Proportional Derivative (PD) Control on the Euclidean Group: Difference between revisions
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| authors = Francesco Bullo and Richard Murray | | authors = Francesco Bullo and Richard Murray | ||
| title = Proportional Derivative (PD) Control on the Euclidean Group | | title = Proportional Derivative (PD) Control on the Euclidean Group | ||
| source = | | source = 1995 European Control Conference (Rome) | ||
| year = | | year = 1994 | ||
| type = | | type = Conference paper | ||
| funding = | | funding = | ||
| url = http://www.cds.caltech.edu/~murray/preprints/ | | url = http://www.cds.caltech.edu/~murray/preprints/bm95-ecc.pdf | ||
| abstract = | | abstract = | ||
In this paper we study the stabilization problem for control systems defined | In this paper we study the stabilization problem for control systems | ||
on SE(3) | 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 | algebras) to generalize the classical PD control in a coordinate--free | ||
in a coordinate-free way. For the SO(3) case, the compactness of the group | way. For the SO(3) case, the compactness of the group gives rise to a | ||
gives rise to a natural metric structure and to a natural choice of | natural metric structure and to a natural choice of preferred control | ||
preferred control direction: an optimal (in the sense of geodesic) solution | direction: an optimal (in the sense of geodesic) solution is given to | ||
is given to the attitude control problem. In the SE(3) case, no natural | the attitude control problem. In the SE(3) case, no natural metric is | ||
metric is uniquely defined, so that more freedom is left in the control | uniquely defined, so that more freedom is left in the control design. | ||
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 | control problem on SO(3) x R^3. For the simple SE(2) case, | ||
reported to illustrate the behavior of the different choices. | simulations are reported to illustrate the behavior of the different | ||
discuss the trajectory tracking problem and show how to reduce it to a | choices. Finally, we discuss the trajectory tracking problem and show | ||
stabilization problem, mimicking the usual approach in R^n | how to reduce it to a stabilization problem, mimicking the usual | ||
approach in R^n. | |||
| flags = NoRequest | | flags = NoRequest | ||
| tag = | | tag = bm95-ecc | ||
| id = | | id = 1994n | ||
}} | }} | ||
Latest revision as of 06:20, 15 May 2016
Francesco Bullo and Richard Murray
1995 European Control Conference (Rome)
In this paper we study the stabilization problem for control 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 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. Finally, we discuss the trajectory tracking problem and show how to reduce it to a stabilization problem, mimicking the usual approach in R^n.
- Conference paper: http://www.cds.caltech.edu/~murray/preprints/bm95-ecc.pdf
- Project(s):
