A "Piano Movers" Problem Reformulated

TL;DR

This paper reformulates the classic 'Piano Mover's Problem' to enable CAD-based solutions, demonstrating how problem formulation critically impacts the feasibility of robot motion planning with algebraic methods.

Contribution

It introduces a new problem formulation that allows CAD construction and path determination, highlighting the importance of precise problem modeling for algebraic motion planning.

Findings

A new CAD-compatible formulation of the problem

Analysis of CADs for different problem variations

Insights into problem formulation's impact on CAD feasibility

Abstract

It has long been known that cylindrical algebraic decompositions (CADs) can in theory be used for robot motion planning. However, in practice even the simplest examples can be too complicated to tackle. We consider in detail a "Piano Mover's Problem" which considers moving an infinitesimally thin piano (or ladder) through a right-angled corridor. Producing a CAD for the original formulation of this problem is still infeasible after 25 years of improvements in both CAD theory and computer hardware. We review some alternative formulations in the literature which use differing levels of geometric analysis before input to a CAD algorithm. Simpler formulations allow CAD to easily address the question of the existence of a path. We provide a new formulation for which both a CAD can be constructed and from which an actual path could be determined if one exists, and analyse the CADs produced using this approach for variations of the problem. This emphasises the importance of the precise formulation of such problems for CAD. We analyse the formulations and their CADs considering a variety of heuristics and general criteria, leading to conclusions about tackling other problems of this form.