Bertrand Tondu
email: tondu@insa-toulouse.fr

Mario Piedrahita-Bello

Lionel Salmon

Gábor Molnár

Azzedine Bousseksou

Robust Linear Control of a Bending Molecular Artificial Muscle Based on Spin Crossover Molecules

Keywords: Artificial muscle, Spin crossover, Bending actuator, Open-loop identification, PID control

Bending actuators represent a popular class of artificial muscles for which efficient closedloop control is a remarkable challenge, due to the complexity of physico-chemical phenomena occurring during dynamic contraction. In this study, we investigate an electro-thermally actuated, bilayer bending actuator based on spin crossover molecules. While these artificial muscles can openloop contract in several tens of seconds, we show that a simple closed-loop PID-control is able to reduce the response time to ca.1.5 s, even with loads up to 5 times the actuator weight (i.e. a maximal load of about 343 mg). We demonstrate also satisfactory sine wave tracking performance. The relevance of this linear control approach applied to a nonlinear actuator is the consequence of the high sensibility of the actuator to the current, responsible for the Joule effect. Notably, the actuator roughly behaves like a second-order linear system, whose time "constants" decrease with current.

Consequently, the natural speed improvement peculiar to the PID-controller is further amplified by this nonlinear effect, without any loss in stability of the closed-loop system.

Introduction

Multilayer bending actuators are a class of actuating devices, whose bending movement results from a difference of strain between the constitutive layers. They can be controlled by various external stimuli, but electrical current or tension are the most relevant due to their ease of use. Such actuator devices can be considered as artificial muscles if, by analogy with skeletal muscle, their change of dimension can be controlled in open-loop [START_REF] Tondu | What is an artificial muscle? A systemic approach[END_REF]. In particular, the development of Ionic Polymer-Metal Composite (IPMC) artificial muscles opened the way towards practical applications especially in biomedical devices [START_REF] Jager | Biomedical applications of polypyrrole microactuators: from single-cell clinic to microrobots[END_REF]- [START_REF] Herrlich | Sphincter-like micro actuators based on electroactive polymer[END_REF] and micro-robotics [START_REF] Smela | Electrochemically driven polypyrrole bilayers for moving and positioning bulk micromachined silicon plates[END_REF]- [START_REF] Alici | Performance quantification of conducting polymer actuators for real applications: A microgripping system[END_REF]. However, whatever the chemical composition of multilayer bending actuators, their control remains a considerable problem due to the complex nature of physical phenomena occurring during the bending movements of their multilayer structure.

Attempts have been made for proposing physical models of the multilayer bending of conductive polymers [START_REF] Du | A multilayer bending model for conducting polymer actuators[END_REF]- [START_REF] Annabestani | Nonlinear identification of IPMC actuators based on ANFIS-NARX paradigm[END_REF]. Such models are particularly interesting for understanding the bending phenomenon, but their direct use for control is not obvious. Moreover, although open-loop control is feasible, closed-loop control is essential for using such artificial muscle as an actuator: trajectory to be tracked can be directly specified, time-response of the actuator can be quicker, and physical phenomena peculiar to the materials used can be mastered, such as the so-called "drift"-phenomenon, which is a slow convergence towards the steady-state position [START_REF] Yao | Feedback control of tri-Layer polymer actuators to improve their positioning ability and speed of response[END_REF]. In this context, open-loop positioning stability offers the advantage of making possible an open-loop identification of the dynamic contraction in order to use the identified model as a feedforward element of the closed-loop controller [START_REF] John | Inversion-based feedforward control of polypyrrole trilayer bender actuators[END_REF], [START_REF] Zolfagharian | Bending control of a 3D printed polyelectrolyte soft actuator with uncertain model[END_REF]. The nonlinear nature of multilayer bending actuators often calls for sophisticated nonlinear approaches such as sliding mode control [START_REF] Wang | Adaptive sliding mode control of tri-layer conjugated polymer actuators[END_REF], adaptive control [START_REF] Fang | Robust adaptive control of conjugated polymer actuators[END_REF], [START_REF] Zakeri | Digital control for an IPMC actuator using adaptive optimal proportional integral plus method: Simulation and experimental study[END_REF], self-tuning control [START_REF] Mcdaid | Gain scheduled control of IPMC actuator with 'model-free' iterative feedback tuning[END_REF], [START_REF] Deng | Robust non-linear control design to an ionic polymer metal composite with hysteresis using operator-based approach[END_REF], fuzzy or other intelligent-like control [START_REF] Druitt | Intelligent control of electroactive polymer actuators based on fuzzy and neurofuzzy methodologies[END_REF]. These sophisticated control approaches have for disadvantage their low efficiency. They can be very accurate, but the complex model has to be re-verified if a new sample is used and a high number of parameters (or rules) must be tuned.

When the actuator is considered alone, this long process of control development can be admissible.

It is, however, less obvious if we consider applications involving several actuators acting together as it is the case in robotics. For this reason, although we have a long experience in nonlinear control of pneumatic artificial muscles [START_REF] Tondu | Modeling and control of McKibben artificial muscle robot actuators[END_REF], [START_REF] Ebadzadeh | Computation of inverse functions in a model of cerebellar and reflex pathways allows to control a mobile mechanical segment[END_REF], we also tried to promote simple ways for controlling artificial muscles. Linear PID (proportional-integral-derivative) control is particularly easy to put into work, whatever the linear or nonlinear character of the system to be controlled. This approach was naturally applied to bending actuators with parameters chosen using customary approaches, such as the Ziegler-Nichols rules [START_REF] Richardson | Control of ionic polymer metal composites[END_REF]. Here we discuss an alternative, phenomenological approach of the PID-control applied to artificial muscles: an open-loop linear identification stage is used for highlighting peculiar properties of the dynamic contraction of the artificial muscle, from which simple, linear closed-loop control schemes can be derived. For example, in the case of pneumatic McKibben muscles, we recently proposed an unusual linear I-controller based on the fact that any artificial muscle has its own stiffness and damping, which can respectively play the role of the P and D-actions of the PID [START_REF] Tondu | Robust and accurate closed-loop control of McKibben artificial muscle contraction with a single integral action[END_REF].

In the present work, we propose to put into work a similar phenomenological control approach in the case of a promising, new bilayer bending actuator, based on spin crossover (SCO) particles [START_REF] Molnár | Spin Crossover Nanomaterials: From Fundamental Concepts to Devices[END_REF], which are loaded into a P(VDF-TrFE), poly(vinylidene fluoride-trifluoroethylene) polymer matrix.

To form a bilayer, the SCO@P(VDF-TrFE) layer is topped with a Ag@P(VDF-TrFE) (Ag = silver flakes) conductive layer [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF]. The actuating strain arises from the significant molecular volume change that accompanies the electro-thermally induced spin crossover phenomenon in these materials [START_REF] Shepherd | Molecular actuators driven by cooperative spin-state switching[END_REF]. This bending actuator has two key characteristics. First, it is fabricated as a monolithic object with a common polymer matrix backbone for the two layers, with consequently no delamination process occurring during the bending phenomenon. Second, the resulting monolithic material can be easily cut for different shapes and sizes [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF]. In our previous work, several samples were developed by modifying the composition and microstructure of the SCO@P(VDF-TrFE) active layer -allowing us to adjust the actuator properties (strain, transition temperature, response time, etc.) [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF]. Despite their differences, we have identified several common properties of these samples: They display a high sensibility to the control current, their rising time is rather long (several tens of seconds) and exhibit a pure delay before the movement begins. The latter phenomenon is a direct consequence of the time required to reach the spin transition temperature (see [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF]). From the consideration of these properties, it is clear that one must privilege a low transition temperature to reduce both the initial pure delay and the rising time. For this reason, in the present work, we used a device incorporating 25 wt% of the sample [Fe(4-pentyl-1,2,4-triazole)3](tosylate)2 (actuator 1) with spin transition occurring at 67 °C and 42 °C on heating and cooling, respectively. Although the pure delay was reduced in this sample to about 0.5 s, the rising time still largely exceeds 10 s. As a consequence, the 95% response time -the time beyond which the response keeps between 0.95 and 1.05 times the steady-state position -varies between ~25 s (1.2 A-step) and ~40 s (0.9 A-step). From this rough analysis, one might wonder whether our approach is relevant in terms of coming up with a new technology of bending actuation.

