Feedback Linearization of the Human Head Rotation Dynamics using Tait-Bryan Parameterization

Abstract

Using the well-known Tait-Bryan (TB) Parameterization, we revisit the human head pointing control problem 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 × 3 square system and recent results have shown that this system is feedback linearizable on a suitable neighborhood $\mathscr {N}$ of the state space. In this paper, we estimate a lower bound on the size of $\mathscr {N}$ by computing distance between the Donders’ and the associated Singularity surface, using the TB parameterization. The computed distance function is compared with a similarly computed distance function using Novelia and O’Reilly parametrization. Our surprising result, on the location of the singular points, suggests an advantage of using the TB parameterization for this class of feedback linearization.