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arXiv · 1801.00577

The variational discretization of the constrained higher-order Lagrange-Poincar\'e equations

Anthony Bloch·Leonardo Colombo·Fernando Jiménez
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Abstract

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincar\'e equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincar\'e equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

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Anthony Bloch, Leonardo Colombo, Fernando Jiménez. 2018-01-02. The variational discretization of the constrained higher-order Lagrange-Poincar\'e equations. https://arxiv.org/abs/1801.00577

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