Anatolii Khalin

Rosane Ushirobira

Denis Efimov

Géry Casiez

On Computer Mouse Pointing Model Online Identification and Endpoint Prediction

This paper proposes a new simplified pointing model as a feedback-based dynamical system, including both human and computer sides of the process. It takes into account the commutation between the correction and ballistic phases in pointing tasks. We use the mouse position increment signal from noisy experimental data to achieve our main objectives: to estimate the model parameters online and predict the task endpoint. Some estimation tools and validation results, applying linear regression techniques on the experimental data are presented. We also compare with a similar prediction algorithm to show the potential of our algorithm's implementation.

I. INTRODUCTION

C OMPUTER and information technologies conquered a big part of our everyday lives, and in this way, the optimization, in a smooth and pleasurable manner, of Human-Computer Interaction (HCI) processes has become an important issue. There are different approaches to analyzing and designing HCI algorithms, and some of them are based on the control theory. According to [START_REF] Oulasvirta | Computational interaction[END_REF], it allows the user and computer systems to be linked by describing the communication between the parts and the process dynamics as a whole. In such an approach, the human-computer interaction can be presented as a feedback interconnection. An experimental comparison of different methods demonstrates that control theory applications are up-and-coming in the HCI domain [START_REF] Müller | Control theoretic models of pointing[END_REF]. Most applications of control theory in HCI so far are directed to pointing tasks.

Pointing is a fundamental task, and it has been a research subject for many years in the HCI field. A pointing task consists of guiding some device in the control (also called motor) space for navigating the cursor in the operating space (a display). There are some papers about hardware optimization (for instance, a recent study about the optimal sensor position in the mouse [START_REF] Kim | Optimal sensor position for a computer mouse[END_REF]), but the majority of papers in HCI with regards to pointing concern the so-called interaction techniques [START_REF] Blanch | Semantic pointing: improving target acquisition with control-display ratio adaptation[END_REF], [START_REF] Grossman | The bubble cursor: enhancing target acquisition by dynamic resizing of the cursor's activation area[END_REF], [START_REF] Chapuis | Dynaspot: speed-dependent area cursor[END_REF], [START_REF] Blanch | Rake cursor: improving pointing performance with concurrent input channels[END_REF]. Despite the development of new devices, touchpads and computer mice are the most popular input devices for most users, and they are still widely used for such tasks as pointing, tracking, dragging, targeted-tracking [START_REF] Senanayake | Targetedtracking with pointing devices[END_REF], etc. That is why there are several models for direct and indirect pointing tasks available in the literature, providing different levels of the process description.

Regarding the goals of this work, the considered problem can be divided into two parts: the first one is the modeling of a pointing task and the second one is the endpoint prediction.

The first concept related to the modeling of a pointing task was the Fitts law [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], which states that the pointing time has a logarithmic relation to the ratio of distance and the target's width, also called Index of Difficulty (ID). The Fitts law has been experimentally validated many times [START_REF] Zhai | Speed-accuracy tradeoff in fitts' law tasks-on the equivalency of actual and nominal pointing precision[END_REF], [START_REF] Chapuis | Fitts' law in the wild: A field study of aimed movements[END_REF]. However, Fitts law can model pointing time in relation to the ID of the task, but it does not allow to model the dynamics of the movement. The first attempt to provide a dynamical pointing model was the so-called Surge model [START_REF] Costello | The surge model of the well-trained human operator in simple manual control[END_REF]. It was the first try to model the task in two modes: the constant-coefficient model and the second-order controller model, representing open-loop and closed-loop phases, correspondingly. Later, VITE (Vector Integration to Endpoint) model was introduced in [START_REF] Bullock | Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation[END_REF] as a linear time-invariant system describing the motion, controlled by an agonist-antagonist pair of muscles (wrist rotation). Simultaneously, Meyer et al. [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF] established that the pointing task contains two phases: a ballistic (supposed to be an openloop process of the initial movement towards the target) and a correction phase (that finishes the task by closed-loop visual guidance until the target is reached). This concept was called Meyer's Optimized Initial Impulse model [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF]. Another feature of this model was the possibility of modeling the multiple open-loop phases (also called sub-movements). More recently, based on this principle and the VITE model, a united switched dynamical model was created in [START_REF] Aranovskiy | A switched dynamic model for pointing tasks with a computer mouse[END_REF], verified on experimental data of [START_REF] Casiez | No more bricolage! methods and tools to characterize, replicate and compare pointing transfer functions[END_REF], and whose closed-loop stability has been proven. Another line of research was based on Todorov's optimal feedback control of motor coordination [START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF], [START_REF] Todorov | Optimality principles in sensorimotor control[END_REF], [START_REF] Todorov | A generalized iterative lqg method for locallyoptimal feedback control of constrained nonlinear stochastic systems[END_REF]. The main idea was to apply Stochastic Optimal Control to model a human movement [START_REF] Qian | Movement duration, fitts's law, and an infinite-horizon optimal feedback control model for biological motor systems[END_REF], [START_REF] Rigoux | A model of reward-and effort-based optimal decision making and motor control[END_REF], [START_REF] Izawa | Motor adaptation as a process of reoptimization[END_REF]. The latest result of such an approach can be found in [START_REF] Berret | Stochastic optimal feedforward-feedback control determines timing and variability of arm movements with or without vision[END_REF], where, besides feedback, the authors also consider feedforward control, which models the part of the process with motor planning to generate a command (before the movement begins). Another recent idea was based on the so-called Intermittent Control [START_REF] Martín | Intermittent control as a model of mouse movements[END_REF]. The key feature of the latter framework was a build-in eventbased trigger, which allowed the model to switch between closed-and open-loop control depending on the error between the goal point and the one observed by the user. Such an approach appeared to be compatible with current physiological and psychological theories. A different view on modeling worth mentioning is through the information theory. In the recent work [START_REF] Gori | A feedback information-theoretic transmission scheme (FITTS) for modeling trajectory variability in aimed movements[END_REF], the two-phase process is viewed as a communication problem where the information is transmitted from a source (initial/current point) to a destination (endpoint) over a so-called channel, perturbed by Gaussian noise with a presence of feedback (in a second phase). Despite the diversity of approaches for modeling the pointing task, we believe that searching for a simple, and at the same time, an accurate model is yet a relevant research direction.

The second problem considered in this paper with regards to pointing is the endpoint prediction. Indeed, it is possible to facilitate the interaction and shorten the movement time by predicting the user's desired destination. Several papers have been published regarding endpoint prediction, and few algorithms in this area were proposed. Starting from Lank et al. [START_REF] Lank | Endpoint prediction using motion kinematics[END_REF], the Kinematic Endpoint Prediction (KEP) model was introduced. This model is based on the minimum jerk law, formulated in [START_REF] Hogan | An organizing principle for a class of voluntary movements[END_REF], and the proposed prediction relies on a polynomial curve fitting (based on a fourth-order polynomial). The model showed the best performance at 80% of the path and distances more than 600 pixels. Another idea was presented in [START_REF] Asano | Predictive interaction using the delphian desktop[END_REF], where a simple regression-based extrapolation was used, making the endpoint guess at the peak of the velocity. The prediction technique based on the inverse optimal control model was introduced in [START_REF] Ziebart | Probabilistic pointing target prediction via inverse optimal control[END_REF], where a probabilistic model and machine learning were combined. It showed better performance compared to previously proposed models, especially during the short-distance trials. The Kinematic Template Matching concept was introduced in [START_REF] Pasqual | Mouse pointing endpoint prediction using kinematic template matching[END_REF]. The template matching algorithm compares the velocity profile and decides which distance the user wants to cover based on previous trials. The algorithm showed a better performance for the 2D-task than the KEP model and reasonable general accuracy.

We can formally divide all existing endpoint prediction algorithms into two groups:

group I) algorithms with knowledge about previous trials (memory), and group II) without any knowledge (no memory).

In the first group, we can put the algorithms based on the linear regression [START_REF] Asano | Predictive interaction using the delphian desktop[END_REF], any machine learning tools [START_REF] Ziebart | Probabilistic pointing target prediction via inverse optimal control[END_REF], and the recently introduced KTM algorithm [START_REF] Pasqual | Mouse pointing endpoint prediction using kinematic template matching[END_REF] because all of them require several trials of the same user or on the same setup to identify the endpoint online. The algorithms without memory, such as KEP [START_REF] Lank | Endpoint prediction using motion kinematics[END_REF], rely on more general rules and assumptions about the motion, and they can be applied from the first try.

Another innovative idea for facilitating the target acquisition for the user was presented in [START_REF] Rozado | Mouse and keyboard cursor warping to accelerate and reduce the effort of routine hci input tasks[END_REF]. The authors used a target prediction approach with the eye-gaze device, and the idea was to skip the ballistic phase entirely by cursor warping (moving the pointer directly to the area of users' sight). The results of the experiments showed a significant decrease in the pointing time. In this paper, we will follow the classical pointing task setup, using only the computer mouse for our prediction algorithm, since it is the case for most users.

