To Tilt Your Head or Not To: Potentially

In this paper we study the human head movement, when the head shifts its orientation between two possible heading directions, as a simple mechanical control system. Head movements obey Donders’ constraint (as opposed to the Listing’s constraint for eye movement), which states that the allowed orientations of the head are obtained by rotating a fixed ‘primary heading direction’ by a subclass of rotation matrices. These rotation matrices have their axes of rotation restricted to a fixed surface, called the Donders’ surface. Donders’ Law states that when head moves spontaneously from left to right and back or from top to bottom and back, the head rotation matrix has no torsional component. On the other hand, when the head moves diagonally from the top left to the bottom right and back or from the top right to the bottom left and back, spontaneously, the axis of rotation has a small torsional component. The torsional component effectively rotates the head with respect to the frontal line of ‘heading’. The head appears slightly tilted as a result of the slight torsion. Defining a suitable Riemannian metric, we study dynamic control of head movement when the head orientations satisfy the Donders’ constraint throughout its entire trajectory. Head movements are actuated by choosing a suitable potential function and the oscillations are damped by adding a suitable damping term. An important result of this paper is to show the effect of the torsional component as head is allowed to move between two ‘headings’. We show that when the peak value of the allowable torsion is high, i.e. when the head is allowed to be more tilted, transition time between two headings is shortened.