Locally Optimal Solutions to Constraint Displacement Problems via Path-Obstacle Overlaps

TL;DR

The paper tackles planning in environments where a robot may displace movable constraints to realize a feasible path. It introduces a two-stage method: first, compute a trajectory through movable obstacles by minimizing a problem-specific objective that accounts for overlaps, then compute locally optimal obstacle displacements to remove overlaps and produce a collision-free path. By unifying MCD and MCR within this overlap-displacement framework and validating in 2D planar projections with circle-based bounds, the work demonstrates practical locally optimal solutions aided by MPC and nonlinear optimization. The approach offers a flexible, offline planning tool for cluttered or dynamic spaces, with future directions including handling interacting obstacles, displacement bounds, and tighter integration with manipulation planning.

Abstract

We present a unified approach for constraint displacement problems in which a robot finds a feasible path by displacing constraints or obstacles. To this end, we propose a two stage process that returns locally optimal obstacle displacements to enable a feasible path for the robot. The first stage proceeds by computing a trajectory through the obstacles while minimizing an appropriate objective function. In the second stage, these obstacles are displaced to make the computed robot trajectory feasible, that is, collision-free. Several examples are provided that successfully demonstrate our approach on two distinct classes of constraint displacement problems.

Figures (13)

• Figure 1: (a) The robot at the starting point (green) needs to move to the goal (red) while navigating through movable obstacles. (b) The MCD path is depicted with displaced obstacles shown in cyan. (c) One obstacle has been removed from the environment to create a collision-free path, without the need to displace any obstacles. Our approach solves the classical motion planning problem and computes a path without moving any obstacles. (d) The cost function has been modified to address the MCR problem. The cyan object needs to be removed for a feasible path to exist.
• Figure 2: (a) A rectangle (gray) represented using a single bounding circle. (b) Two circles used to bound the rectangle.
• Figure 3: Illustration of circle-polygon overlap. (a) The overlapped line in red is displaced to a new location. (b) The green square is the displaced location with no overlap with the circle. (c) Multiple circles intersect with a square. The optimization in (\ref{['eq:cpdisp']}) computes a zero overlap location (in green) for the square.
• Figure 4: Example of pure rotations and translations. Additional constraints are added in (\ref{['eq:cpdisp']}) to obtain the desired results. (a) A rectangular rod (in red) than can perform only pure rotations. The displaced rectangle is shown in green. (b) A rectangle than can only perform transnational motions. The overlapping rectangle is translated to a new location (green) with no overlap.
• Figure 5: A rectangle (in blue) and a triangle (in red) are intersecting initially. A new location (in green) is found for the movable triangle with no intersection with the rectangle.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3