Mohamad Hijazi

Mathilde Rieu
email: rieu@emse.fr

Valérie Stambouli

Guy Tournier

Jean-Paul Viricelle

Christophe Pijolat

Ambient temperature selective ammonia gas sensor

Keywords: SnO2, Functionalization, APTES, Ammonia gas, Room temperature detection

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Breath analysis is considered as noninvasive and safe method for the detection of diseases [START_REF] Broza | Combined Volatolomics for Monitoring of Human Body Chemistry[END_REF].

Gas sensors have shown to be promising devices for selective gas detection related to disease diagnosis [START_REF] Adiguzel | Breath sensors for lung cancer diagnosis[END_REF][START_REF] Righettoni | Breath analysis by nanostructured metal oxides as chemo-resistive gas sensors[END_REF]. These sensors can be used to detect the gases emanated from the human body.

For example, ammonia is a disease marker for liver problems. Indeed, ammonia in humans is converted to urea in the liver and then passes in the urines through the kidney, while unconverted ammonia is excreted in breath of 10 ppb for healthy subjects [START_REF] Capone | Solid state gas sensors: state of the art and future activities[END_REF]. The ammonia concentration increases in case of malfunctioning of liver and kidney reaching more than 1 ppm in presence of renal failure [START_REF] Dubois | Breath Ammonia Testing for Diagnosis of Hepatic Encephalopathy[END_REF][START_REF] Güntner | Selective sensing of NH3 by Si-doped α-MoO3 for breath analysis[END_REF].

SnO2 sensors have been well investigated from a very long time [START_REF] Yamazoe | Effects of additives on semiconductor gas sensors[END_REF][START_REF] Lalauze | A new approach to selective detection of gas by an SnO2 solid-state sensor[END_REF][START_REF] Watson | The tin oxide gas sensor and its applications[END_REF], since they can detect many gases with high sensitivity and low synthesis cost [START_REF] Barsan | Metal oxide-based gas sensor research: How to?[END_REF][START_REF] Korotcenkov | Gas response control through structural and chemical modification of metal oxide films: state of the art and approaches[END_REF]. The interactions of SnO2 material with gases have been extensively studied [START_REF] Gong | Interaction between thin-film tin oxide gas sensor and five organic vapors[END_REF][START_REF] Wang | Metal Oxide Gas Sensors: Sensitivity and Influencing Factors[END_REF]. The chemical reactions of target gases with SnO2 particle surface can generate variations in their electrical resistances. SnO2 is n-type semiconductor, in this case, the adsorbed oxygen on particle surface takes electrons from the conduction band at elevated temperature, generating depletion layer between the conduction band and the surface (space-charge region). Reducing gases such as CO are oxidized on the surface, then they consume the adsorbed surface oxygen giving back the electrons to the conduction band. This decrease in the depletion layer decreases the resistance of whole film [START_REF] Barsan | Conduction mechanism switch for SnO2 based sensors during operation in application relevant conditions; implications for modeling of sensing[END_REF]. However, like other metal oxides, SnO2 sensors have lack of selectivity and operate at high temperature (350-500 °C), except if some particular activation with light for example, is carried out [START_REF] Anothainart | Light enhanced NO2 gas sensing with tin oxide at room temperature: conductance and work function measurements[END_REF][START_REF] Comini | UV light activation of tin oxide thin films for NO2 sensing at low temperatures[END_REF]. Many techniques were applied to enhance the selectivity such as (i) the addition of gas filter [START_REF] Tournier | Selective filter for SnO2-based gas sensor: application to hydrogen trace detection[END_REF], or small amount of noble metals [START_REF] Tian | A low temperature gas sensor based on Pd-functionalized mesoporous SnO2 fibers for detecting trace formaldehyde[END_REF][START_REF] Cabot | Analysis of the noble metal catalytic additives introduced by impregnation of as obtained SnO2 sol-gel nanocrystals for gas sensors[END_REF][START_REF] Trung | Effective decoration of Pd nanoparticles on the surface of SnO2 nanowires for enhancement of CO gas-sensing performance[END_REF], (ii) the use of oxides mixture [START_REF] Zeng | Enhanced gas sensing properties by SnO2 nanosphere functionalized TiO2 nanobelts[END_REF][START_REF] Van Hieu | Enhanced performance of SnO2 nanowires ethanol sensor by functionalizing with La2O3[END_REF], or hybrid film of SnO2 and organic polymers [START_REF] Geng | Characterization and gas sensitivity study of polyaniline/SnO2 hybrid material prepared by hydrothermal route[END_REF][START_REF] Bai | Gas sensors based on conducting polymers[END_REF]. Since several years, there is a high request to develop analytical tools which are able to work at temperature lower than 200 °C in order to incorporate them in plastic devices and to reduce the power consumption [START_REF] Camara | Tubular gas preconcentrators based on inkjet printed micro-hotplates on foil[END_REF][START_REF] Rieu | Fully inkjet printed SnO2 gas sensor on plastic substrate[END_REF]. For such devices, researchers are now focused on the development of room temperature gas sensors. The first reported metal oxide gas sensor was based on palladium nanowires for the detection H2 [START_REF] Atashbar | Room temperature gas sensor based on metallic nanowires[END_REF]. Concerning SnO2 gas sensors, the more recent studies at room temperature are related to NO2 detection [START_REF] Anothainart | Light enhanced NO2 gas sensing with tin oxide at room temperature: conductance and work function measurements[END_REF][START_REF] Comini | UV light activation of tin oxide thin films for NO2 sensing at low temperatures[END_REF] or to formaldehyde sensors [START_REF] Tian | A low temperature gas sensor based on Pd-functionalized mesoporous SnO2 fibers for detecting trace formaldehyde[END_REF].

