Viacheslav Khomenko
email: slava.khomenko@gmail.com

Fethi Olivier Bruneau

Ben Ouezdou

Patrick Henaff
email: patrick.henaff@u-cergy.fr

Artem Melnyk

Volodymyr Borysenko

Non-Invasive Low Cost Method for Linear and Angular Accelerations Measurement in Biped Locomotion Mechanisms

A new non-invasive low cost method for accelerations measurement in biped locomotion mechanisms is presented. The aim is to measure all the joint accelerations of the locomotion system during the walking that is difficult because of the nonaccessibility of several joint axes. Our measurement approach is non-direct using two triaxial accelerometers aligned on the main axis of each body, that ensures measurement of five accelerations per joint, two angular and three linear. By considering negligible the vertical rotation of a leg in the absolute frame during walking, the vertical rotational acceleration of the hip is measurable by two accelerometers aligned along the sagittal axis. For tibia and thigh, the accelerometers are aligned along vertical axes. The calculation of accelerations in articulation references is achieved by solving a system of Newton mechanics equations. The theoretical statements were validated experimentally on the robot for vertical flexion-extension movement.

I.

INTRODUCTION

Non-invasive, or body mounted systems for kinematic analysis of polyarticulated mechanisms are mainly developed since '90s of the XX century, with large proliferation of MEMS sensors [START_REF] Liu | Analysis of lower limb segment orientation using triaxial accelerometers[END_REF][START_REF] Channells | Determination of rotational kinematics of the lower leg during sprint running using accelerometers[END_REF][START_REF] Willemsen | Lower extremity angle measurement with accelerometers -error and sensitivity analysis[END_REF][START_REF] Mayagoitia | Accelerometer and rate gyroscope measurement of kinematics: an inexpansive alternative to optical motion analysis sytems[END_REF][START_REF] Yang | A review of accelerometry-based wearable motion detectors for physical activity monitoring[END_REF]. Commercialized inertial sensors dedicated to measure displacements and velocities of legged locomotion mechanisms [START_REF] Yang | A review of accelerometry-based wearable motion detectors for physical activity monitoring[END_REF] exist, but they are generally expensive and their frequency bandwidth is not enough for acceleration measurement that is important for motion analysis and control. Usually it is a combination of linear accelerometers, of gyroscopes and magnetometers. The signals are treated by Kalman filters [START_REF] Moriya | System for three dimensional positioning and tracking[END_REF], but this approach creates integration and derivation errors. Many authors proposed accelerometer-only based approaches [START_REF] Liu | Analysis of lower limb segment orientation using triaxial accelerometers[END_REF][START_REF] Channells | Determination of rotational kinematics of the lower leg during sprint running using accelerometers[END_REF][START_REF] Willemsen | Lower extremity angle measurement with accelerometers -error and sensitivity analysis[END_REF][START_REF] Corey | Measuring Angular Accelerations with Linear Accelerometers[END_REF][START_REF] Krishnan | Measurement of angular velocity and linear acceleration using linear accelerometers[END_REF][START_REF] Qin | Robust design of inertial measurement units based on accelerometers[END_REF][START_REF] Zappa | On the number and placement of accelerometers for angular velocity and acceleration determination[END_REF][START_REF] Cardou | Singularity analysis of accelerometer strapdowns for the estimation of the acceleration field of a planar rigidbody motion[END_REF][START_REF] Cardou | Computing the Rigid-Body Acceleration Field from Nine Accelerometer Measurements[END_REF][START_REF] Yoganandan | Lightweight lowprofile nine-accelerometer package to obtain head angular accelerations in short-duration impacts[END_REF][START_REF] Fong | The use of wearable inertial motion sensors in human lower limb biomechanics studies: a systematic review[END_REF][START_REF] Yang | Measurement of six degree-of-freedom ground motion by using eight accelerometers[END_REF].

