1,607 research outputs found
Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction
In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed
here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such
an approach is that it makes use of the special structure of the system, especially its symmetry
structure and thus it leads rather directly to the desired conclusions for such systems.
Lagrangian reduction can do in one step what one can alternatively do by applying the
Pontryagin Maximum Principle followed by an application of Poisson reduction. The idea of
using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and
Crouch [1995a,b] in a somewhat different context and the general idea is closely related to those
in Montgomery [1990] and Vershik and Gershkovich [1994]. Here we develop this idea further
and apply it to some known examples, such as optimal control on Lie groups and principal
bundles (such as the ball and plate problem) and reorientation examples with zero angular
momentum (such as the satellite with moveable masses). However, one of our main goals is to
extend the method to the case of nonholonomic systems with a nontrivial momentum equation in
the context of the work of Bloch, Krishnaprasad, Marsden and Murray [1995]. The snakeboard
is used to illustrate the method
Poisson reduction for nonholonomic mechanical systems with symmetry
This paper continues the work of Koon and Marsden [1997b] that began the
comparison of the Hamiltonian and Lagrangian formulations of nonholonomic
systems. Because of the necessary replacement of conservation laws with the
momentum equation, it is natural to let the value of momentum be a variable
and for this reason it is natural to take a Poisson viewpoint. Some of this
theory has been started in van der Schaft and Maschke [1994]. We build on
their work, further develop the theory of nonholonomic Poisson reduction, and
tie this theory to other work in the area. We use this reduction procedure
to organize nonholonomic dynamics into a reconstruction equation, a nonholonomic
momentum equation and the reduced Lagrange d’Alembert equations in
Hamiltonian form. We also show that these equations are equivalent to those
given by the Lagrangian reduction methods of Bloch, Krishnaprasad, Marsden
and Murray [1996]. Because of the results of Koon and Marsden [1997b],
this is also equivalent to the results of Bates and Sniatycki [1993], obtained by
nonholonomic symplectic reduction.
Two interesting complications make this effort especially interesting. First
of all, as we have mentioned, symmetry need not lead to conservation laws
but rather to a momentum equation. Second, the natural Poisson bracket fails
to satisfy the Jacobi identity. In fact, the so-called Jacobiizer (the cyclic sum
that vanishes when the Jacobi identity holds), or equivalently, the Schouten
bracket, is an interesting expression involving the curvature of the underlying
distribution describing the nonholonomic constraints.
The Poisson reduction results in this paper are important for the future
development of the stability theory for nonholonomic mechanical systems with
symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In particular, they
should be useful for the development of the powerful block diagonalization
properties of the energy-momentum method developed by Simo, Lewis and
Marsden [1991]
The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems
This paper compares the Hamiltonian approach to systems with nonholonomic constraints
(see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschke
[1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996]
and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the
approaches and each has its own advantages; some structures have been discovered on one side
and their analogues on the other side are interesting to clarify. For example, the momentum
equation and the reconstruction equation were first found on the Lagrangian side and are useful
for the control theory of these systems, while the failure of the reduced two form to be closed
(i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the
Hamiltonian side. Clarifying the relation between these approaches is important for the future
development of the control theory and stability and bifurcation theory for such systems. In
addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see
Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum
phase) control system
Simulating Nonholonomic Dynamics
This paper develops different discretization schemes for nonholonomic
mechanical systems through a discrete geometric approach. The proposed methods
are designed to account for the special geometric structure of the nonholonomic
motion. Two different families of nonholonomic integrators are developed and
examined numerically: the geometric nonholonomic integrator (GNI) and the
reduced d'Alembert-Pontryagin integrator (RDP). As a result, the paper provides
a general tool for engineering applications, i.e. for automatic derivation of
numerically accurate and stable dynamics integration schemes applicable to a
variety of robotic vehicle models
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
Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian)
systems, a generalized formulation of Lagrangian mechanics that can incorporate
degenerate Lagrangians as well as holonomic and nonholonomic constraints. We
refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi
equation. For non-degenerate Lagrangian systems with nonholonomic constraints,
the theory specializes to the recently developed nonholonomic Hamilton-Jacobi
theory. We are particularly interested in applications to a certain class of
degenerate nonholonomic Lagrangian systems with symmetries, which we refer to
as weakly degenerate Chaplygin systems, that arise as simplified models of
nonholonomic mechanical systems; these systems are shown to reduce to
non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian
systems defined with non-closed two-forms. Accordingly, the
Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic
Hamilton-Jacobi equation associated with the reduced system. We illustrate
through a few examples how the Dirac-Hamilton-Jacobi equation can be used to
exactly integrate the equations of motion.Comment: 44 pages, 3 figure
A class of nonholonomic kinematic constraints in elasticity
We propose a first example of a simple classical field theory with
nonholonomic constraints. Our model is a straightforward modification of a
Cosserat rod. Based on a mechanical analogy, we argue that the constraint
forces should be modeled in a special way, and we show how such a procedure can
be naturally implemented in the framework of geometric field theory. Finally,
we derive the equations of motion and we propose a geometric integration scheme
for the dynamics of a simplified model.Comment: 28 pages, 7 figures, uses IOPP LaTeX style (included) (v3: section 2
entirely rewritten
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