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## Geometric homogeneity and controllability of nonlinear systems. available electronically http:www.cds.caltech.edu/˜pvela (2003)

Venue: | Submitted IEEE Conf. Dec. Contr |

Citations: | 1 - 1 self |

### Citations

215 | Nonholonomic mechanical systems with symmetry,
- Bloch, Krishnaprasad, et al.
- 1996
(Show Context)
Citation Context ...d constraints are compatible, we focus on another approach that describe constraints via an Ehresmann connection, A : T Q → V Q, on a fiber bundle Q with bundle projection τ : Q → M and model fiber S =-=[13]-=-. The appropriate vector bundle is E ∼ = M × S × V , where V is the model vector space for the tangent bundle T M, and M × S is the base space. Configuration controllability uses Lie( � Γ, Sym(Y � ), ... |

157 |
A general theorem on local controllability,”
- Sussmann
- 1987
(Show Context)
Citation Context ...ir practical utility, mechanical systems also have inherent structure that simplifies their analysis. Determining controllability for nonlinear control systems is in general quite difficult. Sussmann =-=[2]-=- provides sufficient conditions to determine small-time local controllability, however there is factorial growth in the number of elements to test. Lewis and Murray [3] showed how the affine connectio... |

133 |
Control and Stabilization of Nonholonomic Dynamic Systems”,
- Bloch, Reyhanoglu, et al.
- 1992
(Show Context)
Citation Context ...study for nonlinear control systems. This is especially true given that analogous controllability results have subsequently been found to hold for alternative approaches to mechanical control systems =-=[4, 5, 6]-=-. This paper develops the concept of geometric homogeneity for vector bundles and demonstrates how geometric homogeneity is the connecting link to all of these seemingly related results. Section 2 rev... |

97 | Configuration Controllability of Simple Mechanical Control Systems.
- Lewis, Murray
- 1997
(Show Context)
Citation Context ...nly. Define ψ to be the bijection Y = { X1, . . . , Xm+1 } ↦→ Y. Definition 18 An element P ∈ P r(Y) is called bad if γa(P ) is even for each a = 1 . . . m. If P is not bad then it is good. Theorem 2 =-=[3]-=- Consider the bijection ψ : Y → Y, which sends Xa to Y lift a for a = 1 . . . m, and Xl+1 to Zlift . Suppose that (11) is such that every bad symmetric product in P ∈ P r(Y) has the property that k� E... |

82 | Geometric phases and robotic locomotion,
- Kelly, Murray
- 1995
(Show Context)
Citation Context ...local form of the principal connection, Aloc(r) : TrM → g, only depends on the base space. Configuration controllability is further reduced to analysis of the associated adjoint bundle �g, recovering =-=[14, 15]-=-. Constrained Mechanical Systems with Symmetry For some systems, the constraints do not span the Lie algebra, g, of the Lie group, G, meaning that not all of the fiber is constrained. Let the control ... |

45 |
Tangent bundle geometry for Lagrangian dynamics
- Crampin
- 1983
(Show Context)
Citation Context ... dynamics and control. Controllability results often use dilations [2, 7]. Mc’Closkey and Morin [8] use homogeneity to obtain stabilizing controllers for certain nonlinear systems with drift. Crampin =-=[9]-=- uses homogeneity to study the geometry of Lagrangian systems. We seek to develop the 1 This work was supported in part by the National Science Foundation by Engineering Research Center grant NSF94027... |

27 | Nonlinear Control of Mechanical Systems: A Riemannian Geometry Approach
- Bullo
- 1998
(Show Context)
Citation Context ...suffice for all types of mechanical systems. It would clearly be useful to design control laws from the controllability analysis. This has been done in the simple mechanical system framework by Bullo =-=[16]-=-. In related work, the authors have developed a generalized averaging method for nonlinear systems that recovers the symmetric products and Lie brackets needed for control. This technique generalizes ... |

24 | Geometric Structures in Systems Theory - Crouch - 1981 |

18 | Simple mechanical control systems with constraints and symmetry
- Cortés, Martínez, et al.
(Show Context)
Citation Context ...r space V will be either the Lie algebra g, or its dual g ∗ . The homogeneous analysis will recover the controllability results from both instances. For V = g, this recovers the work of Cortés et al. =-=[4]-=-, and for V = g ∗ this recovers Ostrowski and Burdick [6]. 5 Conclusion This paper explored the role of geometric homogeneity on the evolution of control systems on vector bundles and developed a cont... |

16 | Local Motion Planning for Nonholonomic Control Systems Evolving on Principal Bundles
- Radford, Burdick
- 1998
(Show Context)
Citation Context ...local form of the principal connection, Aloc(r) : TrM → g, only depends on the base space. Configuration controllability is further reduced to analysis of the associated adjoint bundle �g, recovering =-=[14, 15]-=-. Constrained Mechanical Systems with Symmetry For some systems, the constraints do not span the Lie algebra, g, of the Lie group, G, meaning that not all of the fiber is constrained. Let the control ... |

15 |
Geometric homogeneity and applications to stabilization
- Kawski
- 1995
(Show Context)
Citation Context ...in a common framework. 2 Geometric Homogeneity and Vector Bundles Geometric homogeneity has been used in various studies of nonlinear dynamics and control. Controllability results often use dilations =-=[2, 7]-=-. Mc’Closkey and Morin [8] use homogeneity to obtain stabilizing controllers for certain nonlinear systems with drift. Crampin [9] uses homogeneity to study the geometry of Lagrangian systems. We seek... |

13 |
Controllability tests for mechanical systems with constraints and symmetries
- Ostrowski, Burdick
- 1997
(Show Context)
Citation Context ...study for nonlinear control systems. This is especially true given that analogous controllability results have subsequently been found to hold for alternative approaches to mechanical control systems =-=[4, 5, 6]-=-. This paper develops the concept of geometric homogeneity for vector bundles and demonstrates how geometric homogeneity is the connecting link to all of these seemingly related results. Section 2 rev... |

11 |
Time-varying homogeneous feedback: Design tools for the exponential stabilization of systems with drift
- M’Closkey
- 1998
(Show Context)
Citation Context ...etric Homogeneity and Vector Bundles Geometric homogeneity has been used in various studies of nonlinear dynamics and control. Controllability results often use dilations [2, 7]. Mc’Closkey and Morin =-=[8]-=- use homogeneity to obtain stabilizing controllers for certain nonlinear systems with drift. Crampin [9] uses homogeneity to study the geometry of Lagrangian systems. We seek to develop the 1 This wor... |

10 | On the homogeneity of the affine connection model for mechanical control systems
- Bullo, Lewis
- 2000
(Show Context)
Citation Context ...anical Engineering California Institute of Technology, Mail Code 107-81, Pasadena, CA, 91125 pvela@cds.caltech.edu, jwb@robotics.caltech.edu Abstract. We followup on a suggestion from Bullo and Lewis =-=[1]-=- concerning the importance of geometric homogeneity for mechanical systems. It is shown that controllability results for a large class of mechanical systems with drift can be recovered by considering ... |

2 | Control of mechanical systems with drift using higher-order averaging - Vela, Burdick - 2003 |