Angeliki Tataridou
email: angeliki.tataridou@grenoble-inp.fr

Gerard Ghibaudo

Christoforos Theodorou

VERILOR: A Verilog-A Model of Lorentzian Spectra for Simulating Trap-related Noise in CMOS Circuits

Keywords: Low-frequency noise, G-R noise, Lorentzian noise, Random telegraph noise, Noise Variability, Tri-Gate Nanowire MOSFETs, Verilog-A, Circuit noise

This work presents a new method of noise model implementation for circuit simulations, which enables the generation of Lorentzian-type spectra. The importance of Lorentzian noise modeling is explored in contrast to classic 1/f or Random Telegraph Noise (RTN) modeling, in terms of PSD, total noise power, and device-to-device noise variability reproduction, through analysis of experimental data. The Verilog-A modeling methods for both types of Lorentzian noise are demonstrated and validated, using a Lambert-W function charge-based drain current model. Finally, fundamental digital and analog circuits such as the Ring Oscillator are used to showcase the usefulness and applicability of the "VERILOR" model in circuit noise simulations.

I. INTRODUCTION

With the incorporation of new materials and device architectures under the demand of "More Moore" and "More than Moore" technology roadmaps, defect-related effects such as low frequency noise (LFN) are attracting more attention by the semiconductor industry. The LFN level increases with the device downscaling [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF], while there is also the appearance of Lorentzian spectra instead of 1/f-like [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF]. This type of power spectral density (PSD) shape may originate either from oxide trap-related random telegraph noise (RTN) signals (leading to 1/f in large area oxides) or generation recombination (G-R) centers in the MOSFET channel region [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF]. Whatever the origin or PSD form, LFN results to instabilities in both digital [START_REF] Aadithya | SAMURAI: An accurate method for modelling and simulating nonstationary random telegraph noise in SRAMs[END_REF] and analog circuits [START_REF] Wang | Random telegraph signal in CMOS image sensor pixels[END_REF], therefore its accurate modeling is crucial for the reliable design and safe operation of circuits.

While various works have been presented on the timedomain modeling of the RTN effect for circuit simulators [START_REF] Aadithya | SAMURAI: An accurate method for modelling and simulating nonstationary random telegraph noise in SRAMs[END_REF][START_REF] Luo | Compact modeling of Random Telegraph Noise in nanoscale MOSFETs and impacts on digital circuits[END_REF], there are very few that concern the modeling of Lorentzian noise sources directly in the frequency domain [START_REF] Leyris | Random telegraph signal noise SPICE modeling for circuit simulators[END_REF]. The latter could, on one hand, significantly reduce the noise simulation time, and on the other hand enable a direct and accurate simulation of the PSD for circuits, such as the oscillators, where the noise is important to be studied in the frequency domain. Moreover, the pre-existing PSD modules in Verilog-A are limited to 1/f and white noise, therefore the Lorentzian type of noise needs a dedicated module to be developed. In this work, we present a method for the implementation of Lorentzian noise spectra in Verilog-A, in a way that can automatically generate Lorentzian or 1/f noise depending on the trap density and the oxide area, for all bias conditions. After underlining the advantages of Lorentzian modeling with regards to 1/f and time domain modeling of RTN, with the help of a Lambert-W function charge-based drain current model [START_REF] Karatsori | Full gate voltage range Lambert-function based methodology for FDSOI MOSFET parameter extraction[END_REF], we demonstrate our module's applicability in circuit simulators, and how it can enable precise noise variability studies at a circuit level. Finally, some circuit noise application examples are shown, revealing for example how a ring oscillator's (RO) phase noise can be impacted by the presence of a Lorentzian-type PSD.

II. SIMULATION SET-UP AND TEST DEVICES

In order to explain the motivation for our modeling idea, and how it can prove very useful for emerging technology devices, we used experimental results from measurements performed on Tri-Gate Nanowire SOI p-channel FETs, fabricated by CEA-LETI [START_REF] Barraud | Opportunities and challenges of nanowire-based CMOS technologies[END_REF]. They consist of a 145 nm thick buried oxide, a Si film of H = 11 nm and a 2 nm Equivalent Oxide Thickness (EOT) HfSiON gate dielectric with TiN metal gate. Drain current PSD measurements were performed with the programmable bias point probe system NOISYS. Both the device-and circuit-level noise simulations were performed with the Cadence Spectre simulator.

III. LORENTZIAN NOISE MODELING

As already mentioned, the miniaturization of the devices increases the probability of the existence of isolated traps as opposed to large oxide areas. In this case the main LFN source stops being 1/f-like but instead appears a Lorentzian-like dependence: plateau at low frequencies and 1/f 2 dependence above a certain cut-off frequency f c , as described by [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF], where S(0) the plateau at 0 Hz.

𝑆(𝑓) = 𝑆(0) 1 + (𝑓 𝑓 𝑐 ⁄ ) 2 (1)
In many publications RTN is modeled in the time domain [START_REF] Aadithya | SAMURAI: An accurate method for modelling and simulating nonstationary random telegraph noise in SRAMs[END_REF][START_REF] Luo | Compact modeling of Random Telegraph Noise in nanoscale MOSFETs and impacts on digital circuits[END_REF]. The advantage of this method is that the state of a circuit can be accessed at any time and so possible errors due to RTN can be examined. However, concerning variability studies or multiple traps with huge spans of time constants, transient circuit simulations can take very long, while the situation becomes even more difficult when increasing the number of components in a circuit.

To overcome time-consuming issues, but also be able to directly perform noise spectrum simulations, the RTN modeling in the frequency domain becomes crucial. To do so, the authors of some publications use the LFN variability level, σ(log(S)), extracted from experiments or models and add the +/-3σ to the 1/f average spectrum [START_REF] Banaszeski | A Physics-Based Statistical RTN Model for the Low Frequency Noise in MOSFETs[END_REF], in an effort to cover the extreme LFN level variations induced by RTN. By doing this, although the safe design of a circuit in terms of LFN is succeeded, the LFN level itself might be significantly overestimated or underestimated. To outline this phenomenon, we show in Fig. 1 a set of 93 experimentally measured input-referred gate voltage noise spectra, along with the +/-3σ limits added to the average 1/f, with σ(log(S Vg )) extracted at 20 Hz. We chose a very small gate area device so as to increase the noise variability and the probability of RTN traps. As shown in Fig. 1, the +/-3σ spectra fails to accurately estimate the maximum and minimum extreme conditions. Furthermore, what's even more important is not the value of the PSD at a certain frequency, but the total noise power of the fluctuations for each die calculated by integrating the PSD over the whole frequency range, as in (2):

𝜎 𝑉𝑔 2 = ∫ 𝑆 𝑉𝑔 𝑑𝑓 𝑓 𝑚𝑎𝑥 𝑓 𝑚𝑖𝑛 (2)
The maximum and minimum total noise power of the experimentally measured spectra and the ones taken from 1/f +/-3σ were calculated using (2) and are depicted in Fig. 2. The huge difference observed in both the maximum and minimum noise power between data and 1/f model reveals both the importance and the need for a different type of modeling approach for highly downscaled devices. This difference can be nonetheless understood, if we take as an example the expected noise power of a 1/f and a Lorentzian spectrum that coincide at 10Hz (Fig. 3(a)), a typical frequency for the extraction of characteristic noise levels. Combining (1) and ( 2), the Lorentzian integral can be explicitly expressed through [START_REF] Wang | Random telegraph signal in CMOS image sensor pixels[END_REF]:

𝜎 𝑉𝑔,𝐿𝑜𝑟 2 = 𝑆 𝑉𝑔 (0)𝑓 𝑐 (tan -1 𝑓 𝑚𝑎𝑥 𝑓 𝑐 -tan -1 𝑓 𝑚𝑖𝑛 𝑓 𝑐 ) (3)
Concerning the flicker (1/f) noise spectrum, its total power can be easily calculated through (4),

𝜎 𝑉𝑔,𝐹𝑙𝑖𝑐𝑘 2 = 𝑆 𝑉𝑔 (1)𝑙𝑛(𝑓 𝑚𝑎𝑥 𝑓 𝑚𝑖𝑛 ⁄ ) (4)
where S vg (1) the 1/f PSD value at 1 Hz.

Using ( 3) and ( 4), we obtained the total noise power of the Flicker and Lorentzian spectra shown in Fig. 3(a), and plotted in Fig. 3(b) versus a varying bandwidth f-f min . It is worth noting that, while the Lorentzian PSD amplitude is 2 orders of magnitude below 1/f at the lower and the higher edges of the bandwidth, the total noise power of the Lorentzian spectrum becomes significantly higher (1 order of magnitude), even for a bandwidth equal to its cut-off frequency f c .

We proceed to emulate the experimental variability using a PSD level generator, that considers a Lorentzian noise spectrum for each RTN fluctuator and their sum for each die, as already done in [START_REF] Ioannidis | Low frequency noise variability in high-k/metal gate stack 28nm bulk and FD-SOI CMOS transistors[END_REF], through [START_REF] Leyris | Random telegraph signal noise SPICE modeling for circuit simulators[END_REF]: (5

𝑆 𝑉𝑔 (𝑓) 𝑑 = ∑ 4𝛥𝑉 𝑡,𝑘 2 𝐴 𝑘 𝜏 𝑘 1 + (2𝜋𝑓𝜏 𝑘 )
)
where N T,d is the total number of active traps for die 'd', k the index, ΔV t the RTN amplitude, A the space mark ratio, τ c and τ e the mean capture/emission time constants of each trap. N T was chosen for every die from a Poisson distribution with an average <N T > = N t WLt ox ΔE, where N t the oxide trap density per volume per energy and ΔE the total energy bandgap. Moreover, for the calculation of τ c and τ e we implemented a simplified SRH-like approach, as in ( 6):

𝜏 𝑐 (𝑉 𝑔 ) = 𝜏 0 𝑄 𝑖 (𝑉 𝑡 ) 𝑄 𝑖 (𝑉 𝑔 ) , 𝜏 𝑒 = 𝜏 0 𝑒 𝛥𝐸 𝑇 𝑘𝑇 (6)
where τ 0 is a time constant distributed exponentially between 1 ns and 1 ks, and Q i is the inversion charge density calculated using Lambert-W function. ΔΕ T is the difference between the trap and the Fermi energy levels, ranging between -5kT and 5kT, in order to cover all possibly active traps for every bias.

As it is shown in Fig. 4, although the 1/f variability (Monte Carlo results) follows well the experimental data at 20 Hz, where we extracted the σ, it fails to do so at higher frequencies, i.e. 120 Hz, and its prediction is even worse regarding the total noise power, σ 2 Vg , estimated by [START_REF] Aadithya | SAMURAI: An accurate method for modelling and simulating nonstationary random telegraph noise in SRAMs[END_REF]. On the other hand, our model, using ( 5) and ( 6), can reproduce well the experimental variability in all cases, proving its usefulness and accuracy for circuit-level LFN simulations.

