Lionel Santinacci

Maïmouna W Diouf

Maïssa K S Barr

Bruno Fabre

Loïc Joanny

Francis Gouttefangeas

Gabriel Loget

Protected Light-Trapping Silicon Structured by a Simple Process for Sunlight-assisted Water Splitting

Keywords: Black Silicon, Electrochemical Etching, Atomic Layer Deposition, Nanostructuration, Photoanode, Water Splitting, Photoelectrochemical Cells

Macroporous layers are grown onto n-type silicon by successive photoelectrochemical etching in HF containing solution and chemical etching in KOH. This specific latter treatment gives highly antireflective properties of the Si surface. The duration of the chemical etching is optimized to render the surface as absorbent as possible and the morphology of the as-grown layer is characterized by scanning electron microscopy. Further functionalization of such structured Si surface is carried out by atomic layer deposition of a thin conformal and homogenous TiO 2 layer that is crystallized by an annealing at 450°C. This process allows using such surfaces as photoanodes for water oxidation. The 40 nm-thick TiO 2 film acts

indeed as an efficient protective layer against the photocorrosion of the porous Si in KOH, enhances its wettability and enlarge the light absorption of the photoelectrode. The macroporous Si has a beneficial effect on water oxidation in 1 M KOH and leads to a considerable negative shift of onset potential of ~400 mV as well as a 50 % increase in photocurrent at 1 V vs SCE.

INTRODUCTION

The conversion of sunlight into storable and transportable chemical fuels, that can be used for generating electricity, is a very appealing alternative to batteries and could be highly beneficial to circumvent the intermittency of renewable sources of energy. [START_REF] Lewis | Powering the Planet: Chemical Challenges in Solar Energy Utilization[END_REF] In this context, the development of photoelectrochemical cells (PECs) allowing the splitting of water for producing H 2 and O 2 at high rates is a very active field of research. [START_REF] Pinaud | Technical and Economic Feasibility of Centralized Facilities for Solar Hydrogen Production via Photocatalysis and Photoelectrochemistry[END_REF][START_REF] Prévot | Photoelectrochemical Tandem Cells for Solar Water Splitting[END_REF][START_REF] Sivula | Semiconducting Materials for Photoelectrochemical Energy Conversion[END_REF] PECs are generally based on semiconductor electrodes, immersed in an aqueous electrolyte, that can use the sunlight to drive electrochemical reactions such as water oxidation and reduction. [START_REF] Fujishima | Electrochemical Photolysis of Water at a Semiconductor Electrode[END_REF][START_REF] Walter | Solar Water Splitting Cells[END_REF][START_REF] Osterloh | Inorganic Nanostructures for Photoelectrochemical and Photocatalytic Water Splitting[END_REF][START_REF] Kang | Electrochemical Synthesis of Photoelectrodes and Catalysts for Use in Solar Water Splitting[END_REF] Among all the semiconductors that have been studied for PECs, silicon is currently attracting a tremendous scientific interest for application in water splitting for several reasons. [START_REF] Sun | Enabling Silicon for Solar-Fuel Production[END_REF] First, it is the second most abundant element in the earth's crust and therefore it is relatively cheap. Second, it exhibits a suitable electronic structure with well-positioned band edges and a narrow bad gap of 1.1 eV, which is particularly adapted to the solar spectrum. However, Si suffers from two main drawbacks which are: its instability in aqueous media and its high reflectance. This first issue is crucial when Si is used as a photoanode because in this case photocorrosion leads to the passivation of the Si with an insulating SiO x layer and therefore induces its deactivation. Nevertheless, this stability issue can be solved by appropriate protection of the surface. In particular, it has been shown that Si photoanodes can be highly stable in the harsh conditions required for splitting water when protected with thin layers grown by physical vapor deposition (PVD), [START_REF] Kenney | High-Performance Silicon Photoanodes Passivated with Ultrathin Nickel Films for Water Oxidation[END_REF] by atomic layer deposition (ALD) [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF][START_REF] Chen | Atomic Layer-Deposited Tunnel Oxide Stabilizes Silicon Photoanodes for Water Oxidation[END_REF][START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF][START_REF] Hu | Amorphous TiO2 Coatings Stabilize Si, GaAs, and GaP Photoanodes for Efficient Water Oxidation[END_REF][START_REF] Lin | Amorphous Si Thin Film Based Photocathodes with High Photovoltage for Efficient Hydrogen Production[END_REF][START_REF] Wang | Controllable Fabrication of Nanostructured Materials for Photoelectrochemical Water Splitting via Atomic Layer Deposition[END_REF][START_REF] Santinacci | ALD for Photoelectrochemical Water Splitting[END_REF][START_REF] Barr | Engineering a Three-Dimensional, Photoelectrochemically Active P-NiO/i-Sb2S3 Junction by Atomic Layer Deposition[END_REF] or electrodeposition. [START_REF] Hill | An Electrodeposited Inhomogeneous Metal-Insulator-Semiconductor Junction for Efficient Photoelectrochemical Water Oxidation[END_REF][START_REF] Ahn | Single-Nanoparticle Collision Events: Tunneling Electron Transfer on a Titanium Dioxide Passivated N-Silicon Electrode[END_REF] The main optical loss at the Si surface is caused by the high Fresnel reflection of solar light by planar Si, which avoids the absorption of a large part of incident photons that cannot generate electron-hole pairs. This problem has triggered a strong interest in developing light management strategies to increase photon absorption and therefore to improve the Si-based PEC efficiencies. [START_REF] Sun | Enabling Silicon for Solar-Fuel Production[END_REF][START_REF] Boettcher | Photoelectrochemical Hydrogen Evolution Using Si Microwire Arrays[END_REF][START_REF] Liu | Black Silicon: Fabrication Methods, Properties and Solar Energy Applications[END_REF] In this context, micro-and nanostructuration of Si surfaces are efficient approaches for effectively decreasing the reflectivity of crystalline Si and, at the same time, increasing the surface area while keeping the electronic properties of bulk Si. [START_REF] Sun | Enabling Silicon for Solar-Fuel Production[END_REF][START_REF] Boettcher | Photoelectrochemical Hydrogen Evolution Using Si Microwire Arrays[END_REF][START_REF] Liu | Black Silicon: Fabrication Methods, Properties and Solar Energy Applications[END_REF] The antireflective nature of structured Si surfaces arises either from a continuous gradient of refractive index or from multiple internal reflections, [START_REF] Liu | Black Silicon: Fabrication Methods, Properties and Solar Energy Applications[END_REF] when the structures are respectively smaller or larger than the incident wavelength. Several processes have been reported for fabricating structured antireflective or "black" Si. Vapor deposition [START_REF] Boettcher | Photoelectrochemical Hydrogen Evolution Using Si Microwire Arrays[END_REF] and patterned wet, [START_REF] Rey | Fully Tunable Silicon Nanowire Arrays Fabricated by Soft Nanoparticle Templating[END_REF][START_REF] Mavrokefalos | Efficient Light Trapping in Inverted Nanopyramid Thin Crystalline Silicon Membranes for Solar Cell Applications[END_REF] dry, [START_REF] Zhu | Optical Absorption Enhancement in Amorphous Silicon Nanowire and Nanocone Arrays[END_REF][START_REF] Wu | Near-Unity below-Band-Gap Absorption by Microstructured Silicon[END_REF] or electrochemical etching [START_REF] Langner | Molecular-and Nano-Tubes[END_REF] have been used to prepare Si surfaces having very promising optical and electrochemical properties. Nevertheless, these processes require the use of relatively costly vacuum or lithographic equipment. Few cost-effective techniques can be used to fabricate antireflective Si. For instance, alkaline wet etching is commonly employed to prepare random micropyramid arrays. [START_REF] Ding | Designing Efficient Solar-Driven Hydrogen Evolution Photocathodes Using Semitransparent MoQ X Cl Y (Q = S, Se) Catalysts on Si Micropyramids[END_REF] Alternatively, nano-and micro-scale structuration can be performed by simple processes such as metal-assisted chemical etching (MACE) or electrochemical etching on electronic or metallurgical grade Si. [START_REF] Oh | Nanoporous Black Silicon Photocathode for H2 Production by Photoelectrochemical Water Splitting[END_REF][START_REF] Oh | Enhanced Photoelectrochemical Hydrogen Production from Silicon Nanowire Array Photocathode[END_REF][START_REF] Huang | Metal-Assisted Chemical Etching of Silicon: A Review[END_REF][START_REF] Li | Upgraded Silicon Nanowires by Metal-Assisted Etching of Metallurgical Silicon: A New Route to Nanostructured Solar-Grade Silicon[END_REF] Although these structures exhibit low reflectance, if their dimension is too small, it can lead to a low built-in field and a weak charge separation. [START_REF] Santinacci | Electrochemical and Optical Characterizations of Anodic Porous N-InP(100) Layers[END_REF] In addition, those structures usually have high contact angles and narrow pore apertures, which may inhibit solvent penetration and optimal diffusion of ions within the pores. In this article, we report a rapid, inexpensive two-step method for structuring n-type Si (100) surfaces with micron-sized cavities. The process is based on the photoelectrochemical etching (PEE) of the Si surface and its subsequent alkaline etching. This method produces a layer of random macropores over a large area, which renders the Si surfaces antireflective over the visible spectrum. We demonstrate that such surfaces can be coated by ALD of TiO 2 thin conformal layers and that they can be used as stable photoanodes producing enhanced photocurrents under simulated sunlight with respect to their planar counterparts.

EXPERIMENTAL SECTION

Structuration and Characterization of Si Surfaces

The n-type silicon (100) wafers (phosphorus doped, 1-5 Ω cm, double side polished, Siltronix) were cut in 1.5 × 1.5 cm² squares and degreased by sonication (10 min) in acetone (MOS electronic grade, Carlo Erba), ethanol (RSE electronic grade, Carlo Erba) and ultrapure water (resistivity: 18.2 MW cm). The surfaces were then cleaned in 3/1 v/v concentrated H 2 SO 4 (MOS electronic grade, BASF)/30% H 2 O 2 (VLSI electronic grade, Sigma-Aldrich) at 100°C for 30 min, followed by copious rinsing with ultrapure water (Caution: The concentrated aqueous

H 2 SO 4 /H 2 O 2 (
piranha) solution is very dangerous, particularly in contact with organic materials, and should be handled extremely carefully). The surfaces were etched in 5/1 v/v ultrapure water/50% aq. HF (MOS electronic grade, Sigma-Aldrich) for 2 min in order to remove the oxide layer and were dried under an argon stream. For the photoelectrochemical etching, the surfaces were inserted into a home-made open-top O-ring cell (the diameter of the O-ring was 1 cm). The electrical contact was achieved by pressing an aluminum foil at the back of the Si surface and the counter electrode was a Pt ring (diameter of 1 cm) placed above the silicon surface. The cell was filled with 4 mL of electrolyte solution containing absolute ethanol (VWR) and 50% aq. HF at a 1/1 v/v ratio. The Si surface was irradiated with a solar simulator (LS0106, LOT Quantum Design) and a current of 40 mA was applied for 10 min with a power supply (LAB-SMP, ET System). The surface was then rinsed with absolute ethanol and dried under an argon flow. The electrochemically-etched samples were then directly exposed to a freshly prepared 1 M KOH (Sigma-Aldrich) aqueous solution containing 10 %vol. ethanol for 10 min.

