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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 rightangled 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.
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.
Original language  English 

Title of host publication  Proceedings of SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing 
Publisher  IEEE 
Pages  5360 
Number of pages  8 
DOIs  
Publication 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 

Country/Territory  Romania 
City  Timisoara 
Period  23/09/13 → 26/09/13 
Keywords
 Cylindrical Algebraic Decomposition
 Robot Motion Planning
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 1 Finished

Real Geometry and Connectedness via Triangular Description
Davenport, J., Bradford, R., England, M. & Wilson, D.
Engineering and Physical Sciences Research Council
1/10/11 → 31/12/15
Project: Research council