54 research outputs found
Discrete nonholonomic LL systems on Lie groups
This paper applies the recently developed theory of discrete nonholonomic mechanics to the
study of discrete nonholonomic left-invariant dynamics on Lie groups. The theory is illustrated
with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin
sleigh. The preservation of the reduced energy by the discrete flow is observed and the
discrete momentum conservation is discussed
The Lyapunov-Malkin Theorem and Stabilization of the Unicycle with Rider
This paper analyzes stabilization of a nonholonomic system consisting of a unicycle with rider. It is shown that one can achieve stability of slow steady vertical motions by imposing a feedback control force on the rider’s limb
Flat Nonholonomic Matching
In this paper we extend the matching technique to a
class of nonholonomic systems with symmetries. Assuming
that the momentum equation defines an integrable
distribution, we introduce a family of reduced
systems. The method of controlled Lagrangians is then
applied to these systems resulting in a smooth stabilizing
controller
Controlled Lagrangian Methods and Tracking of Accelerated Motions
Matching techniques are applied to the problem of stabilization of uniformly accelerated motions of mechanical
systems with symmetry. The theory is illustrated with a simple model-a wheel and pendulum system
Nonholonomic Dynamics
Nonholonomic systems are, roughly speaking, mechanical
systems with constraints on their velocity
that are not derivable from position constraints.
They arise, for instance, in mechanical systems
that have rolling contact (for example, the rolling
of wheels without slipping) or certain kinds of sliding
contact (such as the sliding of skates). They are
a remarkable generalization of classical Lagrangian
and Hamiltonian systems in which one allows position
constraints only.
There are some fascinating differences between
nonholonomic systems and classical Hamiltonian
or Lagrangian systems. Among other things: nonholonomic
systems are nonvariational—they arise
from the Lagrange-d’Alembert principle and not
from Hamilton’s principle; while energy is preserved
for nonholonomic systems, momentum is
not always preserved for systems with symmetry
(i.e., there is nontrivial dynamics associated with
the nonholonomic generalization of Noether’s
theorem); nonholonomic systems are almost Poisson
but not Poisson (i.e., there is a bracket that together
with the energy on the phase space defines
the motion, but the bracket generally does not satisfy
the Jacobi identity); and finally, unlike the
Hamiltonian setting, volume may not be preserved
in the phase space, leading to interesting asymptotic
stability in some cases, despite energy conservation.
The purpose of this article is to engage
the reader’s interest by highlighting some of these
differences along with some current research in the
area. There has been some confusion in the literature
for quite some time over issues such as the
variational character of nonholonomic systems, so
it is appropriate that we begin with a brief review
of the history of the subject
The energy–momentum method for the stability of non-holonomic systems
In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit
both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for
holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top
Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System
Matching techniques are developed for discrete
mechanical systems with symmetry. We describe new phenomena
that arise in the controlled Lagrangian approach for mechanical
systems in the discrete context. In particular, one needs
to either make an appropriate selection of momentum levels or
introduce a new parameter into the controlled Lagrangian to
complete the matching procedure. We also discuss digital and
model predictive control
Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems
The method of controlled Lagrangians for discrete mechanical systems is
extended to include potential shaping in order to achieve complete state-space
asymptotic stabilization. New terms in the controlled shape equation that are
necessary for matching in the discrete context are introduced. The theory is
illustrated with the problem of stabilization of the cart-pendulum system on an
incline. We also discuss digital and model predictive control.Comment: IEEE Conference on Decision and Control, 2006 6 pages, 4 figure
Matching and stabilization of discrete mechanical systems
Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems
with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in
the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for
matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers
Matching and stabilization of the unicycle with rider
In this paper we apply matching techniques for controlled Lagrangians to
the stabilization problem of a nonholonomic system consisting of a unicycle with rider.
We show how generalized matching results may be applied to the Routhian associated
with this nonholonomic system
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