1,045 research outputs found
Virtual Holonomic Constraints for Euler-Lagrange systems under sampling
In this paper, we consider the problem of imposing Virtual Holonomic Constraints to mechanical systems in Euler-Lagrangian form under sampling. An exact solution based on multi-rate sampling of order two over each input channel is described. The results are applied to orbital stabilization of the pendubot with illustrative simulations
On the Lagrangian Structure of Reduced Dynamics Under Virtual Holonomic Constraints
This paper investigates a class of Lagrangian control systems with
degrees-of-freedom (DOF) and n-1 actuators, assuming that virtual
holonomic constraints have been enforced via feedback, and a basic regularity
condition holds. The reduced dynamics of such systems are described by a
second-order unforced differential equation. We present necessary and
sufficient conditions under which the reduced dynamics are those of a
mechanical system with one DOF and, more generally, under which they have a
Lagrangian structure. In both cases, we show that typical solutions satisfying
the virtual constraints lie in a restricted class which we completely
characterize.Comment: 23 pages, 5 figures, published online in ESAIM:COCV on April 28th,
201
Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics
I extract some philosophical morals from some aspects of Lagrangian
mechanics. (A companion paper will present similar morals from Hamiltonian
mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology:
Lagrangian mechanics provides a level of description of phenomena which has
been largely ignored by philosophers, since it falls between their accustomed
levels--``laws of nature'' and ``models''. Another main moral concerns
ontology: the ontology of Lagrangian mechanics is both more subtle and more
problematic than philosophers often realize.
The treatment of Lagrangian mechanics provides an introduction to the subject
for philosophers, and is technically elementary. In particular, it is confined
to systems with a finite number of degrees of freedom, and for the most part
eschews modern geometry. But it includes a presentation of Routhian reduction
and of Noether's ``first theorem''.Comment: 106 pages, no figure
Momentum and energy preserving integrators for nonholonomic dynamics
In this paper, we propose a geometric integrator for nonholonomic mechanical
systems. It can be applied to discrete Lagrangian systems specified through a
discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and
a (generally nonintegrable) distribution in TQ. In the proposed method, a
discretization of the constraints is not required. We show that the method
preserves the discrete nonholonomic momentum map, and also that the
nonholonomic constraints are preserved in average. We study in particular the
case where Q has a Lie group structure and the discrete Lagrangian and/or
nonholonomic constraints have various invariance properties, and show that the
method is also energy-preserving in some important cases.Comment: 18 pages, 6 figures; v2: example and figures added, minor correction
to example 2; v3: added section on nonholonomic Stoermer-Verlet metho
Spaces with torsion from embedding and the special role of autoparallel trajectories
As a contribution to the ongoing discussion of trajectories of spinless
particles in spaces with torsion we show that the geometry of such spaces can
be induced by embedding their curves in a euclidean space without torsion.
Technically speaking, we define the tangent (velocity) space of the embedded
space imposing non-holonomic constraints upon the tangent space of the
embedding space. Parallel transport in the embedded space is determined as an
induced parallel transport on the surface of constraints. Gauss' principle of
least constraint is used to show that autoparallels realize a constrained
motion that has a minimal deviation from the free, unconstrained motion, this
being a mathematical expression of the principle of inertia.Comment: LaTeX file in src, no figures. Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Paper also at
http://www.physik.fu-berlin.de/~kleinert/kleiner_re259/preprint.htm
Control of a Bicycle Using Virtual Holonomic Constraints
The paper studies the problem of making Getz's bicycle model traverse a
strictly convex Jordan curve with bounded roll angle and bounded speed. The
approach to solving this problem is based on the virtual holonomic constraint
(VHC) method. Specifically, a VHC is enforced making the roll angle of the
bicycle become a function of the bicycle's position along the curve. It is
shown that the VHC can be automatically generated as a periodic solution of a
scalar periodic differential equation, which we call virtual constraint
generator. Finally, it is shown that if the curve is sufficiently long as
compared to the height of the bicycle's centre of mass and its wheel base, then
the enforcement of a suitable VHC makes the bicycle traverse the curve with a
steady-state speed profile which is periodic and independent of initial
conditions. An outcome of this work is a proof that the constrained dynamics of
a Lagrangian control system subject to a VHC are generally not Lagrangian.Comment: 18 pages, 8 figure
- âŠ