Higher-Order Constrained Variational Problems on Principal Bundles with Applications to Optimal Control of Underactuated Systems

Colombo, Leonardo; Gupta, Rohit; Bloch, Anthony M.

doi:10.1016/j.ifacol.2015.10.219

IFAC-PapersOnLine

2015

DOI: 10.1016/j.ifacol.2015.10.219

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Higher-Order Constrained Variational Problems on Principal Bundles with Applications to Optimal Control of Underactuated Systems

Leonardo Colombo

Rohit Gupta

Anthony M. Bloch

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2019

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“…We focus on systems whose phase space is of higher-order, i.e. T (k) Q [20], [33], [34], and moreover is invariant under the action of symmetries. The Euler-Lagrange and Lagrange-Poincaré equations for these systems were introduced by F. Gay-Balmaz, D. Holm and T. Ratiu in [32].…”

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confidence: 99%

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

Bloch¹,

Colombo²,

Jiménez³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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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-Poincaré 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-Poincaré 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.2010 Mathematics Subject Classification. Primary: 37K05 ; Secondary: 37J15; 37N05; 49M25; 49S05; 49J15.

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mentioning

confidence: 99%

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

Bloch¹,

Colombo²,

Jiménez³

2019

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

View full text Add to dashboard Cite

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Higher-Order Constrained Variational Problems on Principal Bundles with Applications to Optimal Control of Underactuated Systems

Cited by 1 publication

References 10 publications

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

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