domain: math info spi

text:
Dorian Verdel
email: dorian.verdel@ens-paris-saclay.fr

Simon Bastide
email: simon.bastide@universite-paris-saclay.fr

Nicolas Vignais
email: nicolas.vignais@universite-paris-saclay.fr

Olivier Bruneau
email: olivier.bruneau@ens-paris-saclay.fr

Bastien Berret
email: bastien.berret@universite-paris-saclay.fr

An Identification-Based Method Improving the Transparency of a Robotic Upper Limb Exoskeleton

Keywords: Control laws, Identification, Human/Robot interaction, Transparency indices, Motor control

Over the past decade, research on human-robot collaboration has grown exponentially, motivated by appealing applications to improve the daily life of patients/operators. A primary requirement in many applications is to implement highly "transparent" control laws to reduce the robot impact on human movement. This impact may be quantified through relevant motor control indices. In this paper, we show that control laws based on careful identification procedures improve transparency compared to classical closed-loop position control laws. A new performance index based on the ratio between electromyographic activity and limb acceleration is also introduced to assess the quality of human exoskeleton interaction.

Introduction

Over the past decade, exoskeletons encountered a large success, mainly for handling and rehabilitation applications. Exoskeleton-type assistive devices aim at providing efforts to one or several segments in order to assist the user in achieving specifics tasks. Amongst the different types of exoskeletons, active robotic devices are especially promising because they offer a great versatility that cannot be equalled by passive devices. They may be useful to support injured or disabled people in recovering and performing the movements required in daily living activities. [START_REF] Huo | Adaptive proxy-based controller of an active ankle foot orthosis to assist lower limb movements of paretic patients[END_REF] They may as well reduce joint efforts and prevent the development of musculoskeletal disorders at work. [START_REF] Bogue | Robotic exoskeletons: A review of recent progress[END_REF][START_REF] De Looze | Exoskeletons for industrial application and their potential effects on physical work load[END_REF] For instance, improvements of rehabilitation protocols were envisioned by using upper limbs powered exoskeletons. [START_REF] Frisoli | Positive effects of robotic exoskeleton training of upper limb reaching movements after stroke[END_REF][START_REF] Frisoli | Arm Rehabilitation with a Robotic Exoskeleleton in Virtual Reality[END_REF] Benefits in terms of prevention of workers' musculoskeletal disorders [START_REF] Sylla | Ergonomic contribution of ABLE exoskeleton in automotive industry[END_REF] and metabolic energy savings [START_REF] Mooney | Autonomous exoskeleton reduces metabolic cost of human walking during load carriage[END_REF] could also be expected with this type of powered assistance. However, despite these potential benefits, the concrete usability of powered assistive exoskeletons is hampered by the absence of evidence for in situ efficiency. Indeed, meta-analyses do not yet highlight any clear consensus on the positive effects of robot-assisted therapies compared to conventional ones. [START_REF] Veerbeek | Effects of robot-assisted therapy for the upper limb after stroke: A systematic review and meta-analysis[END_REF] a These authors contributed equally.

Furthermore, the possible industrial applications may be challenged by the evidence of an average 30% movement slowdown when wearing a robotic "transparent" exoskeleton (see below) of the upper limb, [START_REF] Bastide | Interacting with a 'Transparent' Upper-Limb Exoskeleton: A Human Motor Control Approach[END_REF][START_REF] Jarrassé | A methodology to quantify alterations in human upper limb movement during co-manipulation with an exoskeleton[END_REF] which could result in a critical decrease in productivity. Besides hardware and technical issues, their lack of concrete usability is mainly due to the fact that only few exoskeleton devices allow a symbiotic interaction with humans. Exoskeleton research and development have recently been interested in evaluating exoskeletons from a more human-centred point of view, by taking into account the human response to interaction with an exoskeleton as a critical feature of human/exoskeleton interactions (HEI). [START_REF] Di Natali | Design and evaluation of a soft assistive lower limb exoskeleton[END_REF][START_REF] Ercolini | A novel generation of ergonomic upperlimb wearable robots: Design challenges and solutions[END_REF][START_REF] Huang | Robotic neurorehabilitation: A computational motor learning perspective[END_REF][START_REF] Jarrassé | Connecting a human limb to an exoskeleton[END_REF][START_REF] Jin | Effects of Exoskeleton Weight and Inertia on Human Walking[END_REF][START_REF] Martelli | The effects on biomechanics of walking and balance recovery in a novel pelvis exoskeleton during zero-torque control[END_REF][START_REF] Xiloyannis | Physiological and kinematic effects of a soft exosuit on arm movements[END_REF] Indeed, how humans adapt their motor control to the interaction forces induced by wearing a robotic exoskeleton implementing some control laws should drive the development of new and more intuitive control laws.

The ability of an exoskeleton not to change or influence human movement is called transparency. [START_REF] Proietti | Upper-limb robotic exoskeletons for neurorehabilitation: A review on control strategies[END_REF] Theoretically, a perfect transparency would be reached if the interaction forces between the exoskeleton and the user are equal to zero. In reality, achieving perfect transparency is impossible, especially because human movements cannot be fully predicted. [START_REF] Jarrassé | How can Human Motion Prediction Increase Transparency?[END_REF] Getting as close as possible to this situation is desired in many applications [START_REF] Ajoudani | Progress and prospects of the human-robot collaboration[END_REF] as it may improve the quality of the interaction and contribute to better master task-dependent control laws. In general, a transparent exoskeleton should be able to compensate its own weight, friction, inertia and, in the case of load carrying, the weight and inertia of the mass being carried. Alternatively, if a sufficient degree of transparency is not achieved in baseline experiments without loads, an acceptable compensation of interactions with external objects will not be possible and the user will have to manage a combination of flawed compensations of the exoskeleton and of the carried load. In other words, a lack of transparency means that unwanted or uncontrolled interaction forces will appear, which may induce a motor adaptation from the user and/or prevent the acceptability of the device. Therefore, transparency is a primary and crucial step towards both industrial and medical applications in the sense that it provides a baseline setting for more advanced scenarios such as compensation for the weight of the user's arm or any other control law modifying the interaction forces. Overall, we argue that transparency is a necessary condition towards more symbiotic HEI.

State of the Art

Assess and improve transparency

The assessment of transparency has generally been done by comparing movement indices under natural conditions (without wearing exoskeletons) and while wearing the exoskeleton. [START_REF] Jarrassé | A methodology to quantify alterations in human upper limb movement during co-manipulation with an exoskeleton[END_REF][START_REF] Pirondini | Evaluation of the effects of the Arm Light Exoskeleton on movement execution and muscle activities: A pilot study on healthy subjects[END_REF] Often, these studies have focused on movements involving many degrees of freedom such as pointing movements performed with the whole upper limb thereby making it difficult to understand non-linear phenomena that may occur during the interaction in such complex movements. Moreover, this approach tends to contrast with an important part of the literature on human motor control, which has largely focused on simpler movements involving one [START_REF] Atkeson | Kinematic features of unrestrained vertical arm movements[END_REF][START_REF] Gaveau | Energy-related optimal control accounts for gravitational load: comparing shoulder, elbow, and wrist rotations[END_REF] or two degrees of freedom. [START_REF] Flash | The coordination of arm movements: An experimentally confirmed mathematical model[END_REF][START_REF] Morasso | Spatial control of arm movements[END_REF] Human motor control studies have often focused on simple movements because of the variability that can appear between subjects when achieving complex movements. This variability makes the derivation of explanatory models far more complicated. Moreover, the interpretation of physiological measures for complex movements, such as electromyographic (EMG) signals, can be very challenging. As our objective is to design new control laws that impact as little as possible the human motor control (i.e., that maximise transparency), it appears relevant to investigate their effects on well-known and widely documented human movements. Therefore, our approach is to reduce the complexity of the studied movement to a simple movement involving only one degree of freedom to carefully assess the quality of different transparent control laws.

Traditional methods employed to improve transparency have primarily focused on the mechanical design of exoskeletons' joints [START_REF] Garrec | Screw and Cable Acutators (SCS) and Their Applications to Force Feedback Teleoperation, Exoskeleton and Anthropomorphic Robotics[END_REF][START_REF] Garrec | ABLE, an Innovative Transparent Exoskeleton for the Upper-Limb[END_REF] and connections with the user. Indeed, the system {human + exoskeleton} is highly hyperstatic with simple connections and therefore implies unwanted interaction forces. Transparency and comfort have been improved by adding passive degrees of freedom, that release hyperstatic constraints, in the connections' design. [START_REF] Jarrassé | Connecting a human limb to an exoskeleton[END_REF][START_REF] Mallat | Human-Exoskeleton Joint Misalignment: A Systematic Review[END_REF][START_REF] Schiele | Kinematic design to improve ergonomics in human machine interaction[END_REF] Furthermore, approaches based on human movement prediction are promising for previously known [START_REF] Jarrassé | How can Human Motion Prediction Increase Transparency?[END_REF] or repetitive movements, [START_REF] Peternel | Adaptive control of exoskeleton robots for periodic assistive behaviours based on EMG feedback minimisation[END_REF] but their extension to a wider range of behaviours is still a challenge due to the difficulty of predicting the user's intended movement in general settings. In another approach, the use of force sensors could be relevant to allow the exoskeleton to follow the user's movements by directly minimising the interaction forces. [START_REF] Jarrassé | How can Human Motion Prediction Increase Transparency?[END_REF][START_REF] Carignan | Development of an Exoskeleton Haptic Interface for Virtual Task Training[END_REF] Nevertheless, this method is not often used because few exoskeletons include force sensors as standard equipment. More often, the robot's control is based on a closed-loop position control [START_REF] Huo | Adaptive proxy-based controller of an active ankle foot orthosis to assist lower limb movements of paretic patients[END_REF][START_REF] Bastide | Interacting with a 'Transparent' Upper-Limb Exoskeleton: A Human Motor Control Approach[END_REF][START_REF] Jarrassé | How can Human Motion Prediction Increase Transparency?[END_REF][START_REF] Pirondini | Evaluation of the effects of the Arm Light Exoskeleton on movement execution and muscle activities: A pilot study on healthy subjects[END_REF] as most robotic devices use incremental encoders as on-board sensors. Thus, the most direct solution for a better transparency appears to be controlling the robot so that it compensates its own dynamics. To make this solution reliable, a coherent model of the robot has to be designed and its parameters have to be known as precisely as possible. Yet, few studies addressed the role of identification to improve exoskeletons' transparency.

Our premise is that an accurate identification of the robot, possibly introducing directiondependent models, may improve transparency. Indeed, direction-dependent models allow to take into account phenomena that are not described by classical rigid body dynamics. [START_REF] Hamon | Dynamic Identification of Robots with a Dry Friction Model Depending on Load and Velocity[END_REF] Nevertheless, this type of model is very rarely studied in the literature. This approach is quite different, but complementary, from current methods adopted in the literature such as identifying the parameters of the interaction between humans and robots instead of identifying the robot itself. [START_REF] Franken | Bridging the gap between passivity and transparency[END_REF]

Identification methodologies

Identification is a large field of robotics that has been extensively studied over the past decades. A great variety of models and experiments can be used in order to identify a robot, but all of them can be classified into one of the two methods, direct identification of all parameters or step-by-step identification of groups of parameters. [START_REF] Khalil | Modélisation, identification et commande des robot[END_REF] Then, two approaches co-exist, one non-parametric and one parametric. [START_REF] Moberg | Modeling and Control of Flexible Manipulators[END_REF][START_REF] Östring | Closed-loop identification of an industrial robot containing flexibilities[END_REF] The first approach consists in considering the system as a black box following a general structure to identify, such as the general linear model structure. [START_REF] Östring | Closed-loop identification of an industrial robot containing flexibilities[END_REF][START_REF] Ljung | System Identification: Theory for the User[END_REF] One of the limitations of this approach is the lack of interpretation that can be made after the identification, which is why the present study is not based on it. The second approach consists in identifying a model arising from classical mechanical models such as direct or inverse geometrical models to improve the robot positioning. [START_REF] Gong | Nongeometric error identification and compensation for robotic system by inverse calibration[END_REF][START_REF] Hayati | Improving the absolute positioning accuracy of robot manipulators[END_REF][START_REF] Renders | Kinematic calibration and geometrical parameter identification for robots[END_REF] Currently, identifications often need to describe the dynamics of robots, therefore Newton's equations are often a basis for identification models. [START_REF] Gautier | Dynamic Identification of Robots with Power Model[END_REF][START_REF] Geffard | On the Use of a Base Force/Torque Sensor in Teleoperation[END_REF] The main limit of this approach is that in order to take into account all the possible phenomena, the number and observability of parameters can be an issue. Nevertheless, as our study focuses on one degree of freedom, this approach is the one retained in the present study. Newton's equations involve mainly consensual physical interactions and parameters (inertia, Coriolis effect,...), but most of the the time friction is taken into account to model robots. The models used to describe friction are far more diverse and directly depend on the application. [START_REF] Gäfvert | Friction models and friction compensation[END_REF] The most widely spread model is a non-linear model combining a Coulomb dry friction and a viscous friction proportional to velocity. [START_REF] Jarrassé | How can Human Motion Prediction Increase Transparency?[END_REF][START_REF] Gautier | Dynamic Identification of Robots with Power Model[END_REF][START_REF] Pham | Identification of Joint Stiffness with Bandpass Filtering[END_REF][START_REF] Vuong | Dynamic model identification for industrial robots[END_REF] This model is simple and involves very few parameters, however, it can be inaccurate in certain applications. More complex models, dependent on the load, the sign of velocity and the output power have then been developed. [START_REF] Hamon | Dynamic Identification of Robots with a Dry Friction Model Depending on Load and Velocity[END_REF] These dependencies can, for instance, capture the changes of contact surfaces and loads in the transmission when the sign of the power changes. The drawback is that such models involve more parameters and are, consequently, more difficult to identify. Nevertheless, they can be better suited when working with systems that are highly backdriveable and flexible. As presented in the next section, the exoskeleton used in the present study belongs to this category, therefore, models taking into account unusual dependencies such as the sign of velocity are potentially relevant to model it accurately.

The rest of this paper is hence organised as follows: In Section 3, the identification methodology, adapted to the characteristics of the ABLE upper limb exoskeleton, [START_REF] Garrec | ABLE, an Innovative Transparent Exoskeleton for the Upper-Limb[END_REF] is outlined and the obtained results are analysed. Then, in Section 4, the experimental protocol testing the influence of the proposed control laws on human motor control is described and a new performance index based on EMG signals and accelerations is introduced. Finally, the results of these experimental tests are exposed and discussed in Section 5.

Identification

Presentation of ABLE

This study is conducted on an ABLE upper limb exoskeleton (see Fig. 1) with four actuated joints and one free translation that cannot be directly monitored (in terms of position and velocity). These four actuated joints correspond to the three rotations of the human shoulder (internal/external rotation, flexion/extension and abduction/adduction) and to the rotation of the human elbow (flexion/extension). The free joint is the last one of the chain and takes the form of a slider (i.e., a prismatic joint) moving on a rail to adapt the distance between the robot elbow and the human wrist during the movement. The presence of this joint is motivated by ergonomic considerations. The release of translations along the main axis of the limb segment prevents the development of a hyperstatic force. [START_REF] Jarrassé | Connecting a human limb to an exoskeleton[END_REF] This robot has promising capacities in terms of transparency [START_REF] Garrec | Screw and Cable Acutators (SCS) and Their Applications to Force Feedback Teleoperation, Exoskeleton and Anthropomorphic Robotics[END_REF][START_REF] Garrec | ABLE, an Innovative Transparent Exoskeleton for the Upper-Limb[END_REF] because of high levels of backdriveability. Nevertheless, these high levels of backdriveability can lead to complex processes of identification of the dynamics of the robot, starting with friction.

Geometric and dynamic models

The need for high levels of transparency implies, in our approach, a high quality of identification of the dynamic parameters of the robot. The objective is to provide an accurate compensation of the robot's dynamics. Nevertheless, this step is often neglected to the benefit of a closed-loop "black box" control.

General dynamic approach. In ABLE's case, the transmission is designed to be particularly compliant [START_REF] Garrec | Screw and Cable Acutators (SCS) and Their Applications to Force Feedback Teleoperation, Exoskeleton and Anthropomorphic Robotics[END_REF] and, therefore introduces a lot of flexibilities and assembly gaps that generate a lot of friction differences between the different directions of movement. As described in Section 2.2, a lot of friction models already exist in the literature. In particular, the model developed in ref. [32] seems adapted to our situation as it takes into account the sign of the output power and the load. Nevertheless, this model is particularly complex to identify as its identification requires multiple pre-identified external payloads and many trials. In order to limit the number of different coefficients to identify, and therefore improve their observability, and to limit the number of trials, a differentiated identification is performed for upward and downward movements. Finally, the classical model τ f = sign( q)τ C + ν q and its static part expressed as τ f = τ adh , are used for this study because of their limited number of coefficients. As this model does not take into account all dependencies of friction in ABLE, differences of identification between upward and downward movements will be accounted for by the other coefficients, such as the gravitational torque as described in Section 3.4.

Dynamical effects other than friction also affect the transparency, particularly the gravity-related efforts and the inertia of the system. Therefore, the generic dynamic model 45 presented in Eq. ( 1) is identified by considering two sets of the parameters, based on the sign of the velocity, as described previously: where M(q) is the inertia matrix, C(q, q) is the vector containing the Coriolis/centripetal terms, G(q) is the vector containing the gravitational torques, τ C is the Coulomb dry friction torque, ν is the viscous friction coefficient, τ m is the actuators torques vector, q is the joint accelerations vector, q is the joint velocities vector and q is the joint positions vector. As a conclusion, our approach is a new hybrid approach between a non-parametric identification and a parametric identification (see Section 2.2) in the sense that the values held by the identified coefficients are not the "real values" but the form of the model itself arises from Newton's equations.

M(q) q + C(q, q) + G(q) + sign( q)τ C + ν q = τ m , (1)
Identification restrictions. In agreement with previous motor control studies, human elbow flexion/extension movements are of special interest in the present study (see Section 2.1 for the rationale). As a consequence, we shall focus on the identification of the model of ABLE's elbow, which is dependent on the position x of the slider {5} (see Fig. 2). The position of this slider is the main identification problem regarding the present ABLE exoskeleton. As this position is not measurable by internal means, an external measure is needed. The coupled kinematic model integrating both human upper limb and ABLE is presented in Fig. 2.

In this model, G 45 is the centre of gravity of the group {4 + 5}. This kinematic model is used to write two models, a geometric model that is used to estimate the position of the point C in the frame (O, -→ x 4 , -→ y 4 ), which is the robot frame, and a dynamic model that contains all parameters influencing the transparency as described in Section 3.1. The geometric model is written in Eq. ( 2); the two parameters of this model (x and d) should be identified to estimate the position of the slider:

X = x cos(q 4 ) -d sin(q 4 ) Y = x sin(q 4 ) + d cos(q 4 ), ( 2
)
where X and Y are the coordinates of the point C in the frame (O, -→ x 0 , -→ y 0 ), which is the optoelectronic system frame of reference translated to the point O and q 4 is the angular ABLE's elbow position.

The dynamic model derived from Eq. ( 1) and Fig. 2 is written as in Eq. ( 3) to describe the behaviour of ABLE's last actuated joint:

τ m,4 = m 45 g (x 45 (x) cos(q 4 ) + y 45 (x) sin(q 4 )) + ν 4 q4 + J eq (x) q4 + sign( q4 )τ dr y,4 1 r 4 , ( 3
)
where τ m,4 = k t i 4 is the torque of the fourth motor estimated with the motor current i 4 and the motor constant k t . The first parameters that must be identified are m 45 x 45 (x) and m 45 y 45 (x). They will allow to design an accurate gravity compensation taking into account the position of the slider. Then, the following parameters must be identified: τ dr y,4 for the Coulomb friction compensation, ν 4 for the viscous friction compensation and J eq (x), which is the equivalent inertia of segments {4} and {5} at the joint level, for the inertial compensation. Finally, r 4 is the reduction ratio given by the constructor for the last actuated axis. The inertial coefficient J eq (x) is dependent on the slider position by a second degree relation arising from the Huygens theorem. Therefore, this moment of inertia is described by three coefficients, as written in Eq. ( 4).

J eq (x) = a 0 + a 1 x + a 2 x 2 . ( 4
)

Identification methodology

The identification methodology that is applied is a step-by-step methodology. The main advantage of this methodology is to ensure the best possible observability of the different parameters on a mono-axis identification by isolating them and identifying them in different trials. This method also allows to work on more configurations of the device than just the classical sinusoidal and trapezoidal motions and, therefore, to describe other possible behaviours. The inconvenient of this type of method is the risk of accumulating errors on each step of identification, therefore multiple identifications of each coefficients are achieved. A three-step protocol is implemented to conduct the identification. The first step of this protocol consists in acquiring current and position measures while holding static positions in order to achieve a first identification of the gravity torque and Coulomb adherence coefficients. The second step is the acquisition of current, position and velocity measures during constant velocities movements in order to identify again the gravity torque model, to minimise errors accumulation, and identify the Coulomb and viscous frictions compensations. Finally, a wave excitation of the last axis is implemented and measured in order to identify the inertial coefficients presented in Eq. ( 4). Another method, that is not retained in the present study, is to identify all the parameters in one or two trials (as described in Section 2.2). This method gives slightly less accurate results on our robot but reduces drastically the number of trials. As our study focuses on one degree of freedom, the solution giving the better results is retained. As explained in Section 3.2, the position x of the slider during the trials must be known in order to obtain a fine identification of ABLE. As this position cannot be measured by internal means, an optoelectronic system, composed of eight infrared cameras and 3 mm reflective markers placed on the robot, has been used to estimate it ( c Qualisys, Göteborg, Sweden).

The different protocols implemented are described below.

• Static task (position control loop implemented with a Proportional Integral (PI) correction at 1 kHz):

• Twenty positions distributed equally across the whole angular capacity of the last joint for six different positions of the slider distributed equally along the rail to cover the whole working space. • Each position is maintained for 5 s in order to collect large amounts of data and to make sure that the system has reached a stable position. • Constant velocities and dynamic tasks (velocity control loop implemented with a PI correction at 1 kHz):

• Five different control velocities distributed equally across the interval [1, 11] (rev/s) and 25 s of sinusoidal wave excitation for the same positions of the slider as in the static experiment. • Each constant velocity is used for one upward and one downward movement.

The sinusoidal wave used for the identification of the inertia follows the expression presented in Eq. ( 5):

q 4 (t) = A sin(ω(t)t) , where ω(t) = 1 + 3 t T , ( 5
)
where the amplitude is A = 0.5 rad, the duration is T = 25 s and the pulsation ω(t) is expressed in rad/s. For all the tasks, the motor current and the angular position of the motor are measured at 1 kHz. Additionally, the angular velocity and acceleration are estimated by numerical differentiation and the position of the slider is estimated thanks to optoelectronic measures (179 Hz). Least squares methods are used to estimate the axis of rotation of the elbow. All the current measures provided by ABLE's controller and all position measures provided by the optoelectronic system are filtered by a low-pass filter (based on a sliding mean on n values with n = 100 for the current measures and n = 40 for the position measures). All the estimations of parameters are then performed thanks to the "lsqnonlin" function from the Optimization Toolbox of c MATLAB (MathWorks, Natick, MA, USA).

Methodology validation.

As described previously, our approach consists in a dissociated identification according to the sign of velocity. This choice is made to simplify the model to identify (and therefore simplify the identification process) while describing accurately the behaviour of the robot.

To support this methodology, we conduct an analysis on 18 trials carried out under 3 conditions. The 6 sinusoidal trials and the 6 constant velocities trials previously performed for the identification (see the tasks description above) are used and 6 trials at low constant velocity (10 upward and 10 downward movements each at 1 rev/s) with 6 different positions of the slider are added to the analysis. Then, for each trial, the mean absolute model error is computed without dissociation and with dissociation as presented in Eq. ( 6),

e = 1 N N j=1 |i 4, j -î4, j |, ( 6
)
where e is the mean absolute model error in amperes, N is the number of measures of the trial, i 4, j is the jth measure of the motor current, î4, j is the jth estimation of the motor current and "4" is the number of the considered joint (elbow). The estimation of the motor current is obtained with Eq. ( 3) and measurements of the position, velocity and acceleration of the robot.

After gathering data on these trials, a statistical analysis is conducted. Classical indices are computed and a paired t-test is carried out between the 2 sets of 18 values of the model error (one set with dissociation and one set without). The significance value of the t-test is fixed at p < 0.05.

Results

Validation. As expected given the friction model presented previously, [START_REF] Hamon | Dynamic Identification of Robots with a Dry Friction Model Depending on Load and Velocity[END_REF] differences appeared between the models identified for upward and downward movements. Due to the choice of model made in Section 3.1, these differences are passed on to all the identified coefficients. Furthermore, an identification based only on one model without dissociation between upward and downward movements would not be satisfying. The results of the methodological validation are presented in Table I.

The result of the comparison of the two sets with a paired t-test shows a significant difference (t = 5.422, p = 4.572 × 10 -5 ). A boxplot representing these data and an example of the improvement in the identification accuracy with the direction-dependent identifications are given in Fig. 3.

All these results show that our method greatly improves the accuracy of the identification process. Indeed, overall, the mean model error is approximately divided by a factor 3.4 and the reduction of model error was significant on 18 trials. Moreover, these results show that a classical identification without dependency on movement direction results in a relatively large model error, making a purely compensatory law unreliable. Our method could also be applied with a different approach regarding the step-by-step procedure. Indeed, all coefficients could be identified at once as long as the system is solicited in both movement directions with numerous different velocities and accelerations.

Identified values. All the identification parameters obtained through the previous process are reported into the Table II below.

As it can be seen in Table II, our approach increases the number of parameters compared to a non-dissociated approach. As previously stated, this does not increase the number of trials to achieve the identification. Nevertheless, it does increase the complexity of the implementation because of the number of parameters to take into account. For a n-degrees of freedom serial robot with p parameters to identify in each joint ( p being the number of parameters with dissociated identification) and by considering coupling between all axes and directions, the total number of parameters to identify would be,

N = p n-1 i=0 2 i = p(2 n -1), ( 7
)
where N is the total number of parameters and 2 n -1 is the number of directional sets to identify. This exponential result seems acceptable for upper limb exoskeletons that often have a maximum of seven degrees of freedom but it may be a limiting factor for whole-body robots where dozens of degrees of freedom are used.

Experimental Protocol

Materials and task

Participants. Six healthy right-handed young adults (three females, three males, 25 ± 1.3 years, 175.5 ± 5.6 cm height, 63.5 ± 6.1 kg weight) participated in this study. All the participants were naive to the purpose of the experiment. Written informed consent was obtained from each participant as required by Helsinki declaration. [START_REF]World Medical Association Declaration of Helsinki. Ethical principles for medical research involving human subjects[END_REF] The ethical committee for research (Université Paris-Saclay, 2017-34) approved the experimental protocol.

Materials. The experimentations are conducted on the ABLE exoskeleton identified in Section 3. The Qualisys optoelectronic system is used to track the human and robot limbs during movements and therefore to extract the kinematical characteristics of the movement. Reflective markers are placed as Task. The task consisted of point-to-point reaching movements in the sagittal plane involving elbow flexion-extension only. Goals are displayed as targets on a large screen in front of participants as described in Fig. 4(c). A pointer representing the human index forearm line projection is also displayed in real time. Participants are asked to position and keep the pointer on the target location for around 2 s. Once this condition is satisfied, the current target disappears and a new target is displayed. The 0 degree reference is 30 degrees below the horizontal. Five amplitudes of movements are imposed: 12, 24, 36, 48 and 60 degrees rotations around the elbow axis. Movements are performed under four different conditions. Both conditions and amplitudes are randomised accross participants.

