Minimum energy optimal external torque control of human binocular vision
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RESEARCH ARTICLE
Minimum energy optimal external torque control of human binocular vision Bijoy K. Ghosh1,2 · Bhagya Athukorallage1 Received: 13 August 2020 / Revised: 10 September 2020 / Accepted: 11 September 2020 / Published online: 23 November 2020 © The Author(s) 2020
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 L2 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. Keywords Binocular vision · Riemannian geometry · Optimal control · Human eye movement
1 Introduction 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. Typically we assume that the eyes are on a stationary head, and rotate to inspect point targets that are located in 3D. The gaze directions are always constrained to pass through a point and the goal of the sensing mechanism is to initially start with a target fixed in its * Bijoy K. Ghosh [email protected] Bhagya Athukorallage [email protected] 1
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409‑1042, USA
School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
2
view and to switch to an alternate target in the visual field in a fixed interval of time, assumed to be [0, 1]. The pa
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