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

On the integration of singularity-free representations of SO(3) for direct optimal control

In this paper we analyze the performance of different combinations of: (1) parameterization of the rotational degrees of freedom (DOF) of multibody systems, and (2) choice of the integration scheme, in the context of direct optimal control discretized according to the direct multiple-shooting method. The considered representations include quaternions and Direction Cosine Matrices, both having the peculiarity of being non-singular and requiring more than three parameters to describe an element of the Special Orthogonal group SO(3). These representations yield invariants in the dynamics of the system, i.e., algebraic conditions which have to be satisfied in order for the model to be representative of physical reality. The investigated integration schemes include the classical explicit Rungeâ\u80\u93Kutta method, its stabilized version based on Baumgarteâ\u80\u99s technique, which tends to reduce the drift from the underlying manifold, and a structure-preserving alternative, namely the Rungeâ\u80\u93Kutta Munthe-Kaas method, which preserves the invariants by construction. The performances of the combined choice of representation and integrator are assessed by solving thousands of planning tasks for a nonholonomic, underactuated cart-pendulum system, where the pendulum can experience arbitrarily large 3D rotations. The aspects analyzed include success rate, average number of iterations and CPU time to convergence, and quality of the solution. The results reveal how structure-preserving integrators are the only choice for lower accuracies, whereas higher-order, non-stabilized standard integrators seem to be the computationally most competitive solution when higher levels of accuracy are pursued. Overall, the quaternion-based representation is the most efficient in terms of both iterations and CPU time to convergence, albeit at the cost of lower success rates and increased probability of being trapped by higher local minima

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

Lectures on Mechanics

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule

Park City Lectures on Mechanics, Dynamics, and Symmetry

In these ve lectures, I cover selected items from the following topics: 1. Reduction theory for mechanical systems with symmetry, 2. Stability, bifurcation and underwater vehicle dynamics, 3. Systems with rolling constraints and locomotion, 4. Optimal control and stabilization of balance systems, 5. Variational integrators. Each topic itself could be expanded into several lectures, but I limited myself to what I could reasonably explain in the allotted time. The hope is that the overview is informative enough so that the reader can understand the fundamental ideas and can intelligently choose from the literature for additional details on topics of interest. Compatible with the theme of the PCI graduate school, I assume that the readers are familiar with the elements of geometric mechanics, including the basics of symplectic and Poisson geometry. The reader can find the needed background in, for example, Marsden and Ratiu [1998]

Methods of applied dynamics

The monograph was prepared to give the practicing engineer a clear understanding of dynamics with special consideration given to the dynamic analysis of aerospace systems. It is conceived to be both a desk-top reference and a refresher for aerospace engineers in government and industry. It could also be used as a supplement to standard texts for in-house training courses on the subject. Beginning with the basic concepts of kinematics and dynamics, the discussion proceeds to treat the dynamics of a system of particles. Both classical and modern formulations of the Lagrange equations, including constraints, are discussed and applied to the dynamic modeling of aerospace structures using the modal synthesis technique