Nayla Farouki

Philippe Grangier

The Einstein-Bohr debate: finding a common ground of understanding ?

After reminding the main issues at stake in the famous Einstein-Bohr debate initiated in 1935, we tentatively propose a way to get them closer, thus shedding a new light on this historical discussion.

Realism in physics is often understood as the idea that all physical objects are built up by assembling smaller (or more elementary) objects, and that all objects carry their own properties, whatever their level in this construction. This idea works fine in classical physics, but is clearly more problematic in quantum physics, where the nature of objects becomes more elusive, and where it is dubious that a single object carries its own properties, especially when it is considered to be part of an entangled set.

Unhappy with this elusive character of quantum mechanics (QM), Einstein, Podolsky and Rosen (EPR) expressed their own view of realism [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF] in a stronger and more definite statement, saying that, "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity".

Yet, Niels Bohr disagreed. In his answer, written also in 1935 [START_REF] Bohr | Can Quantum-Mechanical Description of Physical Reality be Considered Complete?[END_REF], he wrote: "EPR's criterion of physical reality contains an ambiguity in the meaning of 'without in any way disturbing a system'. Of course there is no mechanical disturbance of the system under investigation, but there is an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. These conditions constitute an inherent element of the description of any phenomenon to which the term 'physical reality' can be properly attached". These statements by Bohr were mostly accepted, but not really understood: what are these "very conditions" quoted as inherent elements to speak about physical reality? We will try to clarify this issue below.

In the 1960's, Bohr gave his own full-fledged definition of what physics should be [START_REF] Bohr | The Unity of Human Knowledge[END_REF]: "Physics is to be regarded not so much as the study of something a priori given, but rather as the development of methods of ordering and surveying human experience. In this respect our task must be to account for such experience in a manner independent of individual subjective judgement and therefore objective in the sense that it can be unambiguously communicated in ordinary human language". But this latter statement sounded extreme, and renouncing the very idea of a physical object is a hard price to pay for most physicists -this is even more the case nowadays, given the impressive progress achieved in getting full control of individual quantum objects.

In 1964, an important advance took place [START_REF] Bell | On the Einstein-Podolski-Rosen paradox[END_REF]. Theoretical and experimental work, initiated by John Bell, showed that the EPR view could not work as such, and must at least be amended [START_REF] Aspect | Closing the Door on Einstein and Bohr's Quantum Debate[END_REF]. Quantum reality does not abide -at least not entirely -by Einstein's conditions. As Bohr's arguments were either hard to accept or equivocal, the debate went back to its initial state, and new questions and theories about the "real objects" involved in a quantum description kept showing up [START_REF] Laloë | Do We Really Understand Quantum Mechanics?[END_REF].

B. Physical realism and contextual objectivity.

Yet, if absolute certainty and total predictability could not be the conditions required to attain quantum reality, perhaps, a different set of rules could make this reality convincing again? This is what is proposed here. First, we define physical realism with the more consensual statement that "the purpose of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules". Universality and intelligibility are accepted criteria for scientific realism, perhaps not as strong as full certainty and predictability, but they may be more efficient in the QM realm. This being done, would it be possible to generalize the argument for universality and intelligibility so as to admit at least parts of Einstein's and Bohr's positions? In an attempt to define what is real in QM, and to get Einstein and Bohr closer, let us first take into account the evidence that can be gathered from experience, while provisionally putting aside the mathematical apparatus through which this experience is described.

The answer we propose [START_REF] Auffèves | Contexts, Systems and Modalities: a new ontology for quantum mechanics[END_REF] relies on the simple observation that in quantum physics, like in classical physics, the values of some physical properties can be measured repeatedly, and the results can be predicted with certainty. But in quantum physics, this can generally be obtained only by keeping constant all the relevant elements of the measurement procedure, that we call a context. If we admit that this complete classical specification of the context is just the "very condition" required by Bohr, then the following statement, paraphrasing Einstein's, is perfectly fine, also from a quantum point of view: "If, without in any way disturbing a system, nor changing the context, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity".

This statement satisfies EPR's idea to look for certainty in scientific knowledge, as a criterion for reality [START_REF] Auffèves | Violation of Bell's inequalities in a quantum realistic framework[END_REF]. The "element of physical reality" that Einstein sought is present in a deterministic way, but it is submitted to a condition not perceived in classical physics: the statement is true as long as the context is not changed [9]. So the element of quantum reality belongs jointly to the (quantum) system and to the (classical) context, and is real as EPR would have wanted, but its reality comes with a twist: it permanently associates a quantum system and a classical context. To distinguish it from other forms of classical realities, we shall call it a modality.

When associated with quantization -postulated as an intrinsic property of quantum objects [START_REF] Auffèves | Contexts, Systems and Modalities: a new ontology for quantum mechanics[END_REF] -a change of context will produce a random change affecting the modalities of a quantum object, when it shows up in a different context [START_REF] Auffèves | Contexts, Systems and Modalities: a new ontology for quantum mechanics[END_REF][START_REF] Auffèves | Violation of Bell's inequalities in a quantum realistic framework[END_REF]. This randomness appears in the form of probabilities, and it can be shown that they can be computed according to the usual quantum formalism [START_REF] Auffèves | Recovering the quantum formalism from physically realist axioms[END_REF][START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF]. In this approach, the modality is a real phenomenon involving a system and a context as physical objects, and it is thus ontologically distinct from the state vector or projector, which is a mathematical object used to calculate probabilities. Mixing up these two notions -the real phenomenon on the one hand and the mathematical object used to describe it on the otherhas been generating a lot of confusion, which is removed within the quantum realism framework proposed here.

C. What did we learn ?

Our contextual description also tells something about Einstein's and Bohr's respective positions. It does agree with Einstein's realism in terms of certainty, necessity, and therefore physical reality. At the opposite, it clashes with his classical conviction that physical properties should be independent of the context in which they are defined. We have also a good agreement with Bohr's intuition that the conditions of the measurement must be taken into consideration: some properties do vary randomly when the context is changed. We attribute this random change to the fact that elementary physical properties are quantized, implying that randomness is indeed essential to understand quantum physical behavior. This randomness, famously, was refused by Einstein. Though a final reconciliation between Einstein and Bohr seems out of scope, both Einstein's objectivity and Bohr's contextuality do find a common ground in our approach. Does this new understanding answer the major challenge of defining quantum reality? We believe that this is the case, since certainty exists, quantum probabilities make sense due to quantization, and Bohr's intuition that we ought to consider the experimental conditions as an inherent part of the object under study is given its full meaning. This only requires that we describe objectivity in a new more humble way: there is no such a thing as absolute objectivity, i.e. the apprehension of the Universe, taken as a whole, with all its logical and ontological necessity as Newton and Einstein conceived it. Contextual objectivity [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF] may provide us with true and efficient knowledge about the world we live in.

Acknowledgements. The authors thank Alexia Auffèves for continuous contributions, and Franck Laloë & Roger Balian for many useful discussions.


Sylvie Bégot
email: sylvie.begot@univ-fcomte.fr

Muluken Getie

Alpha Diallo

François Lanzetta

Magali Barthès

Michel De Labachelerie

A novel model and design of a MEMS Stirling cooler for local refrigeration

In this paper, we present a new model design and parametric studies of a miniature Stirling cooler machine for on-site refrigeration. The MEMS (Microelectromechanical systems) technology is investigated to design this machine. The concept could be used to provide cooling at chip scale and mitigate hot spots in electronic devices. Whereas numerous works deal with Stirling engines at a macroscopic scale, only a few works concern miniaturized Stirling engines. Therefore, a model analysis giving insights of the impact of the technological choices and downsizing of the machine is needed. A base design model is presented. The model results lead to a cooling power of 10 mW and a Coefficient Of Performance of 1.45. A parametric study is conducted for operational and design parameters. Compared to macro-scale design, the same trend is observed for the influence of the thermal performance regenerator. Different trends from macroscopic engines were observed for hysteresis losses importance, and the choice of the working gas. The raise in power due to the raise in frequency expected for micro-scale devices is counterbalanced by the degradation of the COP due to the increase in thermofluidic losses. Squeeze film damping and finite speed losses can be neglected at this scale.