There are many works concerning measuring of angular accelerations exclusively with linear mono-axis accelerometers that date the second part of the previous century [START_REF] Corey | Measuring Angular Accelerations with Linear Accelerometers[END_REF][START_REF] Krishnan | Measurement of angular velocity and linear acceleration using linear accelerometers[END_REF]. One of the first [START_REF] Corey | Measuring Angular Accelerations with Linear Accelerometers[END_REF] is limited only to in-plane movements. In-space measurements were presented in [START_REF] Krishnan | Measurement of angular velocity and linear acceleration using linear accelerometers[END_REF], but this method uses disks rotating with constant angular frequency. More recent researches consider measurement systems without rotating parts with five, six and more accelerometers axes placed in orthogonal way on the body [START_REF] Liu | Analysis of lower limb segment orientation using triaxial accelerometers[END_REF][START_REF] Qin | Robust design of inertial measurement units based on accelerometers[END_REF][START_REF] Zappa | On the number and placement of accelerometers for angular velocity and acceleration determination[END_REF][START_REF] Cardou | Singularity analysis of accelerometer strapdowns for the estimation of the acceleration field of a planar rigidbody motion[END_REF][START_REF] Cardou | Computing the Rigid-Body Acceleration Field from Nine Accelerometer Measurements[END_REF][START_REF] Yoganandan | Lightweight lowprofile nine-accelerometer package to obtain head angular accelerations in short-duration impacts[END_REF][START_REF] Fong | The use of wearable inertial motion sensors in human lower limb biomechanics studies: a systematic review[END_REF][START_REF] Yang | Measurement of six degree-of-freedom ground motion by using eight accelerometers[END_REF]. Their biggest problems are the singularities in determination of the solutions [START_REF] Zappa | On the number and placement of accelerometers for angular velocity and acceleration determination[END_REF][START_REF] Cardou | Singularity analysis of accelerometer strapdowns for the estimation of the acceleration field of a planar rigidbody motion[END_REF]. Avoiding singularities is achieved using 9 or 12 axis accelerometer setups, when accelerometers are placed for example along the three axes of a cube, and one in its center. Stability analysis [START_REF] Cardou | Computing the Rigid-Body Acceleration Field from Nine Accelerometer Measurements[END_REF] proves that this method gives very good results and can be used for calibration purposes, in biomechanics [START_REF] Yoganandan | Lightweight lowprofile nine-accelerometer package to obtain head angular accelerations in short-duration impacts[END_REF][START_REF] Fong | The use of wearable inertial motion sensors in human lower limb biomechanics studies: a systematic review[END_REF] and robotics [START_REF] Cardou | Computing the Rigid-Body Acceleration Field from Nine Accelerometer Measurements[END_REF]. But in practical applications such redundancy is not always needed and less quantity of information is usually acceptable, e.g. the 8 accelerometers setup was used in 6-DOF accelerations measurement in earthquake analysis [START_REF] Yang | Measurement of six degree-of-freedom ground motion by using eight accelerometers[END_REF]. However, all these methods require a large number of accelerometers that must be positioned in places of the bodies which are not necessarily accessible without modifying the mechanical conception of the mechanism [START_REF] Yang | A review of accelerometry-based wearable motion detectors for physical activity monitoring[END_REF]. Moreover, there are very few works focused on applications for mechanisms with more than one moving body. For polyarticulated systems the number of sensors grows proportionally to the number of bodies. That affects the reliability and maintenance costs, thus making the measurement system very expensive and not attractive for practical implementation.

An interesting approach for a locomotion mechanism was proposed in [START_REF] Liu | Analysis of lower limb segment orientation using triaxial accelerometers[END_REF]. This method is based on the use of two aligned triaxial accelerometers on the thigh, but the drawback is that the hip rotations are not measured and only linear accelerations of the hip relative to the global coordinate system are taken into account. The models and experimental results are given for different two-segment systems (one human leg and one motorized arm).

Our approach is based on the same technique but extended to the entire biped locomotion system (Fig. 1). We use four accelerometers per leg and two on the hip. Thus we can calculate five accelerations for each articulation included the accelerations of the hip around vertical and frontal axes. This paper is divided into four sections. After this introduction, the second section presents the methodology of our approach. The mathematical model corresponding to the proposed sensor combination is given and the measurement algorithm is described. The third section gives experimental results for vertical flexion-extension movement of a biped robot. The paper is terminated with conclusions and further work.

II. METHODOLOGY

A. Mathematical Model of Measurement

We consider the biped mechanisms hip as the nodal part of a body (Fig. 2). Due to the symmetry of a biped construction in relation to its hip, we present the mathematical model for the hip and one leg (left or right). The inertial frame is denoted as {0, x, y, z}, three presented local frames are denoted as: {h 0 , x h , y h , z h } for the hip, {th 0 , x th , y th , z th } for the thigh and {ti 0 , x ti , y ti , z ti } for the tibia. One leg model is presented for more clarity.

The measured hip movements are: 3 linear (along x, y, z axes of the inertial frame) and 2 rotational (which we consider in the local frame of the hip around vertical and frontal axes y h , z h ). The thigh is connected to the hip; it can rotate around its frontal and sagittal axes (x th , z th ). The tibia is connected to the thigh and it has only 1 axis of rotation (z ti ).

The model was constructed in assumption that there is an ideal contact between foot and floor (the rotation of foot relative to the floor is considering negligible) and walking is normal (e.g. stable movement on a surface). The accelerations given by accelerometer sensors positioned on the mechanism as shown on the Fig. 2 were received in general form based on Newton formalism: For simplification, we detail here only the determination of the analytical solution of the first equation of the system (1) corresponding to two triaxial accelerometer sensors mounted on the hip. It was found that in general case its determinant is zero and thus the solution cannot be found. But, in particular case of a biped locomotion system, the rotational acceleration along one axis can supposed zero, and then the unique solution can be achieved because of cross-axis terms elimination in the vector 2 , 1 h h r α ×

( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ × × + × + + ⋅ ⋅ + × × + × + + × × + × + = × × + × + + ⋅ ⋅ + × × + × + = ⋅ + ⋅ + × × + × = , ; ; 2 , 1 2 , 1 * * * 0 2 , 1 2 , 1 2 , 1 * * 0 2 , 1 * 0 2 , 1 2 , 1 2 , 1 ti ti ti ti ti ti th h h th ti ti th th ti th th h h th h h ti th th th th th th h h th th h h th h h th h h h h h h h h h q q q q q q q q q q q q r ω ω r α g ρ ω ω ρ α ρ ω ω ρ α a a r ω ω r α g ρ ω ω ρ α a a g a r ω ω r α a (
. For the hip, the accelerations are: ( )