In the present paper, therefore we will extend and deepen the preliminary analysis of the control of 1 conducted in ref. [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF] so as to grasp how a simple PID closed-loop control approach can enable this actuator to contract in closed-loop in less than 2 seconds. The originality of our approach consists in assuming that a simple linear PID-control can derive benefit from certain nonlinear properties of the artificial muscle for largely improving its contraction performances. After a first open-loop identification step proposed in Section 3.1, we will justify the choice of a PID-control in Section 3.2, and finally we will emphasize the robustness of the considered actuator when carrying loads and highlight its performance during sine-wave tracking in Sections 3.3 and 3.4, respectively.

Experimental

The bilayer actuators were fabricated as described in ref. [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF]. They were cut in a 'U-shape' and clamped at one end (see Fig. 1). The moving part of the actuator has a lateral dimension of ca. 3 × 1 cm 2 , with a 0.2 cm gap in the middle to create a conductive circuit. The thickness of the bilayer is roughly 150 µm and the weight of the moving part is ca. 70 mg. We note that this choice of shape makes easy to embed a glued load. A computer-controlled current source (Keithley 2420) is used for heating the actuator via Joule effect, whereas the resulting bending movement is controlled by means of a laser triangulation position sensor (Micro Epsilon OptoNCDT 1750), pointing on the tip of the actuator. Open-and closed-loop control is achieved by means of the LabView-software (see Fig. 1). In its actual form, we have not tried to include a force sensor in the experimental set-up. This is especially difficult to do in the case of a bending actuator; this is why we preferred to estimate the force ability of the actuator by considering known embedded masses at its tip.

Results and discussion

Linear identification of the artificial muscle

The stable open-loop positioning character of the artificial muscle points the way towards open-loop identification, especially from recording its positioning in response to various step-wise excitations.

The identification problem can be tackled using various methods such as the ANFIS-NARX paradigm, recently applied to IPMC (Ionic Polymer Metal Composite) [START_REF] Annabestani | Nonlinear identification of IPMC actuators based on ANFIS-NARX paradigm[END_REF] to achieve an advanced non-linear model of the full bending actuator. In a more conventional way, classical techniques of linear system identification applied to the time variation of the sample can be used. In ref. [START_REF] Yao | Feedback control of tri-Layer polymer actuators to improve their positioning ability and speed of response[END_REF], the authors propose a sixth-order transfer function with a constant gain to identify their polypyrrole trilayer actuator. Our approach differs in the sense that it is based on a preliminary observation of the open-loop step-response of the bending actuator. From Figs. 2a and2b, it appears that the initial slope is close to zero and, whatever the control current, no large oscillation can be observed during the rising phase towards the steady-state. We remark also that no overshoot of the steady-state value occurs. However, a pure delay exists whatever the current value. Since it appears to be limited to less than 0.5 second, it will be neglected in our linear identification. As a consequence, a simple overdamped second-order model with two real poles will be considered according to the following transfer function:

( )/ ( ) = /(1 + )(1 +
), where xM denotes the tip position of the bending artificial muscle, I the control current, K the gain of the system, T1 and T2, the time "constants" of the system, supposed to be overdamped, and s designates the variable of Laplace formalism.

In our open-loop identification experiments, the control current value is varied in a limited range between 0.7 and 1.

( / )) ( ) ( ( 0 t x t x t x Mean L L t t horizon - ≤ ≤
, where thorizon is a 'time horizon' we have taken equal to 70 s. Similar accuracy was obtained in the whole positioning range (0.11 mm for 0.7 A, 0.07 mm for 0.8 A, 0.08 mm for 0.9 A, 0.06 mm for 1 A, 0.08 mm for 1.1 A and 0.10 mm for 1.2 A), which suggests a satisfactorily identification. We must underline the absence of 'drift', which often makes the convergence to the steady-state slower than expected for a linear system. This phenomenon is frequently observed in multilayer bending actuation devices especially for the higher step control values [START_REF] Yao | Feedback control of tri-Layer polymer actuators to improve their positioning ability and speed of response[END_REF]. The fact that T1, T2 and the gain K are changing with control current emphasizes the nonlinear character of the actuator.

However, two remarks can be made:

-The damping factor z is almost constant, approximatively equal to 1. In open-loop, the actuator behaves like a critically damped linear system, which means that the system returns to the equilibrium as fast as possible without overshooting. This naturally optimal damped character of the actuator should make the tuning of a PID closed-loop control easier;

-The ωn-parameter appears to increase with rising current, from the 0.9 A-step value. This means that, above this threshold-value, the higher is the control current, the faster is the bending. This is another very positive point for a linear closed-loop control: when the actuator will be far away from its goal, the effect of the PID will be amplified by the natural actuator behavior, and this effect will be moderated at the goal's approach. Moreover, the fact that the gain K appears to increase slowly with increasing current intensity also participates in this amplification effect of the PID controller. In the next section, we show how we can derive benefit of this identification for a simple, but relevant, linear PID-control of this nonlinear system.

PID control

The pseudo second-order dynamic behavior of the considered actuator, with its two real poles, suggests a simple closed-loop control by means of a PI-controller, which we can write as:

( ) = [ ( ) + ( ) ( ) ] (1)
where I is now the closed-loop control current, ε(t) is the desired position minus the real position, and

, are the corresponding proportional and integral parameters of the PI. The corresponding transfer

function of the open-loop system can be written as:

( ) / ( ) = (1 + )/[(1 + )(1 + ) ] (2)
It is clear that the choice = ( , ) leads to eliminate the dominant pole of the system to be controlled, which is apparently always in our case. By doing = in Equ. ( 2), the following function transfer of the closed-loop system results -where ! is the desired position:

! ( )/ ( ) = /[ + (1 + ) ] (3)
It is then possible to choose the G-gain by means of the following relationship:

= " #$ % & % ( 4
)
where z is the damping factor of the closed-loop system. Typically, z = 0.7 realizes the best compromise between low rising time and overshooting. Fig. 3a shows the simulation of a closed-loop step-response for a target position of 5 mm, which is compared to the experimentally measured openloop response with a similar target position (5.2 mm). We can observe here the classical speed improvement obtained by the PI-controller: the response time is diminished from ~31 s to ~14 s.

Let us consider now a PID written in the form:

( ) = [ ( ) + ! + (t) + ( ) ( ) ] (5)
where ! corresponds to the derivative parameter, in complement with the already defined gain and integral parameters, applied to the real sample. As shown in Fig. 3a, a simple manual tuning of parameters has led to an experimental step-response with a response time of about 1.6 s, which is an order of magnitude smaller than the value predicted by the simulated PI-controller. To explain this huge difference, we shall examine the electrical energy transmitted to the actuator during its contraction in the case of the real open-loop response and the real closed-loop response (Fig. 3b).

