Two-input control systems on the euclidean group  SE (2)

Ross M. Adams; Rory Biggs; Claudiu C. Remsing

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 4, page 947-975
  • ISSN: 1292-8119

Abstract

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Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrisations for all integral curves are exhibited.

How to cite

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Adams, Ross M., Biggs, Rory, and Remsing, Claudiu C.. "Two-input control systems on the euclidean group  SE (2)." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 947-975. <http://eudml.org/doc/272839>.

@article{Adams2013,
abstract = {Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space &#x1d530;&#x1d522; (2)*. These reduced Hamilton − Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrisations for all integral curves are exhibited.},
author = {Adams, Ross M., Biggs, Rory, Remsing, Claudiu C.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {left-invariant control system; (detached) feedback equivalence; Lie − Poisson structure; energy-casimir method; Jacobi elliptic function; detached feedback equivalence; Lie-Poisson structure; energy-Casimir method},
language = {eng},
number = {4},
pages = {947-975},
publisher = {EDP-Sciences},
title = {Two-input control systems on the euclidean group  SE (2)},
url = {http://eudml.org/doc/272839},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Adams, Ross M.
AU - Biggs, Rory
AU - Remsing, Claudiu C.
TI - Two-input control systems on the euclidean group  SE (2)
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 4
SP - 947
EP - 975
AB - Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space &#x1d530;&#x1d522; (2)*. These reduced Hamilton − Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrisations for all integral curves are exhibited.
LA - eng
KW - left-invariant control system; (detached) feedback equivalence; Lie − Poisson structure; energy-casimir method; Jacobi elliptic function; detached feedback equivalence; Lie-Poisson structure; energy-Casimir method
UR - http://eudml.org/doc/272839
ER -

References

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