[ ] [ ] ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ - ⋅ - ⋅ - - = - ⋅ - ⋅ - - = - ⋅ - ⋅ - - = - - = - - - = = , ) ( ; ) ( ; ) ( ; ; ; 0 1 1 1 0 1 1 1 0 1 1 1 0 2 1 2 1 z x h
( ) . ) ( ) ( ) ( ) ( ) ( ) ( ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ - ⋅ ⋅ - ⋅ - ⋅ ⋅ ⋅ - ⋅ ⋅ - ⋅ - ⋅ ⋅ ⋅ - ⋅ ⋅ - ⋅ - ⋅ ⋅ = = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = × × =
⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⋅ - ⋅ = - ± = . ; , , 1 2 , , 2 1 , , 0 , 1 , 2 , x h z y x h x h z y x h x h z y x h x h y z h y z h z y h r a r a r a r a a α (4)
The principle showed above is applied for the thigh and tibia, but the bodies' mutual rotations in concordance with (1) should be taken into account. The detailed solutions for them are not presented in this paper because the volume is limited.

B. Algorithm of Measurement

The algorithm consists of three phases: initialization; calculation of bodies' angular and linear accelerations and quaternion estimation.

The initialization phase is necessary to: 1) parameterize sensors sensitivity coefficients and offsets according to the individual sensors calibration data usually provided by manufacturer; 2) determine mutual orientation of sensors due to possible errors of positioning. Thus, during the initialization phase the biped mechanism should be on the standing posture. The thigh and tibia should be aligned with their main vertical axes, and with the vertical axis of a hip. Equations ( 1) and ( 2) should be used on the second phase to determine the bodies' accelerations. The quarternions can be calculated knowing actual orientations of mechanism bodies. It is usually achieved by sensing on the motor level or by estimating angles by the gravity vector in quasi static conditions with accelerometers and then applying a narrow-band low-pass filter.

III. EXPERIMENTAL RESULTS

Experimental validation is carried out on the biped anthropomorphic robot of Fig. 1 for vertical flexion-extension movements with frequencies of movements appropriate to the considered mechanism (0.3 -0.7 Hz). All the algorithms are embedded in an industrial electronic real time unit that controls the robot. The principle of sensors alignment allows the differential calculation of accelerations which eliminates the centrifuge terms in the kinematic equations for one body. By combining kinematic equations of two connected bodies one can express five accelerations on the articulation joint axes.

The experimental results are depicted on Fig. 3 for the thigh angular and linear accelerations. For angular accelerations, the results are compared to ones based on usual derivation techniques. It shows that our method produces better sensitivity, less noise and reduced time delay because of larger frequency bandwidth of accelerometers comparing to gyroscopes and inclinometers.

Due to better measurement performances of the accelerometer scheme it is possible to observe parasite oscillations whose origin is in the robot mechanical transmissions elasticities and backlashes. It is necessary to measure these phenomena precisely to decrease their influence on the walk of a robot. The advantage of the accelerometer approach is more evident when the higher resolution is needed (that corresponds to movements with the lowest frequency of f fe =0.3Hz on the Fig. 3). For higher accelerations (f fe =0.7Hz) the difference between accelerometer and gyroscopic methods in angular accelerations measurements is not so essential. The limiting factor in this case is the narrower bandwidth of a gyroscope. Inclinometer method does not gives good results in both cases.

IV. CONCLUSION

A non-invasive method for angular and linear accelerations measurements in poly-articulated mechanisms is proposed. The approach uses the rigid body dynamic theory for a number of triaxial acceleration sensors placed in a specific way on the mechanism. The proposed method was first modeled using Cyberbotics Webots TM software and then validated experimentally on the biped robot ROBIAN, LISV, France, for flexion-extension movements.

The resulting price of the proposed sensor system is about 4000 € using ten industrial-grade CXL04GP3 accelerometers. This is a price of one multi-DOF industrial-grade IMU, but at least 1 IMU per mechanisms body is needed. Also, the most of market-presented IMU does not permit to measure body-frame accelerations with acceptable bandwidth. The proposed solution offers observation of main accelerations in a biped locomotion mechanism with sufficient precision, noise level and bandwidth for motion analysis and control. Especially, the method allows to measure accelerations on virtual rotational axes of articulations.

In future, additional analysis of the sensors misalignment on the resulting precision will be made. We are making validation measurements during the walk of a biped robot ROBIAN.

Figure 1 .

 1 Figure 1. The biped robot. Two aligned accelerometers per body allow five acceleration measurements per articulation.

Figure 2 .

 2 Figure 2. Reference systems. The hip, thigh and tibia.One leg model is presented for more clarity.

  As one can see from[START_REF] Channells | Determination of rotational kinematics of the lower leg during sprint running using accelerometers[END_REF], accomplishing the condition of 2 sensors aligning along sagittal axis of the hip (

Figure 3 .

 3 Figure 3. Comparison of three measurement methods for two different frequencies of flexion/extension movement: 1 -double derivation of an inclinometer (SCA102T); 2 -derivation of a gyroscope (LPR530AL); 3-proposed accelerometer setup (CXL04GP3).

ACKNOWLEDGMENT

The authors thank to Pr. Y. Alayli, Director of "Laboratoire d'Ingénierie des Systèmes de Versailles, LISV", France, for his useful advices.

This work is supported by EGIDE PHC France-Ukraine program Dnipro-2011 and AUF (University Agency of Francophony).