This energy (Joule heating) has the form RI 2 , where I is the control current and R is the electrical resistance of the actuator, measured by a four-wire measurement setup, simultaneously with the actuation of the device. From these measurements, it appears clearly that the real PID-control can boost the system response by increasing the current intensity to reach, temporally, its maximum 3 Avalue, which is significantly higher than the one required for maintaining the sample in its steadystate. This suggests that when the current intensity is high, but limited by the 3 A-bound, the actuator response accelerates. In other words, its equivalent natural frequency, which was shown to increase between 0.9 A and 1.2 A (Fig. 2b), must continue to increase beyond 1.2 A. To justify this assumption, we developed an original identification of the actuator system from the knowledge of its real closedloop step response. While the second-order open-loop identification suggested the use of a PI-controller for the closedloop control, it appeared in practice that a PID with a non-zero value for the derivative term gives a better result than a pure PI-controller with, however, a relatively low value for the 'D'-term. Fig. 4a displays the real closed-loop step responses obtained for four target positions (2, 3, 4 and 5 mm). The three parameters of the PID were manually tuned in order to get the best achievable 95% response time. For each target position, the optimized PID values ( , ! , ) are very similar: (2.5, 0.1, 0.7),

(2.2, 0.15, 0.4), (1.8, 0.1, 0.5) and (1.8, 0.1, 0.4) for ! = 2, 3, 4 and 5 mm, respectively. We can notice that the value ' ! ', representing the ratio between the derivative component ' ! ' and the integral component ' / ' varies between 0.04 and 0.07 (s 2 ), indicating much greater influence of the integral component over the derivative component. In practice, a first real PI-controller was tuned in order to get the best response time of the closed-loop system, before adding a slight derivative component in order to reduce the observed oscillating behavior of the actuator during its rising movement. For reasons of clarity, the presentation was limited to the sole best PID results. In these conditions, it appears that the overshooting can be kept under control in such a way that the step-response does not leave the range [0.95 ! , 1.05 ! ] as soon as it is inside. On the other hand, the 95% response times are very similar, with values between 1.3 and 1.6 s. These two findings suggest that, despite the pure delay, which now cannot be neglected, the whole closed-loop system globally behaves like a linear system. Consequently, since the PID is a linear controller, the actuator system alone behaves as a linear system, but with a much faster response than the open-loop system. This is this "equivalent" linear system of the actuator, brought about by the PID-controller, which we decided to identify. For doing that, we consider the following transfer function:

/( ) = 0 (1 -s)/(1 + s)(1 + 0 )(1 + 0 ) (6)
This model now considers four parameters: 0 , 0 , 0 , . The first three parameters have the same meaning that the initial parameters K, , of the previously considered linear model of the actuator.

On the other hand, the τ-parameter is supposed to take into account the pure delay of the "equivalent" actuator in the form of the sub-system, inspired by the first-order Padé-approximation [START_REF] Bhattacharyya | Linear Control Theory: Structure, Robustness, and Optimization[END_REF]:

(1 -s)/(1 + s). It is however important to note that the τ-parameter cannot be considered as an estimation of the pure delay of the global system due to the too close values for , 0 and 0 . This linear identification was deduced from the comparison between the simulated step-response of the transfer function /( )6 7( )/(1 + /( )6 7( )), where PID(s) is the transfer function of the PID, whose parameters are the real parameters considered for each desired position. The resulting simulated PID-control of the identified, "equivalent" actuator linear model is drawn in dashed lines in Fig. 4a and compared to the real PID-response. Fig. 4b shows the corresponding model parameters compared to the parameters of the original model of the actuator that we used for open-loop identification. Two main points result from this comparison. First of all, the time "constants" ( 0 , 0 ) of the equivalent actuator linear model are more than one order of magnitude lower than the initial ones ( , ). Furthermore, the parameters 0 , 0 , 0 do not significantly differ for different target positions (between 2 -5 mm). These findings lead us to the surprising conclusion that the PID control is able to derive benefit from the nonlinear character of the actuator for considerably improving its response time. When going from the open-loop to the closed-loop, this response time can be diminished by a factor of 10. This phenomenon seems to be the consequence of the improvement in the responsiveness of the actuator itself when the control current increases.

Remarkably, to some extent, the PID gives rise to the linearization of the actuator, denoted by the fact that the response time becomes roughly independent of the desired position, albeit in a limited range between 2 -5 mm. It is however to be noted that the about 0.5 s muscle delay still exists. Most artificial muscles exhibit a pure delay, which can be reduced by the use of nonlinear control methods [START_REF] Braikia | Improved control strategy of 2-sliding controls applied to a flexible robot arm[END_REF], but at the price of greater complexity in the number and the tuning of controller parameters. In the case of this study, we voluntarily limit our closed-loop controller to a sole linear PID. Neglecting the muscle delay in the closed-loop controller is also motivated by the fact that, in the case of an actuation device, tracking performances are often more important than step response performances, and that initial delay has less effect in tracking performances than in step response performances. Crucially, the tuning of the PID for a rapid step-response does not deteriorate the stability. A simple way for evaluating the stability and the robustness in the case of our closed-loop system can consist in determining the corresponding couple (phase margin, gain margin). We propose to derive these two parameters from the knowledge of the "equivalent" actuator linear model. We obtained, respectively, for the target positions 2, 3, 4 and 5 mm the following couples: (55°, 7 dB), (57°, 10 dB), (58°, 9 dB) and (58°, 10 dB). One must note that a satisfactory phase margin is supposed to be greater than 45° and the gain margin is supposed to be equal to ca. 10 dB [START_REF] Bhattacharyya | Linear Control Theory: Structure, Robustness, and Optimization[END_REF]. In particular, the phase margin of 55°-58° in the case of the equivalent actuator model, largely beyond the usual "comfort value" of 45°, suggests a robustness of the proposed control. In the following, this point will be verified for movements with embedded loads.

Performance with embedded loads

From a practical point of view, any actuator must be able to embed a load [START_REF] John | Inversion-based feedforward control of polypyrrole trilayer bender actuators[END_REF], [START_REF] Hashimoto | Predictive control of piezoelectric actuators with friction drive mechanism[END_REF]. Yet, most studies dealing with the control of multilayer bending actuators do not report experiments with loads. In the case of a bending actuator, the load can be placed at the tip of the cantilever, as shown in Fig. 1d. It is important to note that the ability of lifting a load is made possible by the relative stiffness of the proposed actuator. We have analyzed this ability both in step response and trajectory tracking. The relevance of our control approach is the most evident when we consider the same step responses with embedded load by keeping, in the full range of loads, the same values for the PID-parameters than those tuned for the controller without any load (see Fig. 5). As it can be seen, in all reported experimental step responses, the maximum value of 3 A for the control current is used during a short time, leading to heating of the sample at about 120 °C. Under open-loop conditions, it is impossible to use the system to perform rapid motion as overheating occurs very quickly above 140 °C, and leads to melting of the material after some seconds. For this reason, as already mentioned, currents above 1.3 A are never used in open-loop conditions, but in closed-loop, it is possible to use such temporary overheating. Moreover, it is important to note that current peaks essentially occur in the case of a step response while they are not present during tracking responses, as considered further, due the continuous character of the trajectory to be tracked. We have found the actuator to be able to operate continuously over 2 weeks and 35000 thermal cycles with no observed change in the actuators performance [START_REF] Piedrahita-Bello | Colossal expansion and fast motion in spin-crossover@polymer actuators[END_REF] as the PID controller effectively prevents the material from going above 120 °C while still allowing access to the fast motion granted by brief use of 3A currents.

Sine wave tracking performance

The closed-loop step experiments discussed above are favored by the high sensibility of the actuation device to react quickly to temperature changes when it has reached the spin transition temperature [START_REF] Molnár | Spin Crossover Nanomaterials: From Fundamental Concepts to Devices[END_REF]. However, a reverse movement requires cooling, which takes more time. In order to determine the practical consequences of this non-symmetrical behavior between warming and cooling, peculiar to all Joule-effect actuation systems, we have analyzed the dynamic behavior of our artificial muscle in response to multiple sine waves. Fig. 6 shows selected sine wave tracking experiments. Two cases have been considered. First, a sequence of three sine waves with successive 30s-30s-20s periods and, respectively, 2-4 mm, 3-5 mm and 2-5 mm target position changes was programmed. These periods are sufficiently long to make possible the cooling of the sample during the reverse movement imposed by the sine wave. It is important to note that, in all reported tracking experiments, the same set of PID-parameters ( = 1.8, ! = 0.1, = 0.5) was used. Through this choice of a constant set of PIDparameters, we aimed for analyzing the possibility of an efficient trajectory tracking with a pure linear control, whatever the frequencies and amplitudes of considered sine waves. As it can be seen, the tracking error, defined as the difference between the target and real position, remains between -0.2 and +0.15 mm when no load is embedded, whereas this range slightly increases [-0.25 mm, +0.2 mm] for a 204 mg-load. One shall note that the actuator reacts very well when the direction of the sine wave is changing, which is remarkable in comparison with other types of artificial muscles exhibiting frictional-like phenomena or other adverse effects when the actuator has to move back.

