Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smooth Feedback Motion Planning with Reduced Curvature

Aref Amiri, Steven M. LaValle

Abstract

Feedback motion planning over cell decompositions provides a robust method for generating collision-free robot motion with formal guarantees. However, existing algorithms often produce paths with unnecessary bending, leading to slower motion and higher control effort. This paper presents a computationally efficient method to mitigate this issue for a given simplicial decomposition. A heuristic is introduced that systematically aligns and assigns local vector fields to produce more direct trajectories, complemented by a novel geometric algorithm that constructs a maximal star-shaped chain of simplexes around the goal. This creates a large ``funnel'' in which an optimal, direct-to-goal control law can be safely applied. Simulations demonstrate that our method generates measurably more direct paths, reducing total bending by an average of 91.40\% and LQR control effort by an average of 45.47\%. Furthermore, comparative analysis against sampling-based and optimization-based planners confirms the time efficacy and robustness of our approach. While the proposed algorithms work over any finite-dimensional simplicial complex embedded in the collision-free subset of the configuration space, the practical application focuses on low-dimensional ($d\le3$) configuration spaces, where simplicial decomposition is computationally tractable.

Smooth Feedback Motion Planning with Reduced Curvature

Abstract

Feedback motion planning over cell decompositions provides a robust method for generating collision-free robot motion with formal guarantees. However, existing algorithms often produce paths with unnecessary bending, leading to slower motion and higher control effort. This paper presents a computationally efficient method to mitigate this issue for a given simplicial decomposition. A heuristic is introduced that systematically aligns and assigns local vector fields to produce more direct trajectories, complemented by a novel geometric algorithm that constructs a maximal star-shaped chain of simplexes around the goal. This creates a large ``funnel'' in which an optimal, direct-to-goal control law can be safely applied. Simulations demonstrate that our method generates measurably more direct paths, reducing total bending by an average of 91.40\% and LQR control effort by an average of 45.47\%. Furthermore, comparative analysis against sampling-based and optimization-based planners confirms the time efficacy and robustness of our approach. While the proposed algorithms work over any finite-dimensional simplicial complex embedded in the collision-free subset of the configuration space, the practical application focuses on low-dimensional () configuration spaces, where simplicial decomposition is computationally tractable.

Paper Structure

This paper contains 23 sections, 8 theorems, 6 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Potential Fields, Navigation and Harmonic Functions
  4. Feedback from Cell Decompositions and Bump Functions
  5. Region-Based and Optimization-Based Planning
  6. Problem Formulation
  7. Smooth Feedback on Simplicial Decompositions
  8. Fundamental Definitions
  9. Essential Theorems
  10. Heuristic Vector Field Alignment Method
  11. Heuristic Alignment of Cell Vector Fields
  12. Assignment of Face Vector Fields and Proof of Validity
  13. Blending the Vector Fields
  14. Constructing the Maximal Star-shaped Chain of Simplexes
  15. Geometric Criteria for Star-shaped
  16. ...and 8 more sections

Key Result

Theorem 1

All integral curves generated by the smoothly interpolated cell and face vector fields within any intermediate cell reach the designated exit face of that cell in finite time. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Geometric construction of valid vector fields for 2-simplex (triangle). The conical region, constructed by boundary vectors $b_{i,j}$, illustrates the set of valid constant cell vector fields $V_{c,i}$ that satisfy the conditions of Definition \ref{['def: def2']}. The diagram visualizes the geometric constraints for face vector fields (Definition \ref{['def: def3']}), showing the face inward normal $n_{\text{in},f}$, the exit face outward normal $n_{x}$, and the hyperplane normal $n_{b_f}$. The dashed lines are the GVD surface.
  • Figure 2: Geometric test for expanding the star-shaped region $\mathcal{R}$ with an adjacent simplex $\Delta_T$ (in 2D). (Left) A valid addition where the new vertex $v_\text{new}$ lies within the visibility cone from $x_g$ (Criterion \ref{['cri: cri2']}). (Right) An invalid addition where visibility is obstructed.
  • Figure 3: Qualitative comparison of integral curves for the baseline and proposed methods across three environments. The maximal star-shaped region is highlighted in brown. Notably, in the Bug Trap environment, Steiner points were introduced to generate closer to equilateral triangles and avoid skewed triangles.
  • Figure 4: Qualitative comparison of paths for the baselines across the maze and sparse environments.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof
  • Proposition 1
  • ...and 5 more