Despite all these achievements, the current state of the art can still be improved in many ways. For example, since all the algorithms mentioned above do not utilize a dynamical pointing model representing both sides of the process, the results are less general and less accessible for analysis. Algorithms based on machine learning or probability techniques do not explain the nature of the movement well, and their results are unreliable in unusual setups and situations.

Both problems described above are usually separated and not considered together in a single model-based prediction algorithm, which could be more advantageous for description and analysis. We fill the indicated gap in the present work, and an endpoint prediction algorithm is presented, based on a new simplified version of the dynamical model given in [START_REF] Aranovskiy | A switched dynamic model for pointing tasks with a computer mouse[END_REF]. The aim of simplifying the model from [START_REF] Aranovskiy | A switched dynamic model for pointing tasks with a computer mouse[END_REF] is to improve the possibility of online parameter identification and allow the process characteristics to be adjusted or modified online. Another advantage of the presented model is its portability since we do not suppose to know the used surfaces that the computer mouse operates on, the Pointing Transfer Function (PTF) of the operating systems (which will be introduced later in the text), or other varying setup conditions. A recent adaptive observer technique [START_REF] Wang | Finite-time parameter estimation without persistence of excitation[END_REF] is used to adjust the model parameters and identify everything on the fly. It is suggested how to modify the PTF once the endpoint coordinates are derived. Experimental data with a computer mouse [START_REF] Casiez | No more bricolage! methods and tools to characterize, replicate and compare pointing transfer functions[END_REF], [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF] is applied for this work's model development and validation.

II. PROBLEM STATEMENT

The problem considered in this paper is modeling and parameter identification of an indirect pointing task.

The main idea here is to use the automatic control theory to design the pointing model, where the user is a controller in the closed-loop, whose behavior and decision we do not know exactly, but we can measure and estimate it indirectly and predict the desired position. The interaction and related input and output signals are presented in Fig. 1 for the xdirection displacement (there is also the y displacement in the 2-dimensional case). We suppose that the user positions the cursor on the screen having x and/or y as an input and transforms it with his decision into a wrist movement, producing the force that translates the mouse, which has the position x m and y m on the plane. So, the mouse output will be its velocity V m = V 2 mx +V 2 my , where ẋm = V mx and ẏm = V my are the velocity projections on x and y axes, respectively. The term of mass was omitted: we suppose that the mass of a mouse and a hand on it is a unit, and all other parameters appearing in an equation are normalized by the mass. The basic equations for this process can be presented as follows:

ẋm (t) = V mx (t), Vmx (t) = F fr (t) -F input (t), (1)
ẋ(t) = V mx (t) V m (t) PTF(V m (t)), ∀t ≥ 0
where F fr is the mouse friction force, and F input is the force generated by the user, PTF : R → R is a pointing transfer function (see [START_REF] Casiez | No more bricolage! methods and tools to characterize, replicate and compare pointing transfer functions[END_REF]), that transforms the mouse velocity V m into the cursor velocity on the screen V c (t) = ∂ ∂t x 2 (t) + y 2 (t), ∀t ≥ 0 (usually, through look-up tables that realize a nonlinear function in all existing operating systems [START_REF] Casiez | No more bricolage! methods and tools to characterize, replicate and compare pointing transfer functions[END_REF]), the ratio

di f f m [m]; b) t [s] vs V mx and V di f f mx [m/s]; c) t [s] vs a di f f x [s] V mx
V m is used to project the PTF effort on the corresponding movement direction. We assume that the human force F input is proportional to the endpoint position T x (in the ballistic phase) or the current error between x and T x (in the correction phase). Remark 1. Our idea is to estimate the decision process online, predict the reference position T x (which we originally do not know, only the user does), and improve PTF using the estimated value T x of the endpoint T x . To this end, we introduce an additional feedback in the PTF block to adjust the navigation, shorten the pointing time, and possibly reduce the error rate (usually present in all known tests). Thus, we add a correction term ρ(x -T x ) with some tuning parameter ρ > 0, so the last equations in (1) takes the form:

ẋ(t) = V mx (t) V m (t) PTF(V m (t)) -ρ x(t) -T x (t) , ∀t ≥ 0.
It is also worth mentioning that we still do not know how a user will react to such a modification of PTF, will it be more optimal or not, but we will provide the related experiments to check it in a later future. An example of how users can react to a similar approach can be found in [START_REF] Hourcade | Pointassist for older adults: analyzing sub-movement characteristics to aid in pointing tasks[END_REF], where the endpoint prediction was used to help the elderly navigate on the screen.

For 2-D tasks, the equations for the vertical components y m , V my , y, T y , T y can be presented in the same way.

So the motivating technical issue of this work is to estimate the cursor's wanted position T x and T y as fast as possible using the dataset obtained in the experiment [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF].

III. DATA DESCRIPTION AND PROCESSING

According to [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF], a mouse with a frequency of 1 kHz and 2000 CPI was used, and all measurements are provided at discrete instants of time t k [s] with a varying sampling period h k-1 = t k -t k-1 for k = 1, 2, . . . and t 0 = 0 (the sampling is not exactly 0.001 [s] due to the event-based nature of the operating system). We have an optical sensor inside the mouse, that measures the integer value increment of the mouse displace-

ment ∆x k m = x k m -x k-1 m
in counts (with a measurement noise that is skipped for brevity), where x k m = x m (t k ), which can be converted into velocity in [m/s] for the conformity, considering the mouse CPI and inches to meters ratio ι = 0.0254: The mouse position in [m] is obtained from the calculated velocity (scaled mouse displacement) by each time step accumulation:

V mx (t k ) = V k mx = ι ∆x k m CPI h k-1 , k = 1, 2, . . . .
x k m = x k-1 m + h k-1 V k mx .
The cursor position on the screen

x k = x(t k ) in [pixel]
is available at the same time instants. Similarly, we have

y k m = y m (t k ), ∆y k m = y k m -y k-1 m , V my (t k ) and y k = y(t k ).
Examples of obtained noisy signals are shown in Fig. 2 and demonstrate a bad velocity estimation due to the noise. To obtain a better velocity profile and the respective acceleration (which we need later for the mode identification) we applied a homogeneous differentiator to the mouse position (see [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], and Appendix A for the used equations). This type of differentiator's main advantage is that it possesses a finite-time convergence rate [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], which is faster than any exponential, and in addition, it is proven to be robust to the disturbances [START_REF] Bernuau | Verification of iss, iiss and ioss properties applying weighted homogeneity[END_REF]. Finding a trade-off between smoothness and convergence accuracy, the parameters are tuned as

λ = 5, ξ = 2, α = -0.2,
then V mx = z 2 and a mx = z 3 can be used for further parameter identification of the model. The filtered mouse position, velocity and acceleration are derived by applying the homogeneous differentiator in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF] to the mouse position presented in Fig. 2. From these plots, we can conclude that this homogeneous differentiator filters well the noise and generates rather smooth estimates of the velocity and acceleration.

Remark 2. The experiments of [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF] included 10 users, which navigate the cursor on the screen using the mouse in a task with 13 targets positioned along a circle, following the norm ISO 9241-9 ( §B.6.2.2). Two different distances (150 and 75 mm), target widths (7 and 2 mm), and two different PTFs were used. Depending on the velocity or position profile, it is possible to detect which user is more experienced with the setup and who is less. An experienced user makes a ballistic movement very close towards the goal, and the motion is realized in one round without sub-movements (without additional ballistic corrections). A less-experienced user needs more attempts to reach the surrounding area and may produce several sub-movements to attain the goal (see Fig. 3 for illustrations). Another way to define experienced users is according to the movement time: the lower their time on completing the trial, the more experienced they are. In this paper, we will rely on the first definition since our prediction algorithm will be generalized to the cases with sub-movements. The performance also depends on the target width and the number of trials made with the current setup. Therefore, it makes sense to evaluate the model on the trials without additional sub-movements because when a user utilizes his/her personal setup (PC and mouse), the movements are much more precise than with an unfamiliar one. This will be done first, and next, the model presented in this paper is generalized in subsection V.D, considering all possible cases.

IV. POINTING TRANSFER FUNCTION

As was mentioned earlier, the PTF provides the gain from the mouse velocity V m to the cursor one V c . Every operating system has its look-up tables with standard gains, and every mouse velocity input is interpolated accordingly. The set of the final gains constitute a nonlinear function. The authors of [START_REF] Casiez | No more bricolage! methods and tools to characterize, replicate and compare pointing transfer functions[END_REF] created a useful library, called Libpointing, which captures raw mouse data and allows one to create and use one's own PTF.