The surfaces were finally rinsed with absolute ethanol and dried under an Ar stream. Scanning electron microscopy (SEM) was performed using a JSM 7100F (JEOL) equipment. SEM picture analysis was performed using the ImageJ software for estimating the morphological parameters of the structured porous layers. Contact angle measurements were performed with an Easydrop measuring instrument (Krüss). Reflectance spectra were acquired on a Cary 100 (Varian) spectrophotometer, equipped with an integrating sphere (DRA-CA-301, Labsphere) referenced with a spectralon standard (Labsphere). The total reflectance was measured with the surface tilted to include the specular reflectance component; in contrast, the diffuse reflectance was measured without at a 0° angle. The specular reflectance was obtained by subtracting the diffuse reflectance from the total reflectance.

Atomic Layer Deposition, Annealing and Characterization of the TiO 2 layers

The TiO 2 thin films were synthesized by atomic layer deposition in a Fiji 200 reactor from Ultratech/Cambridge NanoTech using titanium tetraisopropoxide (TTIP, 98%, Strem chemicals) and ultrapure water (resistivity: 18.2 MW cm). The TTIP canister was maintained at 80°C while the temperature of the reaction chamber was set to 120°C. The ALD cycle consisted of sequential pulse, exposure and purge of Ti precursor and H 2 O, alternatively. The pulse, exposure and purge durations were 2:7:15 s and 0.2:7:15 s for TTIP and H 2 O, respectively. The ALD of TiO 2 was performed on both porous and flat Si substrates in order to facilitate some characterizations. The number of ALD cycles (N) was varied in order to adjust the thickness of TiO 2 layers. After the ALD of TiO 2 , the samples were annealed for 2 hours at 450°C in air to transform the amorphous layer to polycrystalline anatase. In addition to SEM, the interface between the macroporous Si and the TiO 2 coatings was investigated by transmission electron microscopy (TEM) using a JEM 3010 (JEOL) TEM. The samples were cleaved into small pieces of 3×2 mm 2 , polished and thinned by ion milling. Chemical analysis of the porous layers after ALD processes was carried out by energy dispersive X-ray spectroscopy (EDS) using a Xman detector (Oxford Instruments) mounted on Supra 40VP SEM (Zeiss). The crystal structures of the TiO 2 layers were analyzed by X-ray diffraction (XRD) using a Rigaku RU-200 rotating anode X-ray generator (operating power 40 kV/30 mA) equipped with a Xenocs Fox3D Cu 12_INF mirror and a Mar345 image plate detector from Rayonix. The thickness of the TiO 2 thin layers on plane Si surfaces has been measured by spectroscopic ellipsometry using a M-2000V (J. A. Woollam Co.). The curve fitting was performed with the CompleteEase software applying a Tauc-Lorentz oscillator model. [START_REF] Aghaee | Low Temperature Temporal and Spatial Atomic Layer Deposition of TiO2 Films[END_REF]

Photoelectrochemical Measurements

An ohmic contact was established on the backside of the TiO 2 -modified Si surface with a metal wire by first scrubbing the surface with sand paper and then applying InGa Eutectic (99.99%, Alfa Aesar). A layer of silver paste (Electron Microscopy Sciences) was then deposited on the contact. After drying of the silver paste, the copper wire was inserted in a glass capillary and the electrode area (0.3 cm 2 ) was defined with an epoxy-based resin (Loctite 9492, Henkel), that covered all the back of the Si surface as well as the silver paste. The water oxidation experiments were performed in a home-made three neck cell in which were inserted a KCl saturated calomel reference electrode (SCE) and a Pt counter electrode. The cell was filled with a 1 M KOH solution that was deaerated by bubbling Ar for 20 min prior to experiments. The TiO 2 -modified Si surface, sealed in epoxy was disposed in front of a quartz window and was used as a working electrode. The light source was provided by a solar simulator (LS0106, LOT Quantum Design).

Electrochemical measurements were performed with a potentiostat/galvanostat Autolab PGSTAT 302N (Eco Chemie BV). The cyclic voltammograms were recorded at 20 mV/s. The two-step structuration process used in this work is described in Figure 1. First, a photoelectrochemical etching (PEE) was performed to pre-structure the Si surfaces. The n-Si surfaces were held at a constant anodic current (50.9 mA cm -2 ), in an ethanol/ HF electrolyte for 10 min using a front side illumination. As shown in Figure 2a, the PEE step generated a ~30 µm-thick layer of porous silicon (that was luminescent, as observed by naked eyes under λ = 365 nm irradiation) which consisted of a 13 µm-thick layer of a loosely-attached porous "crust" on the top of a dense layer of micro/mesoporous silicon (see Figure S1 in supporting information for a top view). Importantly, SEM observations reveal the presence of macropores at the bottom of the layer, that are typically filled with micro/mesopores. [START_REF] Lehmann | Formation Mechanism and Properties of Electrochemically Etched Trenches in N-Type Silicon[END_REF][START_REF] Albu-Yaron | Morphology of Porous N-Type Silicon Obtained by Photoelectrochemical Etching II[END_REF][START_REF] Lévy-Clément | Morphology of Porous N-Type Silicon Obtained by Photoelectrochemical Etching[END_REF] In the second step, the pre-structured surface was readily chemically etched in an aqueous solution of 1 M KOH comprising 10% vol. of ethanol (to improve the penetration of the solution into the hydrophobic porous layer). KOH solutions are commonly used to etch Si according to the following reaction:

RESULTS AND DISCUSSION

Fabrication and morphological characterization of the macroporous Si surfaces

38 Si + 2 OH -+ 2 H 2 O ® SiO 2 (OH) 2 2-+ 2 H 2 (1)
Due to the very high surface area of micro/mesoporous Si, this material etches at a much faster rate that bulk Si. Thanks to this, micro/mesoporous Si has been previously used as a sacrificial layer for MEMs fabrication [START_REF] Bell | Porous Silicon as a Sacrificial Material[END_REF] and for transferring nanostructures on various substrates. In order to follow the evolution of the porous layer with respect to the etching time, surfaces prepared by PEE and immersed in KOH for different times were observed by SEM. The effect of the KOH etching time is shown in the series of cross-section images in Figure 2. As it can be seen in Figure 2b, the "crust" and the top nanoporous layer readily dissolved during the first minutes of etching, leading to open-top macropores (see Figure S1b for top view). This is totally consistent with a preferential etching of nanoporous Si with respect to bulk Si, [START_REF] Sailor | Preparation of Micro-, Meso-, and Macro-Porous Silicon Layers[END_REF] and is confirmed by the observation of a higher bubble evolution rate during the first minutes of etching. This bubble evolution is caused by a high production of H 2 , as it can be seen in equation 1. After 4 to 5 min, the rate of bubble evolution clearly decreases suggesting that the whole micro/mesoporous layer was etched. However, after this period of time, H 2 evolution does not stop but proceed at a slower rate, suggesting that the etching of bulk silicon takes place.

Figure 2c shows that 10 min of etching produced larger cavities with wider apertures (the top view is shown in Figure S1c). For extended etching times, e. g. 1 h, the macroporosity disappeared and low aspect ratio microwells could be observed on the surface (Figures 2d andS1d). To summarize, the morphological evolution shown in Figure 2 can be easily understood by the preferential etching of the nanostructured Si on the porous layer that quickly dissolved and led to a random template of macropores (Figure 2b). The further etching of this template induced a widening of the cavities (Figure 2c) that leads in fine the removal of the pore walls (Figure 2d).

Interestingly, the surfaces exhibited a clear color evolution during the etching process, which can be observed in the insets of Figure 2a-d Morphological characterization of the so-fabricated samples was performed by SEM analysis.

Figure 3 shows some images of a typical structured Si surface. This figure reveals clearly the presence of a high density of macropores, randomly distributed on the silicon substrate.

Measurements performed on several surfaces allowed to determine a layer thickness of 4.9 ± 1.4 µm (Figure 3a). The SEM picture of Figure 3b shows the presence of single pores but also branched pores connected by the same aperture. This image allows us to estimate a pore density of ~2 × 10 5 mm -2 and a size of aperture in the micrometer range. Although the pore distribution and dimensions are rather regular over the majority of the etched surface, we noticed a decrease in the thickness and aperture size at the border of the etched disc that is attributed to the uneven distribution of the electric field at the edge of the surface during PEE. High-magnification SEM top view reveals some interesting features of the pores. In particular, Figure 3c shows that their bottom comprised (100), ( 101), ( 110) and (111) exposed facets, which is a result of the well-known anisotropic etching of Si by KOH. [START_REF] Bressers | Surface Morphology of P-Type (100)silicon Etched in Aqueous Alkaline Solution[END_REF] Deposition and Characterization of the TiO 2 -Protective Layers.

The structured Si surfaces were then coated with a conformal layer of TiO 2 using atomic layer deposition, which is known to efficiently protect silicon photoanodes in alkaline electrolytes. [START_REF] Hu | Amorphous TiO2 Coatings Stabilize Si, GaAs, and GaP Photoanodes for Efficient Water Oxidation[END_REF][START_REF] Lin | Amorphous Si Thin Film Based Photocathodes with High Photovoltage for Efficient Hydrogen Production[END_REF][START_REF] Chen | Atomic Layer-Deposited Tunnel Oxide Stabilizes Silicon Photoanodes for Water Oxidation[END_REF] The deposition proceeds according to the expected ALD process that was first proposed by Ritala et al. [START_REF] Ritala | Titanium Isopropoxide as a Precursor in Atomic Layer Epitaxy of Titanium Dioxide Thin Films[END_REF] The thickness evolution has been monitored by ex situ spectroscopic ellipsometry (supporting information, Figure S2). It reveals a linear increase from which the growth per cycle (GPC) of 0.28 Å/cycle is deduced. This is slightly higher than initial reports using TTIP [START_REF] Ritala | Titanium Isopropoxide as a Precursor in Atomic Layer Epitaxy of Titanium Dioxide Thin Films[END_REF][START_REF] Aarik | Titanium Isopropoxide as a Precursor for Atomic Layer Deposition: Characterization of Titanium Dioxide Growth Process[END_REF] but it is in line with recent works. [START_REF] Aghaee | Low Temperature Temporal and Spatial Atomic Layer Deposition of TiO2 Films[END_REF] It is then closer to the GPCs of 0.4 and 0.8 Å/cycle measured when the deposition is carried out with TiCl 4 or tetrakis(dimethylamino)titanium (TDMAT), respectively. [START_REF] Ritala | Growth of Titanium Dioxide Thin Films by Atomic Layer Epitaxy[END_REF][START_REF] Xie | Growth Kinetics and Crystallization Behavior of TiO2 Films Prepared by Plasma Enhanced Atomic Layer Deposition[END_REF] The silicon/TiO 2 interface was characterized by SEM as shown in Figure 4. The two pictures reveal that the TiO 2 film is highly conformal and rather homogeneous with a layer thickness ranging from 37 to 41 nm, both on the bottom (Figure 4a) or at the top of the pores (Figure 4b).