Introduction

The development of electronic appliances and their miniaturization raise the concern of cooling methods and technologies for increased high flux devices and non-uniform power dissipation. Moreover, the failure factor of electronic devices increases sharply with their operating temperature [1]. On the other hand, the development of MicroElectroMechanical Systems (MEMS) makes possible to design thermal machines at a micro-scale level. Therefore, in this paper, we investigate the modelling and design of a Stirling micro-cooler. The aimed application is to provide cooling at chip scale and mitigate hot spots in electronic devices. Whereas numerous works deal with Stirling engines or coolers at a macroscopic scale [2][3], only a few works concern miniaturized Stirling machines. The first prototype on a centimetre scale of a Stirling engine was developed by Nakajima et al. in 1989 [4]. It included a 0.05 cm 3 swept volume and delivered a power of 10 mW at 10 Hz operating between temperatures of 273 K and 373 K. In 2002, Moran patented a micro-cooler made up from a Stirling micromachine [5]. Silicon membranes, electrostatically actuated, played the role of the pistons. Several types of regenerators were tested [START_REF] Moran | 2 nd International Energy Conversion Engineering Conference[END_REF]. After experimental tests between 100 and 1000 Hz, no notable difference in temperature between the chambers was observed. In 2013, Guo et al. [START_REF] Guo | [END_REF] proposed a multiphysical modelling of a MEMS Stirling cycle cooler. The expansion and compression volumes were planned to be etched in a silicon wafer. The pistons were replaced by 2.25 mm diameter electrostatic actuated membranes. A regenerator made up of pillars was placed between the 2 volumes. The machine pressure was 2 bar. A Schmidt isothermal analysis supplemented by a loss modelling including pressure drop and regenerator effectiveness showed that the expected COP at 200 Hz was 5.2 for temperatures of 313 K for the hot sink and 288 K for the cold source. In 2015, Formosa et al. proposed and modelled a multiphase membrane machine for thermal energy harvesting [8][9]. It consisted of three coupled elementary phases, each composed of a 5 mm-diameter membrane which acted as a piston and transducer integrating a piezoelectric planar spiral, an expansion chamber, and a compression chamber linked by a regenerator.

To conclude, Stirling machines at macroscopic scale are a well-known technology both in engine and cooler modes, but except for the centimetric engine designed by Nakajima et al., micro-scale Stirling cycles machines are not operating yet. Therefore, it seems appropriate to investigate Stirling coolers at micro-scale by developing and adapting a complete thermodynamic model to the specificity of micro-scale design. This is the main topic of the paper. The first part of the paper presents the micro-machines specific features and configuration. The second part describes the adapted adiabatic model including losses. The last part of the paper presents the base design characteristics and results. Parametric studies investigating the influence of operating parameters, design parameters, and the nature of the working gas are presented. From these results, some design guidelines for miniature engines are provided.

Micro-cooler specific features

The Stirling micro-cooler design has to be adapted for technological reasons. In this section, we describe the initial choices of the design. These choices have an impact on the model equations, for this reason, the model is described in detail after defining the micro-scale design main features. In the first place, we choose an Alpha configuration because the symmetry of the compression and expansion chambers are adapted to batch fabrication. The machine materials that can be used are those adapted to MEMS facilities. Therefore, the chosen base materials are glass and silicon [START_REF] Madou | Fundamentals of microfabrication: the science of miniaturization[END_REF]. Due its higher thermal conductivity, silicon is used where heat exchange is needed, while glass is used when heat exchange is detrimental. The general layout of the machine, the heat exchangers and the regenerator are presented in Fig. 1.

Fig. 1. Micro-cooler layout

Second, the pistons are not practical at microscale. Therefore, the micromachine base design consists of an engine with compression and expansion chambers swept by membranes instead of pistons. These membranes can be electrostatically actuated. This choice was also made by Moran [5] and Guo et al. [START_REF] Guo | [END_REF]. Thus, the membrane is a hybrid membrane made from a centre silicon disk embedded in RTV silicone (Fig. 2). More details on the design and fabrication are given in Diallo et al.'s work [START_REF] Diallo | Contribution to the design and construction of a thermal micromachine with Stirling cycle[END_REF] where the technological feasibility of this membrane was established.

Fig. 2. Hybrid Membranes

The membrane swept volume is only a fraction of the cylinder volume. As in macro-scale design, a clearance volume separates the membrane top dead centre position from the chamber maximal height. The hot and cold heat exchangers are made from ducts etched in the silicon plate closing the chambers. In macro-scale design, the regenerator is usually made from woven screens, metal felts or involute foils. In our case, these technologies are not applicable, therefore we choose to use an array of pillars to fulfil the function of regeneration. Silicon is chosen as the material of the pillars to promote heat transfer, glass is chosen for the walls to limit the heat transfer from the hot part of the machine to the cold part. The fabrication of the regenerator includes a first step where a glass wafer and silicon wafer are assembled by anodic bonding, then the pillars are etched by Deep Reactive Ion Etching (DRIE). More details on the design, fabrication and CFD modelling of the regenerator can be found in references [START_REF] Dellali | 17th International Stirling Engine Conference and Exhibition[END_REF][START_REF] Dellali | [END_REF] where the technological feasibility of this kind of regenerator was established.

Principle

The model is based on the developments presented by Urieli and Berchowitz [START_REF] Urieli | Stirling cycle engine analysis[END_REF]. The "simple" analysis is used and adapted to the specific configuration of micro-coolers. The first concern is the use of membranes instead of pistons. Therefore, shuttle heat losses, mass leakage losses and piston friction losses are not considered. The swept volume is only a fraction of the chamber volume. The second concern is the buffer or bounce space volume.

Due do the miniaturization of the machine and the membrane fragility, a bounce space volume is designed and its volume is kept to a minimum. The third concern is the specific geometry of the regenerator. A specific correlation different from those established for woven screens has to be used. We make the hypothesis that conduction losses are limited to the regenerator walls. Last, we investigated squeeze film damping as the miniaturization may lead to this phenomenon. Indeed, as the volume forces that work on a machine vary in direct proportion to the cube of the length, surface forces that may be negligible at a macroscopic scale may play an important role in a micromachine [START_REF] Bao | [END_REF].

Equations

The cooling power 𝑄 ̇𝑖𝑛 of the micro-cooler is defined as:

𝑄 ̇𝑖𝑛 = 𝑄 ̇𝑘 -𝑄 ̇𝑟𝑙 -𝑄 ̇𝑤𝑟𝑙 (1)
where 𝑄 ̇𝑘 is the net thermal power exchanged at the cold heat exchanger, 𝑄 ̇𝑟𝑙 is the thermal power resulting from the regenerator efficiency, and 𝑄 ̇𝑤𝑟𝑙 is the thermal power due the conduction between the cold and hot parts of the machine [START_REF] Urieli | Stirling cycle engine analysis[END_REF]. The Coefficient Of Performance is computed as the ratio of the cooling power to the absorbed mechanical power:

𝐶𝑂𝑃 = 𝑄 ̇𝑖𝑛 𝑊 ̇𝑚𝑒𝑐ℎ (2)
The absorbed mechanical power is determined as the power yielded by the ideal adiabatic analysis 𝑊 ̇𝑎𝑑 plus the fluid friction losses 𝑊 ̇𝑓𝑟 , the hysteresis losses in the bounce space 𝑊 ̇ℎ𝑦𝑠 , the finite speed losses of the membrane 𝑊 ̇𝑓𝑠 and the squeeze film damping losses 𝑊 ̇𝑠𝑓 :

𝑊 ̇𝑚𝑒𝑐ℎ = 𝑊 ̇𝑎𝑑 + 𝑊 ̇𝑓𝑟 + 𝑊 ̇ℎ𝑦𝑠 + 𝑊 ̇𝑓𝑠 +𝑊 ̇𝑠𝑓 (3)
The fluid friction losses 𝑊 ̇𝑓𝑟 are due to the fluid friction losses in the heat exchangers and in the regenerator. They are computed from the various friction factors adapted to the geometries. For the regenerator, the correlation from Vanapalli et al. [16] is used, for the heat exchangers, the correlation established for flows at steady-state in channels of rectangular cross-section are used. The hysteresis losses are computed as:

