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Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum
Author(s) -
Chaturvedi Nalin A.,
McClamroch N. Harris
Publication year - 2007
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.1178
Subject(s) - center manifold , invariant manifold , manifold (fluid mechanics) , inverted pendulum , kapitza's pendulum , mathematics , exponential stability , control theory (sociology) , lyapunov function , pendulum , nonlinear system , mathematical analysis , physics , computer science , double pendulum , control (management) , bifurcation , engineering , mechanical engineering , hopf bifurcation , quantum mechanics , artificial intelligence
The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed‐loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed‐loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed‐loop that converges to the inverted equilibrium manifold is identified. Copyright © 2007 John Wiley & Sons, Ltd.