As exposed previously, for breath analysis, there is also a high demand for ammonia sensors working at room temperature. Among the current studies, it can be mentioned the sensors using tungsten disulfide (WS2) [START_REF] Li | WS2 nanoflakes based selective ammonia sensors at room temperature[END_REF] or carbon nanotubes (CNTs) [START_REF] Van Hieu | Highly sensitive thin film NH3 gas sensor operating at room temperature based on SnO2/MWCNTs composite[END_REF] as sensing material, and especially the ones based on reduced grapheme oxide (RGO) [START_REF] Sun | Facile preparation of polypyrrole-reduced graphene oxide hybrid for enhancing NH3 sensing at room temperature[END_REF][START_REF] Tran | Reduced graphene oxide as an over-coating layer on silver nanostructures for detecting NH3 gas at room temperature[END_REF][START_REF] Su | NH3 gas sensor based on Pd/SnO2/RGO ternary composite operated at room-temperature[END_REF], or on polyaniline (PANI) [START_REF] Kumar | Flexible room temperature ammonia sensor based on polyaniline[END_REF][START_REF] Bai | Polyaniline@SnO2 heterojunction loading on flexible PET thin film for detection of NH3 at room temperature[END_REF][START_REF] Abdulla | Highly sensitive, room temperature gas sensor based on polyaniline-multiwalled carbon nanotubes (PANI/MWCNTs) nanocomposite for trace-level ammonia detection[END_REF][START_REF] Khuspe | SnO2 nanoparticles-modified polyaniline films as highly selective, sensitive, reproducible and stable ammonia sensors[END_REF].

Molecular modification of metal oxide by organic film is another way to enhance the selectivity and to decrease the sensing temperature (room temperature gas sensors) [START_REF] Matsubara | Organically hybridized SnO2 gas sensors[END_REF][START_REF] Wang | Effect of Functional Groups on the Sensing Properties of Silicon Nanowires toward Volatile Compounds[END_REF][START_REF] Wang | Artificial Sensing Intelligence with Silicon Nanowires for Ultraselective Detection in the Gas Phase[END_REF]. The need of selective sensors with high sensitivity in presence of humidity at low gases concentration pushes the research to modify SnO2 sensing element in order to change its interaction with gases. The modifications with organic functional groups having different polarities could change the sensor response to specific gases (e.g. ammonia) depending on their polarity [START_REF] Wang | Effect of Functional Groups on the Sensing Properties of Silicon Nanowires toward Volatile Compounds[END_REF]. In the literature, a functionalization based on APTES (3-aminopropyltriethoxysilane) combined with hexanoyl chloride or methyl adipoyl chloride was investigated on silicon oxide field effect transistors [START_REF] Wang | Artificial Sensing Intelligence with Silicon Nanowires for Ultraselective Detection in the Gas Phase[END_REF].

These devices were tested under some aliphatic alcohol and alkanes molecules. The functionalized field effect transistors have shown responses to a wide variety of volatile organic compounds like alcohols, alkanes etc. The response of such sensors to gases is derived from the change in electrostatic field of the molecular layer which can generate charge carriers in the silicon field effect transistor interface. Interactions with volatile organic compounds on the molecular layer can take place in two types: the first type is the adsorption on the surface of the molecular layer and the second type is the diffusion between the molecular layers [START_REF] Wang | Effect of Functional Groups on the Sensing Properties of Silicon Nanowires toward Volatile Compounds[END_REF][START_REF] Wang | Artificial Sensing Intelligence with Silicon Nanowires for Ultraselective Detection in the Gas Phase[END_REF].