C. Frequency domain and LFN variability

As an easy and direct way to verify that the VERILOR module can be successfully used in a circuit with many transistors, we compared the total output noise to the sum of the individual device spectra for a CMOS inverter (Fig. 9) with highly scaled devices (only a handful of active traps). Moreover, in circuits like oscillators, the LFN up-converts to phase noise close to the oscillation frequency, f 0 , making the frequency domain noise simulation crucial. Using the VERILOR module, we estimated the impact of Lorentzian noise on a 3-stage Ring Oscillator's phase noise. As can be seen in Fig. 10, while 1/f up-converts to 1/f 3 , a Lorentzian spectrum up-converts to 1/f 2 close to f 0 and then 1/f 4 after the Lorentzian's cut-off frequency, f c . As a result, for a scenario where 1/f coincides with the Lorentzian at 1 Hz, the phase noise becomes significantly higher around f c , while the corner frequency, i.e. where LFN meets the thermal noise level, is shifted upwards by more than one decade. This observation underlines the importance of Lorentzian noise modeling in RF/mixed signal circuits, where the operation bandwidth is a critical Figure of Merit. The opposite (lower phase noise and corner frequency) could also take place, if the Lorentzian has a lower plateau, due to the device-to-device variability as explained in III. Thus, it is important to showcase (Fig. 11) how the VERILOR module can also reproduce the LFN variability effect in a circuit simulator, using the physics-based approach of ( 5)-( 6).

Fig. 1 .

 1 Fig. 1. Measured input-referred gate voltage noise spectra on TriGate Nanowire MOSFETs, plotted together with the log-mean spectrum and the +-3σ 1/f spectra based on the 1/f variability modeling. (Weff = 52nm / L = 30nm, Vg = 1V, Vd = 50mV).

Fig. 2 .

 2 Fig. 2. Experimental minimum and maximum noise power along with the +/-3σ power of 1/f variability model in red.

Fig. 3 .

 3 Fig. 3. 1/f versus Lorentzian noise (with fc at 1kHz), in terms of PSD versus frequency (top) and total noise power versus bandwidth (bottom). The 1/f magnitude coincides with Lorentzian plateau at f = 10 Hz.

Fig. 4 .

 4 Fig. 4. Quantile Plots of measured input-referred gate voltage PSD in two frequencies 20 and 120 Hz (top) and total power (bottom).

Fig. 5 .

 5 Fig. 5. Flowchart of the VERILOR module. NpF: total number of points in spectrum; Ppd: points per frequency decade; fmin: starting frequency

Fig. 7 .

 7 Fig. 7. Area-normalized PSD of Id versus frequency at Vg=0.5V (top) and versus normalized current for f = 10 Hz (bottom), for 4 different W and corresponding NT, along with CNF model. (L = 0.3μm)

Fig. 8

 8 Fig. 8 Drain current transient signal of LW-based MOSFET (top), for direct RTN generator and Transient Noise signal based on the Lorentzian spectrum (bottom), which agrees with the RTN spectrum.

Fig. 9 .

 9 Fig. 9. Output current PSD of LW-based CMOS inverter (inset showing I/O curve) versus frequency for Vin=0.5V, along with separate noise contributions from each MOSFET, and their sum. (W=80nm/L=30nm).

Fig. 10 .

 10 Fig. 10. Phase noise of LW-based 3-stage CMOS ring oscillator (signal in inset) versus offset frequency for 1/f and Lorentzian noise. The 1/f magnitude coincides with the Lorentzian plateau at 1 Hz.

Fig. 11 ,

 11 Fig. 11,. Screenshot of noise variability (60 dies) simulation results in Cadence Spectre, using the VERILOR module. (W=80nm/L=30nm)

ACKNOWLEDGMENT

The authors would like to thank Dr. Sylvain Barraud from CEA-LETI for providing the samples.

IV. PROPOSED "VERILOR" MODEL

A. Automatic noise table method

Since there is no possibility for declaring a noise source in Verilog-A through a PSD function different than 1/f (flicker_noise) or thermal (white_noise) noise, we developed our module using the third option: the noise_table function. The latter takes an array of pairs of values (frequency and PSD) as input, leaving the SPICE simulator to interpolate the in-between regions. Therefore, after the automatic creation of a table with frequency values, with a resolution and bandwidth that can be user-controlled, the table of PSD values actually uses [START_REF] Leyris | Random telegraph signal noise SPICE modeling for circuit simulators[END_REF] and sums up the Lorentzian contributors from all active traps. The calculation of N T , ΔV t , A and τ follows the procedure described in Section III, like in [START_REF] Theodorou | A self-contained defect-aware module for realistic simulations of LFN, RTN and time-dependent variability in FD-SOI devices and circuits[END_REF], where a similar module was presented for RTN modeling in the time domain. In the end, the frequency (F) and PSD (Svg) tables are intertwined (odd/even indexes) together in a new table, FS, that is loaded in the noise_table function and inserted as a noise source at the FET gate. The flowchart of our VERILOR module is shown in Fig. 5.

B. Model validation

In order to confirm the validity of our model and its Verilog-A implementation, we simulated the LFN induced by three individual traps, in a single device. We compared the simulated spectra against (5), using the same trap parameters (x t , τ 0 , E t ). As demonstrated in Fig. 6(top), model and theory are in full agreement in terms of PSD shape. In Fig. 6 (bottom) we also illustrate how a g-r noise source from the channel can also be implemented on top of 1/f, by adding both flicker_noise and VERILOR contributors.

V. SIMULATION EXAMPLES WITH "VERILOR" MODULE

A. Scaling: from Lorentzian to 1/f noise and CNF

We performed additional simulations for different gate areas and corresponding total number of traps. As we demonstrate in Fig. 7 (top), the VERILOR module can automatically reproduce Lorentzian-like spectra in highly scaled areas and 1/f-like in larger ones. On top of that, it is able to generate a 1/f noise level that is in total agreement with the area-normalized carrier number fluctuation (CNF) model from weak to strong inversion (Fig. 7 (bottom)).

B. Time-domain RTN and dynamic variability

When it comes to simulating RTN in circuits, if what matters is the RTN amplitude and not so much its sudden transition events, VERILOR can be combined with the simulator's Transient Noise module, to fairly reproduce the power of the RTN signal, as shown in Fig. 8. Both methods (classic RTN time-domain model [START_REF] Theodorou | A self-contained defect-aware module for realistic simulations of LFN, RTN and time-dependent variability in FD-SOI devices and circuits[END_REF] and frequency domain VERILOR module) lead to the same spectra, and to timedomain signals of the same RMS power. Therefore, the level of dynamic variability can be accurately predicted by VERILOR, even with no RTN module in the time domain.


Marie Daou

Alain Marciano

  L'archive ouverte pluridisciplinaire

1

Conservative Liberalism, Ordo-liberalism, and the State, by Kenneth Dyson, Oxford University Press, New York, 2021, xx + 592 pp. £ 120 (hardback), ISBN 9780198854289.

Who needs to be reminded how turbulent and violent the first decades of the 20th century were?

The local wars that then took place, the terrible massacre that was World War I, the two revolutions (in Russia and Germany) that would directly or indirectly dramatically change the face of the world, the economic crisis in Germany in the early 1920s or the 1929 worldwide economic crisis that was (at least partly) responsible of the second major war of the twentieth century. These events obviously affected everyone who lived through them. Some came to think that capitalism had to be abandoned. Others that it should only be reformed. Among the latter were some liberal thinkers convinced that these events were the "symptoms" of an "existential crisis in liberalism", not a crisis of liberalism (Dyson,2,41,45,47). It was not liberalism that was at fault, but what it had become, a "crony capitalism" based on unregulated competition, weak governments playing as little role as possible or intervening without rigour or discipline. To exit the crisis and offer new perspectives to traumatised individuals, it was urgent, vital indeed, to invent a new form of liberalism that would "offer a moral purpose, one that promise[d] a more prosperous, secure, and humane society, and a more worthwhile life, than its alternatives like laissez-faire liberalism and social liberalism" (3). Neoliberalism-as it was called at the Lippmann Colloquium in 1938-was born. Its "most systematic, coherent and developed" form took the name and shape of Ordo-liberalism.

In the wake of the recent debates about neo-and ultra-liberalism, we have witnessed an increase in the number of books and articles devoted to a doctrine that has become fashionable again. Kenneth Dyson, a professor of politics at Cardiff University, also contributed to this stream of literature, focusing on the "intimate connection between Ordo-liberalism and continental state theory". With Conservative Liberalism, Ordo-liberalism, and the State, he "return[s] to this connection" and proposes a rather comprehensive characterisation of Ordo-liberalism. What is particularly important is that Dyson successfully dispels two major mistakes frequently made about Ordo-liberalism. The tendency, first, to confuse it with a doctrine that promotes a free-market unregulated form of economy, and second-especially "in the English-language world-to analyze and criticize [it] from the narrow perspective of economic theory"(ix).

Dyson makes his demonstration first by carefully and rigorously studying the works of the (German) founders of Ordo-liberalism. He relies on primary sources-canonical and neglected texts-and on archival material and discusses all these writings in an historical perspective. The birth of Ordo-liberalism can no more than any other tradition be understood independently from the socio-political context in which it took place. And the intellectual context as well, including the works of those who inspired ordo-liberals, what he calls the "patron saints" of the program. This completes the first two stages of the analysis, which represent the first two parts of the book. Then, Dyson proceeds by showing that it would be a mistake to reduce Ordo-liberalism to a place and a time. He devotes the third part of the book to connecting the Ordo-liberalism of the origins with different forms of Conservative Liberalism that developed in England, France, Italy, Belgium, the Netherlands, and the United States and which, according to him, all belong to the same ideological family.

The members of this family all agreed that liberalism is "about human flourishing and about the inner life of man" (x), or "the inner development of man's aesthetic and moral capacity" (52).

Ordo-liberalism, Dyson points out, postulates that "[a] liberal society was about the formation of the character and moral sense of individuals" (86). This is perfectly captured by the concept of "social market economy" invented by Alfred Müller-Armack to emphasise that what happens in the economy cannot be separated from what happens in other social spheres. Ordo-liberals stressed on "the vital importance of the nature of the legal, social, and cultural orders, of their independence, and of a clear, coherent, and stable framework of institutions and rules" in order to safeguard "the fundamental human values of freedom, dignity and solidarity" (183).