𝑊 ̇ℎ𝑦𝑠 = √ 1 32 𝜔𝛾 3 (𝛾 -1)𝑇 𝑤𝑏 𝑃𝑘 𝑔 ( 𝛥𝑉 𝑏 𝑉 𝑏 ) 2 𝐴 𝑤𝑔,𝑏 (4)
where 𝑊 ̇ℎ𝑦𝑠 , 𝜔, γ, 𝑇 𝑤𝑏 , 𝛥𝑉 𝑏 , 𝑉 𝑏 and 𝐴 𝑤𝑔,𝑏 denote the hysteresis power losses, the angular speed of the machine, the specific heat ratio, the bounce space wall temperature, the bounce space mean pressure, the variation of the bounce space volume, the total bounce space volume, and the wetted area of the bounce space respectively [START_REF] Urieli | Stirling cycle engine analysis[END_REF]. We make the hypotheses that the bounce wall temperature is the same as the cold source temperature Tcold and that the mean pressure is the same as the machine mean pressure. The finite speed losses 𝑊 ̇𝑓𝑠 are computed following the works of Costea et al. [17]. The losses due to finite speed of the membranes are evaluated from:

𝑊 ̇𝑓𝑠 = [∮ 𝛥𝑃 𝑓𝑠 𝑑𝑉 𝑐𝑦𝑐𝑙𝑒 ] 𝑓 (5)
where f is the frequency of operation and 𝛥𝑃 𝑓𝑠 is computed as:

𝛥𝑃 𝑓𝑠 = 1 2 (𝑃 𝑎𝑣 𝑚,𝑐 𝑐 𝑐 + 𝑃 𝑎𝑣 𝑚,𝑒 𝑐 𝑒 ) ( 6
)
where vm is the membrane mean speed, c is the average molecular speed 𝑐 = √3𝑅𝑇, and 𝑎 = √3𝛾.

Th squeeze film losses analysis aim is to check that during the displacement between the expansion and compression spaces, the gas can escape the chambers. The squeeze number 𝜎 is defined as [START_REF] Bao | [END_REF]:

𝜎 = 12𝜇𝑟 𝑚 2 𝜔 𝑃𝑧 2 (7)
where 𝜇 is the gas dynamic viscosity, rm is the characteristic length of the device, in our case the radius of the membrane, P the mean pressure, and z is the height of the film. In our case, z is chamber height. When 𝜎<<1, the Reynolds equation holds and the squeeze film damping force coefficient D for a circular plate can be written as:

𝐷 = 3𝜋 2𝑧 3 𝜇𝑟 𝑚 4 (8)
Then, the damping force can be computed, and from its expression, we derive the losses:

𝑊 ̇𝑠𝑓 = [ 𝜔 2 𝐷𝑉 𝑠𝑤𝑐 2 𝜋𝑟 𝑚 4 ] 𝑓 (9)
where 𝑉 𝑠𝑤𝑐 is the membrane swept compression volume.

In order to assess the performance of the micro-cooler, the efficiency 𝜂 is defined as the ratio of the COP to the ideal COPi of a machine operating on the Carnot cycle:

𝜂 = 𝐶𝑂𝑃 𝐶𝑂𝑃 𝑖 (10)
with 𝐶𝑂𝑃 𝑖 = 𝑇 𝑐𝑜𝑙𝑑 𝑇 ℎ𝑜𝑡 -𝑇 𝑐𝑜𝑙𝑑 [START_REF] Diallo | Contribution to the design and construction of a thermal micromachine with Stirling cycle[END_REF] where the 𝑇 𝑐𝑜𝑙𝑑 is the cold source temperature, 𝑇 ℎ𝑜𝑡 the hot sink temperature.

Base design characteristics and results

Using this model, a base design is proposed. Each chamber diameter is 5 mm. The membrane thickness is 0.2 mm and its maximal deflection is 0.2 mm. The swept volume is 2.75 mm 3 corresponding to 70% of the cylinder volume. The regenerator length is 2 mm and its porosity 0.8. Heat exchangers are made from 6 ducts of 2.5 mm length. The geometric characteristics and operational parameters are presented in the Table 1. These characteristics lead to a 10.6 mW cooling power machine with a COP of 1.45 (Table 2) at 100 Hz. In order to derive some general analysis, dimensionless losses for the main contributors are computed: mechanical losses values are divided by the mechanical power 𝑊 ̇𝑚𝑒𝑐ℎ and the thermal losses are divided by the cooling power 𝑄 ̇𝑖𝑛 . The resulting dimensionless variables and results are displayed in Table 3. An analysis of the losses repartition shows that the regenerator is a key component of the performance, like in macroscopic machines. Regenerator thermal efficiency is a main contributor to the losses, and so are the thermal conduction losses in its walls. Fluid friction is also a major concern at both scales. Hysteresis losses are very high in this miniaturized Stirling machine, as compared to macroscopic machines due to the minimisation of the bounce space volume. Therefore, careful design of the bounce space should be carefully studied to minimize their impact. Both finite speed and squeeze film damping losses are negligible in this base design scale. 4 Parametric analysis

Effect of operating parameters

In this paragraph, the impact of the operating parameters on the micro-cooler performance is investigated. First, we studied the impact of the cold source temperature. On the considered range 273 K-298 K, the temperature difference between heat source and sink diminishes, thus, the machine cooling power raises from 8.2 to 12.2 mW (Fig. 4a). Therefore, the cooling capacity per cm 2 of cooled surface varies from 41.7 mW.cm -2 to 62.1 mW.cm -2 . The COP also raises due to the decrease in the heat losses that depend on the temperature difference (Fig. 4b). The ideal coefficient of performance COPi sharply raises as the temperature difference increases (Fig. 4c). As the mechanical losses do not diminish with the temperature difference, the micro-cooler efficiency decreases from 0.17 to 0.12.

Fig. 4. Impact of the cold source temperature

Second, the analysis of the performances with the frequency variation was made (Fig. 5). As expected, the cooling power raises but the COP presents a maximum at 75 Hz (Fig. 5a). The raise in regenerator efficiency losses is partly responsible for the decrease in the COP when the frequency raises above 75 Hz (Fig. 5b). Whereas the high frequency leads to an increase in power, it is also detrimental to heat exchange. The fluid friction losses also raise with the increase in frequency and thus in gas velocity. The evolution of the COP with the cooling power presents a maximum at 75 Hz (Fig. 5c). At this frequency, a compromise between COP and efficiency is achieved. Therefore, the machine miniaturization does not lead to a large increase in the optimal operational frequency compared to macroscopic engines that operate at about 50 Hz, and this is mainly due to thermofluidic phenomena.

Third, as in macroscopic machines, a raise in pressure directly leads to an approximately linear raise in cooling power (Fig. 6a). The increase in losses with pressure is mainly due to thermal efficiency of the regenerator (Fig. 6b). However, the machine COP remains stable in the pressure range 1x10 5 Pa and 4x10 5 Pa. Operation above this pressure would lead to technological challenges, therefore the influence of higher pressures was not investigated in this study.

Effect of some design parameters

In this paragraph, the influence of some of the geometric design parameters is studied. As the regenerator efficiency and conduction losses are the first two main losses of the base design model (Table 2), regenerator geometry was studied. First, the impact of the regenerator length is studied. The cooling power increases with the regenerator length (Fig. 7a). The COP variation with the regenerator length presents a sharp raise, a maximum at 3 mm, then a slight decrease. The observation of the losses (Fig. 7b) shows that a short regenerator leads to high regenerator thermal efficiency losses and high conduction losses. Indeed, a short regenerator does not enable all heat exchange between the gas and the regenerator pillars to take place during the cycle. The conduction losses are high when the regenerator length is small. Increasing the length leads to an increase in cooling power. On the other hand, the higher length values lead to an increase in the friction losses. These phenomenon account for the shape or the cooling power and COP curves.

Fig. 7. Impact of the regenerator length

The same analysis was made with the variation of the regenerator porosity. For a 2 mmlong regenerator, the COP slightly decreases of 7% from 1.46 to 1.36 over a porosity range of 0.7 to 0.9 (Fig. 8a). On this range, the power decreases of 14% from 11.2 to 9.6 mW. The observation of the main losses (Fig. 8b) shows that the regenerator thermal efficiency losses raise when the porosity is higher but on the other hand friction losses decrease. This accounts for the shape of the COP and cooling power curves.