It validates the thickness evaluation performed by spectroscopic ellipsometry. The film looks relatively smooth and fully covers the porous structures. The SEM images also clearly show the faceting of the macroporous layer induced by KOH treatment that has been mentioned before. Chemical analysis of the samples has been performed by EDX. The spectrum (shown in supporting information, Figure S3) confirms the presence of TiO 2 on the surfaces, revealed by the respective Ti Ka, Ti Kβ and O Ka peaks at 4.5, 4.9 and 0.5 keV. Interestingly, the surfaces also contained detectable amounts of K and C. It is, therefore, supposed that the incorporation of the first element originated from the etching in KOH while the presence of carbon is attributed either to the incorporation of organics during the ALD process, which has already been reported using TTIP as a precursor [START_REF] Xie | Growth Kinetics and Crystallization Behavior of TiO2 Films Prepared by Plasma Enhanced Atomic Layer Deposition[END_REF] or to a plausible carbon contamination in the SEM. Additionally, EDX mapping for Si, Ti, and O on a cross-section is shown in Figure 5. It confirms the presence of conformal titania and reveals that the TiO 2 layer is located evenly in the porous layer. It is, indeed, observed that while Si could be detected on the whole surface, Ti and O are mostly present in the porous layer. As expected, the Si concentration is found constant for the "bulk" silicon and decreased in the porous area due to the lower density of this material. Conversely, almost no Ti and O are detected on the bulk part but increased considerably in the porous layer, confirming that TiO 2 was deposited on the whole porous layer. After the coating, the surfaces were annealed in air at 450°C for 2 h. XRD analyses performed before and after the thermal treatment are shown in Figure 6. The absence of diffraction peaks on the sample analyzed directly after ALD suggests that the as-grown films were amorphous. Note that the Si (100) peak originating from the substrate is not visible on the XRD diagram because it is out of the scanned angle range. Inversely, the characteristic diffraction peaks of anatase clearly appear after annealing. It indicates that the thermal treatment effectively crystallized the TiO 2 thin film. This was already observed on flat Si after annealing at slightly higher temperatures. [START_REF] Mcdowell | The Influence of Structure and Processing on the Behavior of TiO2 Protective Layers for Stabilization of N-Si/TiO2/Ni Photoanodes for Water Oxidation[END_REF] No peak related to the rutile phase is observed, which is desirable since anatase exhibits superior electronic properties.

Contact angles and optical properties

After the characterization of the morphology and the composition of the bare and the TiO 2 -protected porous silicon surfaces, their physicochemical properties such as their wettability 5 nm -1 b and reflectivity are now discussed. Those parameters are crucial for application in the field of photoelectrochemical water splitting. The wettability of the Si surfaces has been investigated by contact angle measurements, performed directly after each surface treatment, as shown in Figure 8. First, hydrogenated flat Si, exhibits a contact angle of 86°, which is fully consistent with data from the literature. [START_REF] Coffinier | Effect of Surface Roughness and Chemical Composition on the Wetting Properties of Silicon-Based Substrates[END_REF] After PEE, the contact angle increases considerably to 142°, revealing the high hydrophobicity of the micro/mesoporous layer. This value evidences that such small pores on Si surfaces inhibit a good permeability of the structure, which is a significant matter for the targeted application. After alkaline etching, the value decreases to 96°, showing that the larger pores considerably improved the wettability. Such a contact angle value can be explained by the change in surface roughness. Nevertheless, FTIR data of the surface after structuration (supporting information, Figure S4) reveals the presence of Si-H but also Si-O bonds, suggesting that the surfaces were partially oxidized after structuring, which probably have a non-negligible influence on the contact angle value. These data demonstrate that the contact angle of the macroporous samples is considerably lower than the ones obtained for nanostructured Si such as nanoporous Si (142°) or the ones typically reported for Si nanowire arrays (~130°), [START_REF] Piret | Culture of Mammalian Cells on Patterned Superhydrophilic/superhydrophobic Silicon Nanowire Arrays[END_REF] sometimes used as photoelectrodes in water. Finally, the contact angle measurements performed on the structured samples protected with the annealed TiO 2 layer show that the coverage of the hydrophobic Si-H moieties by the hydrophilic TiO 2 film considerably increases the wettability which results in a value of contact angle of 26°. As previously discussed and shown in the insets of Figure 2, the structured surfaces clearly exhibit color changes during their structuring and the resulting macroporous surfaces are matte black, which demonstrates their light trapping properties. In order to gain precise insights on their optical characteristics, reflectance measurements have been performed with flat, bare macroporous and anatase TiO 2 -coated macroporous Si surfaces. Whereas the flat Si surface shows a total reflectance comprised between 95% and 43%, reflectance for the bare structured surface is comprised between 34% and 18% (Figure 9). The reflectance of structured Si is therefore decreased by ~ 60%, which proves the significant effect of the Si structuring on the light absorption. Such a decrease in reflectance is in the same range than that previously reported for microstructured arrays comprising micropyramids [START_REF] Ding | Designing Efficient Solar-Driven Hydrogen Evolution Photocathodes Using Semitransparent MoQ X Cl Y (Q = S, Se) Catalysts on Si Micropyramids[END_REF] or inverted micropyramid arrays. [START_REF] Chen | Enhanced Photovoltaic Performance of Inverted Pyramid-Based Nanostructured Black-Silicon Solar Cells Passivated by an Atomic-Layer-Deposited Al 2 O 3 Layer[END_REF] This data clearly demonstrate the light trapping properties of the macroporous surfaces, induced by multiple internal reflections of the incident light. Measurements of the diffuse reflectance allow extracting the specular component, shown in supporting information (Figure S5). These data indicate that the main reflection contribution is due to scattering of incident light at the rough interface, which induced a large diffuse reflectance (~ 85% of the total reflectance signal). In contrast, the specular reflection is found very low (< 5%, Figure S5). Finally, the TiO 2 thin film has a beneficial effect on light absorption, and decreases the total reflectance by ~ 22% with respect to the bare microstructured surface (Figure 9). This effect is thought to be ascribed to The use of structured n-type Si surfaces protected by TiO 2 layers having thicknesses of few tenths of nanometers for water oxidation has been first reported by Yang et al. [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF] and more recently by Wang et al. [START_REF] Shi | Three-Dimensional High-Density Hierarchical Nanowire Architecture for High-Performance Photoelectrochemical Electrodes[END_REF] , Ao et al. [START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF] and by Lewis et al. [START_REF] Shaner | Si/TiO2 Tandem-Junction Microwire Arrays for Unassisted Solar-Driven Water Splitting[END_REF] , who all highlighted beneficial effects of using structured instead of planar Si surfaces. Therefore, the present structured surfaces have been tested as photoanodes in 1 M KOH solution. Under simulated sunlight, water splitting is expected to occur following the mechanism depicted in Figure 10. [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF][START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF] The UV irradiation absorbed by TiO 2 induces the formation of holes in the valence band that have sufficient energy to oxidize hydroxide ions. The unabsorbed part of the incident light generates electrons in the n-Si conduction band that can reduce water at the counter electrode and holes from the n-Si valence band recombine at the Si/TiO 2 interface with electrons from the TiO 2 conduction band.

In addition to the stability caused by the TiO 2 film, a beneficial effect of this interface is the rejection of holes towards the TiO 2 /water interface for driving water oxidation. The performances of the structured photoanodes, irradiated by simulated sunlight, are investigated and compared to a TiO 2 -coated flat Si surface using linear sweep voltammetry, as shown in Figure 11. In the dark, very low current densities, in the range of 1 µA cm -2 , are obtained for both electrodes. In contrast, photocurrents in the range of 10 to 50 µA cm -2 are measured under illumination, indicating that water oxidation effectively occurs at both surfaces.

The two curves have a very similar shape to the one previously reported [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF][START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF][START_REF] Shi | Three-Dimensional High-Density Hierarchical Nanowire Architecture for High-Performance Photoelectrochemical Electrodes[END_REF] but exhibit lower photocurrents. This can be attributed to the resistivity of our Si substrates. Shi et al. [START_REF] Shi | Three-Dimensional High-Density Hierarchical Nanowire Architecture for High-Performance Photoelectrochemical Electrodes[END_REF] measured, indeed, photocurrents in the range of 1 mA cm -2 while the used Si substrates had a resistivity 10 3

times lower (0.001 -0.005 W cm). However, when the doping concentration was in the same range as that for the present substrates, the measured currents were found to be in the same range. [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF][START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF] Finally, McDowell et al. [START_REF] Mcdowell | The Influence of Structure and Processing on the Behavior of TiO2 Protective Layers for Stabilization of N-Si/TiO2/Ni Photoanodes for Water Oxidation[END_REF] reported a much higher maximum current density but for a photoanode coated with an additional Ni layer that acted as a co-catalyst. Additionally, such differences may be a consequence of both differences in spectra of incident light and features of the deposited TiO 2 layers, for instance, most previous reports used thicker TiO 2 films. 55,56 The curves show clearly that the macroporous silicon had a beneficial effect on water oxidation and led to a 50% increase in photocurrent at 1 V vs SCE. This rise is attributed to the improved absorption of photons in the porous matrix. In addition to this effect, the macroporous surface provided a higher active area that contributes to the photocurrent enhancement and also to the considerable negative shift of onset potential of ~400 mV. [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF][START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF] Although the surface structuring is generally expected to enhance the electron-hole recombination rate, the curves of Figure 11a clearly indicate that this effect is smaller than the gain obtained by the improved light absorption and the increased surface area. Another important benefit of the TiO 2 coating is the stability that it provides to the Si matrix because silicon is usually highly prone to etch in this medium (vide supra). For instance, the etching of uncoated Si surfaces could be readily observed in this electrolyte by the appearance of H 2 bubbles (see Eq. 1), which is not the case for the coated Si surfaces. The controlled-potential electrolysis performed at 1 V vs SCE, shown in Figure 11b, demonstrate the high stability of the microstructured coated Si photoanode that could be operated for a duration of more than 1 hour with no sign of degradation and a constant photocurrent in 1 M KOH. To sum up, these electrochemical results clearly demonstrate that the use of the antireflective macroporous surfaces as photoanodes leads to improvement in terms of photocurrent and overpotential with respect to planar surfaces. In addition, the TiO 2 conformal layer provides an efficient protection against etching and photocorrosion.