For each condition, trials consisted of 10 upward and 10 downward movements per amplitude, resulting in 400 movements per participant in total. Participants take a 2 min pause every 50 movements. Two sets of movements are thus performed per condition. One control condition is carried out without ABLE and the three others with ABLE, involving different control laws (details are given in Section 4.2). The participants are standing straight, their back leaning against the rigid base of the exoskeleton. In case of movements performed with ABLE, the right forearm is attached to the robotic device as described in Fig. 4(a). The participant is wearing a splint to ensure the wrist immobilisation and improve interaction via a contact area augmentation. [START_REF] Jarrassé | Connecting a human limb to an exoskeleton[END_REF]

Transparent laws implemented

Three control laws are implemented. The first law is a closed-loop position control and the second and third laws are open-loop controls purely based on compensations of the dynamic model developed in Eq. ( 3). In the second law, the friction compensation is not taken into account in order to test the effect of an incomplete identification. Only the third law includes a complete compensation of the dynamic model of the last joint of ABLE given in Eq. ( 3). These choices are made to ensure that very different levels of transparency are compared during the experimentations. The first implemented law is a closed-loop position control developed by the constructor (named CLPC in the rest of this paper). The position control is combined with a partial identification of the robot including the Coulomb friction and the gravity torque compensations. The other parameters of the dynamic model are not taken into account, in particular, there are no compensation of the viscous friction, no compensation of the inertia and no difference of compensation between upward and downward movements. This law is used as a representative of basic identifications, without dependency on movement direction, coupled to closed-loop control to compensate model error. Furthermore, the position control is achieved by estimating the next position of the robot based on the current estimation of the velocity derived from encoder positions and the sampling frequency of the control panel (1 kHz). The estimation of velocity derived from encoders is known to be problematic, particularly for low velocities as described in ref. [48], and could therefore lead to a decrease in transparency near movement initiation. This law is used as a basis for augmented reality tasks by the constructor and provides a first level of transparency. A difference in experimental conditions between the experimentations with this law and with the other conditions is the reference position of the robot that could not be locked for this control law because the implementation provided by the constructor does not allow to mix a classic position control for the shoulder and a transparent control for the elbow. In order to reduce the bias linked to this behaviour, the robot is manually maintained in position during the experimentation. This bias is in favour of this law for the comparison because of its tendency to decrease the overall rigidity of the system, and therefore allow small unwanted movements of the participants.

The two other laws rely on pure compensation of the dynamic model of the elbow of the robot. There is no feedback loop to ensure that the efforts estimated by the identified model are consistent with the input of the user like for the CLPC law and its position feedback. In this sense, these laws can be qualified of open loop. They are based on the results of the identification process (see Section 3), one taking into account the friction law identified to compensate its effects (named FC as Full Compensation) and one not compensating friction (named PCFr as Partial Compensation without Friction). For these two laws, the reference position of the upper arm is locked thanks to a position control of the shoulder based on position measurements and a PI correction to avoid static errors. The full compensation of the inertia is not possible without anticipating the compensation thanks to feedforward architectures. [START_REF] Geffard | On the Use of a Base Force/Torque Sensor in Teleoperation[END_REF] Moreover, obtaining a correct estimation of the acceleration derived from the discrete positions of the encoders can be very complicated, particularly for low velocities. Therefore, a control law simulating the inertial effects is applied directly on the velocity, this law is described in Eq. ( 8):

M(x, qmeas ) = J eq (x) qk meas , ( 8
)
where M(x, qmeas ) is the estimated inertia torque appearing in blue in Fig. 5, J eq (x) is the identified inertia of the last joint, qmeas is the articular velocity of the robot derived from position measures, x is the slider position and k is a tuning parameter whose average value is based on participants' sensitivity. For upward movements, the value k = -1 is chosen whereas for downward movements, the value providing best results is k = -1.9. Preliminary tests, involving five participants different from the main experiment, were performed to adjust the value of k. The tuning was based on subjective feelings reported by the tested participants in an attempt to minimise shaking and resistance to movement. Participants of the main experiment reported no discomfort, blockage or vibration with this tuning. The law presented in Eq. ( 8) tends to increase the motor current at low velocities to overcome the inertia and to decrease the motor currents linked to inertia at high velocities when accelerations of the human movements are minimal (see Fig. 6). The control schemes implemented for the different conditions of the experiment are described in Fig. 5.

Data processing

The angular position of the human elbow is computed as the angle of the forearm made with the vertical. Angular velocity and acceleration are obtained by numerical differentiation of the low-pass filtered (fourth-order Butterworth, 3 Hz cut-off frequency) angular elbow position. The start and end of the effective movements are identified as the first and last frame, when the elbow angular velocity is above 5% of maximum velocity. The movement duration is simply computed as the time difference between the first and last frame of the movement. EMGs' signals are filtered using a bandpass filter (fourth-order Butterworth, [20, 450] Hz cut-off frequency), centred and rectified. [START_REF] Potvin | Less is more: High pass filtering, to remove up to 99% of the surface EMG signal power, improves EMG-based biceps brachii muscle force estimates[END_REF] For each subject and each muscle, EMG signals are normalised by the maximal activity reached during the experiment. EMGs signals start and end are advanced of 0.1 s compared to kinematic signals because of the electromechanical delay [START_REF] Cavanagh | Electromechanical delay in human skeletal muscle under concentric and eccentric contractions[END_REF] and to better capture the EMGs burst that causes the instantaneous acceleration.

Transparency assessment

The transparency of the laws presented in Section 4.2 must be assessed in order to determine the influence of the identification process. As a consequence, the impact of the implemented laws needs to be evaluated on the basis of pre-existing indices. [START_REF] Jarrassé | A methodology to quantify alterations in human upper limb movement during co-manipulation with an exoskeleton[END_REF] These indices are based on global kinematic characteristics of one movement, whether based on the duration of the movement, on the symmetry of the velocity profile and on its smoothness.

Overall kinematics. The pointing task performed by the participants has been accomplished without velocity or duration constraints. As previously observed, [START_REF] Atkeson | Kinematic features of unrestrained vertical arm movements[END_REF] typical unrestrained arm reaching movements are characterised by bell-shaped velocity profiles. Then, basic movement parameters such as movement duration and maximal velocity are a first relevant description of the motion kinematics. It is known that individuals have a preferred movement pace which may be optimally determined by a trade-off between energy and time costs. [START_REF] Berret | Why don't we move slower? The value of time in the neural control of action[END_REF] Furthermore, movement duration and maximal velocity are both known to increase with the amplitude of movement. Affine fits capture quite well these dependencies. The fact that movement velocity increases with amplitude to limit the associated increase in movement time is sometimes referred to as the isochrony principle (although there is no exact isochrony). [START_REF] Viviani | Trajectory determines movement dynamics[END_REF] It has been shown that the wearing of a "transparent" exoskeleton tends to slow down the movement whatever the amplitude. [START_REF] Bastide | Interacting with a 'Transparent' Upper-Limb Exoskeleton: A Human Motor Control Approach[END_REF] Therefore, we assume that simple indices such has movement duration and maximal velocity are relevant for a macroscopic approach in order to assess the transparency of the tested control laws and their effects on human movement kinematics. We also assume that the conservation of the "isochrony" principle (i.e., approximately affine relationships between amplitude and duration or velocity) is necessary to deem a control law as "transparent".

Muscle activation-based index. None of the previous indices can assess the link between motor command and the resulting acceleration of the human forearm even though it is a critical point to assess the transparency. Indeed, it is important to know how the wearing of the exoskeleton impacts the ratio conversion between these two variables. It has been shown that a high correlation exists between acceleration duration phase and agonist muscle burst duration for forearm flexion and extension. [START_REF] Cooke | Movement-related phasic muscle activation: III. The duration of phasic agonist activity initiating movement[END_REF] Furthermore, the amount of activation during the two agonist muscle bursts involved in a typical triphasic-pattern is also correlated with the acceleration of a rapid movement. [START_REF] Cooke | Movement-related phasic muscle activation. II. Generation and functional role of the triphasic pattern[END_REF] In addition, the maximal acceleration is a good indicator of the movement plan because motor corrections from sensory feedback have little time to come into play. An EMG-based transparency index is thus defined as the ratio between the agonists Root Mean Square (RMS) activation and the maximal acceleration during the movement phase. We then subtracted and normalised the obtained value with the control condition value (without exoskeleton). A value equal to zero reflects that the RMS activation of agonists muscles create the same acceleration than without exoskeleton. This index is called "EMG/Acc" and reflects the muscle activation yield. This index may provide useful information because it integrates measures that participate in the initialisation of the movement and, therefore, are good indicators of the planning of the movement operated by the participants and of the potential perturbation induced by the exoskeleton. As previously shown, [START_REF] Ingram | A single-rate context-dependent learning process underlies rapid adaptation to familiar object dynamics[END_REF][START_REF] Shadmehr | Adaptive representation of dynamics during learning of a motor task[END_REF] the planning of the movement is strongly influenced by exterior perturbations that are internally modelled by humans such as, in this case, an exoskeleton. Therefore, this index reflects how hard the participants expect the movement to be. The expression of this index is given in Eq. ( 9):

E MG/Acc = 1 C A agonist,RM S qelbow,max -1, (9)
where A agonist,RM S is the mean RMS activation of the agonist muscles (here the biceps and brachioradialis), qelbow,max is the maximal angular acceleration of the elbow and C = A agonist,RM S / qelbow,max is the ratio obtained with the control condition (without exoskeleton). This index is not computed for downward movements because of the gravity working in the same direction as the agonists muscle (i.e., triceps brachial lateral head and triceps brachial long head). Indeed, the extensors activation is not solely responsible of the movement. An inactivation of antigravity muscles is observed initially to accelerate a downward movement. [START_REF] Gaveau | A cross-species neural integration of gravity for motor optimisation[END_REF] Observed bursts were too weak to get exploitable data for these downward movements executed at a self-selected velocity (but this could be circumvented in future studies by asking participants to move faster).

Statistical analysis

Descriptive and inferential statistical analyses are performed on the participants' mean values so that the resulting variability is between participants. The Friedman non-parametric statistical test is performed with studied control laws as the factor. This test gives an estimated χ 2 score which allows to obtain a p-value from a χ 2 table. Post hoc comparisons are conducted using the Nemenyi test. This is a conservative test allowing pairwise comparisons. Both Friedman and Nemenyi statistical tests are performed using the PMCMR R package. [START_REF] Pohlert | The Pairwise Multiple Comparison of Mean Ranks Package (PMCMR)[END_REF] The Nemenyi test is performed separately for upward and downward movement directions in case of the relative duration to maximum velocity.