To reach that goal, a few principles should be respected. First, a strong state but with a strictly limited sphere of action-politicians had to be controlled and tamed. Also needed were "rules of the social game" or a "constitution for the economy"-Ordo-liberalism defends "a rule-based order" (26), hence the use of the label "Ordo". Indeed, Ordo-liberals believed in free and competitive markets but not "in perfectly efficient, self-correcting markets" (81).

Then, Ordo-liberals believed in ethics-a social market economy requires "strength of character and the possession of the moral qualities that engendered trustworthiness" (79). But they no more trusted individuals than they trusted markets. Ordo-liberals were convinced that individuals would not spontaneously follow ethical rules and be ethically responsible, or as strong as markets required them to be. An "external discipline" had to be imposed on individuals. Ordo-liberalism, Dyson insists-this is one of his major claims-, should be defined as a form of "disciplinary" liberalism. Disciplinary and aristocratic. Indeed, Ordo-liberals distrusted masses, but trusted elites to counter their power and govern society-a claim one also finds in the work of Ortega y Gasset that some Ordo-liberals admire. This leads Dyson to claim that Ordo-Liberalism is a form of aristocratic liberalism.

disciplinary conception of liberalism" (234). Indeed, many Ordo-liberals were Lutherans. For example, Walter Eucken, one of the founders and most important Ordo-liberals, claimed: "I could neither live nor work if I did not believe that God existed" ( 217). Yet, one must note an ambiguity in Dyson's story for while he claims that "religion is vital in understanding the founding conservative liberals and Ordo-liberals" (210), he also notes that a few of them were atheists. Louis Rougier, for instance, considered that "[t]he dogmas of Christianity 'flattered the illusion of the intellect'" (233).

The first who put these views together were Germans. However, in spite of its "distinctive German and continental European characteristics" (5), Ordo-liberalism is fundamentally "transnational" as Dyson argues. This had to be expected. Indeed, many countries faced a crisis in the first third of the 20 th century. Liberals were convinced that this was a crisis of liberalism and undertook to repair and rebuild liberalism. Dyson illustrates his point by citing a French economist and future Nobel prize laureate, Maurice Allais, and two Chicago-based economists-Henry Simons and Frank Knight. All these economists shared the same diagnosis about laissez-faire liberalism and all agreed that a market economy could not function properly without being regulated. A "positive program for laissez faire" was required, Simons claimed, anticipating Röpke's "advoca[cy] for a 'positive' economic policy" (25). It is striking to see how Knight and Simons's ideas and analyses were close to those of the founders of Ordo-liberalism (as already stressed in [START_REF] Köhler | Transatlantic Roads to Mont Pèlerin: "Old Chicago" and Freiburg in a World of Disintegration Orders[END_REF]. In that sense, it seems natural to consider intellectuals who are similarly thinking about rules, control of the market, strong state, and moral ethic and orderliness as belonging to the same, Ordo-liberal family. Yet, it is not. Not always, at least. Indeed, important nuances between intellectuals of this family must not be dismissed (see Dekker 2022 about Ralph Hawtrey and Frank Knight). Let us focus on two other cases. Jacques Rueff, to start with. Similarities exist between Rueff's ideas and those of Ordo-liberals, but his methodology might not be as close to that of the Ordo-liberals as Dyson claims. He tells us that Eucken, Hawtrey, Rougier and Rueff "rejected the empiricist method and Cartesian-style a priori reasoning and deductive theory as inadequate foundations for the discipline" ( 182). Yet, in his autobiography, [START_REF] Rueff | OEuvres Compl tes I : Autobiographie[END_REF] admitted having been deeply marked by Descartes's Discourse on the Method. Descartes's essay inspired his epistemological work (1922) and some of his economic analysis: his work on the exchange rate and unemployment was grounded in a deductive and empirical approach, as his monetary theory of the 1920s [START_REF] Rueff | héorie des phénom nes monétaires: statique[END_REF]. In these conditions, can Rueff still be considered an Ordo-liberal?

The same question applies to James Buchanan whom Dyson describe as a "disciplinary liberal" (212). Buchanan might have remarked that "There is no God, but Frank Knight is his prophet" (220), but is that sufficient to make him a Calvinist or a representative of a form of "secularized Protestantism of a fundamentalist kind" (220)? One may doubt it. Buchanan always believed that one could trust individuals, that they would cooperate in certain environments because they would voluntarily follow ethical rules. One of the rare-if not the only-essays, in which Buchanan defended a Puritan ethic is "The Samaritan's Dilemma" (1975), an essay not cited by Dyson. Even if Buchanan feared that individuals might be "afraid to be free" (2005), he never concluded that discipline was important. Buchanan did not seem to have shared this family feature with other Ordoliberals. Since discipline is crucial to Dyson's definition of Ordo-liberalism, one may wonder if that does not jeopardise Buchanan's belonging to the family.

Taking differences into account is the interesting -but tricky -aspect of the concept of "family". When the differences become so important, does it still make sense to talk of a family? The question deserves to be raised. At least, sometimes. For, most of the time, Dyson is convincing in his attempt to show that Ordo-liberalism is a family, and therefore that it is not frozen in space. Dyson also shows that Ordo-liberalism is not frozen in time. Ordo-liberalism evolved as its promoters became involved in political decision-making. Indeed, some Ordo-liberals obtained privileged positions such as Erhard and Müller-Armack who were very active applying principles of Ordoliberalism in German politics.

Ordo-liberalism also played an important role in the building of the European Union. For the better, as in the first years of the process, Ordo-liberalism was important because of its focus on unification. As Dyson argues, "Ordo-liberalism was conceived as being part of a large national unifying myth that rested fundamentally on promoting European political unification" (408). Also, from the 1980s, because of how it views monetary policy-"from the 1980s, [the Economic and Monetary Union] witnesses a correspondence of mainstream monetary theory with Ordo-liberalism around the primacy of monetary policy, the importance of clear and firm rules and their consistent application" (395). But also, for the worse, as in 2011 with the resignation of Jürgen Stark from the executive board of the ECB in reaction to a monetary policy judged too permissive which "illustrated that for some Ordo-liberals adaptative learning has its limits" (359). More generally, Dyson considers that with the financial and economic crisis of 2008, "Conservative liberalism and Ordo-liberalism had an opportunity to re-emphasize how liberalism could be destroyed from within by elites who inflicted self-harm as well as harm on others" (438).

At the end of the book, Dyson discusses the challenges Ordo-liberalism had to face, as the elites changed. A double movement took place: a decline of bourgeois values in the ruling elites (due to the 'massification' of higher education for all social classes from 1960s) and the distancing of the intellectual and cultural elites from market economy. The stakes were high since the new elites "proved less readily receptive to the morally conservative character of conservative liberalism" (249), and "cultural and social criticism had different roots from those of Ordo-liberalism" (405). However, Dyson could have dug deeper in this topic to address the capacity of Ordo-and Conservative liberals to renew themselves.

All in all, Dyson must be complimented for having written a comprehensive and extremely well-documented book. Specialists of Ordo-liberalism might not be surprised by the content of the book and may even disagree with him-which opens up to rich discussions. They will nonetheless find a host of references, and information that will also rejoice non-specialists, who will gain a better view of Ordo-liberalism. It will be useful for those, in particular, among historians of economic thought who are interested in the twentieth century. The book will also delight those who are interested in understanding liberalism and neo-liberalism. They will in particular realise that liberalism is not, or should not be, restricted to economics. As Dysons points out, Ordo-liberalism was devised as an "attempt to transcend the scope of economics, narrowly defined, and see political economy in its larger context" (x-xi). Ordo-liberals were foremost "philosophers-economists" or "philosophers-lawyers" (xii). They were a perfect illustration of Friedrich Hayek's wise remark that "nobody can be a great economist who is only an economist" (1956,463). A lesson that is important to remember.

Marie Daou

Université de Montpellier and MRE, France marie.daou@umontpellier.fr Alain Marciano Université de Montpellier and MRE, France alain.marciano@umontpellier.fr


Marie Daou

Alain Marciano

Commodification: The traditional pro-market arguments 1

Introduction

Commodification has been proposed and defended in the first place because markets or the market as an institution has some virtues and advantages. Which are they? This is the question we try to answer in this chapter. We do not aim at giving an exhaustive view of pro-market arguments. Reviewing the huge literature in favor of markets would be too immense a task to complete. Probably impossible. We rather try to focus on the arguments that are important in the debates on commodification and to which arguments the anti-commodification theorists are responding. Our chapter is divided in three sections. The first two arguments relate to the moral or ethical dimensions of markets-on the one had, markets protect individual liberty and, on the other, they teach individuals moral principles and make them virtuous. The last argument would be efficiency, a major advantage of markets compared to other forms of organizing economies or societies.

Liberty, consent, and property rights

A first argument in favor of markets has to do with liberty. A market system leaves people "free to choose", to use Milton and Rose Friedman's words (1980). Markets respect individual economic freedom and guarantee that individuals consent to or, to put it differently, do not force or coerce them into the transactions they engage in. This was how Buchanan (1954, 340) characterized liberty-"somewhat narrowly as the absence of coercion". With this definition, he was perpetuating the Chicago tradition. Indeed, this was the definition of freedom used by Frank Knight (1941, e.g.) 2 and passed along to the next generation of Chicago economists. Or, even more precisely, "the central feature of the market organization of economic activity is that it prevents one person from interfering with another in respect of most of his activities. The consumer is protected from coercion by the seller because of the presence of other sellers with whom he can deal. The seller is protected from coercion by the consumer because of other consumers to whom he can sell. The employee is protected from coercion by the employer because of other employers for whom he can work, and so on." (1962,(14)(15) For instance, as some of the pro-market scholars note, including Adam Smith himself, the birth of a labor market, that is the commodification of labor, has made it possible to substitute an imposed master-servant relationship for a voluntary and free transaction. The argument was then more broadly also used by the partisans of an extended use of the market beyond its traditional limits about some potentially contested commodities such as blood (see, e.g., [START_REF] Cooper | The Economics of Giving and Selling Blood[END_REF][START_REF] Kessel | Transfused Blood, Serum Hepatitis, and the Coase Theorem[END_REF], organs (Brams, 1977,6) or babies [START_REF] Landes | The Economics of the Baby Shortage[END_REF][START_REF] Posner | Economic Analysis of Law[END_REF]Posner, [1973], 141-142;], 141-142;1979, 138-139;[START_REF] Posner | The Regulation of the Market in Adoptions[END_REF]also, among many others, Pritchard, 1984). To these thinkers, markets are valuable, and market transactions legitimate, because the moment the individual engages in a market transaction, he or she can be taken to have consented to it. Indeed, choosing to engage in a transaction means consenting to it. Put differently, (competitive) markets are by definition characterized by what Debrah Satz calls the "possibility of exit" (2010,25). This means or implies that no one is or can be coerced when engaging into market transactions, since she or he has consented to it.

