Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum

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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/56146/1/1178_ftp.pd

Dynamics and control of a 3D pendulum

Abstract — New pendulum models are introduced and stud-ied. The pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force and control forces and moments. Several different pendulum models are developed to analyze properties of the uncontrolled pendulum. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Rigid pendulum and multi-body pendulum control problems are proposed. The 3D pendulum models provide a rich source of examples for nonlinear dynamics and control, some of which are similar to simpler pendulum models and some of which are completely new. I

Axially excited spatial double pendulum nonlinear dynamics.

Analysis of a 3D spatial double physical pendulum system, coupled by two universal joints is performed. External excitation of the mechanism is realized by axial periodic rotations of the first joint of the pendulum. System of ODEs is solved numerically and obtained data are analyzed by a standard approach, including time series, phase plots and Poincaré sections. Additionally, FFT (Fast Fourier Transform and the wavelet transformation algorithms have been applied. Various wavelet basic functions have been compared to find the best fit, e.g. Morlet, Mexican Hat and Gabor wavelets. The so far obtained results allowed for detection of a number of non-linear effects, including chaos, quasi-periodic and periodic dynamics, as well the numerous and different bifurcations. Scenarios of transition from regular to chaotic dynamics have been also illustrated and studied

Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76737/1/AIAA-41565-705.pd

Stabilization of elastic inverted pendulum with hysteresis

In this paper, we investigate the elastic inverted pendulum with hysteretic nonlinearity (a back-lash) in the suspension point. Namely, the problems of stabilization and optimization of such a system are considered. The algorithm (based on the bionic model) which provides the effective procedure for finding of optimal parameters is presented and applied to considered system. The results of numerical simulations, namely the phase portraits and the dynamics of Lyapunov function, are also presented and discussed