Stable Manifolds of Saddle Equilibria for Pendulum Dynamics on S and SO(3)
نویسندگان
چکیده
Attitude control systems naturally evolve on nonlinear configurations, such as S and SO(3). The nontrivial topological properties of these configurations result in interesting and complicated nonlinear dynamics when studying the corresponding closed loop attitude control systems. In this paper, we review some global analysis and simulation techniques that allow us to describe the global nonlinear stable manifolds of the hyperbolic equilibria of these closed loop systems. A deeper understanding of these invariant manifold structures are critical to understanding the global stabilization properties of closed loop attitude control systems, and these global analysis techniques are applicable to a broad range of problems on nonlinear configuration manifolds.
منابع مشابه
Stable manifolds of saddle equilibria for pendulum dynamics on S2 and SO(3)
Global nonlinear dynamics of various classes of closed loop attitude control systems have been studied in recent years [1]. Closely related results on attitude control of a spherical pendulum (with attitude an element of the twosphere S2) and of a 3D pendulum (with attitude an element of the special orthogonal group SO(3)) are given in [2], [3]. These publications address the global closed dyna...
متن کاملComputer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields
In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the NewtonKantorovich theorem is applied to validate the existence of a so-cal...
متن کاملNonlinear Dynamics of the 3D Pendulum
A 3D 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. 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 t...
متن کاملTwo dimensional heteroclinic attractor in the generalized Lotka-Volterra system
We study a simple dynamical model exhibiting sequential dynamics. We show that in this model there exist sets of parameter values for which a cyclic chain of saddle equilibria, Ok, k = 1, . . . , p, have two dimensional unstable manifolds that contain orbits connecting eachOk to the next two equilibrium pointsOk+1 andOk+2 in the chain (Op+1 = O1). We show that the union of these equilibria and ...
متن کاملSaddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B
When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way, provided the corresponding genericity conditions. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbit...
متن کامل