$ that is equivalent to $\mathcal{C}$, motion plannin

Figures (7)

  • Figure 1: (a) A representative discrete configuration space with the start and goal configuration cells colored in green and red, respectively. Other colored cells represent obstacle regions, while uncolored cells represent free space. (b) The sampled obstacle region is colored in blue. This partially constructed configuration space is sufficient to establish motion planning infeasibility since there exists no path from the start to the goal. (c) Segmented representation of the configuration space shown in (b). There are two regions denoted by 1's and 2's, respectively, separated by the obstacle region indicated by 0's. Since the start configuration belongs to region 1 and the goal configuration belongs to region 2, motion planning is infeasible.
  • Figure 2: (a) 2-link robot in its workspace. (b) The configuration space $\mathcal{C}$ with the start and goal configurations shown as green and red dots, respectively. The different colors represent the correspondence between obstacles in the workspace and obstacles in the C-space. The $\mathcal{C}$ has three connected components. (c) An equivalent C-space with 36$\times$36 resolution. (d) C-space with 18$\times$18 resolution. (e) C-space with 12$\times$12 resolution.
  • Figure 3: Different types of robots employed for the 2D experiments. The start configuration of the robot is indicated by green color and the end configuration is denoted by red color. (a) $S_1$: 3-DOF rectangular robot scene. (b) $S_2$: 4-DOF articulated robot scene. (c) $S_5$: A 5-DOF articulated robot scene.
  • Figure 4: Experiment scenes involving the 4-DOF Kinova arm: (a) $S_3$: The Kinova arm attempting to reach inside the frame. (b) $S_4$: The robot arm trying to grasp the cyan block from a position outside the shelf. The red blocks segment the start and goal configurations into different connected components, preventing any path.
  • Figure 5: (a) Modified version of $S_3$ with an obstacle volume 2.4 times larger than in $S_3$. The increased obstacle size results in a broader C-obstacle region, facilitating sampling but potentially leading to unnecessary computations. (b) Obstacle volume reduced to $4/11$ of that in $S_3$. The thinner obstacle makes sampling from the C-obstacle more challenging, leading to higher runtime.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof