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 S0(3) or on a suitable submanifold of 50(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 S0(3) is not required. Finally, in the appendix, we describe a recently introduced potential control problem to address the regulation problem.