Sandwich Approach for Motion Planning and Control

TL;DR

This work tackles motion planning in obstacle-rich environments by transforming the obstacle-laden motion space $\mathcal{P}$ into an obstacle-free planning space $\mathcal{C}$ using obstacle sandwiching with two $\psi$-boundaries, enabling collision-free planning via non-singular $x,y$ to $\phi,\psi$ mappings. A* search performed on $\mathcal{C}$ yields shorter trajectories than traditional planning in $\mathcal{P}$, aided by a geodesic-distance grid and boundary-consistent PDE solutions obtained through finite-difference methods. The trajectory-tracking layer uses MPC with linear safety constraints to ensure the quadcopter follows the planned path while maintaining collision avoidance. The approach is demonstrated on a simulated quadcopter, showing a 6.21% improvement in path length and robust safety-constrained tracking, indicating practical benefits for real-time planning and control in cluttered spaces.

Abstract

This paper develops a new approach for robot motion planning and control in obstacle-laden environments that is inspired by fundamentals of fluid mechanics. For motion planning, we propose a novel transformation between motion space, with arbitrary obstacles of random sizes and shapes, and an obstacle-free planning space with geodesically-varying distances and constrained transitions. We then obtain robot desired trajectory by A* searching over a uniform grid distributed over the planning space. We show that implementing the A* search over the planning space can generate shorter paths when compared to the existing A* searching over the motion space. For trajectory tracking, we propose an MPC-based trajectory tracking control, with linear equality and inequality safety constraints, enforcing the safety requirements of planning and control.

Figures (9)

• Figure 1: Transforming the "motion" space (left) to the "planning" space (right).
• Figure 2: Dividing an example navigable space $\mathcal{S}$ into five navigable channels $\mathcal{S}_1$ through $\mathcal{S}_5$.
• Figure 3: Mapping between the navigable channel $\mathcal{S}_j$ and $j$-th rectangle $\mathcal{C}_j$ in the planning space.
• Figure 4: Mapping between $\partial \mathcal{S}_{j,b}$ and $\partial \mathcal{C}_{j,b}$ ($j=1,\cdots, p$ and $b=1,\cdots,4$).
• Figure 5: Top: Motion space. Bottom: Planning space.

Theorems & Definitions (2)

• Remark 1
• Definition 1