Fig. 8. Impact of the regenerator porosity

For the base design, hysteresis losses in the bounce space are also one of the main losses (Table 2). As they depend on the bounce space volume relative variation 𝛥𝑉 𝑏 /𝑉 𝑏 during the cycle (4), their amplitude is large when this volume relative variation is high. In macroscopic engines, the bounce space volume is usually designed large, and as a consequence the relative variation is small. However, as our aim is to have a miniature machine, this will not be necessarily the case in our design. Thus, we studied the COP and cooling power of the microcooler with the bounce space volume variation. The cooling power remains stable with the relative variation, but the machine COP decreases of 34 % with 𝛥𝑉 𝑏 /𝑉 𝑏 (Fig. 6a). We observe that, for the base design, the hysteresis losses overcome the fluid friction losses for a volume variation of 6% (Fig. 6b). A large bounce space volume leading to small relative variations during the cycle is beneficial, therefore, when designing a micro-cooler, the bounce space volume should be carefully studied.

Effect of the working gas

Finally, we simulated the performance of the micro-cooler using helium and hydrogen as working gases instead of air. Helium is a frequent working gas for macroscopic engines due to its high thermal conductivity. Hydrogen properties are also interesting and miniature engines need a small mass of hydrogen reducing hydrogen safety concerns. The cooling power and COP for the three gases are plotted in Fig. 10a. Helium appears to be the better choice regarding the cooling power, but its COP is reduced due to high hysteresis losses (Fig. 10b) linked to its high specific heat ratio γ. Air presents the better COP but its cooling power is reduced by high regenerator efficiency losses. Hydrogen present intermediate performances in cooling power and COP.

Fig. 3 .

 3 Fig. 3. Regenerator

Fig. 5 .Fig. 6 .

 56 Fig. 5. Impact of the frequency

Fig. 9 .

 9 Fig. 9. Impact of the bounce space volume

Fig. 10 .

 10 Fig. 10. Impact of the working gas

Table 1 .

 1 Base design main parameters

	Parameter	Symbol	Unit	Values
	General
	Gas	-	-	Air
	Mean pressure	𝑃	MPa	0.15
	Hot sink temperature	𝑇 ℎ𝑜𝑡	K	313
	Cold source temperature	𝑇 𝑐𝑜𝑙𝑑	K	288
	Frequency	𝑓	Hz	100
	Phase angle	𝛼	degrees	90
	Chambers
	Membrane diameter	𝑑 𝑚	mm	5
	Membrane stroke	𝑠	mm	0.4
	Chamber height	𝑧	mm	0.23
	Swept volume	𝑉 𝑠𝑤𝑐	mm 3	2.75
	Bounce space
	Volume	𝑉 𝑏	mm 3	13.7
	Volume variation	𝛥𝑉 𝑏 /𝑉 𝑏	%	20
	Height	-	mm	0.7
	Cold and hot heat
	exchangers
	Number of ducts	-	-	6
	Length	𝐿 ℎ,𝑘	mm	2.5
	Width	-	mm	0.4
	Height	-	mm	0.075
	Regenerator
	Length	𝐿 𝑟	mm	2
	Porosity	𝜓	-	0.8
	Internal width	-	mm	0.5
	Internal height	-	mm	0.1
	Wall thickness	-	mm	0.05
	Wall thermal conductivity	𝑘 𝑟	𝑊. 𝑚 -1 . 𝐾 -1	1

Table 2 .

 2 Base design main results

		Symbol	Unit	Values
	Power and efficiency
	Cooling power	𝑄 ̇𝑖𝑛	mW	10.61
	Mechanical power	𝑊 ̇𝑚𝑒𝑐ℎ	mW	7.32
	COP	COP	-	1.45
	Efficiency	𝜂	%	12.6
	Losses
	Regenerator efficiency	𝑄 ̇𝑟𝑙	mW	2.53
	Regenerator conduction	𝑄 ̇𝑤𝑟𝑙	mW	1.05
	Pressure drop	𝑊 ̇𝑓𝑟	mW	0.58
	Hysteresis	𝑊 ̇ℎ𝑦𝑠	mW	2.32
	Membrane finite speed	𝑊 ̇𝑓𝑠	mW	1.48x10 -4
	Squeeze film damping	𝑊 ̇𝑠𝑓	mW	5.60x10 -6

Table 3 .

 3 Base design dimensionless losses

	Dimensionless losses	Symbol	Values
	Regenerator efficiency	𝑄 ̇𝑟𝑙,𝑑𝑙𝑒𝑠𝑠 = 𝑄 ̇𝑟𝑙 /𝑄 ̇𝑖𝑛	0.24
	Regenerator conduction	𝑄 ̇𝑤𝑟𝑙,𝑑𝑙𝑒𝑠𝑠 = 𝑄 ̇𝑤𝑟𝑙 /𝑄 ̇𝑖𝑛	0.10
	Pressure drop	𝑊 ̇𝑓𝑟,𝑑𝑙𝑒𝑠𝑠 = 𝑊 ̇𝑓𝑟 /𝑊 ̇𝑚𝑒𝑐ℎ	0.08
	Hysteresis	𝑊 ̇ℎ𝑦𝑠,𝑑𝑙𝑒𝑠𝑠 = 𝑊 ̇ℎ𝑦𝑠 /𝑊 ̇𝑚𝑒𝑐ℎ	0.32

Conclusion

In this paper, we investigate the modelling and design of a Stirling micro-cooler that could be used to provide cooling at chip scale and mitigate hot spots in electronic devices. This MEMS technology leads to the use of specific materials and geometries that are not common in macroscopic machines. After adaptation of the model to these specific characteristics, we proposed a base design machine. We performed a parametric analysis in order to analyse the performance of the machine with the variation of operational parameters, some geometric parameters and working gas. Compared to macro-scale design, the same trend was observed for the influence of the thermal performance regenerator. However, different trends from macroscopic engines were observed. The raise in power due to the raise in frequency expected for micro-scale devices is counterbalanced by the degradation of the COP due to the increase in thermofluidic losses. Hysteresis losses in the bounce space are also of high amplitude due to the small size if this volume. The choice of the working gas between air, helium and hydrogen has to be adapted to the objectives in cooling power or COP. Both finite speed and squeeze film damping losses are negligible for the investigated machine size.

This work has been supported by the EIPHI Graduate School (contract ANR-17-EURE-0002) and the Region Bourgogne-Franche-Comté.


Benjamin Mauz

Guillaume J Laurent

Redwan Dahmouche

C Édric Cl

Micrometer Positioning Accuracy with a Planar Parallel Continuum Robot

Keywords: soft robots, continuum robots, Cosserat-rod, Kirchhoff-rod, parallel robots, micro-positioning, robot calibration, accuracy

Parallel Continuum Robots (PCR) have several advantages over classical articulated robots, notably a large workspace, miniaturisation capabilities and safe human-robot interactions. However, their low accuracy is still a serious drawback. Indeed, several conditions have to be met for PCR to reach a high accuracy, namely: a repeatable mechanical structure, a correct kinematic model, and a proper estimation of the model's parameters. In this article, we propose a methodology that allows reaching a micrometer accuracy with a PCR. This approach emphasizes the importance of using a repeatable continuum mechanism, identifying the most influential parameters of an accurate kinematic model of the robot and precisely measuring them. The experimental results show that the proposed approach allows to reach an accuracy of 3.3 µm in position and 0.5 mrad in orientation over a 10 mm long circular path. These results push the current limits of PCR accuracy and make them good potential candidates for high accuracy automatic positioning tasks.

INTRODUCTION

Parallel Continuum Robots (PCR) are a recent and rising type of robots introduced in [START_REF] Bryson | Toward parallel continuum manipulators[END_REF]. They are composed of flexible slender elements arranged in parallel and linked to a platform.

Resulting robots present a higher rigidity than soft robots while keeping most of their advantages. This structure provides higher safety than common industrial robots thanks to the flexibility of the PCR's limbs.

For instance, Campa et al. proposed a planar PCR to perform safer collaborative robot applications [START_REF] Campa | A 2 Dof Continuum Parallel Robot for Pick & Place Collaborative Tasks[END_REF]. The flexibility of the limbs provides large continuous deformations that allow for a large workspace. Yang et al. proposed a continuum version of the Delta robot whose continuous joints allow larger rotations than usual joints [START_REF] Yang | Continuum Delta Robot: A Novel Translational Parallel Robot with Continuum Joints[END_REF]. Flexible elements allow to reduce the number of joints and eliminate the mechanical plays [START_REF] Black | Parallel Continuum Robots: Modeling, Analysis, and Actuation-Based Force Sensing[END_REF]. Reducing the number of joints and mechanical parts also facilitates the miniaturization of the designed robot. For example, the continuum Steward-Gough platform introduced in [START_REF] Bryson | Toward parallel continuum manipulators[END_REF] had its dimensions reduced in order to create a surgical PCR whose platform (a gripper) is around 10 mm diameter [START_REF] Orekhov | Modeling parallel continuum robots with general intermediate constraints[END_REF]. This small prototype illustrates the interest and the capability of PCR to perform applications inside confined space (like the human body) where instrumentation and sensor-based controls are difficult to implement. To execute those applications, an accurate PCR could be a solution. However, the reachable accuracy of those structures is still an open question which needs more investigations. The objective of this paper is to propose a methodology to reach a high accuracy with PCR by measuring, identifying and understanding the influence of the different parameters, and have an estimate of a typical level of accuracy that a PCR can reach.

