1,608 research outputs found
Dissipation and Controlled Euler-Poincaré Systems
The method of controlled Lagrangians is a technique for stabilizing underactuated mechanical systems which involves modifying a system’s energy and dynamic structure through feedback. These modifications can obscure the effect of physical dissipation in the closed-loop. For example,
generic damping can destabilize an equilibrium which is closed-loop stable for a conservative system model. In this paper, we consider the effect of damping on Euler-Poincaré (special reduced Lagrangian) systems which have been stabilized about an equilibrium using the method of controlled Lagrangians. We describe a choice of feed-back dissipation which asymptotically stabilizes a sub-class of controlled Euler-Poincaré systems subject to physical damping. As an example, we consider intermediate axis rotation of a damped rigid body with a single internal rotor
Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem
We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane
A Unification of Models of Tethered Satellites
In this paper, different conservative models of tethered satellites are related mathematically, and it is established in what limit they may provide useful insight into the underlying dynamics. An infinite dimensional model is linked to a finite dimensional model, the slack-spring model, through a conjecture on the singular perturbation of tether thickness. The slack-spring model is then naturally related to a billiard model in the limit of an inextensible spring. Next, the motion of a dumbbell model, which is lowest in the hierarchy of models, is identified within the motion of the billiard model through a theorem on the existence of invariant curves by exploiting Moser's twist map theorem. Finally, numerical computations provide insight into the dynamics of the billiard model
Hydrodynamic synchronization of flagellar oscillators
We survey the theory synchronization in collections of noisy oscillators.
This framework is applied to flagellar synchronization by hydrodynamic
interactions. The time-reversibility of hydrodynamics at low Reynolds numbers
prompts swimming strokes that break symmetry to facilitate hydrodynamic
synchronization. We discuss different physical mechanisms for flagellar
synchronization, which break this symmetry in different ways.Comment: 15 pages, 3 figures; accepted for publication in EPJ Special Topics
Issue,Lecture Notes of the Summer School "Microswimmers -- From Single
Particle Motion to Collective Behaviour'', organised by the DFG Priority
Programme SPP 1726 (Forschungszentrum J\"ulich, J\"ulich, 2015
Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping
For pt.I, see ibid., vol.45, p.2253-70 (2000). We extend the method of controlled Lagrangians (CL) to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems. The CL method deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline
Orbital stability in static axisymmetric fields
We investigate the stability of test-particle equilibrium orbits in
axisymmetric, but otherwise arbitrary, gravitational and electromagnetic
fields. We extend previous studies of this problem to include a toroidal
magnetic field. We find that, even though the toroidal magnetic field does not
alter the location of the circular orbits, it enters the problem as a
gyroscopic force with the potential to provide gyroscopic stability. This is in
essence similar to the situation encountered in the reduced three-body problem
where rotation enables stability around the local maxima of the effective
potential. Nevertheless, we show that gyroscopic stabilization by a toroidal
magnetic field is impossible for axisymmetric force fields in source-free
regions because in this case the effective potential does not possess any local
maxima. As an example of an axisymmetric force field with sources, we consider
the classical problem of a rotating, aligned magnetosphere. By analyzing the
dynamics of halo and equatorial particle orbits we conclude that axisymmetric
toroidal fields that are antisymmetric about the equator are unable to provide
gyroscopic stabilization. On the other hand, a toroidal magnetic field that
does not vanish at the equator can provide gyroscopic stabilization for
positively charged particles in prograde equatorial orbits.Comment: 11 pages, 3 figures, submitted to Celestial Mechanics and Dynamical
Astronom
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]
On the critical nature of plastic flow: one and two dimensional models
Steady state plastic flows have been compared to developed turbulence because
the two phenomena share the inherent complexity of particle trajectories, the
scale free spatial patterns and the power law statistics of fluctuations. The
origin of the apparently chaotic and at the same time highly correlated
microscopic response in plasticity remains hidden behind conventional
engineering models which are based on smooth fitting functions. To regain
access to fluctuations, we study in this paper a minimal mesoscopic model whose
goal is to elucidate the origin of scale free behavior in plasticity. We limit
our description to fcc type crystals and leave out both temperature and rate
effects. We provide simple illustrations of the fact that complexity in rate
independent athermal plastic flows is due to marginal stability of the
underlying elastic system. Our conclusions are based on a reduction of an
over-damped visco-elasticity problem for a system with a rugged elastic energy
landscape to an integer valued automaton. We start with an overdamped one
dimensional model and show that it reproduces the main macroscopic
phenomenology of rate independent plastic behavior but falls short of
generating self similar structure of fluctuations. We then provide evidence
that a two dimensional model is already adequate for describing power law
statistics of avalanches and fractal character of dislocation patterning. In
addition to capturing experimentally measured critical exponents, the proposed
minimal model shows finite size scaling collapse and generates realistic shape
functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science
for the special issue in honor of Victor Berdichevsky, 201
A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks
This paper presents a stability test for a class of interconnected nonlinear
systems motivated by biochemical reaction networks. One of the main results
determines global asymptotic stability of the network from the diagonal
stability of a "dissipativity matrix" which incorporates information about the
passivity properties of the subsystems, the interconnection structure of the
network, and the signs of the interconnection terms. This stability test
encompasses the "secant criterion" for cyclic networks presented in our
previous paper, and extends it to a general interconnection structure
represented by a graph. A second main result allows one to accommodate state
products. This extension makes the new stability criterion applicable to a
broader class of models, even in the case of cyclic systems. The new stability
test is illustrated on a mitogen activated protein kinase (MAPK) cascade model,
and on a branched interconnection structure motivated by metabolic networks.
Finally, another result addresses the robustness of stability in the presence
of diffusion terms in a compartmental system made out of identical systems.Comment: See http://www.math.rutgers.edu/~sontag/PUBDIR/index.html for related
(p)reprint
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