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This paper introduces a method to reduce unnecessary bending in feedback motion planning over cell decompositions, which often leads to inefficient robot motion. They achieve this by aligning local vector fields within a simplicial decomposition using a novel heuristic and constructing a maximal star-shaped chain of simplexes around the goal to create a "funnel" for direct control. Simulations show a 91.40% reduction in total bending and a 45.47% reduction in LQR control effort compared to existing methods, while maintaining time efficacy and robustness.
Drastically reduce robot motion bending and control effort by creating a "funnel" directly to the goal within a simplicial decomposition.
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.