In this work, we assume that the shape of the PTF is unknown to develop a platform-independent approach. Therefore, it has to be reconstructed from the measured data, and our idea is that a polynomial of the mouse velocity can approximate the PTF function in [START_REF] Oulasvirta | Computational interaction[END_REF]:

PTF(V m (t)) = V c (t) = q ∑ i=0 c i V i m (t), ∀t ≥ 0
where the nonlinear gain is a polynomial of order q with unknown coefficients c 0 , ...c q . Taking into account the discretetime nature of the measurements, we have:

V k c = q ∑ i=0 c i V i m (k),
where

V k c = V c (t k ) and V m (k) = V m (t k ).
The cursor velocity can be approximated by a simple Euler formula

V k c = d k -d k-1 h k-1 ,
where

d k = x 2 k + y 2
k is the distance on the screen from the origin corner, or by applying the homogeneous differentiator in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF] to d k . Next, a simple linear regression estimation technique [START_REF] Ljung | System identification: theory for the user[END_REF] can be used to estimate the parameters c 0 , . . . , c q : c 0 . . . c q = A -1 k ψ k provided that the matrix A k is not singular, where

A k = δ A k-1 + ω k ω k , ψ k = δ ψ k-1 + ω k V k c , ω k = V 0 m (k) . . . V q m (
k) with the regression vector ω k ; ψ k and A k are auxiliary matrix variables, and δ ∈ (0, 1] is a forgetting factor.

After some trials, we obtain the polynomial function of the current PTF, which can be used for modeling and prediction. The higher is the polynomial order, the more accurately the final coefficient set will represent the current PTF in future trials. So, it can be helpful to utilize the model with continuous functions, avoiding the usage of look-up tables.

V. POINTING TASK MODELING AND IDENTIFICATION

A pointing model has to reflect both muscular and perception, human feedback following the current position on the screen. The task is to estimate the desired position of the cursor given by the constants T x (and T y ), based on the dataset of x and V mx (y and V my , respectively). As usual, the real-world data is noisy, and a challenging estimation issue is that we have no good mouse velocity and acceleration measurements in the experiments. In this section, with a small ambiguity of notation, we will use the signals x m , V mx and a mx gathered after applying the homogeneous differentiator, as previously explained. Remark 3. Although we estimate the horizontal and vertical cursor position dynamics separately, the general pointing task is two-dimensional. As observed in the experimental data provided in [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF], the correcting (tracking) phase usually starts approximately in about 1 5 of the maximal amplitude of V mx on its downhill. In the sequel, we use this rule to define the threshold for commutation between the modes.

A. Correction phase

In (1), we choose the input force as a simple linear feedback

F input = b 2 (x -T x ),
with a parameter b 2 > 0, which is unknown and can be different for each trial (such a choice is often assumed in the so-called Equilibrium-point hypothesis [START_REF] Feldman | Once more on the equilibrium-point hypothesis (λ model) for motor control[END_REF]). We can select the dynamic friction force in a usual form

F fr = -b 1 V mx ,
where b 1 > 0 is the friction coefficient, which is also unknown and trial-sensitive. Usually, b 1 > 0 is dependent on the mouse and the surface mouse operated on, but it also depends on the users' hand position, which influences the mass and makes this coefficient not constant, despite operating the same surfaces. If the parameters b 1 and b 2 are non-negative, then the resulting dynamics is stable, which corresponds to our intuition about the pointing process. Thus, estimation techniques can be applied to this model:

Vmx (t) = -b 1 V mx (t) -b 2 (x(t) -T x ), ∀t ≥ 0
and dividing it by b 2 to decouple the main parameter we want to estimate here, T x , we can present this equation in a suitable form for the linear regression:

x(t) = - 1 b 2 Vmx (t) - b 1 b 2 V mx (t) + T x = ω (t)θ ,
where ω(t) = -a mx (t) -V mx (t) 1 is the regression vector, which contains measured signals, and

θ = 1 b 2 b 1 b 2
T x is the vector of unknown constant parameters to estimate. Next, a linear estimation technique (as for PTF previously) can be applied. However, the Persistence of Excitation (PE) condition [START_REF] Bitmead | Persistence of excitation conditions and the convergence of adaptive schemes[END_REF] needs to be satisfied for the convergence of the parameter estimation error, which is not confirmed in our setting (the system is stable, and the external input is constant; therefore, V x and a mx approach zero asymptotically, while x(t) goes to the constant T x as t goes to infinity). Since we mainly need to estimate only one parameter, i.e., the desired position T x , then a strong excitation providing the PE property for all three parameters in θ is not needed. It is useful to separate the estimation variables for better identification. An efficient and recently introduced DREM procedure [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], [START_REF] Korotina | On parameter tuning and convergence properties of the drem procedure[END_REF] (see Appendix B) allows the coefficients in θ to be evaluated separately. This procedure can also provide a finite-time convergent estimation even without the satisfaction of PE condition [START_REF] Wang | Finite-time parameter estimation without persistence of excitation[END_REF] (interval excitation, which is presented in pointing tasks, is enough). In this work, we selected the auxiliary filters as delays: H i (s) = e iτs for i = 0, 1, 2 for some basic delay τ > 0, then using the steps described in Appendix B, we can get three scalar equations with separated parameters:

Y i (t) = ϕ(t)θ i + v(t), ∀t ≥ 0 ( 2
)
where v is the error variable related to the measurement and differentiation noise, and ϕ is the new scalar regressor, which leads to the adaptation algorithm

˙ T x (t) = -γ 3 ϕ(t)(ϕ(t) T x (t) -Y 3 (t)),
with γ 3 > 0 a tuning parameter and T x the estimate of T x . The noise is unknown, but we can detect when its influence is too strong and cut the prediction in corresponding regions.

A simple way to do it would be to introduce an additional condition on the estimate T x . To reduce the big influence of the noise, we can bound T x under the assumption that it is supposed to be close to the endpoint after the ballistic phase. Although the screen resolution naturally bounds the endpoint position, we want to avoid using the introduced model's setupdependent data. We can restrict our prediction estimation using only the cursor position data obtained from the current trial. Because the correction phase serves for the final navigation of the pointer to the goal, it should ideally operate around the current position value. Choosing the approximate evaluation (bound) as 1 10 of the distance already traveled, we can introduce the condition as follows:

T x (t) -T x (t -) < 0.1 |x(t)|
where T x (t -) denotes the estimate of T x at the previous instant of time. If it is satisfied, we update T x . Otherwise, we keep the previous value, cutting out the noisy region affecting the estimation and producing numerical instability.

B. Ballistic phase

In this phase, we assume that the whole movement is realized intuitively or automatically in the motor space based on the averaged PTF gain sensed by the user (i.e., it is an openloop deterministic motion), and the mouse position follows the curve:

x m (t) = T x 2c (1 -cos(ηt)) , ∀t ≥ 0,
where η > 0 is a user-dependent parameter, which regulates the velocity of displacement, and the movement amplitude is proportional to the goal position T x divided by an averaged PTF gain c > 0. This coefficient represents the ratio between the motor and operational spaces, and for the user, it is expected that x(t) = cx m (t).

In such a case x m π η = T x c and x π η = T x . Note that this gain c can be easily estimated if the PTF function is known (if it has been already identified as explained above):

c = V -1 max V max 0 PTF(s)ds,
where V max is the maximal admissible computer mouse velocity for the user (can be extracted from the data). Nevertheless, as we demonstrate below, we do not need this parameter to estimate the endpoint. Thus, if the ballistic movement is wellrealized, and the human approximation of the PTF gain c is adequate, then the cursor has to fall into a vicinity of the desired position T x . Simple calculations show that

ẋm (t) = V mx (t) = T x η 2c sin(ηt), Vmx (t) = a mx (t) = T x η 2 2c cos(ηt), hence, Vmx (t) = -η 2 x m (t) - T x 2c = - η 2 c x(t) - T x 2 ,
and the obtained model can be related to the one proposed previously for the correction phase by imposing F fr ≡ 0 and

F input = b 2 x -T x 2 with b 2 = η 2 c
. Consequently, in the ballistic phase, it is assumed that the friction force can be neglected during such a fast transient and the goal position for the ballistic phase is just half away from the desired set-point T x (this is an interpretation in terms of visual user feedback).

To estimate the goal position T x using this model, let us observe that if for some t ≥ 0, we have a mx (t ) = 0 then x(t ) = T x 2 . So, a basic estimation algorithm during ballistic phase is:

T x (t) = 2x(t), if a mx (t) •V mx (t) ≥ 0 T x (t -), otherwise , ∀t ≥ 0,
where T x (t -) stands for the estimated value of T x at the previous instant of time.

The idea of doubling the position until the peak of velocity for endpoint estimation is not new. In [START_REF] Keuning-Van Oirschot | Cursor displacement and velocity profiles for targets in various locations[END_REF], observing the experimental data, the authors concluded that it could be an option to obtain the users' goal during the movement. We derived this conclusion here from the equations presented for the ballistic phase of the switched model (thus suggesting a mathematical explanation for [START_REF] Keuning-Van Oirschot | Cursor displacement and velocity profiles for targets in various locations[END_REF]).

