Optimal control and tracking with eye movement dynamics with and without the Listing´s constraint

In this paper we study the human oculomotor system as a simple mechanical control system. Most of the time, such as during `smooth pursuit´, eye movements obey Listing´s constraint, which states that the movements consist of rotation matrices for which the axes are orthogonal to the normal gaze direction. Eye movements fail to satisfy the Listing´s constraint either as a result of abnormality or when the gaze direction is sufficiently oblique, as would typically be the case when the angle of rotation is large. Listing´s constraint is also not satisfied during vestibulo-ocular reflex when the eye moves to compensate for the head movement. The purpose of this paper is to formulate the eye movement as an optimal control problem with and without the Listing´s constraint in order to subsequently make comparisons between the two scenarios. In order to do this, we parameterize the space in such a way that the Listing´s constraint is recovered as a special case. The optimal control problem is posed as a two point boundary value problem.