Thanks to the market, the individual satisfies his or her needs while preserving his or her freedom, and the freedom of others. Thus, coerced market transactions are, by definition, impossible. Posner extended the idea of a consent to its maximum by defining consent as "ex ante compensation" (1980, 492). 3 This means that, whatever the outcome of the transaction (even in the event of an accident or loss), the individuals have discounted and capitalized the future possible losses. When they engage in the transaction, they are already compensated for the losses because the price they pay include them.

That individuals necessarily consent to the transactions they engage inand therefore that markets promote freedomdepends on property rights. 4 Markets can hardly work without such rights. The relationship between markets (consent) and property rights work in two ways. First, individual property rights have to exist to allow uncoerced transactions. Free markets cannot function without property rights. This means that consent on the marketplace presupposes property rightsfor consent to be meaningful, individuals need to have ownership in the goods they trade. Reciprocally, because it is assumed or claimed that markets guarantee voluntary transactions, it can be said that a market transaction also guarantee individual property rights. Indeed, if people have a right to their property premarket exchange, and people consent to all transactions, the new allocation of property will reflect what people rightfully own. Consequently, when a market transaction takes place, the parties' property rights are respected.

In the liberal economic tradition, freedom is closely associated with the two institutions, namely the market and individual (private) property rights. For liberal thinkers, liberty consists in the unhindered use of one's property. This is how property rights give to people the right to do whatever they want with what they own, including trading it on a market. Therefore, to prohibit, to limit or to undermine market transactions in one way or another prevents individuals to use their property rights and accordingly reduces their liberty. This brings us to the specific case of self-ownership. Indeed, if individuals have rights, then "there are things no person or group may do to them (without violating their rights)" (Nozick, 1984, ix). Furthermore, if markets are based on consent, no transaction can take place to which an individual has not consented (and any transaction to which he or she has consented is admissible). Finally, if consent cannot exist without being in possession of the property of oneself, then having property or ownership rights over oneself should also be a necessary condition for the existence and functioning of a system of free markets. Indeed, self-ownership is defined by the fact "that there are things that may not be done to a person without their consent, but which may be done with consent." 5 Therefore, market participants are self-owners. Or, put differently, markets (and consent) without self-ownership are impossible. The absence of self-ownership empties consent and voluntarism from any meaning.

In addition to a certain circularity around the self-ownership argument (the property supposes self-ownership which is necessary to engage in transactions and at the same time the (2016,11). Unlike Rothbard''s analysis, they introduce the moral factor and do not directly discuss the question of inalienability of self-ownership when they formulate their principle. However, this one is debated later in their book (with the examples of right to vote and slavery). 5 It is an extract of Stanford Encyclopedia of Philosophy: https://plato.stanford.edu/entries/libertarianism/ self-ownership is used to justify the notion of ownership), it is an extremely complex and difficult concept to grasp. From an ontological perspective, it could be argued that individuals may well have property rights on objects but no right over themselves because only an entity can claim an ownership right over human beings. For example, Christian religion considers that self-ownership does not exist: "the Lord owns the earth and all it contains, the world and all who live in it" (in the Bible Psalm 24 of David). From a philosophical perspective, another difficulty comes from the nature of what is owned. Cartesian dualism-or any form of dualism, for that matter-between the owner (the subject) and ownership (the object/body), makes self-ownership inconsistent. Thus, owning objects and owning a human being cannot be put on the same footing. This is what Emmanuel Kant, for instance, explained when he wrote:

Man cannot dispose over himself, because he is not a thing. He is not his own property-that would be a contradiction; for so far as he is a person, he is a subject, who can have ownership of other things […] it is impossible, of course, to be at once a thing and a person, a proprietor and a property at the same time. (1997,157) In view of the limits previously presented, self-ownership therefore appears to be a strong and debatable requirement to defend markets. However, if one leaves aside the difficulties about the notion of self-ownership and admits that human beings can own objects and themselves at the same time, then one question becomes important from the perspective of a discussion on commodification: can a market transaction-to which the parties have consented-allow individuals to abandon self-ownership? Can human beings or, even, can body parts be sold on a market? Is slavery, or voluntary enslavement, possible? At first sight, liberty/consent and self-ownership seem to be complementary, rather than rival: property involves consent and disposing his freedom, his own body and his work. However, it seems there is a case where liberty and property are in competition when the ownership right of some legitimizes the deprivation of freedom of others or, put differently, when slavery exists. Thus, the materialist vision of slavery manifests itself in the denial of the right to freedom. In this specific case, property takes precedence over freedom and that is why Locke rejected the slavery. Actually, Locke put forward a twofold, and seemingly contradictory, reason to oppose slavery. First, he insisted that "every man has a property in his own person" (1824, 116), and consequently, no one can thus be deprived from the property of oneself. Second, he also noted that a human being does "not hav[e] the power of his own life" and therefore "cannot by compact or his own consent enslave himself to anyone, nor put himself under the absolute, arbitrary power of another to take away his life when he pleases" (1824: 143; italics added).

Other pro-market scholars also oppose the selling of oneself into slavery. From this perspective, one of the most interesting positions is Murray Rothbard's. A libertarian, Rothbard developed an inalienability theory on self-ownership upon which rests his strong opposition to slavery-"slavery, and even "voluntary slavery," as we have seen, cannot be enforced on the free market because of the inalienability of personal will." (2004,348). His point helps to clarify a pro-market argument about commodification. To Rothbard (1998, 40-41), "[t]he concept of 'voluntary slavery' is an oxymoron, "a contradictory one". Indeed, "so long as a laborer remains totally subservient to his master's will voluntarily, he is not yet a slave since his submission is voluntary; whereas, if he later changed his mind and the master enforced his slavery by violence, the slavery would not then be voluntary." That slavery cannot be voluntary means that it can result only from violence. Rothbard again: "The interpersonal relation under slavery is… one of command and obedience, the commands being enforced by threats of violence. Thus, slavery… is defined as a system in which one must labor under the orders of another under the threat of violence." (2004,82) Thus, from this perspective, property rights over oneself are inalienable. Harold Demsetz made this point when discussing the importance of property rights, and that he made a connection with "slave labor": "A law which establishes the right of a person to his freedom would necessitate a payment on the part of a firm or of the taxpayer sufficient to cover the cost of using that person's labor if his services are to be obtained" (1967,349). 6On the contrary, Nozick's position on this particular matter was anti-Lockean For instance, Nozick asked "whether a free system will allow [an individual] to sell himself into slavery", he replied, "I believe that it would" (1974,331). Even if he specified that the individuals were not forced to accept the transaction; the decision to enslave oneself would have to be voluntary. And to Nozick, individuals are not coerced if they are led to choose an option that improves their situation. Thus, for instance, an individual could sell his lifetime labour to avoid starving and live a little longer.

The same argument as Nozick is introduced by Walter Block (2003) who questions the idea of inalienable right on one's own person and supports the possibility that "everything should be legally alienable or commodifiable" (41), thus the "claim is that if I really own my liberty, then I should be free to dispose of it as I please, even if, by so doing, I end up no longer owning it" (44). From such a perspective, self-ownership does not impose any limit to the use of markets but consent, including the consent to abandon a further right to consent.

Markets, morality, and social cohesion

That markets-and market-based societies-could be vicious or virtuous has largely been emphasized by those who believe that everything should not be for sale, as Debra Satz put it (2010; see also [START_REF] Anderson | Value in Ethics and Economics[END_REF][START_REF] Sandel | What Money Can't Buy: The Moral Limits of Markets[END_REF][START_REF] Sandel | Market Reasoning as Moral Reasoning: Why Economists Should Re-Engage With Political Philosophy[END_REF]. For instance, Michael Sandel insisted that "some market transactions are objectionable" because of "the tendency of market practices to corrupt or crowd out nonmarket values worth caring about… nonmarket norms of moral importance" (2013, 123), such as love, or "attentiveness and thoughtfulness." (124) Although made in the frame of a contemporary debate, such statements are not new.

"There is a long tradition" Ian Maitland pointed out, that "go[es] back to Aristotle" and that consists in "viewing commerce as hostile to the virtues" (1997,17) and in believing that "the market frees individual acquisitiveness from moral, social and/or religious constraints." (18)

The very fact that markets lead individuals to focus narrowly on their own interests necessarily works to the detriment of their social obligations and moral duties. In this respect, markets do not only undermine the moral foundations of the good society, but also the moral foundations of market itself. Thus, wrote Fred Hirsch (1976, 122, 124), "the full permeation of an individualistic calculus", that characterizes a free-market economy, weakens the social morality that is necessary to guarantee an efficient functioning of markets. Markets do not only work against morality. By promoting individual values and immoral behaviors, they are also working against themselves.

Against such arguments, pro-market scholars have developed different analyses that do not always complement each other but that all contribute to defend the thesis that markets do not destroy morality and moral values.

A first line of defense can be found in the famous Fable of the Bees (1714) of Bernard de Mandeville. Self-interest-vices, in his words-are certainly not detrimental to the common good-or "public benefits", still his words. Much to the contrary. The pursuit of self-interest, and the adoption of vicious behaviors, contribute to generate a wealthy and prosperous society:

The root of evil, avarice, that damned ill-natured baneful vice, was slave to prodigality, that noble sin; whilst luxury employed a million of the poor, and odious pride a million more: envy it-self, and vanity, were ministers of Industry; their darling folly, fickleness, in diet, furniture and dress, that strange ridiculous vice, was made the very wheel that turned the trade. […].

[However] vice nursed ingenuity, which joined with time and industry, had carried life's conveniences [...] Pleasures, comforts, ease, [...] nothing could be added more. [START_REF] Mandeville | The Fable of the Bees: or, Private Vices, Publick BenefitsOxford[END_REF](Mandeville, [1716]], 24-25-26) Thus, to Mandeville, markets encourage immoral individuals to behave morally, and work for the common good. Markets do not, however, transform individuals, do not improve their morality. Individuals remain self-interested, greedy, and vain. They put their "vices" at the service of the society.

A more ambitious pro-market argument consists in arguing that markets transform individuals. The thesis could already be found in the works of Montesquieu's doctrine of the "doux commerce". The following quotation perfectly illustrates what the doctrine means: "Commerce cures destructive prejudices, and it is an almost general rule that everywhere there are gentle mores, there is commerce and that everywhere there is commerce, there are gentle mores [...] Commerce […] polishes and softens barbaric ways" (2015 [1748], 338-339).

Here, one understands the difference with Mandeville's view: commerce, or put differently market exchange changes individuals; it is not only a matter of inciting them to behave morally but to become moral individuals. Here, market exchange is defended for moral grounds.

