358 research outputs found
What makes nonholonomic integrators work?
A nonholonomic system is a mechanical system with velocity constraints not
originating from position constraints; rolling without slipping is the typical
example. A nonholonomic integrator is a numerical method specifically designed
for nonholonomic systems. It has been observed numerically that many
nonholonomic integrators exhibit excellent long-time behaviour when applied to
various test problems. The excellent performance is often attributed to some
underlying discrete version of the Lagrange--d'Alembert principle. Instead, in
this paper, we give evidence that reversibility is behind the observed
behaviour. Indeed, we show that many standard nonholonomic test problems have
the structure of being foliated over reversible integrable systems. As most
nonholonomic integrators preserve the foliation and the reversible structure,
near conservation of the first integrals is a consequence of reversible KAM
theory. Therefore, to fully evaluate nonholonomic integrators one has to
consider also non-reversible nonholonomic systems. To this end we construct
perturbed test problems that are integrable but no longer reversible (with
respect to the standard reversibility map). Applying various nonholonomic
integrators from the literature to these problems we observe that no method
performs well on all problems. This further indicates that reversibility is the
main mechanism behind near conservation of first integrals for nonholonomic
integrators. A list of relevant open problems is given.Comment: 27 pages, 9 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
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
Symplectic integrators for index one constraints
We show that symplectic Runge-Kutta methods provide effective symplectic
integrators for Hamiltonian systems with index one constraints. These include
the Hamiltonian description of variational problems subject to position and
velocity constraints nondegenerate in the velocities, such as those arising in
sub-Riemannian geometry and control theory.Comment: 13 pages, accepted in SIAM J Sci Compu
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