Although quaternions are singularity-free in modeling and analysis of rigid bodies in three-dimensional motion, description of torques may lead to unbounded response of a quaternion-based model. This paper gives theorems on the conditions of torque-induced singularity in four coordinate systems: inertial frame, body frame, Euler basis, and dual Euler basis. According to the theorems, torques applied in an inertial frame or a body frame or a Euler basis will never cause unbounded motion; torques applied in a dual Euler basis, however, may lead to unbounded motion.

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