Abstract
In this paper, we consider eyes from the human binocular system, that simultaneously gaze on stationary point targets in space, while optimally skipping from one target to the next, by rotating their individual gaze directions. The head is assumed fixed on the torso and the rotating gaze directions of the two eyes are assumed restricted to pass through a point in the visual space. It is further assumed that, individually the rotations of the two eyes satisfy the well known Listing’s law. We formulate and study a combined optimal gaze rotation for the two eyes, by constructing a single Riemannian metric, on the associated parameter space. The goal is to optimally rotate so that the convergent gaze changes between two pre-specified target points in a finite time interval [0, 1]. The cost function we choose is the total energy, measured by the L^2 norm, of the six external torques on the binocular system. The torque functions are synthesized by solving an associated ‘two-point boundary value problem’. The paper demonstrates, via simulation, the shape of the optimal gaze trajectory of the focused point of the binocular system. The Euclidean distance between the initial and the final point is compared to the arc-length of the optimal trajectory. The consumed energy, is computed for different eye movement chores and discussed in the paper. Via simulation we observe that certain eye movement maneuvers are energy efficient and demonstrate that the optimal external torque is a linear function in time. We also explore and conclude that splitting an arbitrary optimal eye movement into optimal vergence and version components is not energy efficient although this is how the human oculomotor control seems to operate. Optimal gaze trajectories and optimal external torque functions reported in this paper is new.
Highlights
The class of problem that we consider in this paper is how the human eyes switch their focused points between two point targets in the visual space
The eyes rotate with three degrees of freedom [8, 9], rendering the eye movement system, a relatively simple mechanical control system compared to other complex human movement systems [10]
The cause for this property is due to the symmetry between the two eyes and the initial and final target positions
Summary
The class of problem that we consider in this paper is how the human eyes switch their focused points between two point targets in the visual space. The main contribution of this paper is to extend the Riemannian geometric formulation to binocular control problems. As described in our earlier papers [21], to every gaze direction of the eye there corresponds a circle of rotation matrices. This ambiguity is resolved by imposing a Listing’s Law on each of the two eyes. We assume that the gaze directions of the two eyes always meet at a point in the visual space R3 , i.e., the eyes are always focused (see Fig. 1). The Donders’ constraint during the specific eye movements have been studied in [32] and we would
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