554 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
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
Generalized Hamilton-Jacobi equations for nonholonomic dynamics
Employing a suitable nonlinear Lagrange functional, we derive generalized
Hamilton-Jacobi equations for dynamical systems subject to linear velocity
constraints. As long as a solution of the generalized Hamilton-Jacobi equation
exists, the action is actually minimized (not just extremized)
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