CONCLUSION

We have presented a simple and fast two-step wet process, requiring less than 30 min to microstructure n-type Si. This method consists in pre-structuring the flat Si by photoelectrochemical etching and then etching the surface in an alkaline solution, which generates arrays of random macropores. It is shown these structures give light-trapping properties to the silicon surface. Using ALD, the antireflective surfaces are coated with nanometer-thick conformal layers of polycrystalline anatase TiO 2 layers that further improved the wettability and the light absorption of the porous surfaces. These TiO 2 -protected Si microstructured surfaces were highly stable in strongly alkaline solutions and could be used as photoanode for a prolonged time under simulated sunlight. Such photoanode surfaces showed 50% photocurrent enhancements and ~400 mV negative shift of onset potential without any co-catalysts, demonstrating their high potential for solar energy conversion applications. We

Table of Contents Graphic

FIGURE 1 .

 1 FIGURE 1. Scheme of the two-step process used for the fabrication of the antireflective

40

 40

FIGURE 2 .

 2 FIGURE 2. SEM pictures showing cross-sections of the porous Si surface submitted to different

  photoluminescence for the KOH-etched samples (as observed by naked eyes under a λ = 365 nm

FIGURE 3 .Figure 1 (

 31 FIGURE 3. SEM pictures showing a) the cross-section, b) a low magnification top view and c) a

FIGURE 4 .

 4 FIGURE 4. Cross-section SEM pictures showing a) the bottom and b) the top parts of the

FIGURE 5 .FIGURE 6 .

 56 FIGURE 5. a) SEM cross section of a porous Si layer after ALD of TiO 2 (N = 1400 cycles) and

49

 49

  TEM imaging, shown in Figure7a, clearly shows the sharp interface between the substrate and the deposit. The presence of a ~3 nm thick amorphous SiO 2 layer between the TiO 2 and the Si can be easily noticed, which was likely formed during the first hydrolysis cycles of ALD.Crystalline rows are visible on the TEM image. It confirms the polycrystalline nature of TiO 2 layer when it has been annealed. The selected-area electron diffraction (SAED) pattern presented in Figure7bdisplays the typical spots corresponding to the single-crystalline anatase phase. It indicates that though the TiO 2 layer is polycrystalline, the crystallite size is in the range of the film thickness (i. e. 20 to 40 nm). These findings are in line with the previous observations reported by MacDowell et al.[START_REF] Mcdowell | The Influence of Structure and Processing on the Behavior of TiO2 Protective Layers for Stabilization of N-Si/TiO2/Ni Photoanodes for Water Oxidation[END_REF] for TiO 2 films grown onto plane Si from TTIP.

FIGURE 7 .

 7 FIGURE 7. a) TEM cross section view of the TiO 2 /Si interface after annealing in air at 450°C

FIGURE 8 .

 8 FIGURE 8. Evolution of the contact angle as a function of the structuration of the surface. Insets

FIGURE 9 .FIGURE 10 .

 910 FIGURE 9. Total reflectance spectra obtained for planar Si (black), bare macroporous Si (blue)

FIGURE 11 .

 11 FIGURE 11. a) Linear sweep voltammograms in the dark (dashed lines) and under simulated

  [START_REF] Hwang | High Density N-Si/n-TiO2 Core/Shell Nanowire Arrays with Enhanced[END_REF][START_REF] Ao | Black Silicon with Controllable Macropore Array for Enhanced Photoelectrochemical Performance[END_REF]

-220 -112 --112 -112 -112 --004 -

ACKNOWLEDGMENT Dr. F. Tessier and Dr. F. Cheviré (University of Rennes 1) are acknowledged for their help with the optical measurements. D. Paris (University of Rennes 1) is acknowledged for fabricating the etching cell and Dr. Ahmad Fahs (University of Toulon) is acknowledged for EDS measurements. M. Barr and M. Diouf are indebted to the Conseil Régional Provence-Alpes-Côte d'Azur, the CNRS and Encapsulix SAS for the PhD grants.

think that this report opens a way for further research on this type of Si microstructures and we are confident that higher performances will soon be obtained by optimizing the Si microstructures, the TiO 2 layer and decorating these arrays with affordable co-catalysts. ASSOCIATED CONTENT Supporting Information. Additional SEM pictures of the macroporous Si, Evolution of the thickness of the TiO 2 layers with the number of ALD cycles, EDX spectrum of the TiO 2 -coated porous Si, FTIR spectra of the macroporous Si surface, and total, diffuse and the specular reflectance of the surface at various stages of the surface structuring. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Authors

*gabriel.loget@univ-rennes1.fr; lionel.santinacci@univ-amu.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


Swaminath Venkateswaran
email: swaminath.venkateswaran@ls2n.fr

Damien Chablat
email: damien.chablat@cnrs.fr

Pol Hamon
email: pol.hamon@ls2n.fr

An optimal design of a flexible piping inspection robot *

This article presents an optimization approach for the design of a piping inspection robot. A rigid bio-inspired piping inspection robot that moves like a caterpillar was designed and developed at LS2N, France. By the addition of tensegrity mechanisms between the motor modules, the mobile robot becomes flexible to pass through the bends. However, the existing motor units prove to be oversized for passing through pipe bends at 90 • . Thus, three cascading optimization problems are presented in this article to determine the sizing of robot assembly that can overcome such pipe bends. The first problem deals with

INTRODUCTION

Manual intervention of pipelines in Nuclear or Chemical industries is cumbersome and risky as it not only leads to the loss of human life but also causes long term radiation effects. Piping inspection robots play an important role in such situations as they can perform a given task with better accuracy and also ensures safety to human health. Generally, in-pipe inspection robots can be classified [START_REF] Choi | In-pipe robot with active steering capability for moving inside of pipelines[END_REF] viz: Pig type [START_REF] Okamoto | Autonomous system for oil pipelines inspection[END_REF], Wheel type [START_REF] Okada | A three-wheeled self-adjusting vehicle in a pipe, ferret-1[END_REF], Caterpillar type [START_REF] Roman | Pipe crawling inspection robots: an overview[END_REF], Wall-press type [START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF],

Walking type [START_REF] Neubauer | A spider-like robot that climbs vertically in ducts or pipes[END_REF], Inchworm type [START_REF] Fukuda | Rubber gas actuator driven by hydrogen storage alloy for in-pipe inspection mobile robot with flexible structure[END_REF] and Screw type [START_REF] Horodinca | A simple architecture for in-pipe inspection robots[END_REF]. A bio-inspired piping inspection robot was designed and developed at LS2N, France [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF]. This robot accomplishes the locomotion of a caterpillar in six-steps to move inside a pipeline. Using a slot-follower leg mechanism, the robot attains static postures to establish contact with pipeline walls. The leg mechanisms are capable of working between 40-94 mm diameter straight pipelines [START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF]. However, the robot is a rigid model and is thus limited to straight pipelines. Also, the spindle drive used in the prototype allows the robot to move at a very low speed of 0.43 mm/s [START_REF] Chablat | Dynamic model of a bio-inspired robot for piping inspection[END_REF]. By the addition of tensegrity mechanisms between the motor modules, the robot was made flexible [START_REF] Venkateswaran | A new inspection robot for pipelines with bends and junctions[END_REF]. The dimensions of the existing motor units are oversized and might restrict the flexible robot to overcome pipe bends or junctions.

In this article, an optimization approach is followed to determine the optimum sizing of the entire robot assembly inside a pipeline having 90 • pipe bend where the tensegrity mechanism operates in a passive mode. Optimization algorithms can be classified into deterministic and heuristic approaches [START_REF] Guessasma | Comparing heuristic and deterministic approaches to optimise mechanical parameters of biopolymer composite materials[END_REF]. The deterministic optimization approach solves problems in a structured manner that leads to efficient solutions with respect to design variables and constraints. On the other hand, heuristic/evolutionary algorithms operate on randomness [START_REF] Bennis | Nature-Inspired Methods for Metaheuristics Optimization[END_REF]. Based on the complexity of the de-sign problem, the heuristic approach requires lesser computational times over the deterministic approach [START_REF] Rao | A new optimization algorithm for solving complex constrained design optimization problems[END_REF]. However, there are possibilities where both the algorithms can lead to solutions trapped in a local minimum. While applying such algorithms to real-world engineering problems, the design engineer can select a local minima through careful definition of constraints or bounds for the optimization problem. In the field of robotics, optimization techniques are generally employed for solving problems such as path planning. An interesting example is the research work of Zhang et al. [START_REF] Zhang | Robot path planning in uncertain environment using multi-objective particle swarm optimization[END_REF], where a heuristic approach using the Particle Swarm Optimization (PSO) was employed to determine a collision-free path for the robot. Also, optimization techniques find their application in the design of parallel robots for determining a robust design with maximum operating workspace [START_REF] Caro | Multiobjective design optimization of 3-prr planar parallel manipulators[END_REF] as well as in determining the optimum position for placement of the robot [START_REF] Caro | Workpiece placement optimization for machining operations with a kuka kr270-2 robot[END_REF]. This article focuses on the optimum design of the flexible piping inspection robot proposed in [START_REF] Venkateswaran | A new inspection robot for pipelines with bends and junctions[END_REF] through three different problems. The robot assembly is comprised of multiple sub-systems such as tensegrity mechanisms, motor units and leg mechanisms. Each sub-system is subject to separate set of constraints and objectives under a given working condition. By the incorporation of a multi-objective optimization problem for the entire assembly, the complexity of the system increases and the selection of the solution candidate from the Pareto front is more subjective [START_REF] Caro | Multiobjective design optimization of 3-prr planar parallel manipulators[END_REF].

So, a cascading approach is being implemented through three different sub-problems to identify an optimal design of the robot assembly. The first optimization problem deals with the determination of optimal design parameters of the 4-SPS-U tensegrity mechanism [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF] by a metaheuristic approach. Using Genetic Algorithm in MATLAB, the total length of the mechanism and the free length of the spring are determined by ensuring the static stability of the tensegrity mechanism.

Followed by that the second optimization problem is carried out using the f mincon function in MATLAB. This problem helps in determining the sizes of motor unit of each module without the presence of leg mechanisms. From the results of the second optimization problem, the motors and gear units are identified from catalogue products of Maxon based on dimensions, velocity and force factors. Also, the second optimization problem permits to set the diameter range of pipelines for the robot. For studying the area occupied by the leg mechanisms, the third optimization problem is then carried out which helps in determining the geometry of the slot-follower leg mechanism.

