Semiglobal Feedback Linearization of the Human Head and Binocular Eye Tracking Dynamics Satisfying Donders’ Law

Abstract

In this paper, we revisit the head and binocular eye pointing control problems as a constrained dynamics on $SO(3)$ from the point of view of a nonlinear multi input multi output (MIMO) system. The constraint, proposed by Donders, is that for every human head rotating away from its primary pointing direction, the rotational vectors are restricted to lie on a surface called the Donders’ surface. This paper assumes that the Donders’ surface is described by a quadratic equation on the coordinates of the rotation vector. The inputs to the MIMO system are three external torques provided by muscles rotating the head. The 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. Thus we have a $3\times 3$ square system and recent results have shown that this system is feedback linearizable on a suitable neighborhood $\mathcal{N}$ of the state space. In this paper, we estimate a lower bound on the size of $\mathcal{N}$ by computing a lower bound on the distance between the Donders’ and the Singularity surface. For the eye pointing control, the Donders’ surface is replaced by Listing’s plane whereas the two eyes are constrained by a coplanarity condition. The binocular eye pointing control system is described as a MIMO system on SO $=SO(3)\times SO(3)$ with six inputs and six outputs. Semiglobal feedback linearization, on a suitable neighborhood $\mathcal {B}$ of SO is proposed and an analogous lower bound on this neighborhood is obtained. Binocular tracking control using semiglobal feedback linearization, described in this paper, is new.