Stability Analysis of a Rigid Body with a Flexible Attachment Using the Energy-Casimir Method

We consider a system consisting of a rigid body to which a linear extensible shear beam is attached. For such a system the Energy-Casimir method can be used to investigate the stability of the equilibria. In the case we consider, it can be shown that a test for (formal) stability reduces to checking the positive definiteness of two matrices which depend on the parameters of the system and the particular equilibrium about which the stability is to be ascertained

Motion Control and Coupled Oscillators

It is remarkable that despite the presence of large numbers of degrees of freedom, motion control problems are effectively solved in biological systems. While feedback, regulation and tracking have served us well in engineering, as useful solution paradigms for a wide variety of control problems including motion control, it appears that nature gives prominent roles to planning and co-ordination as well. There is also complex interplay between sensory feedback and motion planning to achieve effective operation in uncertain environments, for example, in movement on uneven terrain cluttered with obstacles. Recent investigations by neurophysiologists have brought to increasing prominence the idea of central pattern generators -- a class of coupled oscillators -- as sources of motion scripts as well as a means for coordinating multiple degrees of freedom. The role of coupled oscillators in motion control systems is currently under intense investigation. In this paper we examine some unifying themes relating movements in biological systems and machines. An important insight in this direction comes from the natural grouping of degrees of freedom and time scales in biological and engineering systems. Such grouping and separation can be treated from a geometric viewpoint using the formalisms and methods of differential geometry, Lie groups, and fiber bundles. Coupled oscillators provide the means to bind degrees of freedom either directly through phase locking or indirectly through geometric phases. This point of view leads to fresh ways of organizing the control structures of complex technological systems

Coordinated Orbit Transfer for Satellite Clusters

We propose a control law which allows a satellite formation to achieveorbit transfer. During the transfer, the formationcan be either maintained or modified to a desired one.Based on the orbit transfer control law proposed by Chang, Chichka and Marsden forsingle satellite,we add coupling terms to the summation of Lyapunov functions forsingle satellites. These terms are functionsof the difference between the mean anomalies (or perigee passing times) of formation members. The asymptotic stability of the desired formationin desired orbits is proved

Cayley Transforms in Micromagnetics

Methods of numerical integration of ordinary differential equations exploiting the Cayley transform arise in a variety of contexts, ranging from the classical mid-point rule to symplectic and (almost) Poisson integrators, to numerical methods on Lie Groups. In earlier work, the first author investigated the interplay between the Cayley transform and the Jacobi identity in establishing certain error formulas for the mid-point rule (with applications to coupled rigid bodies). In this paper, we use the Cayley transform to lift the Landau-Lifshitz-Gilbert equation of micromagnetics to the Lie algebra of the group of currents (on a compact magnetic body) with values in the 3-dimensional rotation group. This follows an idea of Arieh Iserles and, we use the lift to numerically integrate the Landau-Lifshitz-Gilbert equation conserving automatically the norm of the magnetization everywhere

A Model for a Thin Magnetostrictive Actuator

In this paper, we propose a model for dynamic magnetostrictive hysteresisin a thin rod actuator. We derive two equations that representmagnetic and mechanical dynamic equilibrium. Our model results from an application of the energy balance principle.It is a dynamic model as it accounts for inertial effects and mechanicaldissipation as the actuator deforms, and also eddy current lossesin the ferromagnetic material. We show rigorously that the model admits a periodic solution thatis asymptotically stable when a periodic forcing function is applied.(Proc. Conf. Information Sci. and Systems, Princeton, March 1998

Formation Dynamics under a Class of Control Laws

A system of two earth satellites is analyzed as a controlled mechanical system. The orbit of an earth satellite can be represented by a point inthe vector space of ordered pairs of angular momentum and Laplace vectors. Control laws are obtained by introducing a Lyapunov function on this space. Formations of two satellites are achieved asymptotically by the controlled dynamics

A Novel Non-Orthogonal Joint Diagonalization Cost Function for ICA

We present a new scale-invariant cost function for non-orthogonal joint-diagonalization of a set of symmetric matrices with application to Independent Component Analysis (ICA). We derive two gradient minimization schemes to minimize this cost function. We also consider their performance in the context of an ICA algorithm based on non-orthogonal joint diagonalization

The Berry-Hannay Phase of the Equal-Sided, Spring-Jointed, Four-Bar Mechanism: A Detailed Story

In this work we apply the moving systems approach developed by Marsden, Montgomery, and Ratiu to a free-floating, equal-sided, spring-jointed, four-bar mechanism that is being slowly rotated about its central axis and derive a formula for the induced geometric phase. We investigate the phase for a few specific systems using both analytic analysis and simulation

A Lyapunov Functional for the Cubic Nonlinearity Activator-Inhibitor Model Equation

The cubic nonlinearity activator-inhibitor model equation is a simpleexample of a pattern-forming system for which strong mathematical resultscan be obtained. Basic properties of solutions and the derivation ofa Lyapunov functional for the cubic nonlinearity model are presented.Potential applications include control of large MEMS actuator arrays.(In Proc. IEEE Conf. Decision and Control, December 16-18, 1998

A Simple Control Law for UAV Formation Flying

This paper presents a Lie group setting for the problem of control of formations, as a natural outcome of the analysis of a planar two-vehicle formation control law. The vehicle trajectories are described using planar Frenet-Serret equations of motion, which capture the evolution of both the vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The set of all possible (relative) equilibria for arbitrary G-invariant curvature controls is described (where G = SE(2) is a symmetry group for the control law). A generalization of the control law for n vehicles is presented, and the corresponding (relative) equilibria are characterized. Work is on-going to discover stability and convergence results for the n-vehicle problem. The practical motivation for this work is the problem of formation control for meter-scale UAVs; therefore, an implementation approach consistent with UAV payload constraints is also discussed