Bullo, Francesco and Murray, Richard M. (1995) Proportional Derivative (PD) Control on the Euclidean Group. California Institute of Technology . (Unpublished) http://resolver.caltech.edu/CaltechCDSTR:1995.CITCDS95010

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Abstract
In this paper we study the stabilization problem for control systems defined on SE(3) (the special Euclidean group of rigidbody motions) and its subgroups. 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 proportional derivative (PD) control in a coordinatefree way. For the SO(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. In the SE(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 SO(3) approach to the whole of SE(3) or by breaking the problem into a control problem on SO(3) x R^3. For the simple SE(2) case, simulations are reported to illustrate the behavior of the different choices. We also discuss the trajectory tracking problem and show how to reduce it to a stabilization problem, mimicking the usual approach in R^n. Finally, regarding the case of underactuated control systems, we derive linear and homogeneous approximating vector fields for standard systems on SO(3) and SE(3).
Item Type:  Report or Paper (Technical Report)  

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Group:  Control and Dynamical Systems Technical Reports  
Record Number:  CaltechCDSTR:1995.CITCDS95010  
Persistent URL:  http://resolver.caltech.edu/CaltechCDSTR:1995.CITCDS95010  
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ID Code:  28018  
Collection:  CaltechCDSTR  
Deposited By:  Imported from CaltechCDSTR  
Deposited On:  18 Oct 2002  
Last Modified:  18 Mar 2015 23:19 
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