A Geometric Approach to Head/Eye Control

In this paper, we study control problems that can be directly applied to controlling the rotational motion of eye and head. We model eye and head as a sphere, or ellipsoid, rotating about its center, or about its south pole, where the axes of rotation are physiologically constrained, as was proposed originally by Listing and Donders. The Donders´ constraint is either derived from Fick gimbals or from observed rotation data of adult human head. The movement dynamics is derived on SO(3) or on a suitable submanifold of SO(3) after describing a Lagrangian. Using two forms of parametrization, the axis-angle and Tait-Bryan, the motion dynamics is described as an Euler-Lagrange´s equation, which is written together with an externally applied control torque. Using the control system, so obtained, we propose a class of optimal control problem that minimizes the norm of the applied external torque vector. Our control objective is to point the eye or head, toward a stationary point target, also called the regulation problem. The optimal control problem has also been analyzed by writing the dynamical system as a Newton-Euler´s equation using angular velocity as part of the state variables. In this approach, explicit parametrization of SO(3) is not required. Finally, in the appendix, we describe a recently introduced potential control problem to address the regulation problem.