For the "EMG/Acc" index, analyses are performed only for upward movements for reasons explained in Section 4.4. For all indices, the tests are carried out on the average values over the amplitudes. For all tests the level of significance is fixed at p < 0.05.

Results and Discussion

Overall kinematics. The first observation that can be done is that the global profile of vertical arm movements previously observed [START_REF] Atkeson | Kinematic features of unrestrained vertical arm movements[END_REF] is preserved when wearing an exoskeleton. Indeed, the bell-shaped velocity profiles are present and the trajectories and acceleration profiles are similar, as exemplified in Fig. 6. Then, as detailed in Section 4.4, the average durations of all movements (all amplitudes and participants pooled together) is tested through the control laws. These mean durations are reported together with standard deviation, minimum and maximum values and median in Table III. At first sight, the FC law seems to affect the execution duration and its standard deviation less than the other two laws. This main observation is confirmed by a statistical analysis. Statistical differences were observed on movement duration data (χ 2 = 14.4, p = 0.002). The pairwise comparison showed no significant difference between the Ctrl and the FC laws. Nevertheless, statistical differences appeared between the Ctrl and the two other laws, for these two laws the result is p = 0.037. The same statistical differences (i.e., p = 0.37) appeared between the FC law and these two laws. The conclusion given by this statistical analysis is that, in terms of movement execution durations, the FC law has no significant impact contrary to the PCFr and CLPC laws. This result confirms that, as previously introduced, [START_REF] Jarrassé | A methodology to quantify alterations in human upper limb movement during co-manipulation with an exoskeleton[END_REF] the duration of the whole movement can be a good first-sight performance index to estimate the achieved level of transparency.

The second pre-existing performance index that can be used to evaluate transparency is the conservation of the maximum velocity of the participants. The results obtained with the different laws are compiled in Table IV. The Friedman test is significant (χ 2 = 15.4, p = 0.015) which means that statistical differences were observed on the movement maximal velocity data. The pairwise comparison showed no statistical difference between the Ctrl and the FC laws ( p = 0.808). Furthermore, statistical differences appeared between the Ctrl and the two other laws. The result is p = 0.019 for the PCFr law and p = 0.004 for the CLPC law. No statistical difference is observed between FC and PCFr laws (i.e., p = 0.183) and between FC and CLPC laws ( p = 0.066). Finally, according to this index, the FC law has a lower impact on the human movement than the two other laws because there is no significant difference between this condition and the Ctrl condition as opposed to the PCFr and CLPC conditions.

The third kinematic index that can be tested is the conservation of the isochrony principle. If this index is not verified, the tested control law cannot be qualified as transparent because it would imply that the movement cannot be seen as unrestrained, which is the objective of a transparent law.

Despite differences in movement velocity across control laws, the affine relation between amplitude and mean velocity is preserved (see Fig. 7). As can be seen in Table V, participants averaged relationships are strong and coefficients of determination (R 2 ) are comparable to those obtained in the motor control literature. [START_REF] Berret | Vigour of self-paced reaching movement: Cost of time and individual traits[END_REF] Hence, even if there are some minor differences on the slopes and intercepts across conditions it is important to note that this fundamental human movement feature is present with the three control laws. Conclusion on pre-existing indices. Overall, we can conclude that the three implemented laws can be qualified of transparent because they preserve the isochrony principle and the overall shapes of the position, velocity and acceleration profiles. For the FC law, neither the mean duration nor the mean maximal velocity of movement are significantly different from the Ctrl law which is not verified for the other two laws. Therefore, the FC law appears to be the most transparent according to global conservation of the kinematic characteristics related to the velocity of the human movement.

EMG/Acc index. The new index introduced in this paper, described in Section 4.4, is the "EMG/Acc" index. This index is computed on the data retrieved during the experimentations and the results are available in Fig. 8. The mean results displayed in Fig. 8 show an improvement of the muscular activation/acceleration yield with the use of the FC law compared to the two other laws. The CLPC law seems to be the least transparent according to the EMG/Acc index. In order to confirm these observations, the same statistical analysis as previously has been conducted. The Friedman test showed significant differences between the conditions (χ 2 = 12, p = 0.002) and the pairwise comparison post hoc showed a significant difference between the FC law and the CLPC law ( p = 0.001). The statistical difference between the FC law and the PCFr law is not proven by the available dataset ( p = 0.193). These results show that with the FC law, participants are more likely to produce a greater acceleration with the same amount of muscular activation than with the CLPC law. This supports the fact that the EMG/Acc index can be used to express the level of transparency of a law because it shows quantifiable differences between the different conditions, in the same trend of pre-existing indices. This also confirms that the implemented FC law gives a better level of transparency than the other two laws on average and than the CLPC law via a statistical analysis.

Conclusion

The first aim of this study was to improve the transparency of a powered upper limb exoskeleton. [START_REF] Garrec | ABLE, an Innovative Transparent Exoskeleton for the Upper-Limb[END_REF] The transparency at the elbow axis level is enhanced by using a direction-dependent model for identification inspired by the direction-dependent friction model previously introduced [START_REF] Hamon | Dynamic Identification of Robots with a Dry Friction Model Depending on Load and Velocity[END_REF] and by using a control law trying to compensate the whole dynamic of the exoskeleton forearm. Then, we evaluated the behaviour of our new control law on six participants and compared it to the behaviour of two other laws and to the behaviour of participants without the exoskeleton. We also made sure that the behaviours obtained on upward and downward motions followed the same trends and, therefore, confirmed the consistency of our identification procedure. Our results on pre-existing indices show that an accurate identification is a significant step towards a better transparency of exoskeletons. Moreover, the approach based on a dissociation of the identified models for upward and downward movements could be beneficial for this type of devices. Indeed, in order to be usable in the medical or industrial fields, exoskeletons need to reach high levels of backdriveability that can only be achieved by reducing the overall rigidity and precision of the internal guidances. In this paper, we showed that this drawback can be mitigated by this a relatively advanced identification.

The second objective of this study was to introduce a new performance index to evaluate transparency. This index is based on the ratio between agonist muscle activation (measured by EMG) and peak acceleration of the human limb after movement initiation. It takes the form of a normalised ratio of these two features of the human movement and is called EMG/Acc. This index showed significant differences between two of the different implemented transparent control laws. As the results followed the same trend as those obtained with pre-existing indices, this index may be a relevant indicator of transparency. The main advantage of this index is to assess the level of transparency with regard to the efficiency of the conversion between muscle activity and peak acceleration, as well as the effects of wearing an exoskeleton. Therefore, this index is relevant in both industrial and medical applications. Furthermore, this index could be used in any one-degree-of-freedom task to validate or study different types of laws, such as transparent or assistive laws. Nevertheless, the generalisation to n degrees of freedom seems to be complex as the treatment of EMG of n agonists muscles acting simultaneously can be very challenging.

As closed-loop position-based control laws have shown limited capacities in terms of improving the level of transparency, future studies will be conducted after the implementation of a force sensor at the level of the wrist. This force sensor will allow the quantification of the interaction efforts and the development of a closed-loop control law based on its measurements that should improve the level of transparency further. Another measure that could be included is EMG which could lead to a predictive control law, anticipating the direction and the starting force of the movement. This transparent law would then be used as a basis to study the human motor control when wearing an exoskeleton.

Fig. 1 .

 1 Fig. 1. Picture of the 4D ABLE exoskeleton used in this study.

BFig. 2 .

 2 Fig. 2. Kinematic model of the last axis of ABLE with human forearm.

Fig. 3 .

 3 Fig. 3. Results of the validation procedure. (a) Boxplot representing the distribution of errors across trials for both methods. (b) Motor current for a variable velocity command. The red line shows an identification improvement.

Fig. 4 .

 4 Fig. 4. Description of the experimental setup. (a) Markers on the forearms (human and robot). (b) EMGs on the human arm. (c) Target pointing example.

Fig. 5 .

 5 Fig. 5. Control schemes of ABLE implemented. Red: K p = 0 for the FC and PCFr laws. Blue: M(x, qmeas ) = 1 for the CLPC law. Orange: ν = 0 for the FC law. Green: τ dry = 0 for the PCFr law.

Fig. 6 .

 6 Fig. 6. Averaged position, velocity and acceleration on upward 36 • movements for one participant. Ctrl: Control condition without wearing the exoskeleton. FC: Full Compensation based on identification. PCFr: Partial Compensation based on identification without Friction compensation. CLPC: Closed-loop position control.

Fig. 7 .

 7 Fig. 7. Isochrony principle verification for the different conditions. Equations of affine functions are given with V the mean velocity and A the amplitude. Ctrl: Control condition without wearing the exoskeleton. FC: Full Compensation based on identification. PCFr: Partial Compensation based on identification without Friction compensation. CLPC: Closed-loop position control. (a) Upward movements. (b) Downward movements.

Fig. 8 .

 8 Fig. 8. Agonist muscle transparency index for the different conditions and for upward movements. FC: Full Compensation based on identification. PCFr: Partial Compensation based on identification without Friction compensation. CLPC: Closed-loop position control.

Table I .

 I Statistics on mean absolute error.

	Method	Mean Standard deviation	Min	Median Max
	Without dissociation (A) 0.159	0.0712	0.0713 0.142	0.268
	With dissociation (A)	0.0473	0.0252	0.0288 0.0429 0.131

Table II .

 II Results obtained through the identification process (x: slider position in m).