Stéphane Vautier
email: vautier@univ-tlse2.fr.

Michiel Veldhuis

Emilie Lacot

Nadine Matton

Octogone

The ambiguous utility of psychometrics for the interpretative foundation of socially relevant avatars

Keywords: classical test theory, item response modeling, psychometrics, measurement, assessment

The persisting debates that measurement in psychology elicits can be explained by the conflict between two aspiration types. One, the epistemologic aspiration, resting on the search for scientific truth, and two, the social aspiration, resting on the demonstration of a capacity to contribute to psychological assessment problems in particular. Psychometrics answer essentially to psychology's demand for social utility, leading to the quasi-exclusive attribution of importance to quantitative interpretation. For psychology to be considered an empirical science, it has to establish its capacity for the measurement of psychological phenomena, even if this means that it recognizes that these phenomena are essentially qualitative.

The Ambiguous Utility of Psychometrics for the Interpretative Foundation of Socially Relevant

Avatars

The persistence of critical publications about measurement in psychology bears witness to a profound crisis at the heart of the discipline (e.g., [START_REF] Borsboom | Conceptual issues in psychological measurement[END_REF][START_REF] Cliff | Abstract measurement theory and the revolution that never happened[END_REF][START_REF] Dooremalen | Metaphors in psychological conceptualization and explanation[END_REF][START_REF] Essex | Between numbers and notions: A critique of psychological measurement[END_REF][START_REF] Haig | On the conceptual foundations of psychological measurement[END_REF][START_REF] Hood | Validity in psychological testing and scientific realism[END_REF][START_REF] Johnson | Pseudo-mathematics in the mental and social sciences[END_REF][START_REF] Lumsden | Test theory[END_REF][START_REF] Michell | An introduction to the logic of psychological measurement[END_REF]1997;[START_REF] Michell | Normal science, pathological science and psychometrics[END_REF]2003a;[START_REF] Michell | The logic of measurement: A realist overview[END_REF]2008;[START_REF] Porter | Measurement, objectivity, and trust[END_REF][START_REF] Trendler | Measurement theory, psychology and the revolution that cannot hapen[END_REF][START_REF] Zumbo | Responsible modeling of measurement data for appropriate inferences: Important advances in reliability and validity theory[END_REF]. We interpret this crisis as the result of a conflict between two types of aspirations. The first aspiration type is epistemologic: psychology needs to be edified as an empirical science, implying that it respects the empirical facts it observes. This includes the facts that imply a definitive absence of empirical laws in a given descriptive reference system (Vautier, 2011a;[START_REF] Vautier | The operationalisation of general hypotheses versus the discovery of empirical laws in Psychology[END_REF], as for example the absence of a conjoint order relatively to a set of test items [START_REF] Bertrand | D'une échelle ordinale de Guttman à une échelle de rapports de Rasch[END_REF][START_REF] Guttman | A basis for scaling qualitative data[END_REF]. The second aspiration type is political: psychology hopes to become an academic discipline that is capable of not only satisfying but also creating a social demand, implying that it respects the way social problems, particularly psychological assessment problems, are proposed by the employers of psychologists. It becomes ever more difficult to ignore the respective divergent implications of these two aspiration types.

Whereas the scientific aspiration leads to the abandonment of the dream of discovery of authentic quantitative psychological properties [START_REF] Michell | The quantitative imperative: Positivism, naïve realism, and the place of qualitative methods in psychology[END_REF][START_REF] Michell | The place of qualitative research in psychology[END_REF][START_REF] Trendler | Measurement theory, psychology and the revolution that cannot hapen[END_REF], the social utility aspiration leads to the defense of the evaluation of these properties as the basis of psychologists' work [START_REF] Brown | The definition of a profession: The authority of metaphor in the History of intelligence testing, 1890-1930[END_REF][START_REF] Danziger | Constructing the subject: Historical origins of psychological research[END_REF][START_REF] Martin | La mesure de l'esprit : origines et développements de la psychométrie[END_REF][START_REF] Porter | Measurement, objectivity, and trust[END_REF]. Effectively the social demand rests on anthropological simplifications based on value scales. This in order (i) to be able to think that human beings can be ordered according to their psychological properties (e.g. the benefit of a course or sexual offenders' recidivism), and (ii) to be able to objectify their evolution (e.g. therapeutic progress). These practical exigencies create the necessity of psychological anthropometrics, i.e. a science of construct measurement [START_REF] Cronbach | Construct validity in psychological tests[END_REF][START_REF] Maraun | Measurement as a normative practice: Implications of Wittgenstein's philosophy for measurement in psychology[END_REF]. The promotion of this science has, from the start of the 20th century, been assured by psychotechnic engineering [START_REF] Brown | The definition of a profession: The authority of metaphor in the History of intelligence testing, 1890-1930[END_REF][START_REF] Martin | La mesure de l'esprit : origines et développements de la psychométrie[END_REF].

