Minimum distance and optimal transformations on SO(N)
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The group of special (or proper) orthogonal matrices, SO(N), is used throughout engineering mechanics in the analysis and representation of mechanical systems. In this paper, a solution is presented for the optimal transformation between two elements of SO(N). The transformation is assumed to occur during a specified finite time, and a cost function that penalizes the transformation rates is utilized. The optimal transformation is found as a constant-rate rotation in each of the principal planes relating the two elements. Although the kinematics of SO(N) are nonlinear and governed by Poisson's equation, the solution is found to be a linear function of the generalized principal angles. This is made possible by the extension of principal-rotation kinematics from three-dimensional rotations to the general SO(N) group. This extension relates the N-dimensional angular velocity to the derivatives of the principal angles. The cost of the optimal transformation, the square root of the sum of the principal angles squared, also provides a useful measure for the angular distance between two elements of SO(N). © 2007 Springer Science+Business Media B.V.
author list (cited authors)
Sinclair, A. J., Hurtado, J. E., & Junkins, J. L.