C. Switched model identification algorithm

The procedure for updating the endpoint estimation T x using the proposed simplified switched model is divided into three modes: a) Ballistic phase before velocity peak (before a mx reaches zero). We look for the velocity V mx peak during this part of the ballistic phase by comparing the current absolute value to the previous maximum. While V mx grows, T x can be found as: b) Ballistic phase after velocity peak. If the current velocity V mx is lower than its maximum, then T x is constant and defined by its previous value at the peak.

T x (t) = 2x(t).
c) The correction phase. Here, we use the DREM estimation procedure as described before.

Moreover, we can generalize such an approach to the situation when the user's estimation of PTF is not adequate.

D. Prediction of trials with sub-movements

Sometimes during the trial, the user cannot reach the goal with a first try. It is a well-known problem in the pointing task. The additional ballistic phases, which appear before the goal is reached (it is easy to recognize them in the velocity profile shown in Fig. 3), are called sub-movements, and they make the endpoint prediction task much more complicated. The switching model presented in this paper allows us to change the mode at any point in time. Therefore, we can generalize the prediction algorithm presented above by considering possible sub-movements.

The generalized idea for updating the endpoint prediction is presented in Fig. 4. Basically, we check ∆ T x at each time step, depending on the conditions of the current value of the velocity V mx (i) and acceleration a mx (i).

First, we search for the maximum of the mouse velocity magnitude by comparing it to the current value. Second, we determine in which phase of motion we currently are. It is the same condition described in the previous section: below 1 5 of the maximum velocity, we use the estimation from the model for the correction phase; otherwise, we use the ballistic prediction. If we are in the ballistic phase, we have the following condition to verify: whether the velocity is increasing with regard to the direction of the motion or decreasing (it means that we require current acceleration and velocity have the same sign to adjust the prediction by 2∆x). Through this modified condition from the previous section, we also consider overshooting: when the target has already passed and the user moves the mouse in the opposite direction, ∆ T x is negative and our prediction is decreasing in the same way as with undershoot, the sign is considered. Simultaneously, we have the corrective phase condition to bound the prediction when the data is the noisiest. Since the noise itself is unknown, we cannot compensate for its influence, but we can restrict the estimation to those moments of time when the estimation is leaving the predefined boundaries. It usually signifies exactly when the measurement data is the noisiest. When it is satisfied, we update the T x ; otherwise, we keep the prediction to avoid possible numerical instabilities and make the algorithm more robust to the disturbances, as explained in subsection 5.1.

Remark 4. We have the same diagram for Ty in parallel at each timestep. It is possible that according to the V mx velocity profile, we are still in the ballistic phase, while according to V my , we are in the corrective phase and vice versa. We assume that this situation is possible, based on the observation of the 2-D cursor trajectories from the experimental data.

The generalized algorithm cannot estimate the final desired value in the case of sub-movements after the first velocity peak. However, we can estimate the sub-movement final value earlier by merely accumulating new estimates with the first peak value.

Let us demonstrate the efficiency of the proposed model and the estimation approach using real data and simulations.

VI. RESULTS AND SIMULATIONS

A. Validation and identification of our pointing model

First, the chosen model in the correction phase was evaluated in the simulation, where the model parameters were assigned artificially: b 1 = 4, b 2 = 7, T x = 600 pxl and PTF= arctan (it can be any bounded function according to [START_REF] Aranovskiy | A switched dynamic model for pointing tasks with a computer mouse[END_REF]). The model identification results are presented in Fig. 5, the final position and other coefficients are estimated almost simultaneously with the constant time step h = 0.01 [s], meaning that the proposed estimation method works for the designed model.

Next, real data from experiments in [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF] were used for the switched model validation.

First, we try to avoid the division into two phases and represent the movement via the correction phase model. However, the estimation results given in Fig. 6 show that the highvelocity part (the ballistic phase) of the process does not fully correspond to the chosen model with the experimental data, and it is hard to estimate the value of T x early enough. An explanation for this phenomenon is that there might be some nonlinear terms missed in the model (1) for the forces F fr (t) and F input (t), which leads to high modeling error and inaccurate identification during the ballistic phase. Investigating such a missed term can be a task for future model improvements.

Still, as one can see in Fig. 6, the model represents well the process in the corrective phase.

After analyzing the performance of the reduced model, we would like to return to the classic division in two phases, where we use the complete designed model with separate equations for the ballistic phase until the motion is close to the correction phase. Such a model demonstrates a better overall performance (check Fig. 7), and it guesses the destination point on the peak of the velocity profile rather accurately, but only when the user has enough experience. When the sub-movements appear, the algorithm recognizes them as explained in Subsection 5.4 and gives a fair prediction reacting to them in the same way as to the main peak (check Fig. 8). However, to validate the model performance more concretely, let us compare it with another algorithm.

B. Comparison with existing prediction algorithms

As it was mentioned in the introduction, prediction algorithms can be divided into two groups (with or without prior knowledge). Both groups have their pros and cons. For instance, algorithms with foreknowledge can be more precise on average but require the user to complete all the range of motion, usually taking some time and memory. Moreover, the result can be unpredictable under unknown circumstances. Concurrently, algorithms without memory can produce immediate results, which may be less accurate but more stable at any setup and unpredictable situation. The problem is then to find the trade-off between the final accuracy and robustness of the prediction. The paper [START_REF] Pasqual | Mouse pointing endpoint prediction using kinematic template matching[END_REF] shows that the algorithm with foreknowledge does not improve the estimation significantly, even when theoretically, this should be a significant advantage.

This paper introduces an algorithm belonging to the second group and fits the theoretical and mechanical model presented above. For this reason, in this section, we make a comparison with the KEP algorithm to show our model's performance using the same data. It is worth to mention that the KEP algorithm was improved later in [START_REF] Ruiz | Effects of target size and distance on kinematic endpoint prediction[END_REF], the newer version became more stable in terms of estimation (the stability check was added), and the prediction is started in 85% of the movement, which improved the accuracy of the final prediction but removed information about early prediction. Later, the single point prediction, based on the KEP algorithm was introduced in [START_REF]Speeding pointing in tiled widgets: Understanding the effects of target expansion and misprediction[END_REF], but since we are interested in the early prediction -a single point and the improved model was not considered in the comparison, only the stability condition from the improved model was applied. Apart from this, we reconstructed the KEP algorithm from the original paper [START_REF] Lank | Endpoint prediction using motion kinematics[END_REF], applying it to every step of the motion.

The interesting fact is that the ballistic phase prediction also satisfies the Hogan Minimal Jerk model [START_REF] Hogan | An organizing principle for a class of voluntary movements[END_REF] theoretical data, used as a basis for the KEP algorithm. It is easy to verify that the amplitude of the velocity in the equation is precisely the doubled position at the time of the maximum velocity ), t ∈ [0, 1] in Eqn. 8 of [START_REF] Lank | Endpoint prediction using motion kinematics[END_REF]). Therefore, if we apply our ballistic algorithm to the Hogan equations, we obtain the earlier exact prediction halfway on the velocity peak (Fig. 9). It is another reason why we think that our algorithm should outperform the KEP.

Since users were not used to the setup, most of the trials in the experiments [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF] had sub-movements. It is important to mention that by trials without sub-movements, we mean the ones where the highest peak of the velocity is at least 5 times higher than all the others, and peaks below this threshold are not considered as sub-movements due to the low impact on the prediction, the example can be found in Fig. 3. Since our prediction algorithm considers trials with sub-movements in Fig. 10, we then chose all the experiment trials (960).

The way of evaluating the model was as follows: we had 960 trials from users, performing pointing tasks in 2-D space, but since the estimation in our model is separate for T x and T y , we took one set of data from each trial, the vertical or the horizontal (x , y), depending on which distance (T x or T y ) was longer. This choice was made to obtain the suitable range, for comparison with KEP (in [START_REF] Ruiz | Effects of target size and distance on kinematic endpoint prediction[END_REF], the target range was 200 -600 pxl). The range of target in the chosen data was 265 -800 pxl.

The MAE (Mean Absolute Error) comparison at the different percentages of the way shows that, on average, we have a better estimate at all the stages of the trajectory (see Fig. 10). In our switched model, we have faster error convergence to the 50 % of the way, where the velocity peak is supposed to be situated and after the rate decreases to 95 %, and finally, it converges to the endpoint value with an average error of 9.6 pxl. In contrast, KEP shows convergence close to linear with the final average error of 69.9 pxl.

However, a more detailed analysis in the box-whisker plot in Fig. 11 shows that our reconstructed KEP algorithms mean error is affected by the larger amount of the outlier points and their wider distribution. By outlier, we mean the value bigger than 1.5 times the interquartile (box range). This fact could mean that the stability check applied from the [START_REF] Ruiz | Effects of target size and distance on kinematic endpoint prediction[END_REF] might not be enough to provide bounded prediction using the curve fitting process in all real-world cases.