A lot of effort has been expended to develop psychometrics. We think it useful to distinguish psychotechnic engineering from psychometric engineering, the development of the latter following chronologically on that of the former. Psychotechnic engineering responded to a need of aggregating a set of qualitative observations into a score, and the optimatisation of this score [START_REF] Cronbach | Construct validity in psychological tests[END_REF][START_REF] Messick | Validity of psychological assessment: Validation of inferences from persons' responses and performances as scientific inquiry into score meaning[END_REF]. The purpose of psychometric engineering, as a discipline of psychotechnic data modeling, is less clear [START_REF] Thissen | Psychometric engineering as art[END_REF]. In this article we support the point of view that psychometric engineering is interesting for construct psychologists, because of the mere conceptual possibility of these constructs, and not because these statistical models based on continuous latent variables confirm some empirical truth.

To respond to the social demand of the comparison of human beings relatively to constructs, the testing psychologist needs his scores to have a significance as construct measures (see Cliff & Keats, 2003, p. 16). However, tests in themselves do not allow one to conclude about the existence of constructs as continuous quantities. A psychometric model associated to a test rests on the explicit imagination of a causal role of the construct that the test is supposed to measure [START_REF] Borsboom | Latent variable theory[END_REF][START_REF] Borsboom | The theoretical status of latent variables[END_REF]. As such the psychometric model constitutes the probabilistic pole of the construct measurement paradigm.

However its probabilistic nature renders it nomothetically sterile -inept at the discovery of laws -as it is tautological, and epistemologically counter-productive. This because psychometric estimations require empirical conditions -inter-individual aggregation -that deny the individual nature of events occurring in the space where these probabilities have their foundations. Which leads to the conclusion that (i) scientific psychology has to put itself at distance from the social demand of the evaluation of human beings, to be able to fulfill its quest for the truth and that (ii) the psychotechnic engineering has to admit that its lack of scientific foundations calls for a political foundation [START_REF] Dagonet | Réflexions sur la mesure[END_REF].

In this article, (i) the production process of psychotechnic data as well as the conceptual problems that the discrete scores cause will formally be discussed. Then (ii) an analysis of how the classical test theory solves these problems is performed. Also (iii) the solution that item response modeling for dichotomic items proposes will be discussed. Then (iv) the unfalsifiable character of the probability of a response given a certain true score or latent trait comes next. The application of a psychometric model demands a leap from the individual level to the aggregate level to permit a latent score estimation; then (v) it will be shown that the estimation of a model is not equivalent to a validity test of the generic interpretation. Finally, (iv) the epistemologic and ethical consequences of this change of empirical phenomenon apprehension level will be analyzed.

From standardized observation to objective quantitative assessment: Psychotechnics

The production of psychotechnical scores can be described as a composite function that allows the description of people via an empirical process of standardized observation (testing) followed by a symbolic process of quantification of this description (numerical coding). These two applications are described in the following (for a more technical analysis, see [START_REF] Vautier | Why test scores may be improper data for scientific psychology: A qualitative analysis[END_REF].

A function is a relation defined from a set, called its domain, on a set, called its codomain, such as any element of the domain is related to one and only one element in the codomain (e.g., [START_REF] Selby | Sets relations functions[END_REF]. We note Ω the domain of the former, i.e., the application of standardized description, and ω an element of Ω. Every element of Ω is a couple (ordered pair) (person, date of observation), that is noted ω = (u, t). The observation at a moment in time of a person ensures the unicity of the empirical objects ω what leads the description to be a function -someone that has been evaluated at two different dates can be described differently. We note M k the codomain of the application, comprising the possible response patterns. M k is a finite set, most of the time the Cartesian product of several (k) descriptive sets �𝑥 𝑖1 , 𝑥 𝑖2 , … , 𝑥 𝑖𝑚(𝑖) � (i = 1, …, k), where the m(i) modalities 𝑥 𝑖 are nominal or ordinal values. For example, in a test of 50 items with scores in (0, 1) every ω can be attributed an 50-tuple in M 50 = (0, 1) 50 .

The standardized description application, that we also call psychometric description, accomplishes a double objectivation, whence the epistemological importance is fundamental for the empirical constitution (and thus scientific) of psychological facts. Firstly, every object ω has necessarily a state in the set M k of possible descriptions. Would this not be the case, the description of ω could not be universal. Secondly, the determination of ω's state does not depend on specific characteristics that identify its observer. The double objectivity of the psychotechnic description permits to attribute to every proposition that specifies the state of ω a trueness value (true or false) that is epistemologically accessible. These are the facts that can be observed in M k that permit to falsify a theory of what is happening -this theory will be called general and falsifiable if it excludes at least the possibility of one state [START_REF] Popper | The logic of scientific discovery[END_REF][START_REF] Roberts | How persuasive is a good fit? A comment on theory testing[END_REF]Vautier, 2011a;[START_REF] Vautier | The operationalisation of general hypotheses versus the discovery of empirical laws in Psychology[END_REF].

The second application is of a symbolical nature in the way that it does not depend on what is happening in the set Ω: it is about the attribution of one and only one score to every state of the set M k of possible descriptions. We call this application a numerical, metaphorical coding.

The composite function, of the standardized description and the numerical coding applications, permits to attribute a single score to every ω. Projected on the scale of test scores, every ω i can be compared to every ω j with the help of the algebraic operator ≤. The composite function of these applications equips the psychologists with the tools to cope with the social problems that demand an objective comparison -i.e., impersonal and consensual (see [START_REF] Porter | Measurement, objectivity, and trust[END_REF] -of people.

The semantic register of this comparison's interpretation is not exactly linked to that of the set M k of possible responses on the test. The interpretation of the comparison mobilizes the semantics of the construct as a total order. Consider a construct as numerical ability for example.

