Integrating control systems defined on the frame bundles of the space forms

J.D. Biggs, William Holderbaum

Research output: Contribution to conferencePaper

3 Citations (Scopus)

Abstract

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.
LanguageEnglish
Pages3351-3356
Number of pages5
Publication statusPublished - 2006
Event45th IEEE Conference on Decision and Control - San Diego, United States
Duration: 13 Dec 200615 Dec 2006

Conference

Conference45th IEEE Conference on Decision and Control
CountryUnited States
CitySan Diego
Period13/12/0615/12/06

Fingerprint

Lie groups
bundles
control system
Control systems
screws
Algebra
curvature
Euclidean geometry
decoupling
algebra
shift
curves
method

Keywords

  • frame bundles
  • control systems
  • space forms
  • space

Cite this

Biggs, J. D., & Holderbaum, W. (2006). Integrating control systems defined on the frame bundles of the space forms. 3351-3356. Paper presented at 45th IEEE Conference on Decision and Control, San Diego, United States.
Biggs, J.D. ; Holderbaum, William. / Integrating control systems defined on the frame bundles of the space forms. Paper presented at 45th IEEE Conference on Decision and Control, San Diego, United States.5 p.
@conference{e89f610e00b746b0b8b2769ee01adfad,
title = "Integrating control systems defined on the frame bundles of the space forms",
abstract = "This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.",
keywords = "frame bundles, control systems, space forms, space",
author = "J.D. Biggs and William Holderbaum",
note = "Proceedings of the 45th IEEE Conference on Decision and Control, Vols 1-14. IEEE Conference on Decision and Control. IEEE, New York, pp. 3849-3854. ISBN 0191-2216 9781424401703; 45th IEEE Conference on Decision and Control ; Conference date: 13-12-2006 Through 15-12-2006",
year = "2006",
language = "English",
pages = "3351--3356",

}

Biggs, JD & Holderbaum, W 2006, 'Integrating control systems defined on the frame bundles of the space forms' Paper presented at 45th IEEE Conference on Decision and Control, San Diego, United States, 13/12/06 - 15/12/06, pp. 3351-3356.

Integrating control systems defined on the frame bundles of the space forms. / Biggs, J.D.; Holderbaum, William.

2006. 3351-3356 Paper presented at 45th IEEE Conference on Decision and Control, San Diego, United States.

Research output: Contribution to conferencePaper

TY - CONF

T1 - Integrating control systems defined on the frame bundles of the space forms

AU - Biggs, J.D.

AU - Holderbaum, William

N1 - Proceedings of the 45th IEEE Conference on Decision and Control, Vols 1-14. IEEE Conference on Decision and Control. IEEE, New York, pp. 3849-3854. ISBN 0191-2216 9781424401703

PY - 2006

Y1 - 2006

N2 - This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

AB - This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

KW - frame bundles

KW - control systems

KW - space forms

KW - space

UR - https://css.paperplaza.net/conferences/scripts/abstract.pl?ConfID=27&Number=963

M3 - Paper

SP - 3351

EP - 3356

ER -

Biggs JD, Holderbaum W. Integrating control systems defined on the frame bundles of the space forms. 2006. Paper presented at 45th IEEE Conference on Decision and Control, San Diego, United States.