Quaternion-Based Representation of Rotation Minimizing Motions in Euclidean 3-space

This paper presents a quaternion-based framework for constructing rotation-minimizing motions in Euclidean 3-space, formulated via quaternion operator. By introducing a novel quaternion operator, we derive angular velocity representations directly from the quaternion derivative and its conjugate, enabling smooth and minimal-rotation motion. The proposed approach generates rotation-minimizing motions whose trajectories are aligned with the orbits of a given spatial curve, and it offers a convenient mechanism to compute the corresponding quaternion representation when the orbit and a spatial position are specified. The effectiveness of the method is demonstrated through numerical experiments involving the spherical indicatricestangent, normal, and binormal-of space curves. Additionally, we provide a geometric characterization of quaternionic helical curves with respect to the tangential image T, highlighting the theoretical and practical implications of the proposed model in motion design and spatial kinematics.