However, these field effect transistor sensors is still lacking of selectivity.

The aim of the functionalization performed in this work was to passivate the surface states on the SnO2 sensors by molecular layer aiming to optimize their interactions with ammonia gas. One SnO2 sensor was coated with molecules having mostly nonpolar (functional) side groups (SnO2-APTES-alkyl) and two others were coated with molecules having mostly polar side groups (SnO2-APTES and SnO2-APTES-ester) in order to discover their interactions with ammonia which is a polar molecule. It was explored by testing the different functionalized sensors under ammonia which is the target gas. The changes in the response of the modified sensors were compared with pure SnO2. Another objective is to reduce the power consumption by decreasing the operating temperature, since the sensor will be used later for smart devices and portable applications.

In the present work, we focus on the change in sensitivity and selectivity of SnO2 sensors after functionalization with amine (APTES), alkyl (CH3), and ester (COOCH3) end functional groups.

The sensors are firstly functionalized by APTES followed by covalent attachment of alkyl or ester end functional groups molecule. Functionalization is characterized with FTIR analysis, and then detection performances of resulting sensors are investigated in regards of ammonia detection.

Experimental

Fabrication of SnO2 sensors

Thick SnO2 films were deposited on alumina substrate by screen-printing technology. A semiautomatic Aurel C890 machine was used. The procedure for preparing the SnO2 ink and sensor fabrication parameters has been described elsewhere [START_REF] Tournier | Selective filter for SnO2-based gas sensor: application to hydrogen trace detection[END_REF]. SnO2 powder (Prolabo Company) was first mixed with a solvent and an organic binder. SnO2 ink was then screen-printed on an alphaalumina substrate (38×5×0.4 mm 3 ) provided with two gold electrodes deposited by reactive sputtering. The SnO2 material was finally annealed for 10 h at 700 °C in air. Film thickness was about 40 microns. SnO2 particles and agglomerates sizes were found to be between 10 nm and 500 nm. A photographic image of the two sensor faces is presented in Fig. 1.

Molecular modifications of SnO2 sensor

The functionalization was carried out by two-step process. In the first step, 3aminopropyltriethoxysilane (APTES, ACROS Organics) molecules were grafted on SnO2 (silanization). Silanization in liquid phase has been described elsewhere [START_REF] Le | A Label-Free Impedimetric DNA Sensor Based on a Nanoporous SnO2 Film: Fabrication and Detection Performance[END_REF]. SnO2 sensors were immersed in 50 mM APTES dissolved in 95% absolute ethanol and 5% of distilled water for 5 h under stirring at room temperature. Hydroxyl groups present on the surface of SnO2 allow the condensation of APTES. To remove the unbounded APTES molecules, the sensors were rinsed with absolute ethanol and dried under N2 flow (sensor SnO2-APTES). In a second step, SnO2-APTES sensors were immersed in a solution of 10 mM of hexanoyl chloride (98%, Fluka, alkyl: C6H11ClO) or methyl adipoyl chloride (96%, Alfa Aesar, ester: C7H11ClO3) and 5 µL of triethylamine (Fluka) in 5 mL of chloroform as solvent for 12 h under stirring. The terminal amine groups of APTES allow the coupling reaction with molecules bearing acyl chloride group.

The sensors were then rinsed with chloroform and dried under N2 flow (sensors: SnO2-APTESalkyl and SnO2-APTES-ester). The functionalization of SnO2 sensors leads to covalent attachment of amine, ester, and alkyl end functional groups. A schematic illustration of the two steps functionalization is reported in Fig. 2.

Characterization of molecularly modified SnO2

Modified molecular layers were characterized by Attenuated Total Reflectance-Fourier Transform Infrared spectroscopy (ATR-FTIR), the sample being placed face-down on the

R = CH3, COOCH3
diamond crystal, and a force being applied by pressure tip. FTIR spectra were recorded in a wavelength range from 400 to 4000 cm -1 . The scanning resolution was 2 cm -1 . The entire ATR-FTIR spectrums were collected using a Golden Gate Diamond ATR accessory (Bruker Vertex 70).

