Application of potent potential functions in eye/head movement control

In this paper we study the human eye movement and the head movement system as a simple mechanical control system. Most of the time, eye movements obey Listing´s constraint, which states that the allowed orientations of the eye are obtained by rotating a fixed `primary gaze direction´ by a subclass of rotation matrices. These rotation matrices have their axes of rotation restricted to a fixed plane perpendicular to the primary gaze direction. Likewise, a spontaneous head movement satisfy Donder´s constraint which is similar to the Listing´s constraint except that the axes of rotation of the head away from a primary head position is restricted to a fixed surface, which is not necessarily a plane. When head is restrained to remain fixed, eye movement satisfies the Listing´s constraint throughout its entire trajectory. On the other hand, when the head is allowed to move (satisfying the Donder´s constraint), the eye satisfies the Listing´s constraint only at the beginning and at the end of a trajectory. Intermediate points of the trajectory, partially guided by the vestibulo-ocular reflex, do not apparently satisfy the Listing´s constraint. In this paper we introduce control strategy that would regulate the eye from an initial to a final gaze position with or without satisfaction of the Listing´s constraint during all the intermediate points of the eye movement trajectory. We study the head movement problem under the assumption that the head always satisfies the Donder´s constraint. The control signals are generated by choosing a suitable potential function and adding to it a suitable damping term. The overall dynamical system is constructed using the well known Euler Lagrange´s equation. The main result of this paper is to compare, using simulations, the total distance and the total time it takes to regulate the eye and the head between two orientations.