Kinematics of N-dimensional principal rotations
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Whereas much work has been done to describe N-dimensional orientations in terms of principal rotations, these developments have not been extended to the descriptions of N-dimensional kinematics. For three-dimensional rotations the angular velocity can be related to derivatives of the principal angle and axis. Following a similar procedure this paper develops the kinematics for N-dimensional rotations in terms of the derivatives of the principal planes and angles. It is shown that the components of the angular velocity lying in the principal planes are related to the derivative of the principal angles and the out-of-planc components are related to the derivative of the vectors spanning the principal planes. Additionally, the kinematic optimal control problem for N-dimensional reorientation is solved while minimizing a quadratic function of the angular velocity. The optimal angular velocity is found to be a constant rate rotation in each of the principal planes relating the initial and final orientations, where optimal refers to minimizing the rotational displacement in a defined fashion.