This phenomenon is consistent with the symmetrical behavior already noticed for ascending and descending steps (Fig. 2.a). Moreover, during tracking, control current, not shown in Fig. 6 for reasons of clarity, stays at a value close to 1A, which is still in accordance with closed-loop step results shown in Fig. 5, where the steady-state current is inside a [0.8 A-1.2 A]-range for all considered target positions.

In a second type of experiment, we looked for the limits of the actuator when it is submitted to a higher frequency. We programmed a sequence of three sine waves, with the same displacements as in the previous experiment, but the periods are now reduced to 20 and 15 s. As it can be seen in Fig. 6, the descending phase in the 15s-period, which requires the fastest cooling of the sample, generates a large deviation from the desired sine wave with a maximum tracking error of ~0.55 mm. We believe this last result stems from the characteristics of the actuation device itself rather than from the limitations of our control approach.

Conclusions

We investigated the PID control of a new type of bending actuator, based on spin crossover molecules.

The justification of using a so common linear controller for a highly nonlinear actuator is based on a two-step process. We started by an open-loop identification, which led us to propose a second-order linear model, whose parameters were shown to vary with the control current. This open-loop identification step suggests that the time "constants" of the second-order model decrease monotonously from a threshold value, leading to an increasing actuator responsiveness when the control current increases. This natural decrease of the actuator response time with increasing current intensity was effectively demonstrated in closed-loop PID control experiments. To verify this amplification effect, in a second step, we identified an "equivalent actuator" linear model derived from the "best-tuned" PID. Time "constants" of this equivalent linear model appear diminished by a factor of ~10 in comparison with the original actuator linear model. Overall, the proposed PID-control is able to generate, in a current range of 0 -3 A, step responses of the actuator tip with displacements in the range of 2 -5 mm associated with a 95% response time between 1.3 and 1.6 s. Remarkably, the latter remains virtually unchanged when loads are embedded (up to 350 mg, which corresponds to ca. 5 times the own weight of the actuator). The sine wave tracking performance of the actuator is also very satisfying with respect to the physical cooling possibilities of the actuator. In a more general manner, we believe that such a linear control method proves that a simple PID-control can be adapted for nonlinear actuators if the PID can derive benefit of these nonlinear effects.

Further work will aim to include a self-sensing process inside the actuator as discussed, for example, in [START_REF] Hofman | Self-sensing with loaded piezoelectric bending actuators[END_REF]. Practical applications of the actuator require a good control of the temperature; this can easily be done in closed-loop by a direct control of the current value, and the time during which higher values of the current can be imposed. A self-sensing of the actuator position, combined with a simple PID closed-loop control, as defended in this article, should make the actuator autonomous and safe for future human-friendly applications. Further improvements in this direction could be also obtained using an active material with lower spin transition temperature.

Fig. 1 .

 1 Fig. 1. Experimental setup: (a) photograph of a sample clamped in its support, b) typical bending of an actuator with a payload, (c) bilayer structure and actuator geometry, (d) control scheme.

  2 A. We have chosen these values because it makes possible to control the tip movement of the sample between ca. 0 and 7 mm, which is just beyond the positioning range we considered in the closed-loop experiments. In practice, the sample can be controlled in a much larger current range of 0 -3 A. However, beyond a current value of ca. 1.2 A, the temperature of the sample becomes too high (ca. 140 °C) leading to the modification and ultimately to the destruction of the sample. The result of the identification of the open-loop step response is shown in Fig.2. The result of the identification of the open-loop step response is shown in Fig.2. One can depict in Fig.2athe response of the actuator for a positive current step of 1.2A and then its reverse movement when a step back to 0 A is applied. The same parameters have been used for the ascending and descending model (dashed line). The good agreement between real response and the model emphasizes a similar behavior of the actuator bending and unbending. This is a very positive attribute for tracking performances requiring movements in each direction. Fig. 2.b shows responses to different positive steps whereas Fig. 2.c shows the current dependence of the damping factor = ( + ) /2, and the natural frequency = 1/ deduced from the identified values of T1 and T2 [31]. The identification accuracy was simply measured by )

Fig. 2 .

 2 Fig. 2. Open-loop identification: (a) Typical response (full line) and corresponding linear model (dashed line) for a 1.2 A step and its return to zero, (b) Step responses for a given input current (full line) with corresponding linear identification (dashed line) and (c) the extracted parameters of the second-order model.

Fig. 3 .

 3 Fig. 3. (a) Simulation of an optimal PI-control for a 5 mm target position (blue line) compared to the experimental open-loop response (green) as well as to the best manually-tuned, experimental PID step response (red). (b) Measured electrical power supplied to the actuator in the open-loop and closed-loop cases.

Fig. 4 .

 4 Fig. 4. Step-responses in closed-loop control and derived identification of the actuator: (a) Time response for desired positions 2, 3, 4 and 5 mm (in full line) and corresponding simulated PID-control applied to the equivalent identified actuator linear model (in dashed line), (b) Comparison between parameters of the initial identified open-loop linear model and those of the equivalent linear model.

Fig. 5 .

 5 Fig. 5. Closed-loop step response with embedded loads: (a) Position and current versus time for the four considered target positions: 2 mm, 3 mm, 4 mm and 5 mm -in each case for 5 different loads. (b) Corresponding 95% response time versus load.

Fig. 6 .

 6 Fig. 6. Sine wave tracking experiments in closed-loop for different target periods and displacements -either with or without payload. The target (resp. real) positions are plotted using dashed (resp. straight) lines. The tracking errors are also shown.

Acknowledgments. We are grateful to Gaspard Cereza for his contribution to the construction of the actuator test bench. We are also very grateful to the referees for their very helpful comments.

Funding This work received financial support from the Federal University of Toulouse/ Région Occitanie (PhD grant of MPB) and from the Agence Nationale de la Recherche (ANR-19-CE09-0008-01).


Christophe Boudry
email: christophe.boudry@chartes.psl.eu

Hassan Al Hajj

Louis Arnould

Frederic Mouriaux

Analysis of International Publication Trends in Artificial Intelligence in Ophthalmology

Keywords: Artificial intelligence, ophthalmology, bibliometrics, journals, glaucoma, retinal diseases

Introduction

Artificial intelligence (AI) in healthcare consists of using complex algorithms and software, in order to emulate human cognition in the analysis, interpretation, and comprehension of complicated medical data. A number of research studies have previously suggested that AI can perform as well as or better than humans at key healthcare tasks, such as diagnosis [START_REF] Wang | AI in Health: State of the Art, Challenges, and Future Directions[END_REF].

AI has demonstrated promising results in medical fields, such as radiology, pathology, and dermatology. These medical fields have striking similarities to ophthalmology, as they are deeply rooted in diagnostic imaging, the most prominent application of AI in healthcare [START_REF] Jiang | Artificial intelligence in healthcare: past, present and future[END_REF].

