Simultaneous Orthogonal Rotations Angle

Angular orientation refers to the position of a rigid body intrinsic coordinate system relative to a reference coordinate system with the same origin. It is determined with a sequence of rotations needed to move the rigid-body coordinate-system axes initially aligned with the reference coordinate-system axes to their new position. In this paper we present a novel way for representing angular orientation. We define the Simultaneous Orthogonal Rotations Angle (SORA) vector with components equal to the angles of three simultaneous rotations around the coordinate-system axes. The problem of non-commutativity is here avoided. We numerically verify that SORA is equal to the rotation vector – the three simultaneous rotations it comprises are equivalent to a single rotation. The axis of this rotation coincides with the SORA vector while the rotation angle is equal to its magnitude. We further verify that if the coordinate systems are initially aligned, simultaneous rotations around the reference and rigid-body intrinsic axes represent the same angular orientation. Considering the SORA vector, angular orientation of a rigid body can be calculated in a single step thus avoiding the iterative infinitesimal rotation approximation computation. SORA can thus be a very convenient way for angular-orientation representation.