The area occupied by the leg mechanism is identified by accommodating the mechanism on the motor sizing determined from the results of the second optimization problem. By assembling the tensegrity mechanism, motor modules and the leg mechanisms, the digital model of the optimized robot design is realized using CATIA software.

The outline of the article is as follows. Initially, the architecture of the bio-inspired robot and the setup of the optimization test bench are presented. Followed by that, the first optimization problem, which is associated with the design of tensegrity mechanism is defined and solved. The subsequent sections deals with the deterministic optimization problems for identifying the optimal size of motor units and the slot-follower leg mechanism. The article then ends with conclusions.

DEFINITION OF THE CASE STUDY

This section focuses on the presentation of the architecture of the robot assembly with the presence of tensegrity mechanisms. The dimensions and construction of the optimization test bench, which will be employed for the second and third optimization problems, are then presented.

Architecture of the robot

The rigid bio-inspired robot developed at LS2N, France comprises three modules that ensure clamping and elongation phases for accomplishing the locomotion of a caterpillar. Each module consists of a DC brushless motor coupled with a spindle drive to convert rotary motion to prismatic movement [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF][START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF]. The front and rear modules have 3 sets of slot-follower leg mechanisms each to have contact with the inner walls of pipelines at all instances of locomotion. The central module takes care of the elongation and retraction phases 1 . By the study of design issues viz: Passive compliance, Active compliance and Tilt limits [START_REF] Venkateswaran | A new inspection robot for pipelines with bends and junctions[END_REF][START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF], a tensegrity mechanism was proposed and introduced as an articulation unit for the bio-inspired robot. Through singularity and workspace analysis, two types of tensegrity mechanisms viz: 3-SPS-U and 4-SPS-U architectures were analyzed in [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF]. These architectures are differentiated by the number of tensions springs employed where the former employs three springs and the latter employs four springs. Based on the results of [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF], the 3-SPS-U architecture is compatible with addressing the issue of active compliance while the 4-SPS-U architecture addresses both compliance issues. As the 4-SPS-U tensegrity mechanism provides higher tilt limits under passive modes [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF], this architecture is coupled along with the bio-inspired robot to make it flexible. The digital model of the existing bio-inspired robot and the 4-SPS-U tensegrity mechanisms are assembled using the CATIA software to have a flexible robot assembly and it is represented in Fig. 1. In Fig. 1 mechanisms that can overcome such bends, an optimization approach is being followed in this article.

Setting up the optimization test bench

The optimization problems are solved inside a test pipeline which is constructed using MATLAB software. The first optimization problem is solved without the test bench. However, the results coming from this problem are taken as fixed design variables for the two successive problems which are solved within the test bench. The second problem is solved for diameters of the test bench ranging from 70 mm to 160 mm. The test bench for the optimization problem consists of a horizontal pipe section, a 90 • elbow and a vertical pipe section. The optimal sizing of the robot modules is determined especially at the bend section wherein collision of modules against the walls of the pipelines is verified. The test bench for the optimization problem is represented below in Fig. 2. The test pipe represented in Fig. 2a is generally manufactured by Numerical Control (NC) bending processes using a roller with its origin at o and flexible mandrels [START_REF] Li | The interactive effects of wrinkling and other defects in thin-walled tube nc bending process[END_REF]. The coordinates of o are given by [1.5d, 1.5d], where d is the diameter of the pipeline. With respect to o, the trajectories of the test bench are classified as the inner portion, centerline radius (CLR) and the outer portion.

0
The bending radius of these trajectories represented in Fig. 2b is given by [r 1 , r 2 , r 3 ] = [d, 1.5d, 2d].

The optimization problems are solved for a fixed diameter value at any instant. The values considered for d are [70,80,90,100,110,120,130,140,150,160] mm. The pipeline geometry is constructed in MATLAB by sub-dividing into straight and bent cross-sections. The coordinate system for the test bench with respect to o for Fig. 2a is given in Table 1. A discretization is carried Table 1. Coordinates system of pipe geometry for Fig. 2a Coordinate Inner (t=1) Center (t=2)

Outer (t=3)

a t r 1 [0,0.5] r 2 [0,0] r 3 [0,-0.25] b t r 1 [1.5,0.5] r 2 [1,0] r 3 [0.75,-0.25] c t r 1 [2.5,1.5] r 2 [2,1] r 3 [1.75,0.75] d t r 1 [2.5,3] r 2 [2,2]
r 3 [1.75,1.5] out between each coordinate wherein the continuous trajectory is being subdivided into discrete counterparts. This discretization is being applied to the walls of the pipelines as well as the CLR and they are represented by blue dotted lines in Fig. 2b. The inner and outer portion discretization is employed to check for collision against modules during the locomotion of the robot. The discretization of the CLR is mainly used to move the robot as well as to rotate each module. The discretization equations for the straight sections are given by:

m = a t + disc(b t -a t ) (1)
m = c t + disc(d t -c t ) (2)
where disc = 0 : 0.01 : 1 and t = 1, 2, 3

In Eqn. (1) and Eqn. (2), m indicates the coordinates of discretized points between the start and end points of the straight sections. The bend section of pipe with the discretized points between (b t -c t ) is constructed using the equation:

m j = r t sin jπ 40 , -r t cos jπ 40
, where j = 0 to 20 and t = 1, 2, 3

FIRST OPTIMIZATION PROBLEM: DESIGN OF THE TENSEGRITY MECHANISM

During locomotion inside the pipeline, the inspection robot can encounter either an elbow at 90 • or a junction. In the event of a junction, the tensegrity mechanism is actuated through cables to tilt the entire assembly along a certain direction. In other words, the mechanism will be controlled from external motor units to follow a given direction. However, in the case of an elbow at 90 • , the mechanism operates passively without the influence of external forces to overcome the bend. This article mainly focuses on the determination of optimal design parameters of the robot assembly where the tensegrity mechanism operates in a passive mode. From the results of [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF], an inverse pendulum configuration was chosen for the tensegrity mechanism which is by nature an unstable configuration. The first optimization problem aims to determine the dimensions of the 4-SPS-U tensegrity mechanism by ensuring static stability.

Architecture of the tensegrity mechanism and geometric equations

The tensegrity mechanism comprises of a passive universal joint and four tensions springs.

The equations for the mechanism can be generated by correlating to a parallel manipulator of type 4-SPS-U [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF] where S indicates spherical joint, P indicates actuated prismatic joint and U indicates universal joint. The representation of the tensegrity mechanism and its correlation to a parallel manipulator at the home-pose is shown in Fig. 3. The tilt angles of the mechanism about the universal joint are given by η and φ. At the home-pose shown in Fig. 3a, the tilt angles are zero. The coordinate frame of the fixed base has its origin at B 0 . The mounting points for the springs are given by B i (i = 1, 2, 3, 4) and they form an imaginary square whose diagonal length is given by 2r f . The vector coordinates for the base mounting points are given by:

B 1 B 2 B 3 B 4 C 1 C 2 C 3 C 4 Σ 0 Σ 1 B 1 , B 2 B 3 , B 4 C 3 , C 4 C 1 , C 2 A -hr f hr f B 1 B 2 B 3 B 4 C 3 C 1 C 2 C 4 L 1 L 2 L 3 L 4 η Ø r f /√2 r f /√2 (a) (b) (c) F 1 , F 2 F 3 , F 4
b i = r f cos iπ 2 , r f sin iπ 2 , -r f h T , with i = 1, 2, 3, 4 (4)
In Eqn. (4), h is a constant that can vary between 0.1 to 1. The coordinate frame of the end-effector has its origin at C 0 . The spring mounting points on the end-effector are given by C i (i = 1, 2, 3, 4).

The vector coordinates of the end-effector mounting points can be calculated using the XY Euler rotation angles about the point A of the universal joint. The coordinates are given by:

R = R x (η)R y (φ) =         cos(φ) 0 sin(φ) sin(η) sin(φ) cos(η) -sin(η) cos(φ) -cos(η) sin(φ) sin(η) cos(η) cos(φ)         (5)
c i = R r f cos iπ 2 , r f sin iπ 2 , r f h T , with i = 1, 2, 3, 4 (6)
In Eqn. ( 5), R ∈ SE(3) represents the spatial transformation matrix obtained from the Euler angles of universal joint and it is employed to determine the end-effector coordinates. The Inverse Kinematic Problem (IKP) which provides the distance between the base and end-effector of the mechanism is calculated by the equation:

L i = (b ix -c ix ) 2 + (b iy -c iy ) 2 + (b iz -c iz ) 2 , with i = 1, 2, 3, 4 (7)

Objective functions and constraint equations

Singularity and workspace analysis of the 4-SPS-U tensegrity mechanism was carried out in [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF] for h= 1 and the tilt limits were found to be ±5π/18 radians. However, the static stability of the mechanism under this configuration was not studied. According to Lagrange, for a moving system, the equation of motion [START_REF] Meirovitch | Fundamentals of vibrations[END_REF] is given by:

τ = d dt ∂T ∂ q - ∂T ∂q + ∂U ∂q where q = [η, φ] T (8)
In Eqn. ( 8), T and U are the kinetic and potential energies of the system. τ represents the generalized torques on the system. Under static modes, the velocity of the system is zero and there exists no kinetic energies. The potential energy of the tensegrity mechanism is contributed mainly by the springs and by the cables that pass through these springs. The total potential energy of the mechanism is given by the equation:

U = U cable + U spring = 4 i=1 F i L i + 4 i=1 k 2 (L i -L 0 ) 2 (9)
In Eq. ( 9), F i is the magnitude of applied force along the cable passing through spring i. The parameter k represents the spring stiffness and L 0 indicates the free length of each spring. Based on the dimensions of the flange used in the existing prototype, the value of r f is considered as 11 mm [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF] for the spring mounting positions. The spring stiffness k is considered as 0.75 N/mm which is close to the standard sizes available at LS2N. As the optimization problems will be solved for passive modes through a 90 • pipe bend, the applied forces (F i ) are taken as zero as there will be no cable actuation. Thus, the total potential energy of the system can be written as:

U = 363 h 2 cos (φ) -181.5 cos (η) -181.5 cos (φ) + 1.5 L 0 2 + 363( h 2 + 1) (10)
From Eqn. [START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF], it is observed that the total potential energy of the system has dependencies on the tilt angles, the parameter h and the free length of the spring L 0 . As the tensegrity mechanism has dependencies on both tilt angles, the following steps are carried out for the stability analysis:

• Generation of the Hessian matrix or the mechanism stiffness

• Estimation of the determinant value of the Hessian matrix

• Estimation of the value of the second-order derivative of total potential energy with respect to one of the tilt angles

The Hessian matrix and its determinant are given by:

