Complexity Results and Fast Methods for Optimal Tabletop Rearrangement with Overhand Grasps (1711.07369v1)

Abstract: This paper studies the underlying combinatorial structure of a class of object rearrangement problems, which appear frequently in applications. The problems involve multiple, similar-geometry objects placed on a flat, horizontal surface, where a robot can approach them from above and perform pick-and-place operations to rearrange them. The paper considers both the case where the start and goal object poses overlap, and where they do not. For overlapping poses, the primary objective is to minimize the number of pick-and-place actions and then to minimize the distance traveled by the end-effector. For the non-overlapping case, the objective is solely to minimize the travel distance of the end-effector. While such problems do not involve all the complexities of general rearrangement, they remain computationally hard in both cases. This is shown through reductions from well-understood, hard combinatorial challenges to these rearrangement problems. The reductions are also shown to hold in the reverse direction, which enables the convenient application on rearrangement of well studied algorithms. These algorithms can be very efficient in practice despite the hardness results. The paper builds on these reduction results to propose an algorithmic pipeline for dealing with the rearrangement problems. Experimental evaluation, including hardware-based trials, shows that the proposed pipeline computes high-quality paths with regards to the optimization objectives. Furthermore, it exhibits highly desirable scalability as the number of objects increases in both the overlapping and non-overlapping setup.

• The paper establishes NP-hardness for non-overlapping and overlapping rearrangement cases while proposing fast algorithms using TSP reductions and ILP optimization.
• It maps non-overlapping scenarios to a Euclidean TSP variant, significantly reducing the end-effector travel distance in robotic path planning.
• Addressing overlapping cases, it leverages dependency graphs and minimum feedback vertex set methods to sequence pick-and-place actions, validated by simulations and hardware experiments.

Complexity Results and Efficient Algorithms for Tabletop Rearrangement with Overhand Grasps

The paper outlined in this paper addresses a specific class of object rearrangement problems commonly encountered in both industrial and domestic settings. The focus is on scenarios where a robotic manipulator, utilizing overhand grasps, must rearrange multiple objects on a non-cluttered, flat surface. The research distinguishes between two primary cases: one where the initial and target arrangements of objects overlap and one where they do not.

For the non-overlapping cases, the problem is framed as a variant of the Euclidean Traveling Salesman Problem (TSP), where the primary objective is to minimize the distance traveled by the end effector of the robot. The complexity of this problem is established through a reduction from Euclidean-TSP, showing NP-hardness. Despite this computational difficulty, the authors successfully propose efficient algorithms that handle the non-overlapping problem instances by leveraging reduction techniques and mapping them to well-explored domains like TSP, yielding high-performance results even with scalability challenges.

The overlapping case presents further complexity, primarily due to dependency cycles among object placements. The fundamental challenge here is modeled using a dependency graph, drawing a parallel to the Feedback Vertex Set (FVS) problem, recognized as NP-hard and APX-hard. The paper details how the resolution of dependencies within these graphs is crucial for determining the optimal sequence of pick-and-place actions, especially when intermediate buffer placements are unavoidable.

To address these varying complexities, the authors propose a comprehensive algorithmic pipeline. This involves solving for minimum feedback vertex sets to break cycles within dependency graphs and adopting integer linear programming (ILP) approaches to optimize the overall movement of the end effector. Extensive simulations validated by hardware experiments authenticate the proposed methods, demonstrating that the solutions not only meet the constraints of the physical setups but also optimize performance metrics significantly.

The implications of this research span both theoretical and practical domains. The theoretical significance lies in the establishment of a formal complexity characterization and efficient algorithmic strategies for a core robotics challenge. Practically, the proposed methodologies promise enhanced efficiency in industrial applications involving object manipulation, potentially translating to improved operational productivity and cost savings.

Future exploration could delve into dynamic rearrangement scenarios where environmental configurations shift unpredictably, extending the current static-focused methods. Additionally, incorporating richer interaction models between multiple robots or enabling non-prehensile actions could further broaden the applicability of these repositioning algorithms in more complex and cluttered environments.

In conclusion, this paper presents a rigorous treatment of tabletop rearrangement problems, offering scalable algorithmic solutions applicable across diverse settings, thus pushing the boundaries of practical robotics manipulation while contributing to foundational computational theory in motion planning and combinatorial optimization.