Human Head and Binocular Eye Movement Control via Feedback Linearization

Abstract

In this paper, eye and head movement control problems are studied as a constrained dynamics on SO(3) from the point of view of a nonlinear multi input multi output system. In the mid-nineteenth century, Donders had proposed that for every human head rotating away from the primary pointing direction, the rotational vectors are restricted to lie on a surface. The same is true for the human eye as well, under the head-fixed condition as proposed by Listing, wherein the Donders' surface would degenerate to a plane, called the Listing's plane. Additionally, when two eyes are gazing at a moving target, their gaze directions have to intersect at all time. Each eye and the head are controlled by a triplet of external torques provided by muscles. For the head (respectively Eye) movement problem, three output signals are chosen as follows. Two of the signals are coordinates of the frontal pointing direction. The third signal measures deviation of the state vector from the Donders' surface (respectively Listing's plane). For the binocular eye movement problem, inputs are the two triplets of external torques that control the two eyes. Output signals from one eye are identical to what we had in the monocular problem. Output signals for the other eye is modified as follows. One of the coordinate of the frontal pointing direction is replaced by a scalar that measures deviation of the gaze direction vectors to meet at a point. Thus we have a 3 × 3 (or a 6 × 6 for the binocular case) square system and the claim in this paper is that the proposed square system is locally feedback linearizable on a suitable neighborhood $\mathscr{N}$ of the state space.