18,773 research outputs found
A Discrete Geometric Optimal Control Framework for Systems with Symmetries
This paper studies the optimal motion control of
mechanical systems through a discrete geometric approach. At
the core of our formulation is a discrete Lagrange-d’Alembert-
Pontryagin variational principle, from which are derived discrete
equations of motion that serve as constraints in our optimization
framework. We apply this discrete mechanical approach to
holonomic systems with symmetries and, as a result, geometric
structure and motion invariants are preserved. We illustrate our
method by computing optimal trajectories for a simple model of
an air vehicle flying through a digital terrain elevation map, and
point out some of the numerical benefits that ensue
Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms
This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a
discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators
Stochastic Variational Integrators
This paper presents a continuous and discrete Lagrangian theory for
stochastic Hamiltonian systems on manifolds. The main result is to derive
stochastic governing equations for such systems from a critical point of a
stochastic action. Using this result the paper derives Langevin-type equations
for constrained mechanical systems and implements a stochastic analog of
Lagrangian reduction. These are easy consequences of the fact that the
stochastic action is intrinsically defined. Stochastic variational integrators
(SVIs) are developed using a discretized stochastic variational principle. The
paper shows that the discrete flow of an SVI is a.s. symplectic and in the
presence of symmetry a.s. momentum-map preserving. A first-order mean-square
convergent SVI for mechanical systems on Lie groups is introduced. As an
application of the theory, SVIs are exhibited for multiple, randomly forced and
torqued rigid-bodies interacting via a potential.Comment: 21 pages, 8 figure
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
Geometric, Variational Discretization of Continuum Theories
This study derives geometric, variational discretizations of continuum
theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the
dynamics of complex fluids. A central role in these discretizations is played
by the geometric formulation of fluid dynamics, which views solutions to the
governing equations for perfect fluid flow as geodesics on the group of
volume-preserving diffeomorphisms of the fluid domain. Inspired by this
framework, we construct a finite-dimensional approximation to the
diffeomorphism group and its Lie algebra, thereby permitting a variational
temporal discretization of geodesics on the spatially discretized
diffeomorphism group. The extension to MHD and complex fluid flow is then made
through an appeal to the theory of Euler-Poincar\'{e} systems with advection,
which provides a generalization of the variational formulation of ideal fluid
flow to fluids with one or more advected parameters. Upon deriving a family of
structured integrators for these systems, we test their performance via a
numerical implementation of the update schemes on a cartesian grid. Among the
hallmarks of these new numerical methods are exact preservation of momenta
arising from symmetries, automatic satisfaction of solenoidal constraints on
vector fields, good long-term energy behavior, robustness with respect to the
spatial and temporal resolution of the discretization, and applicability to
irregular meshes
Nonsmooth Lagrangian mechanics and variational collision integrators
Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.
Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated
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