A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups

Author

Manton, Jonathan H.

Author_Institution

Dept. of Electr. & Electron. Eng., Melbourne Univ., Australia

Volume

3

fYear

2004

fDate

6-9 Dec. 2004

Firstpage

2211

Abstract

Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size.

Keywords

Lie groups; computational geometry; convergence of numerical methods; iterative methods; matrix algebra; theorem proving; Karcher mean; Riemannian gradient descent algorithm; compact Lie group; fuzzy control; geometry; iterative algorithm; mass center; orthogonal matrices; robotics; special orthogonal group; step size; unit determinant; vision; Aging; Convergence; Fuzzy control; Geometry; Iterative algorithms; Noise measurement; Process control; Robots; Signal processing algorithms; Signal to noise ratio;

fLanguage

English

Publisher

ieee

Conference_Titel

Control, Automation, Robotics and Vision Conference, 2004. ICARCV 2004 8th

Print_ISBN

0-7803-8653-1

Type

conf

DOI

10.1109/ICARCV.2004.1469774

Filename

1469774