H =    ∂ 2 U ∂η 2 ∂ 2 U ∂η∂φ ∂ 2 U ∂η∂φ ∂ 2 U ∂φ 2    =    f 11 f 12 f 21 f 22    (11) det(H) = f 11 f 22 -f 2 12 ( 12
)
As the total potential energy is a function of the two pose variables, the following conditions are possible [START_REF] Dawkins | Paul's online math notes[END_REF]:

1. The total potential energy has a relative maximum when det(H) > 0 and f 11 (resp. f 22 ) < 0 2. The total potential energy has a relative minimum when det(H) > 0 and f 11 (resp. f 22 ) > 0 Thus, to have a stable configuration the second condition must be satisfied. The optimal design parameters of the tensegrity mechanism are identified at the home-pose where the tilt angles are zero. The objective function of the first optimization will aim to maximize the secondorder derivative of the potential energy with respect to one of the tilt angles and it is given by:

f un 1 = f 11 = 77 L 0 h 2 -22 h 2 + 11 (13)
This function is subject to two inequality constraints, which ensure that the determinant value of the Hessian matrix and the value of the objective function remains a positive value during each iteration. The constraint equations are given by:

det(H) = 16.5 L 0 h -363 h 3 + 181.5h 2 (14)
g 1 : det(H) ≥ 0 , g 2 : f un 1 ≥ 0 (15)

Problem statement and results of optimization

Using the objective function and the constraint equations, the first optimization problem can be stated as:

Maximize: f un 1 (x)
subject to theconstraints:

g 1 , g 2 where x = [h, L 0 ] T
The optimization problem can be solved using a gradient-descent approach. However, this approach leads to only one solution based on the bounds set for the problem. In order to have multiple solutions distributed over a domain, a metaheuristic approach [START_REF] Yang | Metaheuristic optimization[END_REF] using Genetic Algorithm is carried out in MATLAB with the help of Eqn. ( 13 is close to zero. In other words, it can be said that for these values, a flat plane is obtained for the total potential energy against the tilt angles and it is represented by the flat surface in Fig. 4b. The mechanism is in a stable configuration if h is less than 0.83, and it is in an unstable configuration if h is greater than 0.83 for L 0 = 6 mm. Thus, in order to have a stable configuration, a value of h = 0.8 is considered for L 0 = 6 mm. Under these design parameters, the total potential energy of the mechanism is plotted against the tilt angles at the home-pose and it is represented in Fig. 4b. It could be observed that under the home-pose, a stable configuration of the mechanism is obtained.

Using these results of optimization, the singularity and workspace analysis of the tensegrity mechanism is carried out using the Cylindrical Algebraic Decomposition (CAD) in Maple as investigated in [START_REF] Venkateswaran | Singularity and workspace analysis of 3-spsu and 4-sps-u tensegrity mechanisms[END_REF]. Under passive modes, tilt limits of ±π/4 radians were obtained for the mechanism within the singularity free workspace. As this mechanism will be coupled along with the piping inspection robot to pass through pipe bends, at any instance of locomotion, the orientation of mechanism about the z-axis is unknown. Thus, for further analysis of the robot assembly inside the test bench, the extreme tilt posture of the tensegrity mechanism will be considered. A configuration similar to Fig. 3c is considered where the mounting distance between two springs is given by 2r f . In this posture, when one of the tilt angles reaches a maximum of ±π/4 radians with the other tilt angle at 0 radians, one of the four prismatic springs reaches a limit of 7.8 mm, while the opposing spring reaches 24.6 mm. The remaining two springs have the same values of 16.3 mm. This orientation of the mechanism will be considered for further analysis to determine the optimal sizing of the robot modules. The 2D representation of the tensegrity mechanism at this condition is shown in Fig. 5.

SECOND OPTIMIZATION PROBLEM: DETERMINATION OF MOTOR SIZING

The second optimization problem aims to determine the sizing of motor modules of the inspection robot inside the test bench without the presence of leg mechanisms. The optimization problem is tested for pipeline diameters ranging from 70 to 160 mm. From the results of optimization, a suitable motor-spindle combination unit is identified from the catalogue of Maxon. Apart from the motor sizing, the existing bio-inspired robot has issues associated with velocities and cable management. Based on results obtained from the second optimization problem, the Maxon motors are identified in such a way that the following factors are also addressed:

r f r f 1.6 r f B 1 B 3 B 2 , B 4 C 1 C 2 , C 4 C 3 A A B 1 C 1 B 3 C 3 (a) (b
1. Velocity: An alternate spindle drive unit with a lower gear reduction ratio compared to the existing prototype (455:1) to accomplish faster displacements during locomotion 2. Hall sensors: A DC-motor coupled with Hall sensor units which will be useful for the control phase of the robot while working inside an unknown environment [START_REF] Venkateswaran | Prototyping a piping inspection robot using a beaglebone black board[END_REF] Once the optimal motor unit is identified, the third problem is carried out for the results of the second optimization problem to determine the size of the slot-follower leg mechanism.

Modeling of robot and design variables

The robot is modeled as multi-body planar blocks with a tensegrity mechanism between each module in MATLAB. Each module comprises a Maxon brushless DC-motor coupled with a spindle drive. The digital model of a module is shown in Fig. 6a. The robot geometry considered for the optimization problem is represented in Fig. 6b and each module is constructed using the geometrical equations of a rectangle. In Fig. 6b, the design variables for optimization are considered in the following way: Extract the coordinates of CLR point i closer to frame 1 .

• [l k1 ,

7:

Calculate rotation angle δ 1 from the coordinates of point extracted in Step-6

8:

Apply rotation angle δ 1 to tensegrity mechanism coupled between Modules-1 and 2.

9:

Extract the coordinates of CLR point i closer to frame 2 .

10:

Calculate rotation angle δ 2 from the coordinates of point extracted in Step-9

11:

Apply rotation angle δ 2 to tensegrity mechanism coupled between Modules-2 and 3. 12: end for

Objective function

A mono-objective optimization problem subject to constraints is solved using MATLAB. The objective function of the problem aims at maximizing the area of the motor units. Since the dimensions of the tensegrity mechanism are fixed, their areas are not taken into account. As the central module takes care of elongation and retraction phases, it is assumed for the computation that this module remains in a fully extended phase at all instances of the simulation. The area of the robot at a given position of CLR is given by:

f un 2 = 3 k=1 (l k1 w k1 + l k2 w k2 ) , where k indicates the module (16)
The area of motor units is estimated as per Eqn. [START_REF] Zhang | Robot path planning in uncertain environment using multi-objective particle swarm optimization[END_REF]. The global sum of f un 2 from Eqn. ( 16) is calculated throughout the discretized points of the CLR and this will be maximized.

Constraint equations

For each position of the robot on the CLR, the collision of modules is checked against the pipeline walls. Inequality constraints are defined in the optimization problem, which will ensure that the modules avoid collision during movement inside the test bench, especially at the bends.

A complete discretization of each module is performed. The discretization equation is similar to Eqn. (1) and Eqn. [START_REF] Okamoto | Autonomous system for oil pipelines inspection[END_REF]. The discretized robot modules inside the test bench are represented below in Fig. 7a. For each position of reference frame 0 on the discretized CLR trajectory, the coordinates of points closer and farther on each module from o are extracted. These points can be extracted using the min and max function of MATLAB. The representation of the coordinates closer (s k ) and farther (q k ) from o on a module is shown in Fig. 7b. Followed by that, the coordinate v k on the inner portion closer to s k and the coordinate u k on the outer portion closer to q k are extracted using the min function. With the extracted set of coordinates, the inequality constraints are defined which ensures that there exist no collisions between the modules and the pipeline walls. The constraint equations for the position of the robot assembly on a discretized CLR point are thus defined by:

o q k u k s k v k (a) (b)
g k : o -q k ≤ o -u k (17)
g k+1 : o -s k ≥ o -v k , where k = 1, 2, 3 (18)
Throughout the locomotion sequence, the collision of each module against pipeline walls is checked using Eqn. [START_REF] Caro | Multiobjective design optimization of 3-prr planar parallel manipulators[END_REF] and Eqn. [START_REF] Caro | Workpiece placement optimization for machining operations with a kuka kr270-2 robot[END_REF]. The pseudo-code for the constraint function and its working in MAT-LAB is provided in Algorithm 2. Extract the coordinate (s ik ) on module k closer to o using min 8:

Extract the coordinate (q ik ) on module k farther to o using max 9:

for l = 1 -n do For each discretized point on inner & outer portions 10:

Calculate u l -q ik 11:

Calculate v l -s ik 12:

end for 13:

Extract the coordinate from u ik on outer portion closer to q ik 14:

Extract the coordinate from v ik on inner portion closer to s ik The design variables for each module which were considered as [l k1 , w k1 , l k2 , w k2 ], resulting in 12 variables are reduced to 4 variables which are given by [l 1 , w 1 , l 2 , w 2 ]. This is because an identical Maxon motor-spindle drive unit will be employed in the three modules. The objective function and the constraint equations are solved using the f mincon function in MATLAB. The design variables are subject to lower and upper bounds to have a closer interpretation of the catalogue parts of Maxon. The existing prototype uses a brushless DC-motor of diameter 16 mm coupled with a spindle drive GP 16 S. The overall length of the motor-spindle unit is 58 mm with a screw length of 102 mm. The advantage of using brushless motors is that they offer Hall sensors that can be useful in the control phase of the robot to determine pipeline diameters inside an unknown/closed environment [START_REF] Venkateswaran | Prototyping a piping inspection robot using a beaglebone black board[END_REF]. The lower (lb) and upper bounds (ub) for the design variables of the optimization problem are set as follows:

1. Parameter l 1 : This parameter corresponds to the length of the motor-spindle unit employed in each module. The lower bound is set at 40 mm and the upper bound is set at 60 mm. This is an essential parameter as it not only plays a vital role in avoiding collisions but also accommodates the slot-follower leg mechanism. The lower bound is set at 20 mm while the upper bound is set at 102 mm which is the maximum length offered by Maxon spindle drives. 4. Parameter w 2 : This corresponds to the diameter of the screw. The lower bound is set at 2 mm and the upper bound is set at 5 mm inline with the catalogue of Maxon [START_REF] Motors | Program 2017/18. High precision Drives and Systems[END_REF].

Thus, the lower and upper bounds for the second optimization problem are given by lb = [40, 10, 20, 2] and ub = [60, [START_REF] Zhang | Robot path planning in uncertain environment using multi-objective particle swarm optimization[END_REF]102,[START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF]. Spindle drives of Maxon offer lead screws and ball screws. The existing prototype uses a ball screw type as it has a higher load carrying capacities and this could be useful when the leg mechanisms are coupled along with the drive. The optimization algorithm is executed with a constant value of diameter throughout the section depicted in Fig. 2a. The algorithm will be evaluated one after the other for pipe diameters starting from 70 mm to 160 mm.

