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

Biggs, J. and Holderbaum, W. (2006) Integrating control systems defined on the frame bundles of the space forms. In: 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

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Abstract/Summary

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.

Item Type:Book or Report Section
Divisions:Faculty of Science > School of Systems Engineering
ID Code:14358
Additional Information:Proceedings Paper 45th IEEE Conference on Decision and Control DEC 13-15, 2006 San Diego, CA
Publisher:IEEE

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