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March  2009, 11(2): 233-262. doi: 10.3934/dcdsb.2009.11.233

A geometric analysis of trajectory design for underwater vehicles

Monique Chyba 1, , Thomas Haberkorn 2, , Ryan N. Smith 3, and  George Wilkens 1,

1. 

University of Hawaii, Department of Mathematics, Honolulu, HI 96822, United States, United States
2. 

Université d'Orléans, Laboratoire MAPMO, 45067 Orléans Cedex, France
3. 

University of Hawaii, Ocean & Resources Engineering Department, Honolulu, HI 96822, United States

Received  October 2007 Revised  April 2008 Published  December 2008

Designing trajectories for a submerged rigid body motivates this paper. Two approaches are addressed: the time optimal approach and the motion planning approach using a concatenation of kinematic motions. We focus on the structure of singular extremals and their relation to the existence of rank-one kinematic reductions; thereby linking the optimization problem to the inherent geometric framework. Using these kinematic reductions, we provide a solution to the motion planning problem in the under-actuated scenario, or equivalently, in the case of actuator failures. We finish the paper comparing a time optimal trajectory to one formed by a concatenation of pure motions.
Keywords: autonomous underwater vehicle, Geometric control, affine connection control system, singular extremal, optimal control, decoupling vector field.
Mathematics Subject Classification: Primary: 93C10, 49N90; Secondary: 53A17.
Citation: Monique Chyba, Thomas Haberkorn, Ryan N. Smith, George Wilkens. A geometric analysis of trajectory design for underwater vehicles. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 233-262. doi: 10.3934/dcdsb.2009.11.233
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