Bibliometrics refers to methods that use quantitative analysis and statistics to describe distribution patterns of publications, temporal evolution, and geographical distribution of research in a given field [START_REF] Bar-Ilan | Informetrics at the beginning of the 21st century-A review[END_REF]. Bibliometrics is helpful in mapping the literature related to a research field and can provide useful data, leading to a better understanding of scientific fields. This can thus play a role in governing policymaking [START_REF] King | The scientific impact of nations[END_REF]. Bibliometric analyses are used in various research fields, including ophthalmology [START_REF] Boudry | Trends and topics in eye disease research in PubMed from 2010 to 2014[END_REF][START_REF] Boudry | International publication trends in dry eye disease research: A bibliometric analysis[END_REF][START_REF] Cen | Bibliometric and visualized analysis of global research on fungal keratitis from 1959 to 2019[END_REF]. Some reviews of the literature have been done in AI in ophthalmology [START_REF] Ting | Artificial intelligence and deep learning in ophthalmology[END_REF][START_REF] Kapoor | The current state of artificial intelligence in ophthalmology[END_REF]. A number of bibliometric analyses related to global research on AI [START_REF] Lei | The development of artificial intelligence: a bibliometric analysis[END_REF][START_REF] Niu | Global Research on Artificial Intelligence from 1990-2014: Spatially-Explicit Bibliometric Analysis[END_REF], AI in healthcare [START_REF] Robert | Bibliometric overview of the utilization of artificial neural networks in medicine and biology[END_REF][START_REF] Tran | Global Evolution of Research in Artificial Intelligence in Health and Medicine: A Bibliometric Study[END_REF][START_REF] Guo | Artificial Intelligence in Health Care: Bibliometric Analysis[END_REF] and in some medical fields (e.g. radiology [START_REF] West | Global Trend in Artificial Intelligence-Based Publications in Radiology From 2000 to 2018[END_REF]) have also been performed. However, to the best of our knowledge, none were related to AI in ophthalmology. Consequently, there is a lack of knowledge about the research situation in this field. The present study used bibliometric tools to provide a retrospective and current view of mainstream research on AI in ophthalmology across the international scientific literature.

Materials and Methods

The search for papers to be included in this study was carried out on May 12th 2020 using the PubMed database (http://www.ncbi.nlm.nih.gov/pubmed) developed by the National Center for Biotechnology Information (NCBI) at the National Library of Medicine (NLM).

PubMed was chosen because it is the most widely used in medicine [START_REF] Falagas | Comparison of PubMed, Scopus, Web of Science, and Google Scholar: strengths and weaknesses[END_REF], and allows a search using MeSH, the National Library of Medicine's controlled vocabulary thesaurus (http://www.nlm.nih.gov/pubs/factsheets/mesh.html). where Mesh stands for "Medical Subject Headings," DP "Date of Publication," and PT "Publication Type."

The MeSH term "Eye Diseases" was chosen because it has been shown to have extensive coverage of eye diseases [START_REF] Boudry | Trends and topics in eye disease research in PubMed from 2010 to 2014[END_REF]. The MeSH term "Artificial Intelligence", whose definition in the NLM-controlled vocabulary thesaurus is "theory and development of computer systems which perform tasks that normally require human intelligence" was chosen because it covers all developed AI methods and is used in medicine (e.g. neural networks, or machine learning) (Fig. 1). "Journal Article" includes the following publication types: journal articles, introductory journal articles, and reviews. "Journal article" was used for the analysis because these document types include whole research ideas and results [START_REF] Fu | Mapping of drinking water research: A bibliometric analysis of research output during 1992-2011[END_REF]. The year 1966 was chosen because it corresponds to the beginning of the Medline database. We did not impose any language restriction in the search, in order to analyze publication patterns of all publications related to AI in ophthalmology.

Fig 1.

MeSH trees corresponding to the MeSH term "Artificial Intelligence". This term is located in two different places in the National Library of Medicine-controlled vocabulary thesaurus Data were downloaded from PubMed in Extensible Markup Language (XML) and were processed through developed hypertext preprocessor language (PHP) scripts, then imported to Microsoft Excel 2013 (Microsoft, Redmond, California, USA) for data processing as previously reported [START_REF] Boudry | Eye neoplasms research: a bibliometric analysis from 1966 to 2012[END_REF]. The total number of articles related to AI in ophthalmology per year was fitted to an exponential curve for our search strategy. The average yearly growth rate [START_REF] Deshazo | Publication trends in the medical informatics literature: 20 years of "Medical Informatics[END_REF] of the literature related to AI in ophthalmology was calculated as the mean percentage of the annual growth rate for the period studied using the equation: annual growth rate = current year total number of articles -previous year's total number of articles/previous year's total number of articles. The average yearly growth rate was also calculated for the whole PubMed database. Languages of publication and journals were determined using the "Language" and "Journal" fields in PubMed data for all articles retrieved. An analysis of the MeSH terms used by PubMed indexers was done to classify the articles, calculated by their frequency in the retrieved articles and represented as a word cloud using https://www.wordclouds.com/. Check tags, i.e. MeSH terms obligatorily used by indexers to describe recurrent patterns in medical articles, were excluded from this frequency analysis because of their compulsory and recurrent aspects. They are shown in Supplementary Table 1. Co-occurrence, the simultaneous association of two MeSH terms in the same articles, was calculated for "clinical" MeSH terms retrieved.

In order to access article citations, PubMed's unique identifiers of articles retrieved in the PubMed database were used to conduct searches in the Web of Science Core Collection (WoSCC). Data were downloaded from WoS in "Full record and cited references" and "Comma Separated Values" (CSV) formats. They were then imported to Microsoft Excel 2013 (Microsoft, Redmond, California, USA) for data processing. Citation counts reflect all the papers obtained on May 12th 2020 when the WoS database search process for this study was conducted. Countries, institutions, and subject categories were examined. Institutions were determined using the "Organizations-Enhanced" field. Articles originating from England, Scotland, Northern Ireland, and Wales were reclassified as originating from the United Kingdom (UK). Hong Kong was considered as part of the People's Republic of China (China). Year 2019 Impact Factors (IF) of journals were found on publisher websites.

VOSviewer (Leiden University, Leiden, Netherlands) was used to generate the knowledge map of countries related to AI in ophthalmology.

Results

Global Publication Trends

Overall, 1356 references over the period 1966-2019 were found in PubMed. Among these 1356 references, 84 were reviews and two introductory journal articles. The first articles related to AI in ophthalmology were published in 1985 and the number of publications increased exponentially to 197 in 2019 (Fig. 2). Interestingly, the number of publications grew steadily until 2016, then grew sharply between 2016 and 2019. Using the cumulative number of publications, we calculated the exponential adjustment (y = 6 -132 *e 0,1534x with r² = 0.96). We calculated that the average yearly growth rate was 18.89% over the last ten years. In the entire PubMed database, it was 4.71% over the same period.

Language of Publication and Journals

Eleven different languages of publication were identified in PubMed. The seven most frequent languages were English (n = 1307; 96.4%), German (n =17; 1.25%), Chinese (n = 10; 0.74%), French (n = 7; 0.52%), and Russian (n = 7; 0.52%). All other languages amounted to less than 2 articles. Because the percentage of articles in English in the entire PubMed database for the same period was 87.01%, this shows that the number of papers on AI in ophthalmology published in English is higher than expected compared to the overall body of medical literature.

A total of 1215 articles were retrieved in the WoSCC. These articles were published in 355 journals, and were cited 36046 time with an average of 29.67 citations per article. Of these 355 journals, 198 (55.77%) published only one article. Only 31 (2.47%) journals published more than ten articles on AI in ophthalmology. Table 1 lists the 15 journals with the greatest number of papers published related to AI in ophthalmology.