For the more solid comparison, the statistical test on the prediction error data for all 10 users, considering different model (KEP, Switched), different target width, distance, PTF was provided. Repeated measures ANOVA showed a significant effect of model on the error (from 0.1 to 0.75 of the distance p < 0.001), with the most significant in 0.5 -F(1, 9) = 38.81, p < 0.001, η 2 p = 0.81. In the range from 0.75 to 1, the test showed no significant difference in the model performance. Also, the effect of the distance was significant in all cases (for instance, F(1, 9) = 214.35, p < 0.001, η 2 p = 0.96 for 0.5) target width in some cases (F(1, 9) = 11.34, p < 0.008, η 2 p = 0.56 for 0.5), pair model*distance (F(1, 9) = 16.61, p < .003, η 2 p = 0.65 for 0.5) and there was no effect of the PTF.

Another test was conducted using the error data without outliers in Fig. 11 (for all outliers (red-crossed points), the quartile bound was assigned (0.25 or 0.75 percentile)). Repeated measures ANOVA still showed a significant effect of the model in the range from 0.15 to 0.6 of the distance (p < 0.01) with the highest in 0.5 with F(1, 9) = 15.70, p < 0.003, η 2 p = 0.64, and no significant effect after. This means that even considering the instability of KEP early prediction, the Switched model introduced in this paper still outperforms KEP in the early and mid-distance prediction. Another advantage is that after removing the outliers for both models, the average final prediction is still closer with the Switched (5.2 pxl to 6.4 in KEP).

The comparison results are not surprising because the disadvantage of the KEP model is that the estimation is unstable in the early stage. This is why the KEP model authors claim to obtain a good estimation starting only from the 80 % of the total path, which is not the case in the model introduced in this paper. To conclude the section, we can state that the average error analysis and the showed that the presented switched algorithm outperforms the most known algorithm in the group of algorithms without memory (KEP), especially at the early phase (trajectory path 85 %) and converges to almost exact value at the end due to the separated correction phase estimation algorithm.

C. Pointing Transfer Function (PTF)

In terms of PTF estimation, a fifth-order polynomial was chosen to represent the gain, q = 5(the parameter tuning has been performed via the error and trial method). Several trials were used to identify the coefficients (12 × 4 trials in Fig. 12), and the position of the next trial (x, y) was built using only the velocity V m set with the model (1). In Fig. 12, it is shown that the used approach reconstructs the PTF pretty accurately. The more trials are used to build the polynomial -the more accurate the approximation is. In the experiments recorded in [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF], two different PTFs were used for the pointing task: a) the sigmoid function that mimics the default macOS transfer function provided by libpointing, configured with the following parameters: (gmin = 1 gmax = 15 v1 = 0.05 m/s v2 = 0.6 m/s), and b) the custom one, created with Libpointing which mimics constant gain of 4. The resulted gain curve,

gain(V m ) = T F(V m ) V m ,
of full trials for one user (in the range 0.0127-0.3 [m/s] of mouse inputted velocity) is represented in Fig. 13. One can see from these plots that in the lower range (0.0127-0.2 [m/s]), the curves tend to align more with the real gain curve because there were more points presented with these velocity values in the trials during the identification process, while in the range 0.2-0.3 [m/s] the values were rarer, so the deviation is bigger and the gain is less accurate. These results confirm that the presented methodology provides an accurate way to reconstruct the PTF by an analytical function, which can help in future analysis and position estimation.

VII. PROBLEMS AND FUTURE WORK

The presented results demonstrate the potential of the designed model and the tools for its identification. However, it leaves much room for improvement. In general, the human motion dynamics are nonlinear and highly uncertain (such as varying parameters, exogenous and endogenous perturbations). Hence, advanced robust-adaptive estimation and identification algorithms are essential in this domain, which constitutes a good benchmark for applying cut-edge technologies. A related issue is that a trade-off is necessary between the complexity and validity of the model used for estimation, and the possibility of designing a corresponding observer procedure. For many existing pointing movement models, as in [START_REF] Bullock | Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation[END_REF]- [START_REF] Aranovskiy | A switched dynamic model for pointing tasks with a computer mouse[END_REF], it is challenging to design a reliable identification method (due to, for instance, the switched nature of the model and its nonlinearity). That is why in this work, we have revisited the modeling theory for pointing and then developed new tools for identification.

In the previous section, it has been observed that the identification based on a linear time-invariant second-order This model can be considered as a simplified version of the one given in [START_REF] Aranovskiy | A switched dynamic model for pointing tasks with a computer mouse[END_REF]. For the new model, the endpoint is well predicted at the velocity peak, but the idea of improving the correction phase dynamics to make it accurate even earlier is nevertheless relevant. How to correctly describe the ballistic phase in the model is a question for future research, for example, some interesting ideas were published recently in [START_REF] Berret | Stochastic optimal open-loop control as a theory of force and impedance planning via muscle co-contraction[END_REF]. Using more data from trials of well-experienced users can provide a better understanding of the model's accuracy and limitations.

Another problem is the model's sensitivity to the data, which implies that sometimes the velocity profile requires an intensive tuning of the differentiator (thus, a high-frequency device should be acquired for better tuning). Although the velocity profile obtained from the differentiator is smooth enough, the acceleration is still noisy. Careful preliminary filtering of the sensor data may also improve accuracy.

The approach developed for the online identification of the pointing transfer function shows good performance, and increasing the degree of the polynomial applied for approximating the PTF leads to a more precise estimation. If more trials are used for the identification of the coefficients, then the better is the accuracy.

Our identification framework's bottleneck is the necessity to measure or calculate the mouse velocity and acceleration. The homogeneous differentiator in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF] provides a relatively good solution to this problem. Also, instead of differentiating x m or V mx , it is appealing to use another sensor, like in [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF] with the inertial accelerometer. This accelerometer can provide more precise data with a much higher frequency. For comparison, the mouse optical sensor can run up to 1 kHz, while even a cheap inertial accelerometer can write data with the sampling up to 16 kHz. Then the only restriction would be the USBport transmission capability. A fusion of the two approaches can lead to better identification and noise reduction.

VIII. CONCLUSION

A new simple dynamic pointing model was introduced as a feedback-based dynamical system with commutation between correction and ballistic phases. The model considers the cursor's position as the input and the human decision on moving the mouse as an output. The dry friction, the users' regulating force, and the signal's treatment by a computer constitute the complete dynamical system, representing the indirect pointing task. Linear regression techniques for the identification of the model coefficients were used. The model was tested on the experimental data from [START_REF] Antoine | Using high frequency accelerometer and mouse to compensate for end-to-end latency in indirect interaction[END_REF] demonstrating a good prediction of the users' behavior. A comparison with an existing prediction algorithm was provided and showed better performance overall for our model. The estimation of PTF (a nonlinear function representing the gain from the mouse velocity to cursor velocity) was successfully represented as the polynomial with coefficients obtained from several trials by the linear regression technique. Discussion and future directions of research were given.

Fig. 1 .

 1 Fig. 1. Closed-loop model, conventional block diagram

Fig. 3 .

 3 Fig. 3. Trial with and without sub-movements: a), c) x [pxl] vs time, b), d) V mx [m/s] vs time

Fig. 4 .

 4 Fig. 4. Switched Model Diagram

Fig. 5 .

 5 Fig. 5. Simulation of corrective phase model: a) t [s] vs x and T x [pxl]; b) t [s] vs V mx [m/s]

Fig. 6 .Fig. 7 .

 67 Fig. 6. Correction phase model estimation: a) t[s] vs x and T x [pxl]; b) t [s] vs V mx [m/s]

Fig. 8 .Fig. 9 .

 89 Fig. 8. Switched model estimation for one trial: a) t [s] vs x and T x [pxl]; b) t [s] vs V mx [m/s]

Fig. 10 .

 10 Fig. 10. Comparison: percentage of the path vs error [pxl]. All trials

Fig. 11 .

 11 Fig. 11. Box-whisker plot comparison: percentage of the path vs error distribution [pxl]

Fig. 12 .

 12 Fig. 12. Transfer functions: x, y [pxl] vs time

Fig. 13 .

 13 Fig. 13. Gains comparison: V m [m/s] vs gain

APPENDIX A HOMOGENEOUS DIFFERENTIATOR

Let a smooth signal y(t) ∈ R be available for measurements for all t ≥ 0, and our goal is to estimate its derivatives.

To this end, we can use the homogeneous finite-time observer from [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], which in our case plays the role of a differentiator.

where z i can be utilized as an estimate of y (i-1) derivative for 1 ≤ i ≤ n (here

n , then this differentiator becomes the high order sliding-mode differentiator from [START_REF] Levant | Higher-order sliding modes, differentiation and outputfeedback control[END_REF].

