### Abstract

Given a collection of robots sharing a common environment, assume that each possesses a graph (a one-dimensional complex also known as a roadmap) approximating its configuration space and, furthermore, that each robot wishes to travel to a goal while optimizing elapsed time. We consider vector-valued (or Pareto) optima for collision-free coordination on the product of these roadmaps with collision-type obstacles. Such optima are by no means unique: in fact, continua of Pareto optimal coordinations are possible. We prove a finite bound on the number of optimal coordinations in the physically relevant case where all obstacles are cylindrical (i.e., defined by pairwise collisions). The proofs rely crucially on perspectives from geometric group theory and CAT(O) geometry. In particular, the finiteness bound depends on the fact that the associated coordination space is devoid of positive curvature. We also demonstrate that the finiteness bound holds for systems with moving obstacles following known trajectories. 10.1137/040609860

Original language | English (US) |
---|---|

Pages (from-to) | 1697-1713 |

Number of pages | 17 |

Journal | SIAM Journal on Control and Optimization |

Volume | 45 |

Issue number | 5 |

DOIs | |

State | Published - Dec 1 2006 |

### Fingerprint

### Keywords

- CAT(O) geometry
- Configuration space
- Nonpositive curvature
- Pareto optimality

### ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics

### Cite this

*SIAM Journal on Control and Optimization*,

*45*(5), 1697-1713. https://doi.org/10.1137/040609860