### Abstract

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. These simpler formulations allow CAD to 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.

Language | English |
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Title of host publication | Proceedings of SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing |

Publisher | IEEE |

Pages | 53-60 |

Number of pages | 8 |

DOIs | |

Status | Published - 2013 |

Event | SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing - Timisoara, Romania Duration: 23 Sep 2013 → 26 Sep 2013 |

### Conference

Conference | SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing |
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Country | Romania |

City | Timisoara |

Period | 23/09/13 → 26/09/13 |

### Fingerprint

### Keywords

- Cylindrical Algebraic Decomposition
- Robot Motion Planning

### Cite this

*Proceedings of SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing*(pp. 53-60). IEEE. https://doi.org/10.1109/SYNASC.2013.14

**A "piano movers" problem reformulated.** / Wilson, David; Davenport, J H; England, M; Bradford, R J.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.*IEEE, pp. 53-60, SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Timisoara, Romania, 23/09/13. https://doi.org/10.1109/SYNASC.2013.14

}

TY - GEN

T1 - A "piano movers" problem reformulated

AU - Wilson, David

AU - Davenport, J H

AU - England, M

AU - Bradford, R J

PY - 2013

Y1 - 2013

N2 - 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. These simpler formulations allow CAD to 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.

AB - 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. These simpler formulations allow CAD to 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.

KW - Cylindrical Algebraic Decomposition

KW - Robot Motion Planning

UR - http://www.synasc.ro/

UR - http://dx.doi.org/10.1109/SYNASC.2013.14

U2 - 10.1109/SYNASC.2013.14

DO - 10.1109/SYNASC.2013.14

M3 - Conference contribution

SP - 53

EP - 60

BT - Proceedings of SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

PB - IEEE

ER -