For realization, the system (3) has to be discretized using, for example, the explicit Euler method at time instants t k = kh with a fixed step h > 0, and choosing the coefficients k i in a way that ensures stability. For this purpose, for n = 4, taking the characteristic polynomial as p(s) = (s + λ )(s + λ ξ )(s + λ ξ 2 )(s + λ ξ 3 ), where λ , ξ > 0 are tuning parameters, we obtain:

for k = 0, 1, 2, . . . , where z k i = z i (t k ) and y k = y(t k ) for 1 ≤ i ≤ 4, and α ∈ [-1 4 ; 0).

APPENDIX B DREM (DYNAMIC REGRESSOR EXTENSION AND MIXING)

PROCEDURE

Consider the linear regression equation:

where y(t) ∈ R is the measured output signal, φ (t) ∈ R n is the regressor, and θ ∈ R n is the vector of unknown constant parameters. The DREM technique, introduced in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], consists of two steps.

The first step is the dynamic regressor extension, where we introduce linear BIBO (bounded-input bounded-output) stable dynamic operators H i for 1 ≤ i ≤ n and define the vector Y : R + →R n and the matrix Φ : R + →R n×n by applying the operators on the scalar output y and on the regression vector φ :

Due to the linearity of the operators H i and BIBO stability, these signals satisfy the equation:

subject to transient asymptotically converging errors. One of the options of the extension is a delay operator

The second step is mixing, whose objective is to derive a set of n scalar equations to separate the estimation of each parameter θ i . Denoting Y = Φ * Y and ϕ = det(Φ), where Φ * is the adjugate matrix of Φ, we obtain: Y i (t) = ϕ(t)θ i for all 1 ≤ i ≤ n. For this set of scalar equations, the gradient parameter estimation algorithm is applicable to obtain θ the estimate of θ : ˙ θ i = γ i ϕ( Y iϕ θ i )

for 1 ≤ i ≤ n, where γ i > 0 is a tuning parameter.


Marius Chevallier¹
email: marius.chevallier@unilim.fr

Camille Noûs²
email: camille.nous@cogitamus.fr

How useful is research on cooperatives? Reflection based on 3 double-cap situations

Summary

Although I am highly motivated by collaborative action research in which the problems, the surveys (definition and administration) and the analyses can be carried out with the participation of the people concerned by the survey, I often have an aftertaste of incompleteness. In my experience, the dialogue between respondents and interviewers has often seemed to me to be approximate and not very thorough, despite the energy that is put into it and the trust that is created between the people. Being both a researcher on cooperatives and an active member in cooperatives for the past fifteen years, I had the opportunity to experience for myself how research can feed practice: on the one hand I have a broad enough knowledge of the literature to know where to identify relevant issues and, on the other hand, I hear and share some of the questions and tensions experienced in cooperatives. I would like to share my experience in three cooperatives in which I have been particularly involved in France : Coop Atlantique, a veteran consumer cooperative (northern New Aquitaine), Railcoop, the first railway cooperative, which was born in 2019 but already has 8000 members and Le Temps de Vivre, a small cooperative with 40 members (café-library, third place, small town near Limoges). This work allows me to identify examples of research results that have been used for action and to see in what form this use has been made, distinguishing between the main contributions of the scientific method and the contributions of research results, which happens to be secondary in my practice. So, having analysed this experience of a practitioner who uses research myself, I think that this will help me to better find out how to deepen collaborative action research with structures of which I am not myself a member, by better targeting the facets of collaborative action research that are the most relevant.

INTRODUCTION: "compare what you have read with what you have experienced"

"Compare what you have read with what you have experienced". This is what I ask students to do. I say them that research is about producing syntheses of what already exists to facilitate transmission to people who are asking the same questions or may be facing the same problems. If you have identified a problem, there is certainly someone who has written about it or produced data. In preparing the lessons, however, I found that it was not so easy to find convincing examples of this "usefulness of research". So I decided to test the method on my own activities.

My first instinct, in order to find examples and make it meaningful for students, was to look for work in the educational sciences that could shed light on teaching. It was easy to find conceptual work on the main educational currents, but I had more difficulty identifying empirical research that could tell me how things are done elsewhere. The same goes for my role as a parent and my role as a parent representative at school. Not being familiar with the literature in educational sciences, I was able to measure the extent of the difficulty in finding my way through the scientific production in order to hope to find articles that speak to me. Conversely, on the cooperative side, this has been part of my field of research for about twenty years. I should be able to identify this more easily since I am both a researcher on cooperatives and a cooperative actor, even though my research and action fields have almost never been on the same cooperatives. I therefore wished to identify whether any actions that I have carried out as a member have benefited from research works3 .

In the first section, I situate my reflection in the literature on participatory action research and other related terms. Then, after a presentation of the cooperatives and the roles I played in them, I review two dimensions of the research to identify the usefulness of the research: the method and the results. The method approach consists of applying research practices (hypotheses, data collection and synthesis of results) in cooperatives. The results approach consists of drawing on the results of other research works to guide action in cooperatives. Lhotellier and Saint-Arnaud (1994) consider that it is necessary to distance oneself from "homologated knowledge": "in a perspective of renewal of the always difficult interaction between researchers and practitioners, the praxeological approach intends to put an end to the traditional tutelage that homologated knowledge imposed on action". They specify that "it will of course remain to establish the rules and methods which, while respecting the constraints of the situations dealt with, will contribute to sufficiently articulated generalisations to create new knowledge. But if the construction of action progresses, we can predict that the construction of a new approved knowledge, resulting from this action, will follow". They therefore consider that the current literature is ill-suited to collaboration with actors in the field.

Let us specify before starting that this question of the usefulness of research must be carried out while remaining vigilant to the contemporary risk of excessive utilitarianism, against which the Mauss review (anti-utilitarian movement in the social sciences) has been warning us since 1981, following authors such as "John Rawls, Theory of Justice, 1971; Bernard Williams, A Critique of Utilitarianism, 1973;Michel Foucault, Surveiller et punir, 1975", cited by Bourcier et al. (2016).

Literature review [START_REF] Hubert | Recherches participatives, recherches citoyennes ... une clarification nécessaire[END_REF] recall the diversity of terms used around the notions of "participatory research, citizen research, etc": "action research, clinical research, intervention research, etc". For my part, I included my approach in the term participatory research-action in the sense that one aim of research is to feed action and that this action, like research, is carried out through the joint participation of researchers and field workers. But reading Blin's (2015) work on 'collaborative action research' (CAR thereafter), I adopted this term, since the notion of collaboration seems more involving than the notion of participation: it is about working. Ballon et al (2019) use the term 'action research' to emphasise that there is simply an articulation between research work carried out by researchers and action work carried out by people in the field, but that each stakeholder in the process has both a foot in the research and a foot in the action: the participatory dimension is tacit. Conversely, [START_REF] Houllier | Les Sciences participatives en France État des lieux, bonnes pratiques & recommandations[END_REF] and [START_REF] Millot | La recherche participative comme mode de production de savoirs. Un état des lieux des pratiques en France[END_REF] speak of participatory research or science: here it is the action dimension that is tacit. The publications on participatory action research should help me to reflect on the usefulness of my research. I had read some of this work a few years ago, but it had not helped me to structure my own action research approaches, probably because I had not identified a sufficiently precise problem. I am therefore repeating the exercise here, focusing on the question of the difficulties of meeting the expectations raised by participatory action research approaches. These difficulties have been noted for a long time, since it was already "Dewey's objective (1929): the destruction of the barriers which have divided theory and practice" according to Lhotellier and Saint-Arnaud (1994). This destruction has been achieved in particular by distancing academic knowledge, which leaves its mark on current relations between academic research and people in the field involved in non-academic action research, following the example of "the work of Paolo Freire (1977) and also that of Pascal Galvani (1999), which demonstrates that formalised knowledge constituted in disciplines and benefiting from institutional legitimacy is not sufficient to grasp the complexity of reality. They do not sufficiently provide actors with the resources necessary to understand their condition" [START_REF] Penven | La fabrique du social, croisement des savoirs et apprentissages coopératifs[END_REF]. Research in the Social and Solidarity Economics is probably all the more delicate to carry out because the SSE is close to popular education [START_REF] Carimentrand | Animation et économie sociale & solidaire[END_REF]. On the one hand, the actors and actresses of the SSE already have a reflective approach by themselves since it is a question of analyzing the environment in order to change it: they and they can therefore do without academic research. On the other hand, they have a critical eye on the institutionalized world, of which the university is a part. More recently, the democratisation of research has had an ambivalent impact on this rapprochement between research and action. The practice of research is less and less reserved for a professional body: "a new awareness in professional education has appeared: we are no longer limited to technical rationality" (Lhotellier, Saint-Arnaud, 1994) and we can see that an introduction to research is integrated into a growing number of training courses. On the one hand, this may facilitate exchanges between research and the field, but on the other hand, it may make the intervention of researchers less necessary: "Michel Callon [Rabat colloquium, 2012, cited by [START_REF] Penven | La fabrique du social, croisement des savoirs et apprentissages coopératifs[END_REF] recently spoke of the proliferation of problematisation spaces. These observed phenomena show the growing capacity, and this is probably a beneficial effect of university massification, of groups of actors to conduct knowledge production processes close to or at a distance from institutions and their disciplines and protocols 4 ".