The accuracy corresponds to the difference between commanded poses and the barycenter of attained ones (ISO 9283, 1998) in position and orientation. A high accuracy enables to position the end-effector of the PCR thanks to its model and allows to successfully perform automated tasks. The accuracy of a robot depends on three key elements: its repeatability, the correctness of its model and the knowledge of the model's parameters. The repeatability corresponds to the deviations of the reached poses for a same command, and mainly depends on robot's design, the quality of its mechanical structure and its actuation system.A repeatable robot is essential to have a predictable behavior and thus to model it. The modeling of continuum robots appears especially challenging because of their virtually infinite number of Degrees-of-Freedom (DoF) and their highly nonlinear behavior due to the large deformations. Two modeling approaches of parallel continuum robotics are widespread in the literature: physical-based models and black-box models such as neural networks. Using an artificial neural network, Wu et al. reached a positioning accuracy of 5 mm. One of the drawbacks in using black-box models is that they could not identify why the positioning accuracy was low [START_REF] Wu | Modeling and analysis of a parallel continuum robot using artificial neural network[END_REF]. This is one of the reasons why most of the studies use physical-based models.

We can distinguish two categories of physical-based models: high fidelity distributed parameter models and low fidelity parameter models [START_REF] Rone | Mechanics Modeling of Multisegment Rod-Driven Continuum Robots[END_REF]. The low fidelity parameter models use strong hypothesis to reduce the number of parameters and to facilitate the modeling and the identification of its parameters. For instance, constant curvature models represent continuum deformations by considering that flexible element shapes are circle arcs [START_REF] Lilge | Continuous Structures in Planar Parallel Robots: A Theoretical Analysis[END_REF][START_REF] Nuelle | Modeling, calibration, and evaluation of a tendon-actuated planar parallel continuum robot[END_REF]. Those approaches are interesting for their simplicity but seem not suitable for high positioning accuracy. For this reason, physical-based models with high fidelity distributed parameters and variable curvature assumption are preferred. Depending on the structure, Cosserat-rod-based models [START_REF] Trivedi | Geometrically Exact Models for Soft Robotic Manipulators[END_REF][START_REF] Orekhov | Modeling parallel continuum robots with general intermediate constraints[END_REF][START_REF] Till | Elastic Stability of Cosserat Rods and Parallel Continuum Robots[END_REF][START_REF] Black | Parallel Continuum Robots: Modeling, Analysis, and Actuation-Based Force Sensing[END_REF] or Kirchhoff-rod-based ones (Altuzarra et al., 2019;[START_REF] Altuzarra | Certified kinematics solution of 2-dof planar parallel continuum mechanisms[END_REF][START_REF] Takano | Real-time shape estimation of Kirchhoff elastic rod based on force/torque sensor[END_REF][START_REF] Campa | A 2 Dof Continuum Parallel Robot for Pick & Place Collaborative Tasks[END_REF] gained consensus because of their ability to predict the shape and forces of the rods. In the case of tendon-actuated continuum structures, Rao et al. proposed guidelines to choose a model depending on the targeted application [START_REF] Rao | How to model tendon-driven continuum robots and benchmark modelling performance[END_REF]. Such guidelines do not currently exist for parallel continuum structures.

In addition to the modeling, the measurement and the identification of the model's parameters are also important. Indeed, to get an accurate prediction of the robot's behavior, the model's parameters need to be close to their real values. To address this issue, robot calibration is usually performed. However, as Wu et al. pointed out, the calibration of PCR is more complex than for regular parallel robots [START_REF] Wu | Experimental statics calibration of a multi-constraint parallel continuum robot[END_REF]. One example is the absence of an analytical model. This complexity generally conducts to choose a simpler model to facilitate the identification step [START_REF] Dehghani | Characteristics identification of continuum robots for exact modeling[END_REF] or to measure the parameters rather than identifying them [START_REF] Orekhov | Analysis and Validation of a Teleoperated Surgical Parallel Continuum Manipulator[END_REF]. The chosen model, the measurement and the identification of its parameters have a strong and deeply intertwined impact on the robot's accuracy.

Recent studies quantified the accuracy of several PCR that are gathered in Table 1. Even if it is difficult to compare them because of their different designs, those values give an overview of the expected reachable accuracy of current PCR using different models. To be more representative to the PCR's design, the positioning accuracy is generally expressed in percent of the nominal or the mean dimension of the continuum flexible segment. For example, Orekhov et al. obtained a mean positioning accuracy of 2.8%

(1.19 mm) and a mean orientation accuracy of 3.81 • after identifying extrinsic parameters of a 6-Degreesof-Freedom (DoF) robot [START_REF] Orekhov | Analysis and Validation of a Teleoperated Surgical Parallel Continuum Manipulator[END_REF]. The accuracy is interesting for a spatial PCR even if it is limited by the actuation system whose positioning accuracy is about 0.1 mm. The actuation system is also one of the main uncertainty sources for the positioning accuracy calculated from the data presented in [START_REF] Yang | Continuum Delta Robot: A Novel Translational Parallel Robot with Continuum Joints[END_REF]. In the studies of [START_REF] Wu | Experimental statics calibration of a multi-constraint parallel continuum robot[END_REF][START_REF] Wu | Modeling and analysis of a parallel continuum robot using artificial neural network[END_REF], several reasons were pointed out like the small construction and assembly tolerances or the friction forces that were not took into consideration. Nuelle et al. proposed a study of a tendon-actuated planar PCR and reached an accuracy of 1.4% (1.8 mm) after identifying all model's parameters thanks to a calibration process [START_REF] Nuelle | Modeling, calibration, and evaluation of a tendon-actuated planar parallel continuum robot[END_REF]. The shown accuracy was limited by the constant curvature approach and by the actuation and the robot design, which suffer from gear backlash and static-friction.

To address the challenge of getting accurate PCR, the proposed approach consists in five key elements.

The first one consists in using a repeatable PCR. The robot recently introduced in [START_REF] Mauzé | Nanometer Precision with a Planar Parallel Continuum Robot[END_REF] was able to provide an outstanding repeatability of 9.13 nm in position and 1.2 µrad in orientation. Choosing a planar architecture allows using a very resolute and long range multidimensional measurement system which facilitates the understating of the proposed methodology. Thanks to its mechanical structure, this robot seems suitable to study the accuracy reachable by PCR. This 3 DoF XYΘ planar parallel continuum is illustrated in Fig. 1. The second point is the use of a correct mechanical model. The measurement of the model's parameters is facilitated by the PCR design which also anticipates the calibration process. That is why fiducial markers, third element of the proposed methodology, are introduced. Those fiducial markers, placed at specific locations of the PCR, enable to efficiently measure the nominal values and to estimate the uncertainties of the different models' parameters. The fourth element is a sensitivity analysis which coupled with the uncertainties allows to determine the most influential parameters. The last key point is the calibration process of the robot.

The next section presents the model of the PCR and the sensitivity analysis. This analysis enables to understand the role of each parameter and to identify the most influential ones. Those parameters require a special attention during the design and the calibration processes of the robot. The robot prototype and the experimental setup are described in Section 3. The models' parameters measurement step with an uncertainty analysis and the calibration process are respectively presented in Section 4 and 5. The last section demonstrates the capability of the robot to perform desired trajectories and quantifies the accuracy using the identified parameters.

3-DOF PLANAR PCR MODEL

To study Parallel Continuum Robots' (PCR) accuracy, we considered a high-grade repeatable robot illustrated in figure 1. This robot is composed of three planar kinematic chains that are linked together to a rigid moving platform. Contrary to classical 3-PRR mechanisms which inspired this design, each chain is composed of a prismatic actuator and a flexible rod that deforms continuously. The three flexible rods transmit the actuators' forces to the platform inducing its motions. Using the traditional naming convention, the flexible rod is denoted using the letter 'F' and the resulting PCR design is then a 3-PF robot. Fig. 2 illustrates the kinematic diagram of the 3-PF robot. The actuation stages, through the prismatic joint values q 1 , q 2 , q 3 , push and pull the rods to move the mobile platform. The rods are slender beams capable of continuous and large deformations. The movement of this mobile platform is restricted to planar displacements (x, y translations and θ rotation).