Countries, International Collaborations, and Institutions

A total of 65 countries were found in author affiliations in the 1215 articles studied. Thirty three (2.72%) articles were entered without author information, so without countries of affiliation. Table 2 shows the ten most productive countries in AI in ophthalmology. As expected, the USA accounted for the largest number of articles published, far more than other countries. The ratio "percentage of articles divided by the percentage of citations" was equal to 1.43 for the USA, logically meaning that articles published by researchers located in the USA were cited 1.43 times more than expected. This shows the high scientific impact of research on AI in ophthalmology in the USA. To a lesser extent, this is also the case for Germany and the UK (ratios equal to 1.24 and 1.16, respectively). Conversely, articles from China, Japan, and Australia are less cited than expected, with ratios equal to 0.56, 0.62, and 0.65, respectively (almost half of the citations expected for China). All other countries, with ratios close to one, received the number of citations expected. The number of articles with international collaboration was 316 (26.1%), with Singapore in the lead (72.88% of articles with international collaboration). The analysis of international collaboration (Fig. 3) shows that the USA, Germany, the United Kingdom, and China are hubs of the international collaborative network. Apart from these countries, researchers tended to collaborate with researchers from neighboring countries. For example, most of the countries situated in Europe (The Netherlands, Switzerland, Spain, Portugal, Poland, and Belgium, shown in red in Fig. 3) or in Asia (Singapore, India, and Malaysia (shown in purple in Fig. 3) are in this case. Of the top 10 most productive institutions, four are located in the USA and three are in Singapore (Table 3). The articles published by most institutions located in the USA are cited more often than the average value obtained for articles overall. For example, the articles from the University of Iowa are cited four times more than the average value. This shows that these institutions are not only very productive quantitatively (number of articles), but also qualitatively (number of citations). The number of MeSH terms found in the 84 reviews was 782. Among these 782 MeSH terms, 266 unique MeSH terms were identified. Fig. 5 presents the word cloud of the twenty most frequent MeSH terms found in these 84 reviews.

Discussion

The aim of this study was to perform a bibliometric analysis related to AI in ophthalmology.

This is an original article and not a systematic review or a meta-analysis. To our knowledge, no similar studies have been conducted to date. This analysis was done using PubMed and WosCC databases, allowing the use of both MeSH terms in the query to obtain qualitative data (citations) as well as reliable determinations of authors' affiliations with countries in the articles studied [START_REF] Boudry | Eye neoplasms research: a bibliometric analysis from 1966 to 2012[END_REF]. The usefulness of using controlled thesaurus vocabulary as MeSH terms for indexing and retrieving documents is well known [START_REF] Chang | Searching the literature using medical subject headings versus text word with PubMed[END_REF][START_REF] Ramos | Bibliometric analysis of leishmaniasis research in Medline (1945-2010)[END_REF]. In this study, the MeSH terms Learning or Telerobotics for Robotics). Therefore, using the "Artificial Intelligence" MeSH term as a query in PubMed can exhaustively retrieve, in one shot, articles related to these 13 sub-categories that can have 128 different synonyms. This great number of existing synonyms makes it difficult to build a search query for extracting articles related to AI in ophthalmology in a bibliographic database (e.g. WosCC) without using MeSH. However, only using PubMed did not allow us to determine the country affiliation of authors with precision. This is because before the year 2014, the country affiliation of the author, when present, indicated only one country per article, which fails to identify collaborative research needed to obtain a qualitative evaluation of research output (i.e. number of citations). Thus, for such a study we must use PubMed and WosCC. There are limitations in the present study.

Although PubMed is the most widely used bibliographic database in medicine, it does not contain all biomedical journals and is biased in favor of English-language journals [START_REF] Vioque | A bibliometric study of scientific literature on obesity research in PubMed (1988-2007[END_REF]. Using

PubMed and WoSCC leads to losing some articles because PubMed and WoSCC databases do not have the same coverage. For example, 141 articles (10.4%) were not taken into account for country, institution, and citation analysis because they were not indexed in the WoSCC.

Another potential limitation of this study should be mentioned, as the search was performed in May 2020. This scientific research field is very dynamic, and major publications might not have been included at the time of publication. Despite these limitations, we believe that this study provides a wide view of trends and scientific productivity related to AI in ophthalmology.

With an average yearly growth rate of 18.89% over the last ten years, the growth of AI in ophthalmology publications has been higher than that of the PubMed database as a whole and is exponential. As a comparison, ophthalmology research showed a yearly average growth rate of only 4.27% over the period 2010-2014 with stabilization of the publication rate [START_REF] Boudry | Eye neoplasms research: a bibliometric analysis from 1966 to 2012[END_REF]. This indicates great interest in AI in ophthalmology on the part of the scientific community, "an attractive topic in science" according to Michon et al and Fernandez-Cano et al [START_REF] Michon | The Dynamic Interest in Topics within the Biomedical Scientific Community[END_REF][START_REF] Fernandez-Cano | Reconsidering Price's model of scientific growth: an overview[END_REF]. Our results are in accordance with former studies which have shown that AI is an attractive topic in other research fields (Global AI [START_REF] West | Global Trend in Artificial Intelligence-Based Publications in Radiology From 2000 to 2018[END_REF] and AI related to radiology [START_REF] Gupta | Artificial Intelligence Research in India: A Scientometric Assessment of Publications Output during 2007-16[END_REF]). One must note that the number of publications in AI in ophthalmology grew sharply between 2016 and 2019, concomitant with Microsoft's announcement that 2016 would be the year of AI, and that its new artificial intelligence and research group would consist of more than 5,000 employees [START_REF] Wingfield | Microsoft Reorganizes Its Research Efforts Around A.I[END_REF]. This shows that the interest in artificial intelligence extends well beyond the academic world. The analysis of the top 15 most productive journals showed that most journals publishing articles related to AI in ophthalmology specialized in computer and medical systems. The percentage of articles related to AI in ophthalmology published in the top 15 was very low (less than one percent for each journal) compared with other subspecialties in ophthalmology (e.g. dry eye diseases [START_REF] Boudry | International publication trends in dry eye disease research: A bibliometric analysis[END_REF]). This can result in difficulties for researchers trying to keep up to date with AI in ophthalmology. In contrast, other research fields have some specialized journals that publish a large part of the articles of interest [START_REF] Liang | Worldwide Research Productivity in the Field of Arthroscopy: A Bibliometric Analysis[END_REF].

When analyzing countries, Singapore was found among the top most productive countries in AI in ophthalmology, while it was mentioned among the most productive countries when considering global research in AI [START_REF] Lei | The development of artificial intelligence: a bibliometric analysis[END_REF][START_REF] Niu | Global Research on Artificial Intelligence from 1990-2014: Spatially-Explicit Bibliometric Analysis[END_REF] or in other medical sub-specialties (e.g. radiology [START_REF] West | Global Trend in Artificial Intelligence-Based Publications in Radiology From 2000 to 2018[END_REF]). This position can be explained by the presence of three institutions located in Singapore that were in the top 10 most productive institutions in AI in ophthalmology research and the large number of articles with international collaboration. International collaboration is necessary and is constantly growing [START_REF] Mindeli | Bibliometric Studies of Scientific Collaboration: International Trends[END_REF], given the growth of accumulated knowledge, increased specialization, and the complexity of science [START_REF] Ribeiro | Growth patterns of the network of international collaboration in science[END_REF]. The analysis of international collaboration showed that researchers in Europe and Asia tend to collaborate with researchers from neighboring countries, while the USA, Germany, the United Kingdom, and China, located in collaboration network hubs, tend to collaborate closely with each other [START_REF] Su | Global scientific collaboration in COPD research[END_REF]. One must note that, due to the multiple motives for international collaboration (i.e. culture, language, history, political and economic factors, or distance between countries [START_REF] Mindeli | Bibliometric Studies of Scientific Collaboration: International Trends[END_REF][START_REF] Sakata | Bibliometric Analysis of International Collaboration in Wind and Solar Energy[END_REF]), our results are difficult to interpret more precisely. At the country level, Singapore still attracts attention, having a very high percentage of articles with international collaboration. This country has already been singled out in AI in health/medicine by Tran et al. [START_REF] Tran | Global Evolution of Research in Artificial Intelligence in Health and Medicine: A Bibliometric Study[END_REF] as having one of the highest percentages of papers with international collaboration (66.2%). It is of note that international scientific collaboration has been shown to offer opportunities to small countries [START_REF] Narin | Scientific co-operation in Europe and the citation of multinationally authored papers[END_REF]. While uncertain, it is true that scientists in small countries have far more scientists outside of their country who can cooperate with others, and far fewer inside than scientists in much larger countries [START_REF] Mindeli | Bibliometric Studies of Scientific Collaboration: International Trends[END_REF].

