Proportional Derivative (PD) Control on the Euclidean Group
Francesco Bullo and Richard Murray
CDS Technical Report 95-010
In this paper we study the stabilization problem for control systems defined
on SE(3) (the special Euclidean group of rigid-body 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 coordinate-free 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).
CDS Technical Report
(PDF, 707K, 47 pages)
Downloading and printing FAQ
Richard Murray (murray@cds.caltech.edu)
Last modified: Tue Aug 30 07:42:20 2005