The semantics of numerical ability are such that a sentence like "Paul is (whatever the date we observe him) numerically more able than Jean (whatever the date we observe him)" has a "social" significance.

But the psychotechnical procedure of projecting a person on a scoring scale reveals a logical problem. Suppose that the scoring scale of an ability test is the following sequence (0, 1, …, 50).

The semantics of numerical ability do not specify how it is possible that the ability variations are organized in 51 steps, because the concept of numerical ability is intuitively construed. By default, the non-specification of the construct leads numerical ability to vary on a continuous scale. As such it would be impossible to measure a score of 30.32 points with this test for example. If, as a measurement instrument, the test would measure a continuum, it should operationalize the principle of continuity that it measures. Thus, it is embarrassing to assert that the test measures numerical ability, while the issue at hand is to understand how the test could measure a continuous quantity, whereas it fabrication principle does not permit this continuity to be established [START_REF] Vautier | About validity of conclusions based on multiple linear regression: A commentary on Kupelian et al[END_REF].

The metaphysical meaning of test scores (1): Classical Test Theory Classical test theory proposes the following solution: if the observable scores vary in a ascending sequence (y 0 , y 1 , …, y z ), then the true score varies in the continuum [y 0 , y z ]. But, as we will show, the true score concept does not have an empirical significance.

(1) Write down the time (t). Let someone (u) make the test and save his score. (2) Go back in time to a moment before moment t. (3) At moment t let u make the test again and save his score again. (4) Forget about mortality and repeat operations (2) and (3) until infinity. (5) At the end of infinity calculate the mean score: now you have discovered the true score of (u, t). This score, which is also called a mathematical expectation [START_REF] Lord | Statistical theories of mental test scores[END_REF][START_REF] Steyer | Classical (psychometric) test theory[END_REF], is an empirical impossibility, nothing but a thought experiment [START_REF] Borsboom | Functional thought experiments[END_REF]. As such, the construct "numerical ability, measured with the numerical ability test" has a pseudo-operative significance.

For now, the true score has been construed from an imaginary series of scores. As such, the true score depends logically on the imaginary series of scores. But the construct measurement psychologist looks to comprehend how a score on a test can be interpreted as the result of a measurement process. In this perspective, forget about observed scores and consider that there is a true score: existing independently of the series of test scores. By doing this, a metaphysical postulate is called upon, that permits to surpass the paradox of the non-operational status of something that is only defined by a thought experiment. From now on, the true score is ontologically primary, instead of the mean score of several measures.

Then there is the concrete measure (i.e., the test administration and the calculated score as a function of made observations), that leads to biases. These biases are added to the true score, which leads to the observed score. This has nothing to do with realism, neither has the consideration of a construct as a continuous quantity for that matter. Let us write this in a mathematical fashion: let τ (tau) be the true score, ε (epsilon) the bias, and y the observed score.

𝑦 = 𝜏 + 𝜀.

(1)

Now it has to be shown how τ varies in the continuum [y 0 , y z ]. For this, it suffices to take a step further in the definition of τ. This step will allow us to show how a second spiritual point, corollary to the first, is mobilized: the probability of observing a score knowing τ. Let there be z+1 probabilities: the probability, knowing τ, to obtain the score y 0 , is noted p 0 ; the probability, knowing τ, to obtain the score y 1 , is noted p 1 ; etc. until the probability, knowing τ, to obtain the score y z , is noted p z . These probabilities are nothing but the frequency of every score, when this frequency is calculated with the infinity of measure points that the initial thought experiment provided. We also admit that we will always get a score when the test is administered (we neglect the situation where a participant refuses to respond). As such, the sum of the probabilities of every score equals 1:

𝑝 0 + 𝑝 1 + ⋯ + 𝑝 𝑧 = 1. (2)
With these probabilities, another definition of the true score can be given:

𝜏 = 𝑝 0 • 𝑦 0 + 𝑝 1 • 𝑦 1 + ⋯ + 𝑝 𝑍 • 𝑦 𝑍 . ( 3
)
The appendix shows formally that τ varies in [y 0 , y z ]. Here we will stick with an intuitive approach.

Does τ have a minimum and a maximum? With some possible scores and some probabilities, one can do whatever one wants as long as their sum remains equal to 1. To minimize τ, the probability to have the minimal score has to be maximized, meaning that p 0 has to be 1, implying that all the remaining probabilities equal 0. Which leads to min(τ) = y 0 • 1 = y 0 . Analogous reasoning leads to max(τ) = y z • 1 = y z . In addition to this, τ can take every real value as the z+1 probabilities vary in

[0, 1].
The construction of the true score shows two philosophical problems. The first problem derives from the volte-face that we have to do in order to call a series of scores on a test the true score: after all, nothing proves that this true score is identical to the true score that the psychologist wants to see as the metaphysical cause of the observed score. The second problem is due to the fact that the true score in classical test theory is contingent on the test. As such, the metaphysical psychologist has to elaborate another solution to show how it is possible that true scores linked to different tests can be caused by the same construct.

The metaphysical meaning of test scores (2): Item Response Theory Item response modeling tackles the problem of the relation between the set M k of possible descriptions and the continuum on the item level instead of the test level. For simplicity reasons we only consider dichotomic items (i.e. items varying in {0, 1}).