In order to provide details on article topics and offer additional information on AI in ophthalmology research trends, we analyzed the top 10 JCR categories of journals and MeSH terms for articles. This analysis confirmed that AI in ophthalmology research mainly focuses on engineering and computing and is mainly technical research related to computer methods, not clinical practice. The analysis also confirms that this research topic involves scientists far removed from medicine. In the future, AI could become a powerful tool for improving screening and triage of patients with chronic ocular diseases. It could also help ophthalmologists enhance their ability to diagnose common diseases and facilitate the decision-making process in the fields of retinal disease and glaucoma, for example. In summary, we are faced with a growing healthcare demand and a decreasing number of ophthalmologists, particularly in some rural areas. AI could help by enhancing medical care in this context.

  The objective of the search strategy used was to exhaustively extract all articles related AI in ophthalmology indexed in the PubMed database. The search strategy was built by identifying the keywords listed in the medical subject headings (MeSH) thesaurus (words appearing in the MeSH field [Mesh]), i.e. the vocabulary of medical and scientific terms that are assigned to PubMed documents by a team of trained experts (indexers). The search strategy was : "Artificial Intelligence" [Mesh] AND ("Eye Diseases"[Mesh] OR "Ophthalmology"[Mesh] OR "Eye"[Mesh]) AND Journal Article [PT] AND 1966:2019 [DP]"

Fig. 2

 2 Fig. 2 Growth of literature related to AI in ophthalmology (annual number per 100000 articles in PubMed and total number)

Fig. 3

 3 Fig. 3 Visual map of networks of country collaboration. A minimum of five documents per country was set as the threshold and 37 countries were included in the map. Dots represent countries and their size indicates the number of publications. Colors indicate different clusters with a high degree of collaboration. The thickness of the link between any two countries is indicative of the extent of co-authorship (i.e. collaboration)

Fig. 4

 4 Fig. 4 Word cloud of the twenty most frequent MeSH terms for AI in ophthalmology research. The size of the words is proportional to their frequency. Terms used in the search query were excluded

Fig. 5

 5 Fig. 5 Word cloud of the twenty most frequent MeSH terms for AI in ophthalmology research limited to reviews. The size of the words is proportional to their frequency. Terms used in the search query were excluded

Table 1 : Top 15 most productive journals on AI in ophthalmology research.

 1

	Journal	Impact	No. of articles	No of	Total number of articles
		Factor	related to AI in	citations (%)	published by the journal
			ophthalmology		(percentage of articles
			(%)		related to AI in
					ophthalmology)
	IEEE Engineering in Medicine and Biology Society	N/A	62 (5.1)	444 (1.23)	27109 (0.23)
	Investigative Ophthalmology & Visual Science	3.812	40 (3.3)	2155 (5.98)	22661 (0.18)
	PLOS ONE	2.776	40 (3.3)	320 (0.89)	223296 (0.02)
	Computer Methods and Programs in Biomedicine	3.424	39 (3.2)	906 (2.51)	4462 (0.87)
	IEEE Transactions in Medical Imaging	6.79	38 (3.1) 4809 (13.34)	4821 (0.79)
	Computers in Biology and Medicine	2.286	32 (2.6)	848 (2.35)	3813 (0.84)
	IEEE Transactions on Biomedical Engineering	4.491	28 (2.3)	1063 (2.95)	8395 (0.33)
	Journal of Medical Systems	2.415	25 (2.1)	453 (1.26)	3315 (0.75)
	Scientific Reports	4.011	20 (1.6)	265 (0.74)	103226 (0.02)
	Ophthalmology	7.732	19 (1.6)	653 (1.81)	16920 (0.11)
	Neural Networks: the official journal of the
	International Neural Network Society	5.785	18 (1.5)	502 (1.39)	3259 (0.55)
	American Journal of Ophthalmology	4.483	16 (1.3)	219 (0.61)	13721 (0.12)
	Journal of Biomedical Optics	2.555	16 (1.3)	804 (2.23)	6363 (0.25)
	Biological Cybernetics	1.305	14 (1.2)	239 (0.66)	2436 (0.57)
	Computerized Medical Imaging and Graphics: the
	official journal of the Computerized Medical
	Imaging Society	3.298	14 (1.2)	476 (1.32)	1990 (0.7)
	Surprisingly, our results showed that articles related to AI in ophthalmology are rarely
	published in ophthalmology journals. Only three specialized in ophthalmology (Investigative
	Ophthalmology & Visual Science, Ophthalmology, and the American Journal of
	Ophthalmology) and the percentage of articles related to AI in ophthalmology published in
	each of the top 15 journals comprise less than one percent of the overall articles published.

These top 15 journals of 355, which represent only 4.23% of all journals found in this study, published one-third of the articles on AI in ophthalmology (33.54%; n = 421). These 421 articles were cited 14156 times, corresponding to 39.27% of the overall citations. Therefore, these 15 journals are "core journals", and are of great interest for researchers because onethird of the articles on AI in ophthalmology could be found in this small fraction of journals.

As usual, multidisciplinary megajournals such as PLOS ONE and Scientific Reports were well represented. All other journals specialized in computer methods and medical systems. This clearly indicates that no one journal is specifically dedicated to publishing articles related to AI in ophthalmology. A total of 71 reviews were retrieved in the WoSCC. These reviews were published in 47 journals, and were cited 5151 times with an average of 72.55 citations per article.

Table 2 : Top 10 most productive countries for AI in ophthalmology research.

 2

	Country	No. of articles	No. of citations (%)	Ratio	Average citation	No. of articles
		(%)		"percentage of	per article	with
				citations /		international
				percentage of		collaboration
				articles"		(%)
	USA	419 (34.5)	17785 (49.34)	1.43	42.44	147 (35.08)
	China	169 (13.91)	2812 (7.8)	0.56	16.64	69 (40.83)
	UK	113 (9.3)	3901 (10.82)	1.16	34.52	68 (61.06)
	Germany	88 (7.24)	3231 (8.96)	1.24	36.71	41 (46.59)
	Australia	66 (5.43)	1264 (3.51)	0.65	19.15	40 (60.61)
	Singapore	59 (4.86)	1769 (4.91)	1.00	29.98	43 (72.88)
	India	58 (4.77)	1816 (5.04)	1.05	31.31	21 (36.2)
	Japan	57 (4.69)	1042 (2.89)	0.62	18.28	15 (26.31)
	Spain	49 (4.03)	1496 (4.15)	1.04	30.53	19 (38.78)
	France	48 (3.95)	1564 (4.34)	1.08	32.58	30 (62.5)

Table 3 : Top 10 most productive institutions for AI in ophthalmology research.