A global work frame (W , x W , y W , z W ) is defined with the z W axis perpendicular to the robot's base.

A frame (P , x P , y P , z P ) is attached to the mobile platform. The three rods are clamped to this platform at their distal ends (B i , x B i , y B i , z B i ). The proximal ends of the rods clamped to the actuator are defined by the frames (A i , x A i , y A i , z A i ).

In the following, we describe the model used to simulate the quasi-static behavior of the robot. First, the model of a single rod is detailed. Then, the three models of rods are linked together thanks to the static equilibrium of the platform. All those elements allow to implement the forward and the inverse kinematic models of the robot.

Rod Model

The PCR's model depends mainly on the modeling of the rods and its correctness, that is why a Kirchhoffrod-based model is used. This model supposed that shear and extension can be neglected. It is possible as the used slender elements have their cross-section more than a hundred times smaller than their lengths.

The proposed structure remains in a plane so, only the planar case of this model is considered without losing the generality of the proposed approach which can also be adapted for a spatial robot.

The curvilinear abscissa is represented by the scalar parameter s ∈ [0, l] where l is the stress-free length of a rod. Along its arc length, p(s) = x(s), y(s)

T and θ(s) respectively define the cross-section centroid position and orientation in the frame attached to the proximal end of the rod. Fig. 3 shows the entire model of a limb.

All involved differential equations can be gathered into the following system:

                   dx(s) ds dy(s) ds dθ(s) ds dnx(s) ds dny(s) ds dm(s) ds                    =                  cos(θ(s)) sin(θ(s)) m(s) EI 0 0 n x (s) sin(θ(s)) -n y (s) cos(θ(s))                  (1)
Where n x (s), n y (s) are respectively the X and Y components of the internal force n(s), m(s) is the internal moment, E is the Young modulus of the rod material, and I is the second area moment of the rod cross-section which depends on the rod's diameter d.

Forward and Inverse Kinematic Models

To model the quasi-static behavior of the PCR, a forward and inverse kinematic models are created. Both result from a numerical resolution of the previous equations considering the rod's boundary conditions and the static equilibrium of the platform.

Those boundary conditions describe how the rods link the actuators to the mobile platform. They yield:

W T P = W T A i • A i T B i • B i T P (2)
where W T A i depends on the joints coordinates q i , A i T B i corresponds to the transformation resulting from the integration of the rod's equations B i T P is constant reflecting the rigid-body conditions between the distal ends of the rods and the mobile platform, and W T P depends on the desired position such as:

W T P =   cos θ -sin θ x sin θ cos θ y 0 0 1   (3)
To know the pose of the platform, the static equilibrium conditions are considered:

           3 i=1 [n i (l i )] -f P = 0 3 i=1 [p B i × n i (l i ) + m i ] -p P × f P -m P = 0 (4)
where f P , m P are the external force and moment applied on the platform in the work frame, where n i , m i are the rods' forces and moments applied on the platform in the work frame, l i is the length of the rod i and p B i and p P are the positions of B i and P in the work frame.

Due to the coupling between the rods, there is no analytical solution of those equations considering the previous boundary and equilibrium conditions, thus, a numeral resolution is performed using a shooting method. This shooting method is based on a optimisation problem where the Eq. ( 2) and Eq. ( 4) constitute its residual vector [START_REF] Till | Elastic Stability of Cosserat Rods and Parallel Continuum Robots[END_REF][START_REF] Mauzé | Nanometer Precision with a Planar Parallel Continuum Robot[END_REF].

Sensitivity Analysis of the models' parameters

The PCR accuracy depends on the values of the model's parameters. To estimate the influence of the different parameters, a sensitivity analysis is performed. It also allows for identifying which parameters required more consideration. There are three kinds of parameters: the intrinsic rod parameters (the Young modulus E of its material, its diameter d and its length l), the ones representing the poses of the rods' proximal and distal ends (respectively A x i , A y i , α i and B x i , B y i , β i ), and the parameters γ i which represent the misalignments between the directions of translations and the rods' orientation at the proximal ends.

The principle of the proposed sensitivity analysis is the following. An arbitrary set of 26 joints configurations is chosen in the center of the workspace. For each configuration, partial derivatives of the platform position considering the different parameters are numerically computed using a finite difference method. The maximal values of the partial derivatives among the configurations and the different rods are gathered in Table 2. All parameters are expressed in the SI base unit.

From the table results, three groups of parameters can be defined considering their influence on the model.

The first group is composed of the Young modulus and the diameters of the rod. The Young modulus has no influence on the pose of the platform. It is the same observation for the diameter if the three parameters are equal. This assumption can be considered as true if the different rods are created from the same element.

If one diameter is different than the other, it introduces an asymmetry and thus a small variation of the modeling. This difference of behavior is only visible for the diameters. Indeed, if all lengths (for example) are changed in the same time, the consequences on the model will be more important than if only one length was changed.

The second group is composed of the angular parameters α i , β i and γ i . The influence of the parameter α i is more important than the one of the other angular parameters. It can be explained by its role in the transmission of the actuator's movement. To understand the influence in terms of orientation, considering a 1 mrad uncertainty on a α i and the value of the table, the resulting positioning error for the model is about 1.3 µm. If this value is compared with a case-study of an uncertainty of 1 mrad made on the orientation of a rigid bar of 30 mm, the estimation of the position is about 30 µm at the bar's end. So, it can be said that the influences of those angular parameters are relatively small. The elasticity, the deformation of the rod and the parallel structure reduced their influence.

The last group is composed of the parameters that are the most influential: the length of the rods l i and their end positions A x i , A y i , B x i , B y i . The differences between the direction X and Y are essentially due to the asymmetry of the robot's structure which is induced by the initial orientation of the rods.

This sensitivity analysis give information about the influence of the different parameters but also indicate the threshold of uncertainty where their influence can not be neglected anymore. To give an example of comparison between the parameters' influence, considering the values of the table, an uncertainty of 1 mrad observed on a proximal end's orientation α i has the same influence of an uncertainty of 1 µm observed on the estimation of the X coordinate of the proximal end position A x i .

In order to get an accurate robot, the parameters whose influences are the most important on the model need to be estimated as precisely as possible. The next section will present the PCR design which takes into account the presented results by introducing fiducial markers. The objective of those markers is to reduce the measurement uncertainty of the most influential parameters.

ROBOT DESIGN AND MEASUREMENT SYSTEM

To validate the modeling of the robot, a prototype has been built by taking into account the previous sensitivity analysis. To perform this comparison between the prototype's experimental behavior and its model, the PCR poses and its models' parameters have been measured by two complementary vision measurement systems. This section introduces the robot design and the associated measurement system.

Robot Design

The robot is composed of a mobile platform coupled to three actuated continuum limbs.

Each limb is assembled on a stack of three manual precision stages, two translation stages (Newport SDS-40) and a rotation stage (Newport M-RS40). These manual stages are used to adjust the position and the orientation of each actuator fixed above them.

The actuators are SmarAct (SLC-1730-S-HV) positioning stages. Their repeatability is below 30 nm and their range is 21 mm. Those actuators possess accurate optical sensors and a repeatable behavior that are necessary to get accurate inputs for the forward kinematic modeling of the PCR.

The rods are optical fibers stripped of their plastic part. Those fibers are in fused silica whose Young

Modulus E is about 69 GPa. They are able to do large deformations without viscous effect thanks to their elasticity. They are 125 µm in diameter and about 30 mm in length. Those rods are connected to the actuators and to the platform by the mean of rod mounts.

The rod mounts are the key elements to define precisely the position of the rods' ends and thus their length. Each rod mount is drilled on one side to clamp the rods. On the top face of the mount, a fiducial marker (QR-Code-like pattern) is engraved. Thanks to its manufacturing, the transformation between the center of the fiducial marker and the hole where the rod are inserted is known with a small uncertainty.

Thus, knowing the pose of the fiducial marker will enable to know the relative pose of the proximal and the distal ends of the three rods (more details will be given in the next section). The proximal rod mounts are fixed on the actuators and the distal rod mounts are glued on the mobile platform.

