In the mid-nineteenth century, Donders had proposed that for every human head rotating away from the primary pointing direction, the rotational vectors in the direction of the corresponding axes of rotation, is restricted to lie on a surface. Donders’ intuition was that under such a restriction, the head orientation would be a function of its pointing direction. In this paper, we revisit Donders’ Law and show that indeed the proposed intuition is true for a restricted class of head-orientations satisfying a class of quadratic Donders’ surfaces, if the head points to a suitable neighborhood of the frontal pointing direction. Moreover, on a suitably chosen subspace of the 3D rotation group SO(3), we describe a head movement dynamical system with input control signals that are the three external torques on the head provided by muscles. Three output signals are also suitably chosen as follows. Two of the output signals are coordinates of the frontal pointing direction. The third signal measures deviation of the state vector from the Donders’ surface. We claim that the square system is locally feedback linearizable on the subspace chosen, and the linear dynamics is decomposed into parts, transverse and tangential to the Donders’ surface. We demonstrate our approach by synthesizing a tracking and path-following controller. Additionally, for different choices of the Donders’ surface parameters, head gaits are visualized by simulating different movement patterns of the head-top vector, as the head-pointing vector rotates around a circle.
- Donders’ surface
- Head movement
- transverse feedback linearization