	Movement	x 45 m 45	y 45 m 45	τ adh,4	τ dry,4	ν 4
	direction	(m.kg)	(m.kg) (N.m) (N.m) (m.kg/s)
	Downward 0.98x -0.45 -0.13		0.047	0.018
	Static	0.46x -0.19 -0.069 -0.12
	Upward	1.68x -0.57 -0.36		0.028	0.0052
	Movement	a 0	a 1	a 2
	direction	(kg.m 2 )	(kg.m)	(kg)
	Downward -0.0019	-0.043 -0.081
	Static
	Upward	-0.00063 -0.050 -0.096

Table III .

 III Results obtained on movement durations. Ctrl: Control condition without wearing the exoskeleton. FC: Full compensation based on identification. PCFr: Partial Compensation based on identification without Friction compensation. CLPC: Closed-loop position control.

		Mean	Standard	Min Median Max
	Condition duration (s) deviation (s)	(s)	(s)	(s)
	Ctrl	0.957	0.199	0.782	0.870	1.281
	FC	0.980	0.203	0.793	0.900	1.271
	PCFr	1.150	0.260	0.829	1.065	1.520
	CLPC	1.135	0.283	0.834	1.060	1.494

Table IV .

 IV Results obtained on movements maximal velocity. Ctrl: Control condition without wearing the exoskeleton. FC: Full compensation based on identification. PCFr: Partial Compensation based on identification without Friction compensation. CLPC: Closed-loop position control.

		Mean max Standard
		velocity	deviation	Min	Median	Max
	Condition	(rad.s -1 )	(rad.s -1 ) (rad.s -1 ) (rad.s -1 ) (rad.s -1 )
	Ctrl	1.416	0.326	0.928	1.526	1.770
	FC	1.345	0.255	0.967	1.413	1.655
	PCFr	1.145	0.301	0.774	1.181	1.610
	CLPC	1.116	0.191	0.776	1.110	1.518

Table V .

 V Descriptive statistics of the obtained coefficients of determination (R 2 ) for regression analyses on amplitude and mean velocity relationships. Ctrl: Control condition without wearing the exoskeleton. FC: Full compensation based on identification. PCFr: Partial Compensation based on identification without Friction compensation. CLPC: Closed-loop position control.

				Standard
	Condition Direction	R 2	deviation Min Median Max
	Ctrl	Up Down	0.972 0.949	0.028 0.052	0.930 0.873	0.990 0.977	0.997 0.991
	FC	Up Down	0.942 0.938	0.054 0.037	0.846 0.884	0.957 0.948	0.986 0.982
	PCFr	Up Down	0.963 0.968	0.018 0.034	0.927 0.912	0.967 0.985	0.979 0.995
	CLPC	Up Down	0.961 0.953	0.040 0.046	0.892 0.895	0.971 0.972	0.997 0.997

https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0263574720001459 Downloaded from https://www.cambridge.org/core. IP address: 83.202.188.138, on 03 Feb 2021 at 13:15:07, subject to the Cambridge Core terms of use, available at

Acknowledgements

We thank F. Geffard and P. Garrec for meaningful discussions. This work is supported by the "IDI 2017" project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02. This work is supported by the French National Agency for Research (grant ANR-19-CE33-0009).


domain: spi phys

text:
Léa Krafft

Elena Gofas-Salas

Yann Lai-Tim

Michel Paques

Laurent Mugnier

Olivier Thouvenin

Pedro Mecê

AND Serge Meimon
email: serge.meimon@onera.fr

Partial-Field Illumination Ophthalmoscope: improving the contrast of a camera-based retinal imager

Effective and accurate in-vivo diagnosis of retinal pathologies requires high performance imaging devices, combining a large field of view and the ability to discriminate the ballistic signal from the diffuse background in order to provide a highly contrasted image of the retinal structures. Here, we have implemented the Partial-Field Illumination Ophthalmoscope, a patterned illumination modality, integrated to a high pixel rate adaptive optics full-field microscope. This non-invasive technique enables us to mitigate the low signal-to-noise ratio, intrinsic of full-field ophthalmoscopes, by partially illuminating the retina with complementary patterns to reconstruct a wide field image. This new modality provides an image contrast spanning from the full-field to the confocal contrast, depending on the pattern size. As a result, it offers various trade-offs in terms of contrast and acquisition speed, guiding the users towards the most efficient system for a particular clinical application.

INTRODUCTION

In order to provide a better understanding and early diagnosis of age-related macular degeneration (AMD), it is critical to be able to follow the retinal degeneration, and in particular the loss of photoreceptor cells [START_REF] Curcio | Photoreceptor loss in age-related macular degeneration[END_REF]. However, this task requires high resolution imaging systems, able to provide highly contrasted photoreceptor maps with a wide field of view. The need for distortion-free images is also essential, particularly for longitudinal studies, when distinguishing the morphological characteristics of the retinal structures [START_REF] Kolb | Webvision: The Organization of the Retina and Visual System[END_REF] and their evolution through time should not be hindered by distortion artifacts coming from the imaging system and the ocular motion [START_REF] Salmon | An Automated Reference Frame Selection (ARFS) Algorithm for Cone Imaging with Adaptive Optics Scanning Light Ophthalmoscopy[END_REF][START_REF] Cooper | Effects of Intraframe Distortion on Measures of Cone Mosaic Geometry from Adaptive Optics Scanning Light Ophthalmoscopy[END_REF]. Therefore, there is a need for highly resolved, distortion-free images of the retina, with a high contrast and a wide field of view.

To this end, ophthalmoscopes have to mitigate aberrations, ocular motion [START_REF] Jarosz | High temporal resolution aberrometry in a 50-eye population and implications for adaptive optics error budget[END_REF][START_REF] Mecê | Fixational eye movement: a negligible source of dynamic aberration[END_REF] and a high diffuse background originating from scattering effects in the multi-layered retina [START_REF] Burns | Adaptive optics imaging of the human retina[END_REF]. Regarding aberrations, Adaptive Optics (AO) has enabled high resolution images of the photoreceptor layer in both full-field ophthalmoscopes [START_REF] Liang | Supernormal vision and high-resolution retinal imaging through adaptive optics[END_REF] and Scanning Laser Ophthalmoscopes (SLO) [START_REF] Roorda | Adaptive optics scanning laser ophthalmoscopy[END_REF], revealing structures that were hidden otherwise [START_REF] Burns | Adaptive optics imaging of the human retina[END_REF][START_REF] Godara | Adaptive Optics Retinal Imaging: Emerging Clinical Applications[END_REF][START_REF] Paques | Adaptive optics ophthalmoscopy: Application to age-related macular degeneration and vascular diseases[END_REF].

Regarding ocular motion and background concerns, a tradeoff has been made either in favor of motion handling, with full-field ophthalmoscopes or in favor of contrast [START_REF] Liang | Supernormal vision and high-resolution retinal imaging through adaptive optics[END_REF][START_REF] Gofas-Salas | High loop rate adaptive optics flood illumination ophthalmoscope with structured illumination capability[END_REF] with the SLO. The full-field ophthalmoscope simultaneously acquires the signals coming from all the positions of its wide illumination field (4 • typically), with exposure time below 10 milliseconds, which prevents motion-induced distortions. On the other hand, the confocal pinhole of the SLO spatially filters out multiplyscattered and out-of-focus light hence improving the quality of the images. However, the full-field ophthalmoscopes remain limited in contrast, while the SLO are prone to motion-induced distortion due to their low pixel rate, which clearly illustrates the complementarity in strengths and weaknesses of these techniques.

In this paper, we implemented the Partial-Field Illumination Ophthalmoscope (PFIO), a high pixel rate AO-full-field ophthalmoscope in which the illumination field can be tuned in order to reduce the diffuse background and improve the contrast of photoreceptor cell images. We demonstrate that spatial filtering can be achieved in a full-field ophtalmoscope by partially illuminating the retina (by placing a programmable mask conjugated with the retina in front of the full-field illumination source). By setting the camera pixels associated with non-illuminated areas on the acquired frame to zero, a numerical spatial filtering is performed similarly to using a variable confocal pinhole in SLO.

We establish that the contrast can be improved proportionally to the square root of the surface of the illuminated area. The full field image can then be obtained by this process ( i.e.image acquisition and numerical spatial filtering) for all the complementary patterns required to illuminate the entire field and by combining these successive images. The resulting image therefore has an increased contrast at the cost of reduced acquisition speed.

This partial-field illumination ophthalmoscope can be positioned between full-field and confocal opthalmoscopes. The characteristic illumination patterns of these three ophthalmoscopes are shown in Fig. 1. For sake of comparison, let us consider the same illumination power in the illuminated area, and the same amount of overall power to compute a complete fullfield image. Thus, the acquisition time of the full-field image scales with the number of complementary patterns N used to recontruct the full field image. The full-field ophthalmoscope only uses one pattern (N = 1) to illuminate the entire region of the retina to be imaged. Therefore, it is the fastest system. However, as no spatial filtering is performed, it provides images with the lowest contrast. In contrast, in the confocal system, the illumination beam diameter corresponds approximately to the Airy disk, so it requires scanning of the region of interest with about one million points (N 10 6 ) to illuminate the entire field of view for an image 2000 × 2000 pixels sampled at Nyquist frequency. The acquisition speed is thus the slowest. In the detection path, a physical confocal hole carries out spatial filtering that enables providing the images with the highest contrast. Between these systems, the partial-field illumination ophthalmoscope illuminates the region of interest partially with an intermediate number of complementary checkerboard patterns (e.g. N = 2 in Fig. 1). As a result, the three ophthalmoscopes in Fig. 1 feature an increasing contrast from left to right, and an increasing speed from right to left.

