Asymptotic Smooth Stabilization of the Inverted 3-D Pendulum

The 3-D 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 3-D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equili...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2009-06, Vol.54 (6), p.1204-1215
Main Authors: Chaturvedi, N.A., McClamroch, N.H., Bernstein, D.S.
Format: Article
Language:English
Subjects:
3-D pendulum
Aerodynamics
Almost global stabilization
Angular velocity
Applied sciences
Asymptotic properties
Attitude control
Computer science
control theory
systems
Control system synthesis
Control theory
Control theory. Systems
equilibrium manifold
Exact sciences and technology
Feedback
Force control
Fundamental areas of phenomenology (including applications)
Gravity
gravity potential
Manifolds
Mathematical analysis
Mechanical systems
Orbital robotics
Pendulums
Physics
Rigid-body dynamics
Solid dynamics (ballistics, collision, multibody system, stabilization...)
Solid mechanics
Space vehicles
Stabilization
Studies
Wheels
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Summary:The 3-D 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 3-D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. The contribution of this paper is that two fundamental stabilization problems for the inverted 3-D pendulum are posed and solved. The first problem, asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold, is solved using smooth and globally defined feedback of angular velocity and attitude of the 3-D pendulum. The second problem, asymptotic stabilization of the inverted equilibrium manifold, is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3-D pendulum. These control problems for the 3-D pendulum exemplify attitude stabilization problems on the configuration manifold SO(3) in the presence of potential forces. Lyapunov analysis and nonlinear geometric methods are used to assess global closed-loop properties, yielding a characterization of the almost global domain of attraction for each case.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2009.2019792