Stabilization of Relative Equilibria II

In this paper, we obtain feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We use a notion of stability ‘modulo the group action’ developed by Patrick [1992]. We deal with both internal instability and with instability of the rigid motion. The methodology is that of potential shaping, but the system is allowed to be internally underactuated, i.e., have fewer internal actuators than the dimension of the shape space

Stabilization of relative equilibria

This paper discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We show how to stabilize an internally unstable relative equilibrium using internal actuators. The methodology is that of potential shaping, but the system is allowed to be underactuated, i.e., have fewer actuators than the dimension of the shape space. The theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum

Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo

Contents

The equations of motion for a mechanical system with nonholonomic velocity constraints are derived using the assumption that the generalized force vector must at each instant annihilate the space of all possible virtual displacements. We show how this assumption is in fact a consequence of more elementary axioms, such as Newton's second law (F = ma) for each particle of the system, or the equations of balance of linear and angular momentum for rigid bodies. Our argument relies on a careful consideration of how the generalized force vector depends on the physical forces on the system. Even in problems where the generalized force vector does not annihilate the space of all possible virtual displacements (as in problems involving sliding friction) our methodology gives an e cient way for obtaining the the equations of motion i

Modeling of Constrained Systems

The equations of motion for a mechanical system with nonholonomic velocity constraints are derived using the assumption that the generalized force vector must at each instant annihilate the space of all possible virtual displacements. We show how this assumption is in fact a consequence of more elementary axioms, such as Newton's second law (F = ma) for each particle of the system, or the equations of balance of linear and angular momentum for rigid bodies. Our argument relies on a careful consideration of how the generalized force vector depends on the physical forces on the system. Even in problems where the generalized force vector does not annihilate the space of all possible virtual displacements (as in problems involving sliding friction) our methodology gives an efficient way for obtaining the the equations of motion in generalized coordinates. Contents 1 Introduction 2 1.1 Constrained systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vakonomic Mechanics . . ..

Reduction of Hamilton's variational principle

This paper builds on the initial work of Marsden and Scheurle on nonabelian Routh reduction. The main objective is to carry out the reduction of variational principles in further detail. In particular, we obtain reduced variational principles which are the symplectic analogue of the well-known reduced variational principles for the Euler-Poincare equations and the Lagrange-Poincare equations. On the Lagrangian side, the symplectic analogue is obtained by suitably imposing the constraints of preservation of the momentum map

Reduction of Hamilton's Variational Principle

This paper builds on the initial work of Marsden and Scheurle on nonabelian Routh reduction. The main objective of the present paper is to carry out the reduction of variational principles in further detail. In particular, we obtain reduced variational principles which are the symplectic analogue of the well known reduced variational principles for the Euler-Poincare equations and the Lagrange-Poincare equations. On the Lagrangian side, the symplectic analogue is obtained by suitably imposing the constraints of preservation of the momentum map