Proportional Derivative (PD) Control on the Euclidean Group. Bullo, F. and rray , R. M M.
abstract   bibtex   
In this paper we study the stabilization problem for con trol systems defined on SE (3), the Euclidean group of rigid—body motions. Assuming one actuator is available for each degree of freedom, we exploit geometric properties of Lie. groups (and corresponding Lie algebras) to generalize the classical PD control in a coordinate—free way. For the 50(3) case, the compactness of the group gives rise to a natural metric structure and to a natural choice of preferred control direction: an optimal (in the sense of geodesic) solution is given to the attitude control problem. 1n the 55(3) case, no natural metric is uniquely defined, so that more freedom is left in the control design. Different formulations of PD feedback can be adopted by extending the 50(3) approach to the whole of 55(3) or by breaking the problem into a control problem on 50(3) x R3. For the simple 5E(2) case, simulations are reported to illustrate the behavior of the different choices. Finally, we discuss the trajectory track ing problem and show how to reduce it to a stabilization problem, mimicking the usual approach in Pl".
@article{bullo_proportional_nodate,
	title = {Proportional {Derivative} ({PD}) {Control} on the {Euclidean} {Group}},
	abstract = {In this paper we study the stabilization problem for con trol systems defined on SE (3), the Euclidean group of rigid—body motions. Assuming one actuator is available for each degree of freedom, we exploit geometric properties of Lie. groups (and corresponding Lie algebras) to generalize the classical PD control in a coordinate—free way. For the 50(3) case, the compactness of the group gives rise to a natural metric structure and to a natural choice of preferred control direction: an optimal (in the sense of geodesic) solution is given to the attitude control problem. 1n the 55(3) case, no natural metric is uniquely defined, so that more freedom is left in the control design. Different formulations of PD feedback can be adopted by extending the 50(3) approach to the whole of 55(3) or by breaking the problem into a control problem on 50(3) x R3. For the simple 5E(2) case, simulations are reported to illustrate the behavior of the different choices. Finally, we discuss the trajectory track ing problem and show how to reduce it to a stabilization problem, mimicking the usual approach in Pl".},
	language = {en},
	author = {Bullo, Francesco and rray, Richard M Mu},
	pages = {7}
}
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