The basic idea is very simple, analogous to the true score, albeit a bit more sophisticated. Let there be a continuum and suppose that a person that responds to an item has a position on this continuum, noted θ (theta). No thought experiment is needed, the continuum exists by definition and every object ω = (u, t) has a position on it. Knowing this position on item I i , we admit that the probability p i1 to find event '1' exists (the probability p i0 = 1p i1 to find event '0' as well). This leads to the simple interpretation of the observed response as a Bernoulli trial (e.g., [START_REF] Feller | An introduction to probability theory and its applications[END_REF].

The expected value of the score is thus p i1 (indeed: 𝜏 = 0 • 𝑝 𝑖0 + 1 • 𝑝 𝑖1 ). Formulated differently, the true score on item I i depends on θ. But to the metaphysical psychologist θ is the quantity of interest, not the true score.

The generalization of this conceptualization to any item I i leads to the postulation of the existence of probability p j1 knowing ω, observing event '1' on item I j . An item response model stipulates that the true score (i.e., p i1 ) depends in an ascending manner on θ: the bigger the value on the latent trait, the more the true score approaches 1. The shell game, that needs to be played in classical test theory, consisting of defining the true score as an expected value and then considering that it exists independently of the scores on which it is based, is of no need here. The postulate of the continuum forms the foundation, because there is only one continuum for as many items as are supposed indicating it.

The two approaches (classical test theory and item response modeling) permit one to conceptualize how a discrete event (the score on a test) depends on the latent trait from a continuum. The advantage of item response modeling is that it clarifies the hypostatic state of the continuum at the same time that it makes the true score logically dependent on its indicator. We will now show why propositions from these two approaches can never be falsified.

Unfalsifiability of probabilistic statements about single random experiments

A proposition that can be falsified is a proposition that can be found to be false. In order for a proposition to be able to be falsified, its negation has to be able to be verified. A theoretical proposition founded on an empirical science can be falsified if it permits the possibility to verify an empirical fact which it excludes theoretically [START_REF] Popper | The logic of scientific discovery[END_REF]. In all the preceding theory, can a proposition that can be falsified be found? No, not if the theory saturates the observation reference system, which is the case in all of the above -a theory saturates the reference system if it proposes no theoretical impossibility (cf. Vautier, 2011a;[START_REF] Vautier | The operationalisation of general hypotheses versus the discovery of empirical laws in Psychology[END_REF].

Let us consider the proposition (1): 𝑦 = 𝜏 + 𝜀. Let y be an observed score in (y 0 , y 1 , …, y z ).

A number τ exists in [y 0 , y z ] and a real number 𝜀 so that their sum equals y. This does not exclude any event in (y 0 , y 1 , …, y z ). Thus this proposition cannot be falsified.

If we go a little bit further and examine the proposition

𝑝 0 = 𝑝 1 = ⋯ = 𝑝 𝑧 = 1/(𝑧 + 1) (4)
(and 𝜏 = 1 𝑧+1

• (𝑦 0 + 𝑦 1 + ⋯ + 𝑦 𝑍 ) = (𝑦 0 + 𝑦 𝑍 )/2. Can this proposition be falsified? For a person and an experimental condition, every observable score is compatible with these probabilities because the theory does not limit the domain of definition of 𝜀.

In order for a proposition of this kind to be falsified it should at least stipulate that one of the probabilities equals zero, in which case it would preclude the scores of which the probability of their observation is zero. Another possibility would be the restriction of the domain of definition of 𝜀 (for example, if we predict that the probabilities equal 1/(z+1) and we restrict 𝜀's domain to [-3, 3], then we predict that 𝑦 ∈ � every object ω = (u, t) has a score θ (the continuum postulate is not disguised anymore). While the theory does not give a zero (or certain) probability to an event '1', every proposition that specifies the values of θ and p i1 cannot be falsified (the case where θ tends to infinity does not have intuitive significance). As in classical test theory, item response modeling uses probabilities whereof the scientific significance remains uncertain. This scientific significance is doomed to remain uncertain, as one would have to study the distribution of '0' and '1' while controlling θ, and a measurement instrument that measures θ does not exist (see [START_REF] Trendler | Measurement theory, psychology and the revolution that cannot hapen[END_REF]. In other words to consider the unique response of a person at a certain moment in time as the result of a Bernoulli trial is a thought experiment as well. This is why psychometric models are considered to lack a scientific vocation (in a Popperian way), leaving only an interpretative vocation. Remarkable in this regard is that specialists of latent variable models as [START_REF] Muthén | Mplus: User's guide[END_REF] declare in the introduction of their program's user manual that "The purpose of modeling data is to describe the structure of data in a simple way so that it is understandable and interpretable" (p. 1).

From the individual level to the collective level: The psychometric model as a pre-scientific fiction

The impossibility to study the probability of observed scores of an object ω empirically leads to the abandonment of the individual in favor of the collective scale to fit a model to the data.

Here one needs to pose the question "why fit a model to the data"? Up to this point it has been admitted that a psychometric model permits one to explain how scores can be (probabilistic) measures of a continuous latent quantity. Consequently, it suffices that the model be recognized as a conceptual possibility so that the conceptual reconciliation of the observation reference system (the score scale or M k ) with the construct is finished.