This view was further developed by the enlightenment philosophers and classical political economists 7 (see Hirschman, 1977Hirschman, , 1982;;[START_REF] Maitland | Virtuous Markets: The Markets as School of the Virtues[END_REF]; for a recent and complete analysis, see [START_REF] Storr | Measuring Markets and Morality[END_REF]. Markets "encourage" (Storr and Choi, 2020, 54) and "support moral development" (57). To be more concrete, the impact of markets is twofold.

First, the market is supposed to limit or prevent immoral behaviors. Thus, Adam Smith, for example, believed that markets rest on mechanisms that lead economic agents to moderate the intensity of their passions [START_REF] Paganelli | The Moralizing Role of Distance in Adam Smith. The Theory of Moral Sentiments as Possible Praise of Commerce[END_REF], and temper the importance of their self-love [START_REF] Walraevens | Adam Smith's Economics and the Lectures on Rhetoric and belles lettres: The Language of Commerce[END_REF]. As Walraevens (2014) noted about Smith's analysis of markets: "the competitive market is a place of education in impartiality and self-control through the 7 Among others, Adam Smith, David Ricardo, Robert Torrens or John Ramsey McCulloch and John Stuart Mill. moderation of personal interest that it demands. No one can permanently impose their will on it" (426). Second, markets encourage moral behaviors. More precisely, markets are said to teach individuals how "to be better people" (Storr and Choi, 2020, 43). By which it is meant that markets teach individuals virtues such as honesty, tolerance, generosity, industriousness, frugality, probity, hospitality (Montesquieu). Virgil Storr and Vini Choi (2020, 53) quote Smith's Lectures on Jurisprudence (1762-1763) where one can read that market exchange (commerce) promote virtues such as "probity and punctuality". 8 Crucial in this process is the role "sympathy" played. In Smith's theory, where sympathy is lacking, repeated contracts can generate minimal levels of courtesy: "The market promotes civility because commercial success depends on the courteous treatment of people who have the option of taking their business elsewhere" (Brozen, 1978, 15). Moreover, market participation develops and strengthens the ability to share the feelings or emotions of others. Indeed, in the economic system described by Smith, the division of labor makes the subsistence means of economic agents depend on the successful satisfaction of the needs of others. In other words, the interdependence between economic agents increases with the division of labor. Meanwhile, sympathy helps us anticipate the needs of others and combined with market exchange, contributes to economic success. Indeed, by allowing perspectives to be compared and the adoption of an impartial point of view through mutual concessions until a reciprocal agreement is obtained, the market participates to the social cohesion. It becomes the instrument of expression of sympathy, by supporting cooperation and economic relations.

Then, a third type of defense of markets on moral grounds consists in arguing that "[m]arkets... are socializing spaces." (Storr and Choi, 2019, 134;italics in original). The thesis was defended by economists as different as Léon Walras and Friedrich Hayek, but it tends to see the market as virtuous for a similar reason, for contributing to the promotion of social cohesion. This is thus how [START_REF] Hicks | Léon Walras[END_REF] interpreted Walras's general equilibrium theory, for having the merit of demonstrating the unity of economic life, with the idea that individual ends are made compatible by the market mechanism:

8 "Whenever commerce is introduced into any country, probity and punctuality always accompany it. [...] Of all the nations in Europe, the Dutch, the most commercial, are the most faithful to their word. The English are more so than the Scotch, but much inferior to the Dutch [...]. This is not at all to be imputed to national character, as some pretend. There is no natural reason why an Englishman or a Scotchman should not be as punctual at performing agreements as a Dutchman. It is far more reduceable to self-interest, that principle which regulates the actions of every man, and which leads men to act in a certain manner from views of advantage [...]. A dealer is afraid of losing his character, and is scrupulous in observing every engagement". ([1762-1763] 1982: 588).

One is the realization of the unity of economic life which emerges so forcefully from his pages. Other economists had had a sense of this unity, but none before had shown it so well. For the unity which Walras demonstrated is not a unity of resources being allotted among a single system of ends-the only unity which really appears in Menger-it is a unity of diverse individual ends reconciled through the mechanism of the market. (Hicks, 1934, 347) In this sense, markets bring isolated and independent individuals together, link them to other individuals and therefore contribute to the stability of social relationships.

In the same vein, the economists from the Austrian school and the ordoliberals 9 also support the idea that the market is not simply a place where goods are traded. These economists are convinced that the market order is more than a term used to characterize a commercial society. It is a genuinely social institution that allows individuals to interact peacefully with one another to achieve ends that are not purely economic. One could also refer to Jacques Rueff, a French liberal thinker sympathetic to ordoliberals and Hayek's notion of social order, who argued that far from being a mere institution of transaction, the market is the most appropriate societal organization to achieve "collective ends", 10 for it is the only form of market which enables us to maintain human freedom and social harmony:

[M]arket […] is, in our political structures, the fundamental change which will unite tomorrow, in a common civilization, all individuals and people who want freedom without disorder, and welfare without servitude, while reducing, as far as is humanly possible, inequality and injustice. (Rueff 1958, 9-10)

Markets and efficiency

Markets are supposed to guarantee that individuals consent to the transaction, and therefore that their freedom and sovereignty are respected. As a corollary, markets guarantee that those who want or need to buy or sell a commodity (contested or otherwise) can indeed complete the transaction as they wish. This points to another major argument in favor of extending, generalizing the use of markets: efficiency. Markets are indeed often viewed as the most efficient way to deal with scarce resources. 11 A society in which transactions take place on markets is supposed or claimed to be wealthier than it would be without such markets. To understand the argument, it requires a clarification as to what efficiency means and, as a 9 Ordoliberalism is a liberal doctrine which gives the State the economic mission to create and maintain a framework allowing free competition on the markets. 10 Rueff did not define exactly the term "collective ends". We know only that they are not simply the sovereign functions (police, justice, and army) since they are already ensured by the protection of property rights in Rueff's thinking. 11 Not that other ways of allocating resources would not create wealth too. The argument is not put in absolute but in relative terms. corollary, as to how markets are defined-and even as to how economics itself is described.

Here, two perspectives exist: the static vs the dynamic definitions of efficiency and markets, which corresponds to the standard or mainstream vs the heterodox definitions.

Efficiency in what is now the standard, mainstream, neoclassical and static sense of the word finds its origins in the works of [START_REF] Walras | Éléments d'économie politique pure[END_REF] and Wilfredo Pareto (1906). Walras introduced a general equilibrium approach, according to which, if competitive conditions prevail and when consumers maximize their utility and producers maximize their profits, a situation of general equilibrium can be reached. Put differently, there always exists a system of price that guarantees that all markets simultaneously are in equilibrium. A few years later, [START_REF] Pareto | Manuale d'economia politica[END_REF] went further than Walras. He defined what would be the best possible situation that can be reached in an economy-Pareto optimality-, and also gave the first formulations of what would be known as the two fundamental theorems of welfare economics. These theorems, that would be formally demonstrated by Kenneth Arrow in the 1950s, establish a twofold correspondence between competition and Pareto optimality. The first theorem says that, under certain conditions (given technology, given resources, stable and given preferences, perfect information, perfect competition), the allocation of resources resulting from the equilibrium system of prices is Pareto efficient. In other words, any competitive equilibrium leads to a Pareto efficient allocation of resources. The second theorem states that, with a given endowments of resources, any Pareto optimal allocation can be achieved through a competitive, decentralized market mechanism. In other words, markets lead to social optimality. All the potential gains from trade are exhausted.

Such views on general equilibrium and markets, are very specific. Efficiency can be reached if very strong and restrictive conditions are satisfied. So strong and restrictive that it seems difficult to defend market efficiency on this ground. Yet, it such perfect models are not problematic in themselves, that is as long as they remain used for what they are-modelsand that they are not compared to real-world markets. Unfortunately, this is what became the "perfectly competitive equilibrium", "a normative benchmark to assess real-world markets" (Boettke et al., 2018, 126). Indeed, some pro-market economists did assume that the "static perfection" of competitive general equilibrium and Pareto efficiency "must (somehow) be attainable" and that "real markets" were "breathtakingly close to approximating the efficiency properties of general competitive equilibrium" (Boettke, 1997, 23). The model is then conflated with the reality. Accordingly, some aspects of that reality are missed: that information is not perfect, that individuals have limited cognitive capacities, that human activities are dynamic, that institutions matter. These are the dimensions that are captured by the heterodox views on efficiency.

Such an alternative approach defines markets in terms of processes, as the process of adjustment towards equilibrium: "the classical understanding of the market process was a dynamic process of adjustment, in which factor prices serve as guides to exchange and production" (Boettke et al, 2018, 127-128). From this perspective, a market is not efficient because it settles itself in an equilibrium position but because it allows individuals to learn, adapt themselves to what others do and combine their plans of action with other individuals; that is, to discover what others plan to do. Market is not only an anonymous and intertemporal place which brings together buyers and suppliers and where goods and services are exchanged, but a process during which information is created, disseminated, and adjusted.

From this perspective, a market economy cannot, to put it in other words, be affected by durable and generalized crisis. Here, the disequilibrium is a means of allowing the individuals to learn what was wrong and why such a glut occurred. This approach is later deepened and refined by the Austrian school of economics. The importance of markets as a means of transmitting information to economic agents is central to Austrian tradition. Hayek and Israel Kirzner are certainly the most prominent among the many economists who contributed to developing this perspective. Both insisted on "the unavoidable imperfection of man's knowledge" (Hayek 1945, 91; see also 1937), on the need to learn and on the "need for a process by which knowledge is constantly communicated and acquired" (1945,91). This is a key role markets play. Indeed, as [START_REF] Kirzner | Perception, Opportunity and Profit[END_REF] wrote, the market "emerges as the necessary implication of the circumstances that people act, and that in their actions they err, discover their errors, and tend to revise their actions in a direction likely to be less erroneous than before" (30). For his part, Hayek argued that, in an institutional environment of private property rights, the market price system uses dispersed knowledge in order to align production and supply with consumption and demand. And, indeed, for both Hayek and Kirzner, prices are crucial in this dynamic learning process. They convey information about the changing conditions, giving individuals the possibility to adjust their behavior accordingly. And they also allow the market participants to evaluate the relevant of their past market decisions and thus correct possible mistakes. Ex post, prices reveal the ultimate (un)profitability of economic decision. Efficiency remains a major advantage of markets and of a market system, even if its representation has evolved.

Conclusion

Without trying to be exhaustive, we reviewed the main arguments that are used to legitimate an expansion of markets without limits. These are the arguments that can be opposed to the anti-commodification theorists concerned with contested markets.

