Reference : Geometry and Symmetries in Coordination Control |

Dissertations and theses : Doctoral thesis | |||

Engineering, computing & technology : Multidisciplinary, general & others | |||

http://hdl.handle.net/2268/9544 | |||

Geometry and Symmetries in Coordination Control | |

English | |

Sarlette, Alain [Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation >] | |

Jan-2009 | |

Université de Liège | |

Doctorat en sciences appliquées, spécialité systèmes et contrôle | |

211 | |

Sepulchre, Rodolphe | |

Wehenkel, Louis | |

Lecomte, Pierre | |

Aeyels, Dirk | |

Blondel, Vincent | |

Bullo, Francesco | |

Rouchon, Pierre | |

[en] The present dissertation studies speciﬁc issues related to the coordination of a set of “agents” evolving on a nonlinear manifold, more particularly a homogeneous manifold or a Lie group. The viewpoint is somewhere between control algorithm design and system analysis, as algorithms are derived from simple principles — often retrieving existing models — to highlight speciﬁc behaviors.
With a fair amount of approximation, the objective of the dissertation can be summarized by the following question: Given a swarm of identical agents evolving on a nonlinear, nonconvex conﬁguration space with high symmetry, how can you deﬁne speciﬁc collective behavior, and how can you design individual agent control laws to get a collective behavior, without introducing hierarchy nor external reference points that would break the symmetry of the conﬁguration space? Maintaining the basic symmetries of the coordination problem lies at the heart of the contributions. The main focus is on the global geometric invariance of the conﬁguration space. This contrasts with most existing work on coordination, where either the agents evolve on vector spaces — which, to some extent, can cover local behavior on manifolds — or coordination is coupled to external reference tracking such that the reference can serve as a beacon around which the geometry is distorted towards vector space-like properties. A second, more standard symmetry is to treat all agents identically. Another basic ingredient of the coordination problem that has important implications in this dissertation is the reduced agent interconnectivity: each agent only gets information from a limited set of other agents, which can be varying. In order to focus on issues related to geometry / symmetry and reduced interconnectivity, individual agent dynamics are drastically simpliﬁed to simple integrators. This is justiﬁed at a “planning” level. Making the step towards realistic dynamics is illustrated for the speciﬁc case of rigid body attitude synchronization. The main contributions of this dissertation are I. an extensive study of synchronization on the circle, (a) highlighting diﬃculties encountered for coordination and (b) proposing simple strategies to overcome these diﬃculties; II. (a) a geometric deﬁnition and related control law for “consensus” conﬁgurations on compact homogeneous manifolds, of which synchronization — all agents at the same point — is a special case, and (b) control laws to (almost) globally reach synchronization and “balancing”, its opposite, under general interconnectivity conditions; III. several propositions for rigid body attitude synchronization under mechanical dynamics; IV. a geometric framework for “coordinated motion” on Lie groups, (a) giving a geometric deﬁnition of coordinated motion and investigating its implications, and (b) providing systematic methods to design control laws for coordinated motion. Examples treated for illustration of the theoretical concepts are the circle S^1 (sometimes the sphere S^n ), the rotation group SO(n), the rigid-body motion groups SE(2) and SE(3) and the Grassmann manifolds Grass(p,n). The developments in this dissertation remain at a rather theoretical level; potential applications are brieﬂy discussed. | |

Fonds de la Recherche Scientifique (Communauté française de Belgique) - F.R.S.-FNRS | |

http://hdl.handle.net/2268/9544 |

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