Our goal here is to answer the following question: for a given field of view and a given number of incident photons, in how many patterns should we split them to get the best image? In other words, which is the optimal N value ? We already know that the higher the N value, the higher the contrast. Nevertheless, a lower N means that more pixels are acquired at a time, which leads to less time-consuming acquisition procedure, and hence less motion-induced distortion. The study presented here investigates the possibility to use a lower N value than the confocal ophthalmoscope while keeping the same contrast.

METHOD A. Setup

The experimental setup has been implemented on the AO-fullfield ophthalmoscope, built at the Paris Quinze-Vingts National Ophthalmology Hospital, described in [START_REF] Gofas-Salas | High loop rate adaptive optics flood illumination ophthalmoscope with structured illumination capability[END_REF] and shown in Fig. 2. We included a new illumination path which consists of a homogeneous incoherent illumination at 850 nm and a spatial light modulator that generates and switches the checkerboard patterns. To perform the various checkerboard patterns, we use an off-the-shelf digital micro-mirror device (DMD) (DLP650LNIR, Texas Instrument) for its high switching speed (up to 12.5 kHz) and its versatility in terms of pattern designs. The array of 1280 × 800 digital micro-mirrors is conjugated to the target layer of the retina and each micro-mirror of 10.8 µm pitch is switched to position "ON" or "OFF" to shape the illumination into the selected pattern to be projected. The mirrors in the "ON" position reflect the incoming light towards the optical path, while the mirrors in the "OFF" position reflect the light out of it. Each pattern, leading to illuminated and non-illuminated areas on the retina, has been uploaded to the controller of the DMD, manipulated through a computer. In this configuration, the mirrors, which are used as switches that shape the illumination, are comparable to multiplexed pinholes. Therefore the DMD, which switches from one pattern to the other, behaves similarly to a Nipkow disk used in spinning disk microscopy [START_REF] Stehbens | Imaging Intracellular Protein Dynamics by Spinning Disk Confocal Microscopy[END_REF][START_REF] Enoki | Singlecell resolution fluorescence imaging of circadian rhythms detected with a Nipkow spinning disk confocal system[END_REF].

Fig. 2.

Experimental set-up: AO-Full-field optical bench described in [START_REF] Gofas-Salas | High loop rate adaptive optics flood illumination ophthalmoscope with structured illumination capability[END_REF], to which an illumination path, composed with a DMD, has been implemented to project checkerboard patterns on the target layer of the retina.

In this setup, the DMD is optically conjugated with the camera. The camera image benefits from the adaptive-optics correction to reach a diffraction-limited resolution [START_REF] Gofas-Salas | High loop rate adaptive optics flood illumination ophthalmoscope with structured illumination capability[END_REF], but the DMD patterns projected onto the imaged retinal plane also benefits from the AO correction, because the incident beam is precompensated for ocular aberrations as well. To navigate between the various layers of the retina, a static defocus offset is sent to the AO-loop, which ensures that both camera and DMD pattern remain conjugated.

B. Pattern projection

Patterns were created with Interactive Data Language (Exelis, USA). We used four sets of complementary checkerboard designs, with the same dimension of squares. Each checkerboard design corresponds to N complementary patterns that illuminate a ratio of one out of N squares, with N = 2, 4, 9, 25. We aim at a real-time enhancement of the contrast of photoreceptors, namely, a minimal processing of the raw video frames. To do so, images with a field of view of 4 • × 4 • were acquired at 90 Hz, with an exposure time of 5 milliseconds, while projecting sequentially the complementary patterns on the retinal layer. The checkerboard squares projected were chosen to be around 44.1 µm wide on the retinal plane, which correspond to 60 camera pixels, as the pixel size of the camera is 6.5 µm and the magnification between the retina and the imaging camera is 8.82. The patterns were switched every two frames.

C. Image acquisition and processing

In-vivo retinal imaging was performed on a healthy young subject. The participant followed institutional guidelines and adhered to the tenets of the declaration of Helsinki. Informed consent was obtained from the subject after the nature and possible outcomes of the study were explained. The subject was seated in front of the AO-full-field ophthalmoscope, stabilized with a chin and forehead rest and was asked to stare at the fixation target, which is an image of a blue crosshair with a moving dot. The fixation target is only used to guide the subject's sight, enabling us to explore the retina. Image acquisition sessions were performed in standard conditions with neither pupil dilation nor cycloplegia, in a dark room, leading to the largest accessible natural pupil dilation. As shown in [START_REF] Meimon | Pupil motion analysis and tracking in ophthalmic systems equipped with wavefront sensing technology[END_REF], during the acquisitions, the subject's pupil size remains steady. Moreover, we made sure that the subject's pupil was centered and larger than the pupil of the system. As the pupil of the system is limited by the deformable mirror and corresponds to 7.2 mm at the pupil plane, the Airy disk diameter is thus about 2 µm. The size of one checkerboard square is about 22 Airy disks, that is 22 times bigger than the Airy disk of the illumination path of a confocal ophthalmoscope. The total light power entering the eye from the illumination source and the wavefront sensor source are respectively 2000 µW and 2 µW, which is below the power values stipulated by the ocular safety limits, established by ISO standards for group 1 devices.

In the first experiment, raw images of the same region of the retina were acquired at an exposure time of 5 milliseconds, with the partial-field illumination ophthalmoscope, for values of N corresponding to 1 (full-field), 2, 4, 9 and 25, under the same experimental conditions. As a basis for comparison to typical AO-SLO images, corresponding to N 10 6 , acquisitions of the same region were also performed with the multimodal adaptive optics retinal imager (MAORI) (Physical Sciences, Inc., Andover, MA, USA) at the Quinze Vingts Hospital in Paris, described in [START_REF] Hammer | Multimodal adaptive optics retinal imager: design and performance[END_REF]. Then, one raw image was selected for each of the PFIO and the SLO images. For each raw image, ten 32 µm × 32 µm regions were selected inside bright areas to compute the Michelson contrast, which corresponds to the ratio I max -I min I max +I min , where I max and I min are the maximum and minimum intensities in each of these regions. We derived the mean and standard deviation of the contrast for each pattern from these values.

In the second experiment, a sequence of images corresponding to N=4 was acquired while projecting sequentially four complementary patterns with the DMD, similarly to what is presented in the figure 1 for N = 2. The images were still acquired with an exposure time of 5 milliseconds. Only the sequences which include saccades needed to be discarded, that is less than 10% the sequence. As the DMD and the camera were not synchronized, we had to acquire two frames per pattern and to discard one frame over two. A subsequence of four raw images corresponding to the four consecutive patterns was selected. The images were cropped to get a region of interest of 1024×1024 pixels, the consecutive frames of complementary projected patterns were manually registered. These frames were finally summed. As a result, one 2.54 • × 2.54 • image (0.75 mm ×0.75 mm) of the photoreceptor layer is reconstructed.

RESULTS AND DISCUSSION

The checkerboard designs described in the Method section were projected on the same area of the retina, under the same experimental conditions. From these data, we aim at evaluating the contrast of the photoreceptors with different ratios of illuminated areas over the field of view, namely when changing the parameter N. In Fig. 3, we show the acquired images for N ranging from 2 to 25, over a wide field of view (1024 × 1024 pixels) and over the same ROI of 256 × 256 pixels. We compare them to both the full-field image (N = 1), acquired with the same optical path, and with the confocal image (N ≈ 10 6 ). All images are obtained with the same amount of incident photons. The images are in accordance with our first assessment as we observe a significant enhancement of the imaging contrast as N increases. The haze is a prominent feature of the full-field image, and is also quite visible in the dark areas of the partial-field illumination ophthalmoscope images. We can see that it is gradually reduced when increasing N. As a result, the photoreceptors appear more contrasted and it becomes easier to distinguish them individually.

In Fig. 4, we aim at quantifying this tendency by plotting the average Michelson contrast, corresponding to each illumination pattern. The bright regions where the contrast was computed were selected at different locations in the field of view to mitigate the influence of a possible inhomogeneity of the background noise. The Michelson contrast over the illumination pattern N follows a √ N fit, as displayed in red. We computed a first degree polynomial fit of the data to the equation model p(1) × √ x in Matlab (using the 'polyfit' function).

The appearance of the photoreceptors in Fig. 3 and the plot of the Michelson contrast in Fig. 4 confirm the hypothesis in Fig. 1 that the contrast increases with N, more precisely as √ N. Following this trend, a confocal ophthalmoscope with, say, a ratio N ≈ 10 6 between the field-of-view and the size of the confocal spot ( i.e.one Airy disc in this case), should yield a Michelson contrast √ N ≈ 1000 times higher than the contrast of the full-field ophalmoscope. Our finding is that the contrast gain of AOSLO is less than 1000 times higher then full-field. The experimental Michelson contrast of the confocal image is indeed only 15 to 30 times better, which corresponds to a value of N between 200 For the confocal image (N ≈ 10 6 ), the Michelson contrast was measured at 0.8, which would be obtained with our setup for a value of N between 200 and 1000, according to the fit. and 1000. This means that the contrast improvement with N saturates, or conversely that an image with the same contrast as a confocal image could be obtained with a PFIO requiring 5000 times less illumination patterns than what is currently needed in the confocal system. Therefore, this potential technical improvement could drastically reduce the acquisition time required for the highest imaging contrast and fosters further investigation towards contrast enhancement using the partial-field illumination ophthalmoscope. Now that we have discussed the theoretical optimal value of N (supported by experimental measurements), let us investigate the practical implementation of PFIO to obtain a complete full-field image with an improved contrast. For N = 4 for instance, we compare the images of the same region, acquired with the full-field ophthalmoscope, the partial-field illumination ophthalmoscope and the confocal ophthalmoscope, under the same experimental conditions, with an equivalent emitted photon flux over the resulting illuminated field. In Fig. 5 -b andf, we show the resulting image obtained with the partial-field illumination ophthalmoscope (see Methods section). A residual grid coming from the projected patterns is still visible after reconstruction. Although it can be removed through processing or by using overlapping complementary patterns the residual grid does not hinder the contrast assessment on the image and was kept for a fair comparison between the three systems. While in the full-field image, in Fig. 5 -a andd, the photoreceptors are barely visible in a hazy background, the latter is filtered out in the reconstructed image obtained with the partial-field illumination ophthalmoscope, making photoreceptors appear more defined, brighter, and even individually distinguishable. The spatial filtering performance of our technique can also be witnessed in Fourier space, as Fig. 5 -e, g and i particularly shows the computed Fourier transform of the three images. The photoreceptor frequency footprint, or Yellot's ring [START_REF] Yellott | Spectral analysis of spatial sampling by photoreceptors: Topological disorder prevents aliasing[END_REF] appearing gradually brighter from left to right, clearly demonstrates the contrast enhancement of the photoreceptor mosaic.

