Riemannian geometric approach to optimal binocular rotation, pyramid based interpolation and bio-mimetic pan-tilt movement

Over the past several years, we have been studying the problem of optimally rotating a rigid sphere about its center, where the rotation is actuated by a triplet of external torques acting on the body. The control objective is to repeatedly direct a suitable radial vector, called the gaze vector, towards a stationary point target in IR³. The orientation of the sphere is constrained to lie in a suitable submanifold of SO(3). Historically, the constrained rotational movements were studied by physiologists in the nineteenth century, interested in eye and head movements. In this paper we revisit the gaze control problem, where two visual sensors, are tasked to simultaneously stare at a point target in the visual space. The target position changes discretely and the problem we consider is how to reorient the gaze directions of the sensors, along the optimal pathway of the human eyes, to the new location of the target. This is done by first solving an optimal control problem on the human binocular system. Next, we use these optimal control and show that a pan-tilt system can be controlled to follow the gaze trajectory of the human eye requiring a nonlinear static feedback of the pan and tilt angles and their derivatives. Our problem formulation uses a new Riemannian geometric description of the orientation space. The paper also introduces a new, pyramid based interpolation method, to implement the optimal controller.