The platform is a 100 mm silicon wafer lifted by a 50 mm diameter air bearing (S205001) from the IBS company. This air bearing avoids friction and is preloaded to maintain a stable elevation of the platform (± 5 µm). A manual linear stage (Newport M-DS25-Z) allows to adjust the level of the air bearing and to get the required planarity of the entire PCR.

Vision Setups

The knowledge of the initial configuration of the robot is mandatory for an accurate simulation and position control of the robot. For this purpose, two vision measurement systems have been set up as shown in Fig. 4.

The first one relies on the localization of fiducial markers embedded on the rod mounts. The position of these fiducial markers are tracked over a large field with a camera (IDS UI3008CP-3) mounted with a 50 mm lens. This vision setup is used to get accurate measurements of the different models' parameters of the robot. More details will be given in the next section.

The second vision measurement system is dedicated to the measurement of the Cartesian pose of the platform with a very high resolution. This system is composed of a camera with a microscope tube, a x20 lens from Mitutoyo and a pseudo-periodic pattern glued on the mobile platform of the PCR. This system is able to measure the x, y position and the θ angle of the platform with a sub-nanometer resolution [START_REF] Andre | Sensing one nanometer over ten centimeters: A micro-encoded target for visual in-plane position measurement[END_REF]. Both cameras are supported by a robust gantry in order to minimize mechanical noise.

PARAMETERS MEASUREMENT

After the PCR's model is created, the PCR accuracy will depend on the value of the model's parameters.

The closer the value of the parameters will be, the higher will be the PCR's accuracy. The measurement of those parameters is an essential step. Specially those whose influence, resulting from the sensitivity analysis, is important. This section explains how nominal values and measurement uncertainties are obtained in the aim to have an accurate control of the robot.

The different parameters are related to the rods, the position of the rods' ends and the actuators' direction of displacement. The quantification of the uncertainty will define intervals which will give more information about the potential modeling errors, and help to identify the parameters during the calibration process.

Rod parameters

The diameter of the rods is measured thanks to a calliper which has a measurement uncertainty of 20 µm.

The length of the rods is measured from images taken with the first vision setup. The measurement uncertainty is estimated to 3 pixels. Considering the pixel/metre ratio, this uncertainty is 126 µm.

Considering the uncertainties values and the results of the sensitivity analysis, the rod diameters influence can be neglected while the rod lengths need to be identified carefully.

Rod end positions

The sensitivity analysis points out that the rod end positions are influential parameters. Those positions depend on the rod mounts position and the clamping conditions. All rod mounts positions can be directly measured with the first vision setup and a specific fiducial marker detection process. This algorithm gives the Cartesian coordinates of all fiducial markers in the camera frame. The six rod mounts can be differentiated thanks to missing squares in each fiducial markers (coding principle). The resolution of the measurement is less than 0.25 µm for X and Y translations and 0.5 mrad for rotation.

As the transformation between the center of the fiducial marker and the clamping hole is known, the Cartesian position of the proximal end A and the distal end B of each rod can be deduced from these markers poses. By construction, the transformations between the center of the fiducial marker and the clamping point of the rod are defined by two translations of 14 mm and 250 µm in the X and Y directions.

The machining accuracy is about 6% of this transformation leading to uncertainties of respectively 840 µm and 15 µm. By acquiring an image close to the rod mount and the clamping area and considering the resulting pixel/metre ratio, the uncertainty in the X direction is reduced to 160 µm. Due to the value of the uncertainties and the influence of those parameters, they will need a special cares during the PCR calibration process.

Actuator's direction of displacement

The last parameters that need to be measured are assembling defaults between the actuators and the rod mounts (Fig. 5). The angle of the stage axis in the work frame, γ i , is measured using 1 mm displacement of the stage with a step size of 125 µm. Each pose of the pattern is recorded during this displacement. The angle of the regression line through these points gives a precise measurement of γ i . Then, the difference between this angle γ i and the pattern pose provides the misalignment angle α i . The uncertainty for both angles is estimated to 5 mrad. Considering their small uncertainties, the parameters β i , γ i will be neglected and only the parameters α i will be considered.

Thanks to this measurement process, nominal values of the different parameters and their uncertainties are known. This knowledge is useful to prepare the calibration of the PCR.

ROBOT CALIBRATION

Even with an accurate measurement process, the addition of small uncertainties reduces the PCR's accuracy.

To reduce those errors and obtain a better fitting between the experimental behavior and the model a calibration process has been implemented. In this article, three identification steps are used in this calibration process described in Fig. 6. The principle of one identification is illustrated in Fig. 7. The new values of the parameters are usually obtained thanks to a minimization of an objective function. This section details the different considered parameters, the definition of this objective function, the identification strategy, the experimental results and the obtained position and orientation errors.

Extrinsic and intrinsic parameters

There are two kinds of parameters considered for the identification step of the calibration process: the intrinsic parameters π r (defined in Table 2) which correspond to the model's parameters, and the external parameters which depend on the measurement system (whose uncertainties are at the nanometer level) used to get the pose of the PCR.

The experimental poses are acquired thanks to the second vision setup as shown in Fig. 4. They correspond to the 2D poses of a frame attached to the pseudo-periodic pattern with regards to the camera frame Se T M .

These measured poses are used to get the experimental Cartesian pose of the platform in the work frame W T P as follows:

W T P = W T Se • Se T M • M T P (5)
The transformations W T Se and M T P respectively correspond to the transformation between the work frame and the camera frame, and to the transformation between the pseudo-periodic pattern frame and the platform frame. W T Se and M T P are constant but cannot be measured directly and must be identified. The six corresponding extrinsic parameters are gathered in the vector π ext . The experimental pose W T P defined by X = [x, y, θ] T has to be compared with those resulting from the simulation defined by X = [x, ŷ, θ] T .

Objective Function

The parameters identification is based on a minimization of an objective function using a least squares algorithm. This algorithm is defined by the sum of the squares of the differences between the experimental Cartesian poses of the robot and the simulated ones using the forward kinematic model. Both experimental and simulation poses depend on a set of n joint coordinates Q = [q 1 . . . q n ] where q j = [q j 1 , q j 2 , q j 3 ] T is the j-th joint vector in the set. Considering the measured and simulated poses which depend on q j and π = [π r , π ext ] T , the vector of residuals is defined by F = [F 1 . . . F n ] T where F j is:

F j (π, q j ) = α * ( θj -θ j ) (x j -x j ) (ŷ j -y j ) (6)
where α is a weighting factor. This factor has been chosen such as the residuals in orientation and position were proportional to the repeatability of the robot.

Using this function, the identification problem can be formalized as:

minimize π F T F subject to π ∈ l b , u b (7)
where, l b and u b are the lower and upper bound values of the parameters, provided by the sensitivity analysis.

The minimisation of this optimization problem is performed using the built-in-function lsqnonlin from Matlab software.

Calibration Process

To increase the PCR's accuracy, the parameters can be identified after their measurements. This initial measurement of the parameters enables the reduction of the parameters' uncertainty intervals. Small intervals help to avoid some local optima and ensure consistent parameters during the minimisation.

Moreover, the sensitivity analysis helps to distinguish the most influential parameters that should be identified in priority. Indeed, the parameters could not be identified at the same time if their values are too far from their true values. For this reason, the identification is carried out in three steps using the same experimental data.

The parameters that need to be identified first are the extrinsic parameters π ext . Indeed, those parameters cannot be measured and have only been coarsely estimated. Then, after setting the extrinsic parameters, the most influential robot's parameters can be identified in a second step. From the analysis, those parameters are the length of the rods, the orientation of the translation direction of the actuator and the position of their ends. The two first steps aim at reducing substantially the errors but the values of the parameters may not

be optimal yet, because some parameters could compensate for the uncertainties of the others. In the last step, all parameters are re-identified together, after the identification of the most influential parameters.

This final optimization begins with an initial set of parameters that assumed to be relatively closed to their true values.

Calibration Results

As previously explained, the joint coordinates have been chosen inside the robot's workspace in which it has a nanometer repeatability. These joint coordinates are sent to the robot and the Cartesian coordinates of the platform are recorded. To ensure a good distribution in the available workspace, the joint coordinates have been randomly chosen using a 3-dimensional Poisson-disc sampling. A set of 99 joints coordinates, more than three times the number of considered parameters, is created to perform their identification.