It is important to point out that the numerical spatial filtering of the multiply-scattered light (i.e. putting to zero the pixels in the dark patches) performed to get the PFIO image proves to be a fundamental step in the process. Skipping the latter and keeping the scattering hazes on the images would indeed result in reconstructing an image equivalent to N = 1, that is the full-field image.

Each of the four partial-field images used to compute Fig. 5 b image was obtained with a τ exp = 5ms exposition (corresponding to the fastest frame rate achievable with our current camera), so that the total exposure time amounts to N × τ = 20ms. For higher values of N, such an exposure time would be unpractical to efficiently observe the retina, which is why we did not attempt to produce PFIO complete images for values of N higher than 4. In order to accelerate PFIO image delivery rate and get closer to an optimal value of N which we estimate to be between 200 the movie of the raw images, zoomed in on the same ROI, after discarding blurred images, as described in the section C, in average 1.5 frames have been removed between two consecutive images, the movie has been slowed down to display each image during 2 seconds, so that one could distinguish each of the frames that have been used to reconstruct the full-field image.) c) Registered AO-SLO image of the same region. Second row: a magnified image of 200 × 200 pixels ROI of each raw image d), f), h) and their corresponding Fourier transform images, both obtained from an identically located ROI on each image, e), g), i). and 1000, a more powerful source and a faster camera could be used (this is indeed the case in confocal imaging, for which the illumination power per surface unit is much higher while the exposure time corresponding to each surface unit is much lower than in our apparatus). In addition, such a N value suggests that using a strip-shaped geometry, which covers 1 over N of the field of view for both the illumination and the detection, could be a more practical implementation than the pattern projection proposed in this study.

Taking it one step further, the partial-field illumination ophthalmoscope could spread across an even larger range of imaging techniques. One straightforward application is the use of the partial-field illumination ophthalmoscope approach to carry out dark-field imaging on a camera-based retinal imager [START_REF] Meimon | Manipulation of the illumination geometry on adaptive optics (ao) flood illumination ophthalmoscope (fio) for dark field imaging of the retina[END_REF][START_REF] Gofas-Salas | Near infrared adaptive optics flood illumination retinal angiography[END_REF][START_REF] Mecê | Optical Incoherence Tomography: a method to generate tomographic retinal cross-sections with non-interferometric adaptive optics ophthalmoscopes[END_REF][START_REF] Gofas-Salas | Improvements to multi-offset adaptive optics scanning laser ophthalmoscopy for in vivo imaging of individual retinal ganglion cell layer neurons in humans[END_REF]. Indeed, by setting the pixels corresponding to illuminated regions to zero, instead of non-illuminated regions, dark-field images could be obtained in a post-processing manner. Although such idea seems simple to put into practice, an important illumination power is necessary, so images from complementary areas can be registered and summed to achieve an entire fieldof-view montage. Another interesting application is to couple the partial-field illumination ophthalmoscope with the recently demonstrated structured illumination ophthalmoscope [START_REF] Lai-Tim | Super-resolution in-vivo retinal imaging using structured illumination ophthalmoscopy[END_REF][START_REF] Lai-Tim | Imagerie haute résolution de la rétine humaine par illumination structurée assistée par optique adaptative[END_REF]. Indeed, as in the structured-illumination ophthalmoscope the super-resolution is limited by the contrast, using the partial-field illumination ophthalmoscope would allow one to achieve even higher super-resolution, since the achievable super-resolution directly depends on the SNR [START_REF] Lai-Tim | Jointly super-resolved and optically sectioned Bayesian reconstruction method for structured illumination microscopy[END_REF]. Finally, the partial-field illumination ophthalmoscope approach could also be a valuable concept to full-field optical coherence tomography (FF-OCT) imaging systems applied for retinal imaging [START_REF] Mecê | High-resolution in-vivo human retinal imaging using full-field OCT with optical stabilization of axial motion[END_REF][START_REF] Mecê | Coherence gate shaping for wide field high-resolution in vivo retinal imaging with full-field OCT[END_REF][START_REF] Scholler | Adaptiveglasses time-domain FFOCT for wide-field high-resolution retinal imaging with increased SNR[END_REF]. The use of the partial-field illumination ophthalmoscope, in fact, could reduce cross-talk (in coherent illumination) [START_REF] Auksorius | Crosstalk-free volumetric in vivo imaging of a human retina with Fourier-domain full-field optical coherence tomography[END_REF], and/or incoherent light level detected by the 2D camera, contributing to an improvement of SNR and contrast of full-field OCT images [START_REF] Scholler | Adaptiveglasses time-domain FFOCT for wide-field high-resolution retinal imaging with increased SNR[END_REF].

CONCLUSION

With the partial-field illumination ophthalmoscope, we propose a new modality that lies between the full-field ophthalmoscope and the confocal ophthalmoscope. By sequentially projecting N complementary checkerboard patterns over the entire field of view and digitally cancelling the non-illuminated pixels, the partial-field illumination ophthalmoscope yields a contrast improvement that scales with √ N .The calculation of the Michelson contrast pointed out how the contrast of the confocal system (N ≈ 10 6 ) could be achieved with a value of N several orders of magnitude lower, in other words, at a significantly higher speed. In this way, the partial-field illumination ophthalmoscope holds promise for widening the range of possibility of existing systems.

Fig. 1 .

 1 Fig. 1. Comparing the full-field ophthalmoscope, the partialfield illumination ophthalmoscope and the confocal ophthalmoscope. Schematic drawing of the characteristic patterns utilized in each technique, similar for the illumination and the detection.

Fig. 3 .

 3 Fig. 3. First row : Single raw wide field images acquired with the full-field ophthalmoscope (N = 1), partial-field illumination ophthalmoscope: N = 2, 4, 9, 25 and the confocal ophthalmoscope (N ≈ 10 6 ), with N indicating the number of complementary frames necessary for reconstruction of the complete field of view. Second row: magnified images of 256 × 256 pixels from the first row.

Fig. 4 .

 4 Fig. 4. Plot of the average and the standard deviation of the Michelson contrast values for each illumination configuration and its fit following a power law proportional to √ N. For the confocal image (N ≈ 10 6 ), the Michelson contrast was measured at 0.8, which would be obtained with our setup for a value of N between 200 and 1000, according to the fit.

Fig. 5 .

 5 Fig. 5. Acquired data on AO-full-field ophthalmoscope, partial-field illumination ophthalmoscope and AO-SLO. First row: a) Raw image acquired on the AO-full-field setup with the conventional full field illumination, b) Reconstructed image obtained with the partial-field illumination ophthalmoscope for N = 4 (see in Visualization1 the movie of the raw images, zoomed in on the same ROI, after discarding blurred images, as described in the section C, in average 1.5 frames have been removed between two consecutive images, the movie has been slowed down to display each image during 2 seconds, so that one could distinguish each of the frames that have been used to reconstruct the full-field image.) c) Registered AO-SLO image of the same region. Second row: a magnified image of 200 × 200 pixels ROI of each raw image d), f), h) and their corresponding Fourier transform images, both obtained from an identically located ROI on each image, e), g), i).

1

 1 Fig. 5. Acquired data on AO-full-field ophthalmoscope, partial-field illumination ophthalmoscope and AO-SLO. First row: a) Raw image acquired on the AO-full-field setup with the conventional full field illumination, b) Reconstructed image obtained with the partial-field illumination ophthalmoscope for N = 4 (see in Visualization1 the movie of the raw images, zoomed in on the same ROI, after discarding blurred images, as described in the section C, in average 1.5 frames have been removed between two consecutive images, the movie has been slowed down to display each image during 2 seconds, so that one could distinguish each of the frames that have been used to reconstruct the full-field image.) c) Registered AO-SLO image of the same region. Second row: a magnified image of 200 × 200 pixels ROI of each raw image d), f), h) and their corresponding Fourier transform images, both obtained from an identically located ROI on each image, e), g), i).

FUNDING

The research leading to these results has received funding from the Agence Nationale de la Recherche (ANR) EyeWin (grant number ANR-18-CE19-0010), the RHU LIGHT4DEAF (grant number ANR-15-RHUS-0001) and from ONERA's internal project PRF TELEMAC.

ACKNOWLEDGMENTS

The authors want to thank Kate Grieve who provided helpful comments to improve and clarify this manuscript.

DISCLOSURES

The authors declare no conflicts of interest.

DATA AVAILABILITY

Original images underlying the results presented in this paper are available in Visualization 1 and can be obtained from the authors upon reasonable request.