For faster convergence of the problem and also to satisfy bounds at all iterations, the Sequential Quadratic Programming (SQP) algorithm [START_REF]Chosing the algorithm-MATLAB & Simulink-Mathworks France[END_REF] is employed within f mincon. The tolerance value of the guess variables and the objective function is set at 10 -9 for the optimization problem.

Results and discussions

As the definition of objective function and constraint equations appear simpler, a faster convergence is obtained. The dimensions of motor units obtained from the optimization algorithm for the various pipe diameters are provided below in Table 2. From the results of the optimization provided in Table 2, a gradual increase in all design parameters could be observed with the increase in pipe diameter. During the simulation of the robot in MATLAB, the minimum and maximum lengths of the tensegrity mechanism were found to be 11.5 mm and 17.6 mm. These limits are well within the estimated numerical values of the tensegrity mechanism at ±π/4 radians. Thus, a stacked model is not necessary for performing the simulations with a modified design of the robot assembly. Based on the results obtained in Table 2, a list of possible motors that match with the dimensions are identified from the catalogue of Maxon and it is provided in Table 3. The results obtained for 70-80 mm diameter pipe ranges provide the least possible motor sizing. The nearest catalogue part for this series is the RE 8 DC-Motor (463223) coupled with an 8 mm diameter spindle drive (473645) with a maximum screw length of 56 mm. This combination is not compatible with the other diametrical ranges as they prove to be undersized. Also, this DC-Motor does not have Hall sensors and they require an additional encoder for having a position control algorithm. Apart from that the efficiencies and feed force factors for this series are significantly low [START_REF] Motors | Program 2017/18. High precision Drives and Systems[END_REF] and this might be an issue when the leg mechanisms are accommodated. Thus, the results of 70-80 mm diameter pipelines are not considered for further analysis. For the 90-120 mm diameter range pipelines, two possible motor-spindle combinations are identified. The first solution is the combination of 320178 RE 16 DC-Motor coupled with a 424731 16 mm diameter spindle drive. Another combination uses the 283828 EC-Motor coupled along with the 424731 spindle series. The first solution offers a DC-Motor with a higher velocity however, the disadvantage of employing the 320178 series motor is that it requires an additional encoder if a position control algorithm is incorporated. It has to be noted that the screw length obtained for the 90 mm diameter pipe is 32 mm. This length might not be sufficient enough to accommodate the slot-follower leg mechanism, however, this can only be verified in the third optimization problem. For the 100-120 mm diameter pipeline, the 283828 EC-Motor coupled with 424744 spindle drive is identified. A screw length of 45 mm is chosen for this combination. This combination, however, appears to be undersized for the 130-160 mm diameter pipes. Another combination that uses the 283828 EC-motor along with the 424745 spindle drive with a screw length of 70 mm is identified, which is optimal for the 120-150 mm diameter pipes. From the list provided in Table 3, the last three motor-spindle combinations can be potentially accommodated for 90-160 mm diameter pipes provided the screw length is modified. For the 160 mm diameter pipe, all the combinations prove to be undersized. However, a motor-spindle combination with 90 mm screw length can be compatible for 150-160 mm diameter pipes. In order to have an optimal result that can be suitable for a smaller diameter pipeline ranges, the results of 100-120 mm range is considered. The screw length is thus fixed at 45 mm. The last three solutions from Table 3 are ideal for this diameter range. The feed force, gear ratio and efficiencies of these motor-spindle combinations are provided below in Table 4. The EC-motor series 283828 is chosen for all these combinations. The 424731 spindle drive offers the highest velocity. However, the feed force offered by this series is low. This might pose a problem during static phases, especially when the leg mechanisms are coupled with the screw drive. The 424744 and 424745 series, on the other hand, offers better feed forces. The 424745 series offers a feed force similar to the 424749 series which is used in the existing prototype.

The 424744 series, on the other hand, offers a lesser feed force but this factor will be sufficient to accommodate the leg mechanisms based on experiments conducted on the existing prototype [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF]. Also, the velocity offered by 424744 series is higher than 424745. This will address the issue of low velocity that is associated with the existing prototype [START_REF] Chablat | Dynamic model of a bio-inspired robot for piping inspection[END_REF]. Taking into account the length, efficiency and velocity factors, the 283282 EC-motor coupled with 424744 spindle drive within a working diametre range of 100-120 mm is considered for further analysis. For ensuring a simpler procedure for the control sequence of the prototype, the same motor-spindle combination is chosen for all three modules.

THIRD OPTIMIZATION PROBLEM: DESIGN OF THE SLOT-FOLLOWER LEG MECHANISM

With the optimal motor sizing determined from the second optimization problem, the next step is to determine the geometry of leg mechanisms that could be accommodated within the screw length. The ratio of screw length to the pipe diameter has a greater influence on the geometry of the leg mechanisms. The geometry of the slot-follower leg mechanism and its assembly on the existing robot is represented in Fig. 8 for the leg mechanism were identified in [START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF] by a multi-objective optimization approach. In this article, a simple mono-objective optimization problem is solved wherein the dimensions of the leg mechanisms are identified through a geometrical approach. The maximization of the area occupied by the space of leg mechanism is carried out for determining the leg length that can work inside 100-120 mm diameter pipelines.

Modeling of the leg mechanisms and design variables

The front and rear module assembly comprise three legs assembled at 120 • with respect to each other [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF]. These legs ensure tight contact during static and dynamic phases of the robot.

For a given locomotion instance, the orientation of the robot about the z-axis is unknown. As the optimization problem is solved within a planar test bench, two-dimensional projections of the leg mechanisms at two extremities are possible and these projections are represented in Fig. 9. In the with the walls of the pipeline. In this scenario, the distance between the central axis of the robot and the contact point with the pipeline wall is given by r cos(θ), where θ is the angle between the leg and y -z plane. This representation is shown in Fig. 9c and Fig. 9d. For the third optimization problem, a geometry similar to Fig. 9c is considered such that collision will be verified on either side against the pipeline walls. For the computation, it is considered that r cos(θ) is equal to r.

The geometry of the leg mechanisms that will be assembled on the robot is represented below in Fig. 10. For the third optimization problem, the central module dimensions are carried over

Objective function

The objective function of the third optimization problem aims to maximize the space occupied by the leg mechanisms. The geometry can be split into triangular and rectangular areas which is represented in Fig. 10. The area of the leg mechanisms used in the robot at a given discretized CLR point can be calculated by:

f un 3 = 2 k=1 e k1 -e k3 e k3 -e k2 2 + f k1 -f k3 f k3 -f k2 2 + e k1 -e k3 e k1 -f k1 (20)
where k = 1, 2 indicates Module-1 and Module-3

As only two modules are being considered, the index k goes from 1 to 2, which corresponds to Modules 1 and 3. Similar to Eqn. ( 16), the global sum of f un 3 for Eqn. ( 20) is estimated and it will be maximized.

Constraint equations

Similar to the second optimization problem, the collision of the geometry of leg mechanisms is verified against the pipeline walls at each position of the CLR. A complete discretization of the leg mechanism geometry is performed using an equation similar to Eqn. (1) and Eqn. [START_REF] Okamoto | Autonomous system for oil pipelines inspection[END_REF].

The representation of the discretized leg mechanism assembled along with the optimized robot modules is shown below in Fig. 12a.

l s2 ρ 1 = ∆y 1 -o 1 ∆x 1 (25)
The dimensions of the motor units from the results of second optimization problem for the 100 mm diameter pipe from Table 3 is given by [l 1 , w 1 , l 2 , w 2 ] = [51. [START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF][START_REF] Zhang | Robot path planning in uncertain environment using multi-objective particle swarm optimization[END_REF]45,[START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF] mm. The stroke lengths ρ 1 and ρ 2 are set as 32 mm and 7 mm with reference to the 45 mm screw length of the motor unit. The reduction in stroke length is caused by the flanges and fasteners that will be used in the robot assembly. The two equality constraints for the third optimization problem are thus given by:

h k1 : ρ 1 = 32 (26)
The pseudo-code of the constraint function for the third optimization problem in MATLAB is provided in Algorithm 3.

Problem statement

For the third optimization problem, the diameter d of the pipeline is fixed as 100 mm. The third optimization problem which is used to estimate the size of the leg mechanism is defined by: maximize Calculate l s1 (d) using Eqn. ( 23) and l s1 (c) using Eqn. (

n i=1 f un 3 (z) subject to constraints: g 1 , g 2 , g 3 , g 4 , h 1 , h 2 where z = [∆x 1 , ∆y 1 , l s2 ] T , i = 1..n
Check equality constraints: h k1 as per Eqn. ( 26) and h k2 as per Eqn. (

end for 20: end for convergence issues for a screw length of 45 mm. The bounds for the design parameter l s2 are set in line with the dimension of the existing prototype [START_REF] Chablat | Dynamic model of a bio-inspired robot for piping inspection[END_REF][START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF]. Using the f mincon function with the SQP algorithm, the third optimization problem is solved to determine the geometry of the leg mechanism.

Results and discussions

The dimensions of the slot-follower leg mechanism obtained after optimization are provided below in Table 5. Compared to the existing prototype, the parameters l s1 and l s2 are increased from 57 mm and 7 mm to 62.7 mm and 9.7 mm respectively. In the event of a straight pipeline with no bends, the slot-follower leg mechanism is capable of adjusting its size between 58.6 mm (2∆y 1 ) to 124 mm (2∆y 2 ) diameter range. When the robot is moving inside a pipeline with bends, the modules are capable of passing through such bends, provided the diameter of pipelines is strictly between 100 to 120 mm. The video link for the simulation of the robot inside 100-120 mm diameter pipelines in MATLAB is provided at the bottom of this page 3 . By assembling the tensegrity mechanism, the EC-motor with spindle drive unit identified from the catalogue of Maxon, the optimized slot-follower leg mechanism and standard fasteners such as circlips, bolts, etc., the entire robot is realized in CAD software. The positions of the robot inside a 100 mm diameter during its locomotion are rendered in CATIA and the sequences are represented in Fig. 13a to Fig. 13e.

The flexible robot resembles an "Elephant trunk" during a passive mode and it is demonstrated in Fig. 13f.

CONCLUSIONS

This article mainly focused on the design of a flexible piping inspection robot by an optimization approach. Three optimization problems were defined and solved to arrive at the design of a flexible robot assembly which can work inside pipelines having straight and bent profiles. The first optimization problem dealt with the determination of design parameters of the tensegrity mechanism by ensuring static stability. As the mechanism was proposed to be tested inside a pipeline having 90 • pipe bend, the design parameter h and the free length of the spring L 0 were identified under passive mode by a metaheuristic approach in MATLAB. Followed by that, the second optimization problem was defined and solved by a deterministic approach to identify the sizing of motor modules without the presence of leg mechanisms. This problem was carried out inside a test bench with various pipe diameters. From the results of the second optimization problem, a series of motor-spindle combinations were identified from the catalogue of Maxon motors. Based on the results obtained, an intermediate-range that holds good for 100-120 mm diameter pipe was chosen. The selection of the motor-spindle unit was done based on dimensions, velocity, feed force and efficiency factors, which were not considered in the design of the initial prototype [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF][START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF][START_REF] Chablat | Dynamic model of a bio-inspired robot for piping inspection[END_REF].