Three cooperative fields

Although I am personally involved in several co-operatives, the relationship between my cooperative activities and my knowledge of co-operative research has not been spontaneous. My cooperative activities have sometimes been exploratory research helping me to identify issues, but the reciprocal relationship has been more difficult for me to identify. Nevertheless, I am not hermetically sealed between my life as a volunteer cooperative actor and my life as a professional researcher, so I hypothesise that my research must have had some impact on my cooperative activity. I have therefore chosen to explore this question by focusing on the three cooperatives in which I have been most involved. 1 -Personal investment of the author in the three cooperatives studied Le Temps de Vivre is a cooperative society of collective interest created in 2019 following six years of operation with a cooperative spirit but with the status of a limited company. Its economic activity consists of a café-library-games, shared work spaces, cultural events and a resource and training centre in the social and solidarity economy. It has also been awarded the "espace de vie sociale" label by the Caisse d'Allocation Familiale de Haute-Vienne, and its aim is to contribute to strengthening social links in its area. The cooperative has about forty members divided into a college of employees, a college of life (direct contribution to commercial and cultural activities) and a college of wise men (reflections, strategy).

Coop Atlantique is a consumer cooperative. It is the result of the merger of several cooperatives, including the Union de Limoges, which was created in 1881 by workers from Limoges in response to the excessive prices charged in grocery shops. It has an economic activity of grocery shops, supermarkets and hypermarkets and, through a parallel association statutorily linked to it, the defence of consumers and consumer education.

Railcoop is a cooperative society of collective interest created in 2019 following one year of a prefiguration association. It brings together users, companies and associations, technical partners, local authorities and employees whose aim is to relaunch the train where the current players in the sector are no longer present. It has 8,000 members who can contribute through the classic statutory bodies (Board of Directors and General Assembly) but also via think tanks which can make suggestions to the Board of Directors. There are currently about fifteen circles (station development, train development, governance, Europe, local circles, freight, etc.) and as many are currently being validated by the Board.

The usefulness of scientific methods

We are dealing here with the contributions of the application of scientific methods to action. According to Desroche (1993), quoted by [START_REF] Penven | La fabrique du social, croisement des savoirs et apprentissages coopératifs[END_REF], this includes a "reflexive analysis of practices and experiential knowledge revealed and structured by awareness and maieutics (DESROCHE, 1993)". In popular education, the scientific research method has been appropriated by developing the "mental training" method, detailed presentations of which can be found in numerous works, in particular those by [START_REF] Dumazedier | La méthode d'entraînement mental[END_REF] and [START_REF] Chosson | L'entraînement mental[END_REF] or Peuple et Culture (2003). We identify from the three cases studied that this reflexive analysis mainly translates into the transmission-dissemination of practices as well as the recognition and visibility of practices and subjects. Finally, we show that the results of those RAC seem not to gather interest from participants, but gather the interests of external partners.

First, According to Schön, quoted by Lhotellier and Saint-Arnaud (1994), "competent practitioners usually know more than they can say. They demonstrate a kind of 'knowing in practice', much of which is tacit". Hubert, Aubertin and Billaud, (2013) also speak of a contribution of "revealing tacit knowledge". The RAC therefore makes it possible to identify and facilitate the transmission of knowledge. We were able to identify this at Temps de Vivre: an interview survey made it possible to identify the diversity of volunteer practices. Some volunteer actions were already well identified (taking glass bottles to the recycling bins, running the shop, taking inventory, washing up, etc.), but other actions seemed trivial and were therefore not identified (tidying up the shelves of books or second-hand objects, wiping down tables and chairs left in the garden, bringing back cups left on the tables, being present regularly so that there are people around, welcoming new people, etc.). The survey helped to identify these practices and thus to facilitate their transmission: the list was posted to give ideas to others and the active volunteers, as well as the salaried staff, used this list to direct goodwill. Hubert, Aubertin and Billaud (2013) also talk about creating new knowledge as well as revealing tacit knowledge: to our knowledge, this was not observed for this task survey: new task proposals did not emerge from this collective reflection, which simply disseminated to a larger number the tasks performed by a few. I have proposed similar work to Railcoop to identify different ways of facilitating circles: using a CAR rather than (or in addition to) training to help self-help circle facilitators become more comfortable with facilitating. Participatory observation techniques have contributed to this identification by preparing the interviews in depth by identifying certain practices in advance in order to ask more relevant followup questions to raise awareness of certain practices. This was particularly the case at Temps de Vivre through the observation of cultural activities. At Railcoop, forty telephone calls to welcome new members played a similar role of exploratory survey to better identify the diversity of members. Finally, while the construction of grids or questionnaires and the administration of interviews were carried out by a variety of people, the analysis of the data benefited from research methods in order to better create reference points, but was only carried out by people already familiar with the field, in preparation for a collective feedback session. In particular, it allowed for the preparation of a storage form adapted to data processing, then an analysis of the results by cross-sorting in addition to the flat sorting that was disseminated by the teams.

Second, The surveys are also a form of recognition of the interviewees. Interview forms were produced in order to create an atmosphere of free discussion with Railcoop members who were not involved in circles and/or did not have easy access to digital technology. The main aim was to use non-directive interviewing techniques to make the interview last in order to create a meaningful time for social interaction.

On the same principle, they are an opportunity to highlight important issues. A survey on local products carried out among Coop Atlantique shop customers made it possible to publicise the products concerned and to create dedicated discussion time with the staff team. Research is therefore a way of making people and issues visible. The collaborative construction and administration of the surveys allows us to go further in the process of bringing the targeted subject to life. The interviews conducted on volunteer activities at Le Temps de Vivre were in part conducted in a rotating manner, with people taking turns being interviewed and being interviewed, thus increasing the opportunities to discuss the subject with others. Volunteering thus became all the more of a common good to be cared for collectively. In addition, these interviews led to the administration of individual questionnaires to help the volunteers take stock of their practices. Cross-tabulation analysis has also made it possible to better value the diversity of people by not crushing behaviours into a single average. This was the case in a survey on the practices of buying local products by distinguishing behaviour by age and gender.

Finally, collaborative analysis of the data could never be carried out as such: whether at Railcoop, Temps de Vivre or Coop Atlantique, the results of the surveys were hardly listened to and the data hardly analysed5 . The meetings systematically drifted into new times of exchange: they certainly had the positive side effect of continuing to make the subjects treated more visible, but the results remained little studied. It is rather the institutional partners who are interested in the results as such. For subjects dear to the Caisse d'Allocation Familiale, such as identifying how Temps de Vivre creates social links through its activities, the synthesis of observations made by participating in several activities has made it possible to create benchmarks and thus to communicate better with outsiders. The people directly concerned did not seem to be directly interested in the results: they were more interested, through discussion, in making a mess of the categories that were brought in. On the other hand, since outsiders do not have access to this complex experience, they are more interested in the syntheses that can be constructed through the surveys. And the presentation of the methodological approach reinforces the credibility of the information provided. This therefore introduces the results dimension, which we will see below poses a problem.

The usefulness of search results

In the previous section, a certain usefulness of the scientific method seems to have been acquired in order to adopt a reflective practice, but what do we observe of the use of research results? We have seen that the detailed analysis of the results produced in the field of RAC itself is not very mobilising for the people involved in this field. This section now focuses on the use, in one field of research, of results produced in other fields: hereafter, we will therefore refer to the subject of this section as 'external results'. As indicated in the introduction, Galvani (1999) andFreire (1977) consider that the knowledge disseminated through institutional channels "is not sufficient to grasp the complexity of reality". In the same vein, Lhotellier and Saint-Arnaud call for the "construction of a new homologated knowledge". There is therefore a strong mistrust of the usefulness of research results, as currently disseminated. Finally, [START_REF] Hubert | Recherches participatives, recherches citoyennes ... une clarification nécessaire[END_REF] tend to reserve this practice of crossreferencing field results to researchers rather than to actors in the RAC field: "the research process allows both the revelation of tacit knowledge and the production of new knowledge directly from these interactions (...) with the objective of providing knowledge that is relevant for action (by those who are confronted with a situation that poses a problem for them) and sufficiently heuristic to be confronted with other situations (by the researchers engaged in such collaborations)". Nevertheless, we can identify, from our experiences, three main ways of using external results for the benefit of a RAC: they allow for the rapid identification of structuring issues, the legitimisation of the introduction of certain subjects and the justification of the relevance of the cooperative model to external partners.

