Principal Rotation Representations of Proper NxN Orthogonal Matrices

Three and four parameter representations of 3x3 orthogonal matrices are extended to the general case of proper NxN orthogonal matrices. These developments generalize the classical Rodrigues parameter, the Euler parameters, and the recently introduced modified Rodrigues parameters to higher dimensions. The developments presented generalize and extend the classical result known as the Cayley transformation. Introduction It is well known in rigid body dynamics, and many other areas of Euclidean analysis, that the rotational coordinates associated with Euler’s Principal Rotation Theorem [1,2,3] leads to especially attractive descriptions of rotational motion. These parameterizations of proper orthogonal 3x3 matrices include the four-parameters set known widely as the Euler parameters, or the quaternion parameters [1,2,3], as well as the classical three-parameter set known as the Rodrigues parameters, or as the Gibbs vector [1,2,3,4]. Also included is a recent three parameter description known as the modified Rodrigues parameters [4,5,6]. As we review briefly below, these parameterizations are of fundamental significance in the geometry and kinematics of three-dimensional motion. Briefly, their advantages are as follows: Euler Parameters: This once redundant four-parameter description of three-dimensional rotational motion maps all possible motions into arcs on a four-dimensional unit sphere. This accomplishes a regularization and the representation is universally nonsingular. The kinematic differential equations contain no transcendental functions and are bi-linear without approximation. Classical Rodrigues Parameters: This three parameter set is proportional to Euler’s principal rotation vector. The magnitude is tan(φ/2), with φ being the principal rotation angle. These param-