We have thus insisted on the notion of "consent" and "property rights" because, to pro-market thinkers, market is a place in which individuals consent to the transactions they engage in. More clearly, the argument is circular, since market rests on consent and transactions reveal that individuals consent to the transactions they engage in. Obviously, this works because property rights protect individuals. When it is the case, the claim is that no one can be forced to trade anything on a market-by contrast, the regulation of the trade of a commodity introduces coercion because individuals can no longer buy and sell the commodity as they want to. Thus, markets for new commodities-those commodities that are not usually traded on markets-guarantee that those who want or need to buy or sell a contested commodity can indeed complete the transaction as they wish. May be, such markets should not be dismissed out of hand, particularly when the "proof of the pudding is in the eating". This is all the more profitable that, though market processes, individuals learn how to behave morally and virtuously, as the argument goes.12 These points were repeatedly made by the advocates of an extension of markets to any kind of good. For them, without a free market, there exists a gap between supply and demand that cannot be closed; using a price mechanism would eliminate it. This reasoning applies to babies [START_REF] Landes | The Economics of the Baby Shortage[END_REF] or to organs [START_REF] Posner | Utilitarianism, Economics, and Legal Theory[END_REF][START_REF] Hansmann | The Economics and Ethics of Markets for Human Organs[END_REF][START_REF] Erin | A Monopsonistic Market: Or How to Buy and Sell Human Organs, Tissues and Cells Ethically[END_REF][START_REF] Becker | Introducing Incentives in the Market for Live and Cadaveric Organ Donations[END_REF][START_REF] Elias | Sacred values? The effect of information on attitudes toward payments for human organs[END_REF]. Therefore, the individuals who are on a waiting list-due to lack of organ-could acquire a "good" they would not have the possibility to "buy" otherwise. Complementarily, those who would be ready to sell organs or body parts are not forced against their will not to engage in transactions. Here, as one realizes, efficiency can be viewed in a narrow and static sense or in the broader and dynamic meaning of the concept, that is either as allowing a better allocation of resources or an improvement in the information individuals have.

A first version of this paper was presented at the conference "Contested Markets: Theories and Controversies", Paris, 3-4 June

We thank Élodie Bertrand, Vida Panitch and Aksel Sterri for extremely useful comments.

In "The Meaning of Freedom" (1941, 91), Knight wrote: "freedom is the opposite of coercion".

This definition is arguably narrow and specific[START_REF] Dworkin | Why Efficiency? -A Response to Professors Calabresi and Posner[END_REF] Westen, 2004a Westen, , 2004b)).

This is very close to Brennan's and Jaworski's "Principle of Wrongful Possession" -"If it is inherently morally wrong for someone to possess (do, use) X, then (normally) it is morally wrong for that person to buy or sell X."Commenté [D1]: Répéttion. Je l'enlèverai car on dit la même chose juste avant : "Thus, coerced market transactions are, by definition, impossible"

Another way of putting the argument would be by saying that it is not the selling of one object that is problematic but the conditions in which the object is sold. Which is exactly the point Debra[START_REF] Satz | Markets in Women's Reproductive Labor[END_REF] 2010) made in her discussion of surrogacy.

The defense of markets is made ceteris paribus, so to speak. : Ssince the market is an "institution", one assumes that the virtues of advantages of the existing markets can be replicated to the new markets that are created.


domain: spi sdv spi.auto

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Gait cycle modeling in cerebral palsy condition

Sabrina OTMANI 1,2 , Guilhem MICHON 1 , Bruno WATIER 2 Abstract-Autonomous gait is a fundamental element for access to independent life and avoiding the de-socialization of people with motor disabilities. In this context, this research is part of the EXOKID project which aims at designing a personalized exoskeleton for children with cerebral palsy. For such personalisation, two 9 years old twin sisters, one with spastic cerebral palsy (C) and a healthy one (H) without any impairments, performed several walks with electromyography (EMG), kinematics and force acquisitions. This paper presents a model of the knee and hip's spastic angular displacement of C during a walk using mechanical differential equations. Two models were designed: one based on the timing of the muscular activation, the other where the timing is defined using a genetic algorithm (GA). These models highlight the spastic contributions of the muscles involved in the walk (agonists and antagonists of the joints studied) and their activations. The amplitude of the activations for both models was carried out using GA.

Gait cycles were modeled with a determination coefficient (R 2 ) higher than 84% for both models.

I. INTRODUCTION

One of the most common causes of motor disability, especially during childhood, is cerebral palsy (CP) with an incidence of 2.0 to 2.5 per 1000 live births. It is a "group of non-progressive, but often changing, motor impairment syndromes secondary to lesions or anomalies of the brain arising in the early stages of development" [START_REF] Rosenbaum | Cerebral palsy: what parents and doctors want to know[END_REF]. A wide range of prenatal, perinatal, and postnatal isolated or combined factors are considered the main causes of cerebral palsy: hypoxia, gene abnormalities... The severity of the CP can be quantified using Growth Motor Function Classification System (GMFCS) decomposed into 5 levels from the lowest to the highest severity. In this context, the biomechanical analysis of the CP gait aims to better understand the key features of the kinematics and muscle activation. Different CP subtypes exist like ataxic CP, dyskinetic CP and spastic CP which is the subtype studied in this paper. Spasticity is also the most common disabling condition seen in cerebral palsy [START_REF] Graham | Pendulum test in cerebral palsy[END_REF] [START_REF] Pharoah | Trends in birth prevalence of cerebral palsy[END_REF] characterized by an abnormal increase in muscle tone creating pathological and involuntary reflexes [START_REF] Lance | What is spasticity?[END_REF] [START_REF] Kheder | Spasticity: pathophysiology, evaluation and management[END_REF] or stiffness of muscle, which can impact the walk of the subject. Spasticity is known to be a "hypersensitive, velocity-dependent response to passive muscle stretch" and also position-dependent as some lower-limb configurations cannot be reached due to the spasticity [6][7][8]. This paper will first focus on the modeling and characterization of the effect of spasticity on a gait cycle of a 9 years old girl called C. who has a spastic CP with a GMFCS level of 2. A gait is a succession of several gait cycles which are determined by two consecutive heel strikes of the same foot [START_REF] Armand | Gait analysis in children with cerebral palsy[END_REF]. Thus, right and left gait cycles are defined according to the food considered. To model the gait cycle, a double pendulum will be considered composed of the hip and the knee, the joints studied. This model represents the swing phase of the movement which is responsible for the magnitude of the step movement. In this model, we assume that the ankle does not have mobility. This work is then divided into two steps: an experimental one to extract the customized data (muscular activity (EMG) and kinematic data) and a modeling one to recreate the gait using the previous data and a common genetic algorithm (GA) to find the optimal parameters. To simplify the modelization, the right and left gait cycles will be modelized as those cycles can be repeated to reconstruct the walk. In the end, we aim to determine how a spastic gait cycle can be modeled using a double pendulum model. This paper is part of the EXOKID project which aims to design a customized exoskeleton for a child with a specific spastic cerebral palsy and with the final purpose to determine how it can correct or enhance the gait of this child.

II. RELATED WORK AND CONTRIBUTIONS

1) Double-pendulum system: Spasticity models are often characterized and designed using data obtained from a pendulum drop test. In [START_REF] Le Cavorzin | A comprehensive model of spastic hypertonia derived from the pendulum test of the leg[END_REF], a non-linear function representing a couple modeling the contribution of involuntary spastic movements is developed. In [START_REF] Bajd | Testing and modelling of spasticity[END_REF], spasticity is modeled by a torque representation of muscle hypertonicity and the leg model itself corresponds to a differential equation dependent on the viscoelastic parameters of the agonist and antagonist muscles of the knee. In this continuity, [START_REF] Fee | Neuromuscular modeling of spasticity in cerebral palsy[END_REF] focuses their work on the muscular contribution in the knee angular displacement during a pendulum drop test. The purpose is to clearly indicate that the knee rotation of subjects with spasticity is an active process. They defined additional velocity-feedback torques to emphasize the velocity-dependent aspect of spasticity. An optimization algorithm was used to define the timing and amplitude of those torques without any consideration for muscular activity. This optimization can enhance the performance of the best fitting model but is a source of inaccuracy when it conceals the actual origin of spasticity, which is the involuntary and pathological activation of the muscles [START_REF] Lance | What is spasticity?[END_REF] [START_REF] Kheder | Spasticity: pathophysiology, evaluation and management[END_REF]. All of the cited models studied only the knee joint with a simple pendulum. However, as our final goal is to cor-rect a whole spastic gait, the main contribution of our work is to model a double-pendulum system composed of a thigh/shank/foot system based on the work of [START_REF] Fee | Neuromuscular modeling of spasticity in cerebral palsy[END_REF] and experimental data.

2) Voluntary muscular activations: As we consider a gait cycle and not a simple pendulum drop test, muscular activations torques were added to better represent the normal contribution of the muscles during a movement as in [START_REF] Wang | Spasticity assessment based on the maximum isometrics voluntary contraction of upper limb muscles in post-stroke hemiplegia[END_REF]. Two models were then defined: one using the onsets/offsets of the muscular activations as the period of activation for the torques representing voluntary muscular activations and one in which the period of activation is obtained using GA. The amplitude of the spasticity torques was defined using GA.

3) Position-dependent torques: Moreover, additional position-feedback torques are added for the spastic model to bring forward the position-dependent character of spasticity with the same timing as velocity-feedback torques [START_REF] Wu | Position as well as velocity dependence of spasticityfour-dimensional characterizations of catch angle[END_REF]. I. The masses of the segments were computed using regression equations of [START_REF] Jensen | Body segment mass, radius and radius of gyration proportions of children[END_REF] for children of 9 years old. Before data collection, we obtained informed consent and assent from the parents and the twin, respectively. The parents also have signed an informed agreement that the Declaration of Helsinki is respected during the experiment. 2) Experimental data acquisition: For three-dimensional gait analysis, 48 markers were fixed on subject bone landmarks following the recommendation of the International Society of Biomechanics (ISB) [14][15]. The positions of the reflective markers were recorded by twenty optoelectronic cameras sampled at 200Hz (VICON, Oxford's metrics, Oxford, UK). A 6-dimensional external contact wrench applied to the subject by the ground was recorded by five force plates, 2 AMTI platforms and 3 Sensix force platforms, embedded in the ground and sampled at 2000Hz. (Figure 1 & 2).