The third optimization problem was then carried out to determine the sizing of the leg mechanisms that could be accommodated for the results of the second optimization problem. The dimensions of the slot-follower leg mechanism were found to be increased when compared to the existing prototype. However, with the modified architecture, the robot is capable of working inside 100-120 mm diameter pipeline range that can have straight sections and bends at 90 • . The presence of leg mechanisms and tensegrity mechanism permits to have a flexible "Elephant trunk" model that allows the robot to pass through pipe bends in the event of passive compliance. By assembling the tensegrity mechanisms, motor modules and the leg mechanisms, the 3D model of the flexible robot was realized in CATIA software.

In future works, the optimization algorithm can be modified to accommodate a relation between screw length and pipeline diameter thereby eliminating the need for fixing the pipeline diameters for each computation. This modification might probably lead to the definition of another objective and a metaheuristic approach will be implemented to solve this problem. With the 3D model being assembled using catalog parts, the static force model developed in [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF] can be extended to the flexible robot. This model can help in determining the forces on the tensegrity mechanisms and the clamping forces which can be useful for performing experiments. A simulation of the robot will be performed using Adams software to validate the locomotion procedure followed by which prototyping of the flexible robot will be realized. For the real-time application, suitable control units and a camera will be accommodated on the robot to perform an inspection of pipelines. Based on simulation trails, an investigation will also be done to replace the central module with a spindle drive having higher velocity for accomplishing faster displacements.

Fig. 1 .

 1 Fig. 1. 3D model of the existing bio-inspired robot coupled with the tensegrity mechanisms

Fig. 2 .

 2 Fig. 2. Overview of the (a) test bench for optimization and (b) parametrization of the test bench for d = 100 mm

Fig. 3 .

 3 Fig. 3. Representation of the (a) Tensegrity mechanism at home-pose, (b) 3D view of the correlation to a 4-SPS-U manipulator and (c) 2D view of the manipulator

Fig. 4 .

 4 Fig. 4. (a) Solutions for optimization problem obtained by Genetic Algorithm and (b) Plot of total potential energy versus the tilt angles η and φ for [h, L 0 ] = [0.83 , 6 mm] (flat surface) and [0.8 , 6 mm] (inverted bell-curve) at the home-pose of the tensegrity mechanism

) y z Fig. 5 .

 z5 Fig. 5. Representation of the (a) posture considered for the 4-SPS-U mechanism in optimization problem and (b) associated design parameters of the mechanism

Fig. 6 .Algorithm 1 3 : 4 : 5 :

 61345 Fig. 6. Representation of the (a) Maxon brushless DC-motor and its spindle drive and (b) robot assembly with various design parameters for the second optimization problem

Fig. 7 .

 7 Fig. 7. Representation of the (a) discretized robot assembly and (b) extraction of coordinates from the discretized model for defining constraints

Algorithm 2

 2 Definition of constraint equations for the second optimization problem Inputs: Discretized CLR coordinates, Robot design parameters Outputs: Inequality constraint equations (g 1 ,g 2 ,g 3 ,g 4 ,g 5 ,g 6 ) 1: Initialization 2: for i = 1 -n do n indicates the end point of CLR

15 :

 15 Check inequality constraints (gk ): oq ik ≤ ou ik 16: Check inequality constraints (g k+1 ): os ik ≥ ov ikWith the objective function and constraint equations being defined, the second optimization problem can be stated as: maximize:n i=1 f un 2 (y)subject to the constraints: g 1 , g 2 , g 3 , g 4 , g 5 , g 6 where y = [l 1 , w 1 , l 2 , w 2 ] T , i = 1..n indicates the discretized CLR points

2 . 3 .

 23 Parameter w 1 : This parameter corresponds to the diameter of the motor-spindle unit. Since Maxon offers DC-Motors with Hall sensors starting from 16 mm, the lower bound is set at 10 mm while the upper bound is fixed at 16 mm. Values higher than 16 mm series are not considered as this may affect the length parameters of the motor-spindle drive or the screw length. Parameter l 2 : This corresponds to the length of the screw coming from the spindle drive.

2 .Fig. 8 .

 28 Fig.8. Representation of the (a) geometry of slot-follower leg mechanism and (b) assembly of the mechanism on the existing prototype[START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF][START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF]

Fig. 9 .

 9 Fig. 9. Representation of the front and side views of the two possible orientations of the leg mechanism at the extremities caused by the rotation of the robot about z-axis

2 Fig. 10 .

 210 Fig. 10. Representation of the geometry of the slot-follower leg mechanism on the motor modules

Fig. 11 .

 11 Fig. 11. Representation of the design parameters of the leg mechanism during (a) declamped and (b) clamped phases

Fig. 12 .

 12 Fig. 12. Representation of the (a) discretized leg mechanism on the motor modules and (b) extraction of coordinates from a discretized leg mechanism for defining the constraints

h

  k2 : l s1 (d) = l s1 (c) , where k = 1, 2 indicates Module-1 and Module-3

Fig. 13 .

 13 Fig. 13. Locomotion sequence of the optimized bio-inspired robot inside a 100 mm diameter pipe from (a) to (e) and correlation of the robot to an "Elephant trunk" in (f)

Table 2 .

 2 Results obtained for the second optimization problem in MATLAB

	Pipe diameter d	l 1	w 1	l 2	w 2
	(mm)	(mm) (mm) (mm) (mm)
	70	41.4	12.7	20	2
	80	47.44	14	24.7	2
	90	50.95	16	32	5
	100	52.72	16	45.17	5
	110	55.02	16	58.25	5
	120	58.21	16	67.33	5
	130	60	16	84.4	5
	140	60	16	86.54	5
	150	60	16	98.8	5
	160	60	16	102	5

Table 3 .

 3 Identification of motor-spindle series from catalogue of Maxon[START_REF] Motors | Program 2017/18. High precision Drives and Systems[END_REF] based on results of optimization where OV indicates over-sizing, OP indicates optimal sizing and UN indicates under-sizing

	Type	Series	l 1	w 1	l 2	w 2	D70 D80 D90 D100 D110 D120 D130 D140 D150 D160
			(mm) (mm) (mm) (mm)
	Motor 463223	31.9	8	20	3	OP OP UN	UN	UN	UN	UN	UN	UN	UN
	Spindle 473645
	Motor 320178	50.4	16	32	5	OV	OV	OP	OP	OP	UN	UN	UN	UN	UN
	Spindle 424731
	Motor 283828	46.4	16	32	5	OV	OV	OP	OP	OP	UN	UN	UN	UN	UN
	Spindle 424731
	Motor 283828	51.5	16	45	5	OV	OV	OV	OP	OP	OP	UN	UN	UN	UN
	Spindle 424744
	Motor 283828	62.4	16	70	5	OV	OV	OV	OV	OV	OP	OP	OP	OP	UN
	Spindle 424745

Table 4 .

 4 Technical specifications of motors from Table3for the results of 100 to 160 mm diameter pipes

			Feed force Efficiency Length (Motor+Spindle)
	Type	Series Gear ratio
			(N)	(%)	(mm)
	Motor 283828
		5.4:1	189	87	46.4
	Spindle 424731
	Motor 283828
		29:1	331	79	51.5
	Spindle 424744
	Motor 283828
		104:1	403	71	55.1
	Spindle 424745

  indicates the discretized CLR positionsThe lower and upper bounds of the design parameters for the simulation are given by lb = Definition of constraint equations for the third optimization problem Inputs: Discretized CLR coordinates, Leg mechanism design parameters, Motor dimensions for 100 mm diameter pipe Outputs: Inequality constraint equations (g 1 ,g 2 ,g 3 ,g 4 ), Equality constraint equations (h 1 ,h 2 )

	1: Initialization
	2: for i = 1 -n do	n indicates the end point of CLR
	3:	for k = 1 -2 do	k indicates the module number
	4:	for j = 0 : 0.01 : 1 do	Discretization
	5:	Discretize each face of leg mechanism on each module
	6:	end for
	7:	Extract the coordinate (m ik ) on module k closer to o using min
	8:	Extract the coordinate (n ik ) on module k farther to o using max
	9:	for l = 1 -n do	For each discretized point on inner & outer portions
	10:	Calculate r l -n ik
	11:	Calculate p l -m ik
	12:	end for
	13:

[40,

[START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF][START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF]

, ub = [65,

40,[START_REF] Venkateswaran | A new inspection robot for pipelines with bends and junctions[END_REF]

. The upper bounds for the design parameters ∆x and ∆y are set by considering a factor, which is 60% of the pipeline diameter. A value higher than 60% leads to Algorithm 3 Extract discretized coordinate from r ik on outer portion closer to n ik 14: Extract discretized coordinate from p ik on inner portion closer to m ik 15: Check inequality constraints (g k ): on ik ≤ or ik 16: Check inequality constraints (g k+1 ): om ik ≥ op ik 17:

Table 5 .

 5 Results obtained for the third optimization problem in MATLAB

	Parameters Dimensions (mm)
	∆x 1	60
	∆y 1	29.3
	∆y 2	62
	l s1	62.7
	l s2	9.7

Working principle of the existing rigid prototype : Click here.

Digital model simulation of the leg mechanism on the existing rigid prototype in CATIA : Click here

Simulation of the robot assembly in MATLAB : Click here.

For a given CLR position, the coordinates of the points closer and farther on each leg mechanism from the point o are extracted. This is done using the min and max functions of MATLAB.

The representation of the coordinates closer (m k ) and farther (n k ) on the rear leg mechanism from central coordinate o is shown in Fig. 12b. The coordinates of the point p k on the inner portion which is closer to m k and the coordinates of the point r k on the outer portion which is closer to n k are extracted using the min function. With these extracted points, four inequality constraints are defined in MATLAB which are given by:

In addition to the inequality constraints, two equality constraints are defined for the third optimization problem. As the simulation is performed for fully declamped phase, the length of leg mechanism l s1 can be calculated from Fig. 10 by the Pythagoras theorem and it is given by:

where k = 1, 2 indicates Module-1 and Module-3 [START_REF] Dawkins | Paul's online math notes[END_REF] In Eqn. [START_REF] Dawkins | Paul's online math notes[END_REF], l s1 (d) indicates the leg length during the declamped phase. During the clamped phase, the length l s1 can be calculated using ∆x 2 , ∆y 2 and o 1 by the Pythagoras theorem from Fig. 11b. The equation is given by:

In Eqn.