56 joints coordinates are selected to validate the identified parameters. The position error of the pose j is defined as the root mean square of the difference between the simulated position (x j , ŷj ) and the experimental one (x j , y j ) such as:

e j P = (x j -x j ) 2 + (ŷ j -y j ) 2 (8)
Similarly, the orientation error is:

e j θ = ( θj -θ j ) 2 = | θj -θ j | (9)
Table 3 reports the results of model errors before and after calibration. After calibration, position error is included between 0.66 µm and 15.26 µm while the absolute orientation error is included between 4×10 -3 mrad and 5.61 mrad. In order to better study the influence of the error made on the values of the intrinsic parameters π r , the extrinsic parameters π ext have been identified even for the nominal model (first step). It shows that the calibration process allows to reduce the position error by a factor of 10 and the orientation error by a factor of 13.

Forward Kinematic Model Validation

To check the validity of the identified forward kinematic model, simulations are performed with a validation set of joints coordinates. Fig. 9 shows the obtained results. Position and absolute orientation errors are respectively included between 1.5 µm to 13.88 µm and between 0.1 mrad to 5.57 mrad. Those results, reported in Table 3, have the same orders of magnitude as those for the calibration set. The small reduction of the position errors is explained by a more compact distribution of poses close to the center of the workspace. The sources of these errors will be discussed in the last section.

For both sets, the maximal resulting position modeling error is 15.26 µm corresponding to 0.05% of the 30 mm length of the flexible rods.

ROBOT ACCURACY

This section presents the results of the proposed methodology on the PCR's accuracy. To quantify this accuracy, the robot is controlled in the Cartesian space. This control depends on the inverse kinematic model whose parameters resulted from the calibration process.

Robot Control

The forward kinematic model enables the prediction of the pose of the robot considering the joint coordinates as inputs. Conversely, the inverse kinematic model allows to calculate the joint coordinates corresponding to a given pose in the Cartesian space. A regular scheme for controlling the position of the robot in the Cartesian space using the inverse kinematic model is implemented as illustrated in Fig. 10. The parameters of the inverse kinematic model are those that have been identified for the forward kinematic model.

To validate the capability of the robot to reach commanded poses in the Cartesian space, two trajectories have been considered: a 2 mm side square and a 3 mm diameter circle with a constant null orientation of the mobile platform. Those trajectories are used as an input for the inverse kinematic model which returns the corresponding joints coordinates Q c . Those joints coordinates are sent to the robot and its poses are recorded by the second vision setup (the one composed by a microscope).

Fig. 11(A) and Fig. 12(A) show the desired trajectories (red points) and the experimental trajectories (blue points). Both positions and orientations obtained experimentally are closed to the desired ones. The differences between them are too small to be seen at the same time than the performed trajectory. Those results show that the robot is capable of following a defined trajectory.

Evaluation of the robot accuracy

The robot accuracy evaluates the closeness of agreement between the pose attained by the robot and its commanded pose. The accuracy of a robot is defined by the standard ISO 9283:1998. The positioning accuracy is the difference between the commanded position and the barycenter of the reached positions:

AP P = (x -x c ) 2 + (ȳ -y c ) 2 (10)
The orientation accuracy is the difference between the commanded angle and the average of the reached orientations:

AP θ = ( θ -θ c ) (11)
The bar operator (.) is the barycenter (or the average) of the reached positions (or orientations) after repeating the same pose at least 30 times.

The accuracy of the robot has been evaluated for the two trajectories as shown in Fig. 11 (B-C) and Fig. 12 (B-C). For both cases, the means of positioning accuracy are below 2 µm (1.27 µm and 1.95 µm)

and the worst positioning accuracies are respectively 2.60 µm and 3.28 µm for the square and circular trajectories. In orientation, the worst orientation accuracy is inferior to 0.532 mrad. Table 4 reports all those results. The resulting mean positioning accuracy is 10.9×10 -3 % of the nominal length of a flexible continuum rod (around 30 mm). Considering the performances of the other PCR gathered in Table 1, the presented positioning accuracy is 10 times better than the current relative positioning accuracy.

Discussion

For both trajectories (square and circle), the pose accuracies depend on the pose of the platform. For instance, in the square trajectory, the orientation accuracy is worst in the corners. A deeper analysis on the experimental data of the calibration process shows that the angular errors are linked, with a correlation ratio of nearly one, to the differences between the initial angle and the current angle. Larger the orientation of the platform, higher is the angular error. Correlations between the position and the angular error are less straightforward to establish with typical ratios between 0.52 and 0.74. With local studies on different areas of the workspace, there are some locations near the workspace borders were the position errors are increased. In conclusion, the model is more accurate in the middle of its workspace in both orientation and position.

CONCLUSION

In this article, a methodology to reach micrometer positioning accuracy is proposed.

This micrometer positioning accuracy was reached thanks to the proposed methodology which consists in five key elements. The first one is the use of a repeatable PCR structure to be able to predict the robot's quasi-static behavior. The second element is to use a correct mechanical model. The third one is the use of fiducial markers in the design of the robot. Those markers allow to efficiently measure the nominal values of the model's parameters and to estimate their uncertainties. The fourth element is to conduct a sensitivity analysis to quantify the influence of the different parameters and find the most influential ones. This crucial information is considered in the robot design to be able to efficiently measure those parameters thanks to fiducial markers. The last step is to calibrate the whole robot by identifying all the parameters through three optimization steps. Using this approach on a XY Θ planar Parallel Continuum Robot (PCR), the maximal

FIGURE CAPTIONS

Frontiers Table 1. Mean orientation AP θ and positioning accuracy AP P (in percent of the nominal or mean dimension dimension of the continuum flexible segment) for different PCR designs.

TABLES CAPTIONS

  Fig. 8 presents the result of the calibration process. With the warm-up cycle of the camera and the actuators, the experimental measurement of the calibration process lasts around 6 hours. It shows the errors between the platform positions and orientations simulated with the nominal model (model before calibration) and the model after calibration using the experimental data.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .

 12345 Figure 1. Picture of the XYΘ parallel continuum robot

Figure 8 .

 8 Figure 8. Positioning (A) and orientation (B) errors of the model (respectively e P and e θ ) for the different samples of the calibration set of joints coordinates. The dashed-lines are the mean values of positioning and orientation errors and respectively correspond to 7.19 µm and 1.7 mrad.

Figure 9 .Figure 10 .

 910 Figure 9. Positioning (A) and orientation (B) errors of the model (respectively e P and e θ ) for the different samples the validation set of joints coordinates. The dashed-lines are the mean values of positioning and orientation accuracy and respectively correspond to 5.5 µm and 1.5 mrad.

Figure 12 .

 12 Figure 12. Result of the robot position control for a circular trajectory: (A) commanded poses (in red) and attained with the prototype (in blue), (B) position and (C) orientation accuracies obtained for the different poses. The horizontal dashed-line is the mean value of the positioning accuracy which is 1.95 µm. The black dashed-lines are characteristic points (numbered points) on the trajectories where the AP P or AP θ monotony change.

Table 2 .

 2 Influences of the different models' parameters (expressed in SI base unit) computed as the maximal finite difference for the different configurations and rods.Parameters max i ( ∂x ∂param ) max i ( ∂y ∂param ) max i (

	∂θ ∂param )

Table 3 .

 3 Positioning and orientation errors (respectively e P and e θ ) with the nominal or the identified parameters and for the calibration or the validation configuration set.

	AP P (µm)	|AP θ | (mrad)
	max mean min max mean min
	Square 2.60 1.27 0.47 0.437 0.20 0.032
	Circle 3.28 1.95 0.28 0.532 0.24 0.046

Table 4 .

 4 Positioning and orientation accuracies (respectively AP P and AP θ ) of the robot for two different trajectories

This is a provisional file, not the final typeset article

Frontiers

ACKNOWLEDGMENTS

This work was supported by µSpider project (ANR-19-CE33-0002-02), by Région Bourgogne Franche-Comté and the EIPHI Graduate School (ANR-17-EURE-0002). This work has been partly supported by the French ROBOTEX network and its FEMTO-ST technological facility (ANR-10-EQPX-44-01).

reached positioning accuracy is 3.3 µm in position and 0.5 mrad in orientation over a 10 mm-long circular trajectory.

In addition to the already recognized advantages of PCR (miniaturisation capabilities, lightweight, etc.), the obtained results make also PCR worth to consider for high precision positioning tasks.

CONFLICT OF INTEREST STATEMENT

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.