Bioinspired velocity control of fast gaze shifts on a robotic anthropomorphic head

Realistic 3D human saccades generated by a 6-DOF biomimetic robotic eye under optimal control

We recently developed a biomimetic robotic eye with six independent tendons, each controlled by their own rotatory motor, and with insertions on the eye ball that faithfully mimic the biomechanics of the human eye. We constructed an accurate physical computational model of this system, and learned to control its nonlinear dynamics by optimising a cost that penalised saccade inaccuracy, movement duration, and total energy expenditure of the motors. To speed up the calculations, the physical simulator was approximated by a recurrent neural network (NARX). We showed that the system can produce realistic eye movements that closely resemble human saccades in all directions: their nonlinear main-sequence dynamics (amplitude-peak eye velocity and duration relationships), cross-coupling of the horizontal and vertical movement components leading to approximately straight saccade trajectories, and the 3D kinematics that restrict 3D eye orientations to a plane (Listing's law). Interestingly, the control algorithm had organised the motors into appropriate agonist-antagonist muscle pairs, and the motor signals for the eye resembled the well-known pulse-step characteristics that have been reported for monkey motoneuronal activity. We here fully analyse the eye-movement properties produced by the computational model across the entire oculomotor range and the underlying control signals. We argue that our system may shed new light on the neural control signals and their couplings within the final neural pathways of the primate oculomotor system, and that an optimal control principle may account for a wide variety of oculomotor behaviours. The generated data are publicly available at https://data.ru.nl/collections/di/dcn/DSC_626870_0003_600.

Modelling 3D saccade generation by feedforward optimal control

An interesting problem for the human saccadic eye-movement system is how to deal with the degrees-of-freedom problem: the six extra-ocular muscles provide three rotational degrees of freedom, while only two are needed to point gaze at any direction. Measurements show that 3D eye orientations during head-fixed saccades in far-viewing conditions lie in Listing’s plane (LP), in which the eye’s cyclotorsion is zero (Listing’s law, LL). Moreover, while saccades are executed as single-axis rotations around a stable eye-angular velocity axis, they follow straight trajectories in LP. Another distinctive saccade property is their nonlinear main-sequence dynamics: the affine relationship between saccade size and movement duration, and the saturation of peak velocity with amplitude. To explain all these properties, we developed a computational model, based on a simplified and upscaled robotic prototype of an eye with 3 degrees of freedom, driven by three independent motor commands, coupled to three antagonistic elastic muscle pairs. As the robotic prototype was not intended to faithfully mimic the detailed biomechanics of the human eye, we did not impose specific prior mechanical constraints on the ocular plant that could, by themselves, generate Listing’s law and the main-sequence. Instead, our goal was to study how these properties can emerge from the application of optimal control principles to simplified eye models. We performed a numerical linearization of the nonlinear system dynamics around the origin using system identification techniques, and developed open-loop controllers for 3D saccade generation. Applying optimal control to the simulated model, could reproduce both Listing’s law and and the main-sequence. We verified the contribution of different terms in the cost optimization functional to realistic 3D saccade behavior, and identified four essential terms: total energy expenditure by the motors, movement duration, gaze accuracy, and the total static force exerted by the muscles during fixation. Our findings suggest that Listing’s law, as well as the saccade dynamics and their trajectories, may all emerge from the same common mechanism that aims to optimize speed-accuracy trade-off for saccades, while minimizing the total muscle force during eccentric fixation.

Saccades are rapid eye movements that humans and other animals perform three to four times per second to scan and perceive the environment around them. These movements orient the eye in space with high precision and in a highly stereotyped fashion. Existing studies on animal models advocate that both mechanical and neuronal functions play an important role in the control of the saccades, but some facts are still not fully understood due to difficulties in experimenting and measuring the variables in living animals. Instead, robots are computational and physical models of reality that expose all its variables and can be programmed in interpretable ways. We have built a robotic model of an artificial eye containing the basic ingredients of human eyes: full 3D rotations, viscous friction and 6 muscle-like actuators connected to the eyeball in a geometry similar to the biological system. By synthesizing robotic eye control systems we found that important characteristics of the movements become similar to human saccades when the control relies on few simple fundamental principles: the maximization of saccade accuracy and the minimization of saccade duration, energy in control, and force in the muscles during fixation.