The estimation of a psychometric model consists of the specification of a null hypothesis (meaning a vector v of υ numerical values belonging to a set M υ ) in a way to minimize a measure of the distance between the data and the modeled image of the data. In itself the psychometric model is not a falsifiable theory. It is a model without a falsifiable theory. The model could be useful to falsify a psychological theory if the theory restricted the space of theoretically possible parameters to a strict subset of M υ , in a way that it would always be possible to find, on an empirical basis, a unique solution outside this subset (in which case the theory would be falsified, see [START_REF] Roberts | How persuasive is a good fit? A comment on theory testing[END_REF].

In the psychometric data modeling practice, the psychometric model is taken to be mispecified (e.g., [START_REF] Drasgow | Equivalence of psychological measurement in heterogeneous populations[END_REF]. Consequently, the idea according to which the psychometric model is a theory of a stochastic process of observed data generation is not seriously considered and the null hypothesis is almost unanimously disqualified (e.g., [START_REF] Hambleton | Fundamentals of item response theory[END_REF][START_REF] Van Der Linden | Handbook of modern item response theory[END_REF].

Ethics of psychological evaluation: social and instrumental objectivity and relativism

In practice, the people that use construct measurement in order to characterize individuals at a moment in time consider that observed scores are the best possible measures of a construct, knowing what we call (abusively, see [START_REF] Messick | Validity of psychological assessment: Validation of inferences from persons' responses and performances as scientific inquiry into score meaning[END_REF][START_REF] Thompson | Psychometrics is datametrics: The test is not reliable[END_REF] the tests' psychometric properties.

If psychometrics do not permit to give the status of scientific objects [START_REF] Granger | La science et les sciences[END_REF] to constructs, the testing community has to recognize their constructivist nature [START_REF] Hacking | The social construction of what?[END_REF]. The fact that scientific psychology has not discovered a psychological quantity does not preclude that these quantities are imaginary, particularly if they have social significance. But engineering dedicated to the determination of positions that people take on these quantities at a moment in time has to be recognized as an engineering of avatar fabrication, i.e. representations of people in a virtual world.

This analysis leads us to compare the psychological evaluation to Internet gaming where players are represented by their avatars. The actors of the evaluation game subscribe implicitly to a social contract, in which the evaluated people accept to be represented by their avatars in a simplified universe that is adapted for objective decision making (i.e., impersonal and consensual).

The rationality of the representation in such a universe is not psychological, but institutional, as the decision is optimized on a collective level.

The collective validity of test scores leads to the fact that the psychotechnical determination of individuals in total orders is a process that is relative to (i) a sample and (ii) to the psychotechnical techniques (as a composition of applications of standardized description and numerical coding, Vautier et al., 2011). From an individual point of view, accepting this psychometric evaluation supposes the acceptation of the abandonment of the idea of psychological autonomy for a heteronomic identity, i.e. based on the relationship between the self and others.

Conclusion

In this article, we have shown that psychometric data modeling is not part of a nomothetical research program, as it does not contribute to the discovery of empirical laws that govern the psychotechnically observable phenomena. The contribution of psychometric modeling is fundamentally a political one, as it permits the assimilation of the reality of phenomena that are described in a qualitative way and can at best be partially ordered to a intuitively totally ordered reality, where the social utility rests on the need for comparison of human beings. If psychological research can find its legitimacy more in knowledge ethics than in social utilitarism (or practicalism, [START_REF] Michell | Quantitative science and the definition of measurement in psychology[END_REF], then psychometric modeling appears to be useless.

  The classical test theory does not specify any case of this kind. The probabilities that classical test theory proposes are entities that do not have a Popperian scientific significance. However it does propose a theory of what is measured, based on (1) the disguised postulate of a continuum and (2) on the postulate of the conditional probabilities of observed scores. It is an interpretative theory that can be of use if the sole goal is to obtain an interpretation of a score as an intuitive quantity measure. Now consider the probabilistic modeling of the response on item I i of a test. By definition,

Appendix

Objective Show that the expected value of a numerical random variable admitting a limit is comprised between the inferior and the superior limit of the variable.

Notations and definitions

Let Y be a numerical random variable admitting a limit. It is an application from the set of events Ω to a finite set of numerical values (𝑦 𝑖 ) 𝑛 .

The (𝑦 𝑖 ) 𝑛 notation designates an ascending sequence of n numerical values

The expected value of Y is defined if a probability law governs the realisation of events of (𝑦 𝑖 ) 𝑛 .

Thus we define the probability law (𝑝 𝑖 ) 𝑛 = (𝑝 0 , 𝑝 1 , … , 𝑝 𝑛 ) such that:

-for every i, p i belongs to [0, 1], -the sum of p i equals 1.

The expected value of Y, noted E(Y), is defined by:

Show that E(Y) ≥ y 0

The following property will be used: if the minimal value of the variable equals 0 then all the values are positive and the expected value is positive as well, for all p i are positive.

Let a real number k exists such that y

and -k = y 0 . QED.

Show that E(Y) ≤ y n

The following property will be used: if the maximal value of the variable equals 0 then all the values are negative and the expected value is negative as well, for all p i are positive.

Let a real number l exists such that y n + l = 0. Suppose Y'' = Y + l defined in (y'' 0 , y'' 1 , ..., 0). E(Y'')

≤ 0 because all sums in the definition are negative.