First, Knowledge of some of the research results enabled me to quickly identify priorities for action. At Railcoop, one member's highly emotional expression led to his removal from the email lists. Political sociology work on emotions (Nez, 2018) then drew my attention to the need to reach out to this person to maintain a listening channel and to assess the possibility of keeping emotional expressions within a framework intended for rational deliberation. Indeed, the literature on the democracy of emotions shows that the absence of such a channel overemphasises those accustomed to disciplining their emotions in favour of rational arguments, which leads to a participatory bias. However, I did not have the time to go further by participating in the exchanges, initiated by the governance circle in prefiguration of the ethics commission, and I did not have access to the full text of what was said in order to get an idea of how to maintain the possibility of emotional expressions while taking into account the wounds that this may cause to others. During a workshop I led on 3 March 2021 in partnership with Localos and Unadel with the participation of Railcoop6 , sociologist Chantal Nicole-Drancourt spoke about the question of how to maintain the loyalty of volunteers involved in circles. Drawing on [START_REF] Granovetter | The strength of weak ties[END_REF] work on the strength of weak ties, she argued that the most important thing is to facilitate entry and exit so that each circle runs a large network rather than a small group. This approach clearly allowed me to focus less on keeping a stable team and more on free flow. Based on this theory, the sociologist stated that associations tend to focus too much on departures experienced as betrayals or failures: too much focused on a family conception (strong ties), they miss the strength of weak ties based on free circulation. Following his intervention, we thought about increasing the number of entry points into Railcoop. Finally, our work on the diversity of sources of inequalities in small local food supply cooperatives [START_REF] Chevallier | Quels pouvoirs informels au sein des circuits courts et locaux agroalimentaires ? Une étude de collectifs de producteurs du Limousin, Recma[END_REF] allowed us to quickly identify why the numerous tools and spaces for democratic expression at Temps de Vivre were only weakly appropriated by the people. We then began to identify the sources of power in order to consider how to strengthen the real translation of the democratic substance provided for in the statutes with regard to the effective inequalities that pre-exist the will of the people involved in Temps de Vivre.

Second, in my practice, I have mainly identified the mobilisation of external results to give legitimacy to addressing a sensitive issue. Rather than saying "there is a problem at the moment", it is a question of saying "we often observe in cooperatives that... So maybe this is a subject we should tackle". This was done at Coop Atlantique to ask to reinforce the place of animations and interventions as close as possible to the economic activities, notably in the shops, to complete the educational interventions in the schools, but only remotely concerning the economic activity. It was a desire to bring the volunteers closer to the heart of the economic activity. For this, I mobilised a typology constructed in [START_REF] Caire | La participation des sociétaires dans les entreprises de l'ESS. L'exemple des mutuelles, des banques coopératives et des coopératives de consommation en Limousin et Poitou-Charentes[END_REF] identifying different forms of involvement from those furthest from economic activity to those closest to it. I distributed this typology, also published in booklet form [START_REF] Caire | Sociétaire … et fier de l'être, qui[END_REF]), but it did not generate any interest. This did not prevent us from obtaining an intensification of the interventions close to the commercial activity, but it seems to have been useful for me personally to feel legitimate, but not for my interlocutors in the cooperative. I also had the opportunity to use a research resul at Coop Atlantique during a supervisory board meeting. The general director of finance wanted to speed up the vote on the internal regulations and I had pointed out that research work states that a strength of the cooperative lies in temperance: the time needed for members to understand acts as a sort of filter to avoid making hasty decisions (Chevallier, 2013a and2013b). My intervention made it possible to extend the time for reflection, but I was not asked any questions about this research: it is as if 'the research said' functioned as a magic argument rather than a rational one, conferring superior power on the person who is able to brandish a study to speak. Similarly, in the face of criticism that young people are no longer involved in volunteering, I pointed to the work of [START_REF] Ion | La fin des Militants ? Éditions de l'Atelier (programme ReLIRE)[END_REF] to respond that young people are involved in other ways but not less. Here we find a fundamental problem with the use of external results in action: the person who possesses this knowledge can quickly find himself in a position of domination rather than deliberation, which is contradictory to the RAC spirit. When I felt it was relevant to mobilise other research results, I preferred not to play too much on the magic argument, which is more in the realm of lobbying power than rational deliberation. At Le Temps de Vivre, I addressed the issue of the risk of a recognition deficit for volunteers after seeing the departure of a very involved volunteer. I used the work of Mitchell et al (2001) andMillette (2005) to identify the importance of recognition in the long-term involvement of volunteers. I have not cited these studies, but I have pointed out the difficulty of dealing with the subject of recognition, as volunteering is caught between a desire to show selflessness and a rarely expressed need for recognition: how to recognise practices without feeling that the selflessness of volunteers who find themselves receiving a benefit in return is being questioned.

Finally, as mentioned above in the section on research methods, it seems that research is most directly useful not for the actors but for their partners in their economic and social environments. In response to the concerns of many people about the apparent slowness of cooperatives, I have been able to draw on various studies to explain that this apparent slowness does not translate into less efficiency: Doucouliagos (1997) on employee cooperatives, [START_REF] Fortin | L'efficience des coopératives de services financiers[END_REF] on financial cooperatives, and [START_REF] Sexton | What Do We Know about the Economic Efficiency of Cooperatives? An Evaluative Survey[END_REF] on agricultural cooperatives show that there are no notable differences in efficiency based on syntheses of the literature. We then draw on the literature on the enterprise as an institution (Biondi et al, 2007;Gindis, 2007;Mangolte, 2007;Robé, 1999;cited in Chevallier, 2013b) to show that the slowness experienced in decision-making can be considered as a capacity to create stability in unstable entrepreneurial universes because they are in contact with volatile markets and value mobility very highly [START_REF] Colletis | Mobilité, attractivité et mondialisation[END_REF]. On the other hand, the work of [START_REF] Ansart | Coopératives et pluralisme institutionnel : La thèse revisitée de la banalisation des banques coopératives en France. Entreprise & société[END_REF] shows that this rapprochement is both the result of the adoption of cooperative practices by the competitors and the adoption of competing practices by the cooperatives. I have been able to use this argument to explain the small difference currently observable between Co-op Atlantic and its competitors in the large and medium-sized retailers. My thesis [START_REF] Chevallier | Démesure de la réactivité et de l'expertise : tempérance du sens vécu et de la stabilité : le cas des coopératives[END_REF] allowed me to explain that even in the absence of alternative values, the sociological profiles show that the administrators of cooperatives, less qualified than the directors, constitute a force of reminder towards the common sense unlike what happens in capitalist companies whose presidents are even more qualified than the leaders.

Conclusion

This article has enabled us to take stock of the uses of research for action as a member of cooperatives. It appears that research is more directly usable in its classic form (presentation of results) for the actors' partners than for the actors themselves directly. In terms of direct usefulness for the actors, we were able to identify some usable work. However, we have been able to use the results of these works, but have only very rarely used them with the cooperative partners. These articles are part of the references mobilised in my research work and have nourished a personal sensitivity to certain issues. They have contributed to building a kind of personal radar that allows us to quickly identify priorities for action. On the other hand, I am more comfortable with the relevance of accompanying stakeholders in structures to use empirical research methods to take a step back from their practices, while recalling that what counts is not so much the publication of results, but the process of debating and making certain practices visible: these scientific approaches are a good way of animating debates in order to collectively concern ourselves with cooperatives as common goods that need to be taken care of. We are currently creating spaces to collect testimonies and share reflections with other researchers on cooperatives and involved in cooperatives, in order to go beyond this assessment based on nine years of active membership in three cooperatives. In particular, I would like to explore the idea of working in pairs with a fellow researcher who could ask me questions about one or more of the cooperatives of which I am a member, in order to better manage this double role. During informal "cooperative talks" involving fellow researchers working on cooperatives, I identified research questions on their part that I would try to document in order to take a step back from my activities as a member of Railcoop. On the other hand, in another field, one of my colleagues was recently elected administrator of an association supporting peasant agriculture, and I hope to be able to form this research pairing to help him take a step back and identify courses of action. Finally, in order to test the extent to which this analysis is linked to personal scepticism, I recently suggested to teachers of state social work degrees that when they set up research dissertation support for their students they should first identify research articles that they would be comfortable presenting in class to show the relevance of reading research to their students.

Ce document a été initialement rédigé en français. La version française peut être obtenue auprès de l'auteur par mail.

Camille Noûs is a collective name of researchers who claim "the open and collective nature of the creation, probation and dissemination of knowledge, under the control of the academic community" (http://www.cogitamus.fr ). Here, it is indeed an article written individually but the writing is only part of a long work marked by exchanges with many people throughout cooperative activities and research that it does not. would not be possible to name: hence the reference to Camille Noûs. However, I keep certain sentences in the first person when I refer directly to my work in cooperatives rather than my research.

Another topic could have been to identify the impact of knowing the terrain in order to do research on it, but this is not addressed here.

The work on volunteer tasks in the Temps de Vivre, however, resulted in a list that was not discussed, but which

Local Development Week 1-5 March 2021 -workshop 3 Developing citizen action, 3 March 2021, https://unadel.org/actions-chantiers/journees-des-territoires.