Dynamics of Human Head and Eye Rotations Under Donders´ Constraint

The rotation of human head and the eye are modeled as a perfect sphere with the rotation actuated by external torques. For the head movement, the axis of rotation is constrained by a law proposed in the 19th century by Donders. For the saccadic eye movement, Donders´ Law is restricted to a law that goes by the name of Listing´s Law. In this paper, head movement and saccadic eye movement are modeled using principles from classical mechanics and the associated Euler Lagrange´s equations (EL) are analyzed. Geodesic curves are obtained in the space of allowed orientations for the head and the eye and projections of these curves on the space S² of pointing directions of the eye/head are shown. A potential function and a damping term has been added to the geodesic dynamics from EL and the resulting head and eye trajectories settle down smoothly towards the unique point of minimum potential. The minimum point can be altered to regulate the end point of the trajectories (potential control). Throughout the paper, the restricted dynamics of the eye and the head movement have been compared with the unrestricted rotational dynamics on SO(3) and the corresponding EL equations have been analyzed. A version of the Donders´ Theorem, on the possible head orientations for a specific head direction, has been stated and proved in Appendix I. In the case of eye movement, Donders´ Theorem restricts to the well known Listing´s Theorem. In Appendix II, a constraint on the angular velocity and the angular acceleration vectors is derived for the head movement satisfying Donders´ constraint. A statement of this constraint that goes by the name “half angle rule,” has been derived.