 3

	Institution (country)	No. of	No. of citations	Ratio	Average
		articles (%)	(%)	"percentage of	citations per
				citations /	article
				percentage of
				articles"
	University of California San Francisco (USA)	63 (5.19)	4073 (11.3)	2.18	64.65
	Harvard University (USA)	36 (2.96)	2186 (6.06)	2.05	60.72
	University of London (UK)	36 (2.96)	1066 (2.96)	1.00	29.61
	National University of Singapore (Singapore)	36 (2.96)	1183 (3.28)	1.11	32.86
	Johns Hopkins University (USA)	30 (2.47)	780 (2.16)	0.88	26
	Agency for Science Technology Research
	(Singapore)	27 (2.22)	662 (1.84)	0.83	24.52
	Chinese Academy of Sciences (China)	26 (2.14)	982 (2.72)	1.27	37.77
	University of Iowa (USA)	26 (2.14)	2988 (8.29)	3.87	114.92
	Nanyang Technological University (Singapore)	24 (1.98)	650 (1.8)	0.91	27.08
	Eidgenossische Technische Hochschule Zurich
	(Switzerland)	22 (1.81)	741 (2.06)	1.14	33.68
	4. Analysis of Subject Categories and MeSH Terms

Based on the Journal Citation Reports (JCR) categories, the AI in ophthalmology output data were distributed into subject categories. As shown Table

4

, most focus on engineering and computing, and only two are related to medicine/clinical practice (Ophthalmology, n = 242; 19.81% and Neurosciences, n = 148; 12.18%).

Table 4 : Top 10 JCR categories of journals on AI in ophthalmology research. Web of Science Categories No. of articles (%)

 4 These MESH terms can be aggregated into four main categories: Artificial Intelligence, which is well represented (e.g. Neural Networks Computer, Machine Learning, Deep Learning), Image Analysis (e.g.

	Engineering biomedical	337 (27.74)
	Ophthalmology	242 (19.91)
	Computer science interdisciplinary applications	177 (14.57)
	Neurosciences	148 (12.18)
	Radiology nuclear medicine medical imaging	145 (11.93)
	Medical informatics	142 (11.69)
	Engineering electrical electronic	134 (11.03)
	Computer science artificial intelligence	125 (10.29)
	Mathematical computational biology	101 (8.31)
	Computer science theory methods	87 (7.16)

The number of MeSH terms found in PubMed was 15792 (average of 11.64 MeSH terms per article) among the 1356 articles. Among these 15792 MeSH terms, 1865 unique MeSH terms were identified. The 20 most frequent MeSH terms are presented Fig. 4. Image Interpretation, Computer-Assisted, Image Enhancement), and Method Evaluations (e.g. Sensitivity and Specificity, Reproducibility of Results, ROC Curves). The fourth category (Ophthalmology) is represented by clinical MeSH terms related to the retina and glaucoma (Supplementary Table

2

shows complementary data for each of these four MeSH terms: the five most co-occurring MeSH terms in articles with these four MeSH terms).

  //www.ncbi.nlm.nih.gov/mesh/68001185] related to AI (e.g. Machine Learning or Robotics) located beneath AI in the MeSH tree. Please note that these 13 sub-categories related to AI can be called by 128 different synonyms (e.g. Transfer Learning for Machine

	"Artificial	Intelligence"	allowed	us	to	search	13	sub-categories
	[https:

MH terms FIve most co-occurring MeSH terms in articles with the MeSH term in the first column (% of co- occurrence)

  Among the twenty most frequent MeSH terms retrieved, there were only four related to clinical topics, revealing the retina and glaucoma as the most frequent subjects of interest in AI in ophthalmology. Numerous imaging techniques are available for the diagnosis and treatment of retinal diseases, for instance fundus images and optical coherence tomography (OCT). In the literature, commonly explored topics are segmentation of regions of interest such lesions, diabetic retinopathy, and premature or age-related macular degeneration (AMD). Ultimately, some systems are approved by medical institutions, such as IDx[START_REF] Abràmoff | Pivotal trial of an autonomous AI-based diagnostic system for detection of diabetic retinopathy in primary care offices[END_REF], approved by the United States Food and Drug Administration and OphtAI, which is European

	Check tag Retina	Number of articles indexed Neural_Networks_Computer (45.27) with this MeSH term (%) Algorithms (37.25)
		Artificial_Intelligence (26.65)
	Humans Female Diabetic_Retinopathy Algorithms (69.21) Image_Interpretation_Computer-Assisted (22.64) 1148 (84.66) Models_Neurological (17.19) 302 (22.27)
	Male	300 (22.12) Image_Interpretation_Computer-Assisted (45.02)
	Animals Middle Aged	Sensitivity_and_Specificity (37.7) 242 (17.85) Pattern_Recognition_Automated (31.94) 229 (16.89) Artificial_Intelligence (30.37)
	Aged Glaucoma	164 (12.09) Algorithms (37.69)
	Adult Aged 80 and over	Neural_Networks_Computer (33.85) 163 (12.02) Optic_Disk (28.46) 57 (4.2) Artificial_Intelligence (28.46)
	Young Adult	44 (3.24) Sensitivity_and_Specificity (27.69)
	Adolescent Fundus_Oculi Mice	Algorithms (56.2) 38 (2.8) Neural_Networks_Computer (39.67) 29 (2.14) Diabetic_Retinopathy (36.36)
	Child	19 (1.4) Image_Interpretation_Computer-Assisted (33.88)
	Rats	14 (1.03) Image_Processing_Computer-Assisted (33.06)
	Child Preschool Rabbits Supplementary	14 (1.03) 13 (0.96)
	Infant	10 (0.74)
	Infant Newborn	8 (0.59)
	Cats	6 (0.44)
	Chick Embryo	4 (0.29)
	Cattle	3 (0.22)
	Dogs	2 (0.15)
	Guinea Pigs	2 (0.15)
	Pregnancy	1 (0.07)
	History 20th Century 1 (0.07)

Table 2 : "Medical" MeSH terms for AI in ophthalmology research; top five co-occurring MeSH tems; top five countries, institutions, and journals.

 2

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Authors' contributions

Conformity marked. There is no denial that AI and deep learning will continue to impact the way in which we screen, manage, and monitor the treatment of retinal diseases, resulting in numerous applications and advances that help us to improve public health. It is predicted that the prevalence of glaucoma will increase by almost 50% over the next twenty years [START_REF] Tham | Global prevalence of glaucoma and projections of glaucoma burden through 2040: a systematic review and meta-analysis[END_REF]. Glaucoma care is thus currently suffering from a mismatch between capacity and demand. The burden of glaucoma care will continue to grow, and early diagnosis remains a much-needed strategy. AI could enhance access to large-scale screening and will improve medical support in low-resource countries through refined automated strategies for clinical diagnosis [START_REF] Phasuk | Automated Glaucoma Screening from Retinal Fundus Image Using Deep Learning[END_REF][START_REF] Tan | Glaucoma screening: where are we and where do we need to go?[END_REF][START_REF] Zou | A novel glaucomatous representation method based on Radon and wavelet transform[END_REF].

In conclusion, this study presents a current view of mainstream research on AI in ophthalmology across the world. This study may be useful for researchers in AI in ophthalmology such as clinicians, but also for scientists. They will better understand this research topic, know the main actors in this field (including journals and countries), and, have an informed overview of this research theme. Decision makers may also be interested in this study to help them guide future developments in this research topic. This analysis can also be useful for those who are less familiar with this field but interested in AI applications in ophthalmology.

Supplementary material

Supplementary Table 1: Check tags for AI in ophthalmology research. Check tags: MeSH terms obligatorily used by indexers to describe recurrent patterns in medical articles for type of material studied (e.g. human, animals, in vitro), gender (male/female), age, chronological elements, and type of study (e.g. comparative study).

Supplementary Table 2: "Medical" MeSH terms for AI in ophthalmology research; top five co-occurring MeSH tems; top five countries, institutions, and journals

Declarations

Conflict of interest/Competing interests

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Financial or non-financial interests

The authors have no relevant financial or non-financial interests to disclose.

Supplementary Table 1: Check tags for AI in ophthalmology research

Check tags: MeSH terms obligatorily used by indexers to describe recurrent patterns in medical articles for type of material studied (e.g. human, animals, in vitro), gender (male/female), age, chronological elements, and type of study (e.g. comparative study).