III. MODELING OF THE SPASTICITY

Sixteen wireless EMG electrodes were placed on the bodies (Table II) based on the SENIAM recommendations Linear envelopes for each muscle were obtained by lowpass filtering fully rectified raw EMG signals with a 10Hz lowpass filter (2nd order Butterworth, zero lag). The EMG data was then filtered with a 4th-order bandpass Butterworth filter between 20 and 400Hz. For each participant and each muscle, EMG envelopes were normalized using their mean value over the whole cycle. Muscle activations (onsets and offsets) were determined using an estimation of the proportion p of the signal corresponding to the baseline. The threshold was set as the mean plus three standard deviations of the p% lowest values of the sample multiplied by the signal-to-noise ratio. Only periods in which the signal was above the threshold for more than 100ms were considered activations. Manual adjustments were done to be more precise on certain signals. Normalization of the EMG signals was done to express each as a percentage of a gait cycle. Markers trajectories were filtered by a 4th-order low pass Butterworth filter with a 10Hz cutoff frequency. Gait cycles and events (foot strike and toe-off) were identified manually. The inverse kinematics was performed with OpenSim [START_REF] Delp | Opensim: open-source software to create and analyze dynamic simulations of movement[END_REF]. Three-dimensional joint angles were computed for each trial (ABC Euler sequence) at the lower limbs.

B. Double pendulum modelization:

Once the data are collected, the thigh and shank trajectories are modeled by using double pendulum equations and the equations defined in [START_REF] Fee | Neuromuscular modeling of spasticity in cerebral palsy[END_REF] (Equations 1 and 2). The foot is considered as a point mass at the end of the shank. The modeling of the knee-muscle spasticity is performed using equations developed in [START_REF] Fee | Neuromuscular modeling of spasticity in cerebral palsy[END_REF]. The spasticity model [START_REF] Fee | Neuromuscular modeling of spasticity in cerebral palsy[END_REF] developed for one degree of freedom (DOF) is used (Equation 5).

For the Hip:

Θ ′′ 1 (m 1 d 2 1 + I 1 + m 2 l 2 1 + m 3 l 2 1 )+ Θ ′′ 2 (m 2 l 1 d 2 cos(Θ 1 -Θ 2 ) -m 3 l 1 l 2 cos(Θ 1 -Θ 2 ))+ Θ ′2 2 (m 2 l 1 d 2 sin(Θ 1 -Θ 2 ) + m 3 l 1 l 2 sin(Θ 1 -Θ 2 ))+ g * sin(Θ 1 )(m 1 d 1 + m 2 l 1 + m 3 l 1 ) + T SpasticityM odel + T act = 0 (1)
For the Knee:

Θ ′′ 2 (m 2 d 2 2 + I 2 + m 3 l 2 2 )+ Θ ′′ 1 (m 2 l 1 d 2 cos(Θ 1 -Θ 2 ) + m 3 l 1 l 2 cos(Θ 1 -Θ 2 ))+ Θ ′2 1 (-m 2 l 1 d 2 sin(Θ 1 -Θ 2 ) -m 3 l 1 l 2 sin(Θ 1 -Θ 2 ))+ g * sin(Θ 2 )(m 2 d 2 + m 3 l 2 ) + T SpasticityM odel + T act = 0 (2)
with:

• Θ 1/2 , Θ ′ 1/2 , Θ ′′ 1/2
: joint relative angle, velocity and acceleration of the hip joint (1) and the knee joint (2).

• I 1/2 : moment of inertia of the thigh and shank expressed at the center of the joint, • d 1 , d 2 : distance between the hip joint(1)/ knee joint [START_REF] Graham | Pendulum test in cerebral palsy[END_REF] and the center of mass of the thigh/shank • l 1 , l 2 : length of the thigh and shank • m 1 , m 2 , m 3 : mass of the thigh, shank and foot respectively • T SpasticityM odel : Torques produced by the spasticity model • T act : Torque vectors produced by voluntary muscular activations This model does not consider contact forces as external forces to the system. Only gravity is considered here. T act represents the vectors for each joint of the torques produced by their flexors and extensors. Flexors are considered during flexion and extensors during extension (see Table II for the muscles and their functions). Thus, depending on the joint studied and the phase of the gait (extension or flexion), T act can be defined as follows :

For the hip :

T actHip = [S.T vector , G.M vector , V.L vector , R.F vector ] (3)
For the knee:

T actKnee = [S.
T vector , M.G vector , B.F vector , R.F vector ] (4) They represent the vectors of the corresponding voluntary muscular activations torques (Table II).

In this paper, 2 spastic models are thus defined: a) Model A: In this model, the timing of the muscular activations torques (T act ) corresponds to the real moment of muscular activations obtained from the EMG data. The magnitude of those torques is obtained using GA.

b) Model B: This model corresponds to the optimization of the timing where the muscles are activated : in this case, the torques T act are applied during the entire cycle and their amplitudes are optimized using genetic algorithms.

C. Spasticity model:

The equation of motion of the thigh and the shank are given respectively by equations 1 and 2 for extensors (subscript e, when Θ' < 0) and flexors (subscript f, when Θ' ≥ 0). Thus, extensors are considered during the flexion (spasticity due to the extensor will impact the flexion) and the same reasoning for the flexors.

T SpasticityM odel = B e,f Θ ′ + K e,f Θ + T e,f + BΘ ′ + KΘ (5)
with :

• B e , B f : damping coefficients,

• K e , K f : stiffness coefficients,

• T e , T f : nonlinear stiffness torques defined by [START_REF] Winters | Generalized analysis and design of antagonistic muscle models: Effect of nonlinear muscle properties on the control of fundamental movements[END_REF] as K1 e,f (e -K2 e,f Θ -1) • B = BE or BF : velocity feedback gains for extension (BE) or flexion (BF ). • K = KE or KF : position feedback gains for extension (KE) or flexion (KF ).

BE, BF , KE and KF gains are considered applied during the whole gait without interruption and with an unknown magnitude as quantification of the spasticity in a muscular activation cannot be done easily and accurately. Their values are obtained using GA.

The values of the damping coefficients (B e and B f ) corresponding to the damping part of the muscle fibers, spring coefficients (K e and K f ), and K1/K2 from the non-linear stiffness torques remain unknown for both hip and knee joints and will be determined using GA. GA is a method used to solve constrained but also unconstrained optimization problems based on a natural selection process that imitates biological evolution [START_REF] Holland | Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence[END_REF]. The goal of GA is to minimize a cost function defined by determining some parameters, chromosomes (Figure 3). A chromosome is a vector of parameters to be determined by the GA. In this paper, GA aims to identify 2 chromosomes: one for the hip joint (Equation 6) and one for the knee joint (Equation 7) defined as follows:

C hip = [K e , K f , B e , B f , K1 e , K2 e , K1 f , K2 f , BF vector , BE vector , KF vector , KE vector , T actHip ] (6)
C knee = [K e , K f , B e , B f , K1 e , K2 e , K1 f , K2 f , BF vector , BE vector , KF vector , KE vector , T actHip ]

with T actHip and T actKnee defined in Equations 3 and 4.

Each cycle has its size representing the number of values per cycle considering the sampling frequency. Thus, BF vector , BE vector , KF vector , KE vector , T actHip and T actKnee have the same size as the gait cycle studied. Indeed, during a gait, muscular activations are expected but it is complicated to determine the part of spasticity in these activations. Therefore, we assumed that spasticity is effective during the whole gait cycle without interruption and with unknown magnitudes.

To determine the parameters of each chromosome, the genetic algorithm was computed with the following parameters:

• Number of parameters to determine : (size of the gait) *10 + 16 parameters for the hip and knee. A large number of chromosomes to test at each iteration allows converging quickly to an optimal solution. The number of iterations, the maximum stall generation and tolerance value allow us to consider all possible cases and to stop when there is no significant improvement in the cost function. The parameters presented were chosen after several tests to find a compromise between the accuracy of the results, the convergence of the solution and computing time. The cost function used was based on the difference between modeled angular positions (determined by the resolution of the differential equation using a Newton-Euler method with a time step of 0.05s) and experimental angular positions of the thigh/shank. The cost function used is based on the coefficient of determination (R 2 ). The cost function is defined as follows for each joint:

Cost_F unction = 1 -R 2 = m i=1 (y i -ŷi ) 2 m i=1 (y i -ȳ) 2 (8)
with :

• m: number of experimental measurements, • y i : value of the experimental measurement i,

• ŷi : corresponding modeled data, • ȳ: mean of all the experimental data At each iteration, the chromosomes are initialized randomly with parameters. The cost function is thus computed and the algorithm continues its calculation until finding an optimal solution. The final cost function used corresponds to the mean of the cost function of the hip and the knee.

In this paper, a single gait of C. was chosen as muscular activations and spasticity vary from one walk to another so the results may be significantly different from one walk to another. This gait is divided into 12 right and left gait cycles. For each model, 2 gait cycles are considered: one right and one left gait cycle. For each gait cycle, the right and left sides of the lower limb are modeled. Those gait cycles were chosen for the lack of issues in their acquisitions. To demonstrate the closeness of those cycles with the other ones, the Normalized Root Mean Square Error (NRMSE) is computed to quantify the difference between the gaits chosen and the others. NRMSE corresponds to the Root Mean Square Error (RMSE) divided by the mean of the experimental signal (Equation 9). NRMSE goes from 0 to 1 which corresponds to the worst result.

RM SE = m i=1 (y i -ŷi ) 2 m ; N RM SE = RM SE ȳ (9)
Considering all the right gait cycles in this walk, the right gait cycle chosen has an NRMSE of 0.15 with the other right gait cycles. For the left gait cycle, the left gait cycle chosen has an NRMSE of 0.1332. Those results demonstrate that those chosen gait cycles are close enough to the other ones of the walk and can be representative.

D. Data Processing

1) Angular displacement values: Hip and knee angular displacement obtained after the modelization were filtered using a 4th-order lowpass Butterworth filter with a cutoff frequency of 10Hz .

2) EMG: As a reminder, EMG data were filtered with a-4th order bandpass Butterworth filter between 20 and 400Hz. Onsets and offsets were determined manually after determining the signal baseline and threshold.

E. Onsets and offsets of the muscular activations

Onsets and offsets of the muscular activations are presented in Table III 4 and Table V.

a) Right side: The right side has an R 2 equal to 84%: right hip at 80% and right knee at 87%. For BF , BE vectors, the mean value is around -0.55±8.4 and for KF , KE vectors, the mean value is around 0.14±8.4. For the muscular activation vectors, the mean value is around -0.05±8.7 N.m with a maximum of 13 N.m. For the hip flexors, the mean value is around -0.37 N.m and for the extensors around 0.10 N.m. For the knee flexors, the mean value is around -0.07N.m and for the extensors around -0.37 N.m.

b) Left side: The left side has an R 2 equal to 90%: right hip at 91% and right knee at 89%. For BF and BE vectors, the mean value is around 0.25±8.5 and for KF and KE vectors, the mean value is around 0.21±8.4. For the muscular activation vectors, the mean value is around 0.06±8.6 N.m with a maximum of 14.6 N.m. For the hip flexors, the mean value is around -0.11 N.m and for the extensors around -0.72 N.m. For the knee flexors, the mean value is around 0.63 N.m and for the extensors around -0.11 N.m.

