Minimum distance and optimal transformations on SO(N) |
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Authors: | Andrew J Sinclair John E Hurtado John L Junkins |
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Institution: | (1) Department of Aerospace Engineering, Auburn University, 211 Davis Hall, Auburn, AL 36849-5338, USA;(2) Department of Aerospace Engineering, Texas A&M University, TAMU-3141, College Station, TX 77843-3141, USA |
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Abstract: | 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). |
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Keywords: | Kinematics Principal planes Special orthogonal matrices Trajectory optimization |
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