2) Right Gait Cycle: Results for the right gait cycle for both sides are presented in Figure 5 and Table VI.

a) Right side: The right side has an R 2 equal to 91%: right hip at 94% and right knee at 88%. For BF , BE vectors, the mean value is around -0.06±8.7 and for KF , KE vectors, the mean value is around -0.12±8.6. For the muscular activation vectors, the mean value is around -0.006±8.6 N.m with a maximum of 12.4 N.m. For the hip flexors, the mean value is around -0.35 N.m and for the extensors around 0.43 N.m. For the knee flexors, the mean value is around -0.04N.m and for the extensors around -0.35 N.m.

b) Left side: The left side has an R 2 equal to 94%: right hip at 91% and right knee at 96%. For BF , BE vectors, the mean value is around 1.29±8.6 and for KF , KE vectors, the mean value is around 0.05±8.7. For the muscular activation vectors, the mean value is around 0.43±8.67 N.m with a maximum of 14.9 N.m. For the hip flexors, the mean value is around -0.48 N.m and for the extensors around 1.38 N.m. For the knee flexors, the mean value is around 0.33N.m and for the extensors around -0.48 N.m.

Globally for model A, knee parameters and the mean value of BF and BE vectors are close to the values defined in [START_REF] Fee | Neuromuscular modeling of spasticity in cerebral palsy[END_REF]. Results obtained are higher than 80% for the R 2 . Contrary to the previous model, the timing where the muscular activations torque vectors are applied is not based on the onsets and offsets of muscular activations presented in Tables III and IV: they are determined by the GA as the value of the gains for the muscular activations torque vectors.

B. Model B

1) Left Gait Cycle: Results for the left gait cycle for both sides are presented in Figure 4 and Table VII.

a) Right side: The right side has an R 2 equal to 95%: right hip at 93% and right knee at 96%. For BF , BE vectors, the mean value is around -0.04±5.86 and for KF , KE vectors, the mean value is around 0.34±5.6. For the muscular activation vectors, the mean value is around 0.18±5.6 N.m with a maximum of 9.9 N.m. For the hip flexors, the mean value is around 0.64 N.m and for the extensors around 0.37 N.m. For the knee flexors, the mean value is around -0.44N.m and for the extensors around 0.64 N.m.

b) Left side: The left side has an R 2 equal to 98%: right hip at 98% and right knee at 99%. For BF , BE vectors, the mean value is around 0.48±5.7 and for KF , KE vectors, the mean value is around -0.10±5.5. For the muscular activation vectors, the mean value is around 0.49±5.81 N.m with a maximum of 9.99 N.m. For the hip flexors, the mean value is around 0.67 N.m and for the extensors around 0.77 N.m. For the knee flexors, the mean value is around -0.08N.m and for the extensors around 0.67 N.m.

2) Right Gait Cycle: Results for the right gait cycle for both sides are presented in Figure 5 and Table VIII.

a) Right side: The right side has an R 2 equal to 98% : right hip at 99% and right knee at 97%. For BF , BE vectors, the mean value is around -0.36±8.5 and for KF , KE vectors, the mean value is around -0.27±8.3. For the muscular activation vectors, the mean value is around 0.21±8.4 N.m with a maximum of 14.1 N.m. For the hip flexors, the mean value is around 0.29 N.m and for the extensors around -0.73 N.m. For the knee flexors, the mean value is around -0.16N.m and for the extensors around 0.29 N.m. b) Left side: The left side has an R 2 equal to 94%: right hip at 95% and right knee at 92%. For BF , BE vectors, the mean value is around 0.40±8.5 and for KF , KE vectors, the mean value is around 0.59±8.76. For the muscular activation vectors, the mean value is around 0.01±8.8 N.m with a maximum ok 15 N.m. For the hip flexors, the mean value is around 0.24 N.m and for the extensors around -0.11 N.m. For the knee flexors, the mean value is around -0.04N.m and for the extensors around 0.24 N.m.

In model B, the period of the muscular activations determined by the GA corresponds to the whole gait: when the muscles should not be active considering values presented in Table III and Table IV, the GA found optimal to create muscular activation torques to better fit the gait cycle. The R 2 results permit demonstrating objectively the relevance of those methods. Model B presents better results than model A: the mean difference is located in the definition of the moment of muscular activations between the two models. The fact that in model B considered a full possible activation during the whole gait can hide 2 issues. First, the manual determination of the offset and onsets of the muscular activations can introduce errors that can modify the timing of real muscular activations which can be compensated by the torques representing the voluntary muscular activations. Thus, a lack of information/parameters used in the model of the gait cycle can be compensated by those new torques as the absence of external forces due to contacts. Nevertheless, the two models present good starting points to model efficiently spastic gait cycles by introducing the contribution of the extensors and flexors muscles.

V. CONCLUSIONS

This paper aims to model the left and right gait cycles of a child with spastic cerebral palsy. The modeling of the spasticity using data from EMG, GA and positiondependent/velocity-dependent torques allows us to have similar results compared to the experimental data from C. This study is a case of study and focused on data of two twin sisters who are not representative of a huge population. This point does not represent a limit to our paper as the aim of the project is to create a customized exoskeleton. The protocol can be enhanced with other twins or other children with spastic cerebral palsy. In this model, there is no mobility for the ankle. In this paper, two gait cycles were studied as cycles can be repeated to create a whole gait. Those simplifications allowed us to focus on the definition of the model. Future works will focus on the control of this gait cycle to make it converge to the one of H. For this purpose, a whole exoskeleton will be modeled and controlled and the interaction of the human and the exoskeleton will be created. Other methods of gait modeling will be tested.

  USING A GENETIC ALGORITHM (GA) A. Method 1) Participants: Two 9 years old twin sisters (C and H) performed a clinical gait analysis (CGA). C has a GMFCS of 2. No maximum voluntary contraction was recorded due to the quick fatigue occurring for C. The anthropometry of C. is presented in Table

Fig. 1 .

 1 Fig. 1. Experimental setup view from the VICON software. Platform 4 is the Sensix platform and the others are the AMTI force platforms.

Fig. 2 .

 2 Fig. 2. CREPS laboratory. Cameras are indicated with the red circles and the three platforms with the blue circle

Fig. 3 .

 3 Fig. 3. Flowchart of genetic algorithm (GA).

Fig. 4 .

 4 Fig. 4. Simulated and experimental left gait cycle for the left and the right side of the lower limb for both models.

Fig. 5 .

 5 Fig. 5. Simulated and experimental right gait cycle for the left and the right side of the lower limb for both models

TABLE II LIST

 II OF THE MUSCLES STUDIED IN THE EXPERIMENT AND THEIRFUNCTION.

	Muscles	Acronym	Function
	Medial gastrocnemius	M.G	Knee flexor and Ankle extensor
	Rectus femoris	R.F	Hip flexor and Knee extensor
	Vastus lateralis	V.L	Knee extensor
	Semi-tendinous	S.T	Knee flexor and Hip extensor
	Biceps femoris	B.F	Knee flexor
	Gluteus maximus	G.M	Hip extensor

  for a Left Gait Cycle and in TableIVfor a Right Gait Cycle. The definition of the onsets and offsets is presented.

TABLE V

 V

	[MODEL A]VALUES OF THE 16 FIRST PARAMETERS COMPOSING THE
	FINAL CHROMOSOME FOR BOTH LEFT AND RIGHT SIDES OF THE LEFT
				GAIT CYCLE
	Left Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-2.5	-1.8	-2.6	2.7	-4.5	-3.8	-4.9	-4.3
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-3.6	-1.3	1.0	-1.0	0.9	0.3	1.5	-0.4
	Right Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		4.1	2.8	-3.8	3.0	4.3	3.4	4.9	3.1
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-3.3	-3.1	-1.1	-0.8	0.5	1.3	-1.5	-0.9

TABLE VI [

 VI MODEL A]VALUES OF THE 16 FIRST PARAMETERS COMPOSING THE FINAL CHROMOSOME FOR BOTH LEFT AND RIGHT SIDES OF THE RIGHT Results for the left gait cycle for both sides are presented in Figure

				GAIT CYCLE
	Left Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-3.1	4.5	-0.5	-0.7	-4.1	-3.2	4.4	4.1
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-2.7	-4.4	0.3	-1.9	-1.8	-1.5	-1.9	-0.8
	Right Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-2.6	-3.2	0.9	-2.2	-3.6	-3.8	4.9	3.2
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-4.3	-3.9	1.5	-0.2	-0.7	-0.9	1.6	0.5
	1) Left Gait Cycle:

TABLE VII [

 VII MODEL B]VALUES OF THE 16 FIRST PARAMETERS COMPOSING THE FINAL CHROMOSOME FOR BOTH LEFT AND RIGHT SIDES OF THE LEFT

				GAIT CYCLE
	Left Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-3.8	-4.2	-2.8	0.9	-2.0	-1.3	-3.3	-4.3
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-4.3	-2.9	-0.8	-0.9	-0.5	-1.4	-0.2	-0.2
	Right Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-4.3	0.8	-4.8	-1.6	3.3	1.2	2.5	3.0
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		3.8	-2.8	-1.5	-1.5	0.9	1.8	-0.4	0.2

TABLE VIII [

 VIII MODEL B]VALUES OF THE 16 FIRST PARAMETERS COMPOSING THE FINAL CHROMOSOME FOR BOTH LEFT AND RIGHT SIDES OF THE RIGHT

				GAIT CYCLE
	Left Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		4.4	-3.0	-3.7	-3.0	-4.3	-2.6	-1.0	2.8
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		-4.5	-4.7	0.5	-1.0	-1.6	-0.2	0.6	0.9
	Right Side
	Hip	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		1.9	0.8	-2.5	-1.3	-1.7	-0.7	-2.8	-3.7
	Knee	Ke	Kf	Be	Bf	K1e K2e K1 f K2 f
		0.6	-4.2	-0.6	-1.9	-1.1	1.1	-1.0	-0.2

ACKNOWLEDGEMENT

The authors would like to thank the Région Occitanie and Université Fédérale Toulouse Midi-Pyrénées (UFTMIP) for their financial support.

IV. RESULTS

In this part, figures 4 and 5 present the results for the model A and B copared to the experimental data for a right and left gait cycle and for both sides (right and left leg).

A. Model A

As explained, in this model, the timing where the muscular activations torque vectors are applied is based on the onsets and offsets of muscular activations presented in Tables III andIV.

CONFLICT OF INTEREST STATEMENT

The authors confirm that there are no conflicts of interest regarding the work described in the current manuscript.