Sensing measurements of modified sensors

In the test bench, the sensor was installed in 80 cm 3 glass chamber under constant gas flow of 15 l/h. The test bench was provided with gas mass flow controllers which allow controlling the concentrations of different gases at same time by PC computer. A cylinder of NH3 gas (300 ppm, diluted in nitrogen) was purchased from Air Product and Chemicals Company. In addition, the test bench was equipped with water bubbler to test the sensors under different relative humidity (RH) balanced with air at 20 °C. Conductance measurements were performed by electronic unit equipped with voltage divider circuit with 1 volt generator that permits measuring the conductance of the SnO2 film. Before injecting the target gas in the test chamber, the sensors were kept 5 h at 100 °C, and then stabilized under air flow for 5 h at 25 °C. Gas sensing properties were then measured to different concentrations of ammonia gas balanced with air with different RH. The normalized conductance was plotted to monitor the sensor response as well as to calculate response and recovery times. The normalized conductance is defined as G/G0, where G is the conductance at any time and G0 is the conductance at the beginning of the test (i.e. t=0).

Response/recovery times are defined hereafter as the time to reach 90% of steady-state sensor response. In addition, the curve of relative response (GN-GA/GA, with GN: conductance after 20 min of ammonia injection, and GA: conductance under air with 5%RH) versus ammonia concentration was plotted for different sensors to compare their sensitivity. Sensitivity is defined as the slope of the calibration curve. Limit of detection (LOD) was also evaluated and it corresponds to a signal equal 3 times the standard deviation of the conductance baseline noise.

Values above the LOD indicate the presence of target gas. The selectivity of SnO2-APTES-ester was tested versus acetone and ethanol gases.

Results and Discussions

Characterization of modified molecular layers

Molecular characterization of the grafted films on SnO2 sensors was carried out by ATR-FTIR. Spectra of SnO2 (red curve), SnO2-APTES (green curve), SnO2-APTES-alkyl (blue curve), and SnO2-APTES-ester (black curve) are presented in Fig. 3 between 800 and 4000 cm -1 . All the sensors showed similar features in the range between 1950 and 2380 cm -1 which is not exploitable as it corresponds to CO2 gas contributions in ambient air. With respect to first step of functionalization which is the attachment of APTES on SnO2, most significant absorption bands were found between 800 and 1800 cm -1 (Fig. 3, green curve). The peak at 938 cm -1 is attributed to Sn-O-Si bond in stretching mode. The ethoxy groups of APTES hydrolyze and react with the hydroxyl groups presented on the surface of SnO2 grains. In addition, the hydrolyzed ethoxy groups to hydroxyl react with another hydroxyl of the neighbor grafted APTES molecule, which leads to SnO2 surface covered with siloxane network [START_REF] Kim | Formation, structure, and reactivity of aminoterminated organic films on silicon substrates[END_REF]. This feature is shown by the wide band between 978 cm -1 and 1178 cm -1 which is attributed to siloxane groups (Si-O-Si) from polymerized APTES. The -NH3 + and -NH2 vibrational signals of SnO2-APTES are found at 1496 cm -1 and 1570 cm -1 respectively. Regardless the pure SnO2, the CH2 stretch peaks for all modified sensor founded at 2935 cm -1 are related to the backbone of the attached molecules. Thus, from these peaks, the presence of characteristic features of APTES on the surface of SnO2 is confirmed.

The second step of sensors modification was the attachment of a film on SnO2-APTES, ended with alkyl or ester groups. These modifications were carried out by reaction of amines with acyl chlorides leading to the production of one equivalent acid, which forms a salt with unreacted amine of APTES and diminish the yield. The addition of triethylamine base is to neutralize this acid and leads to push the reaction forward. From the FTIR spectra in Fig. 3 (blue and black curves), the alkyl and ester sensors exhibit two peaks at 1547 cm -1 and 1645 cm -1 which correspond to carbonyl stretch mode and N-H bending mode of amide respectively. An additional broad peak between ~ 3000 and ~ 3600 cm -1 corresponds to N-H stretch of amide. These peaks confirm the success of the reaction between amine group of APTES and acyl chloride groups.

Asymmetrical C-H stretching mode of CH3 for SnO2-APTES-alkyl and for SnO2-APTES-ester appears at 2965 cm -1 . The stretching peak of carbon double bounded to oxygen of ester group of SnO2-APTES-ester is found at 1734 cm -1 (Fig. 3, black curve). These results show that SnO2 sensors are modified as expected with alkyl and ester end groups.

As a conclusion, FTIR analysis confirms that functionalization is effectively achieved on SnO2 by showing the presence of attached APTES molecules on SnO2 after silanization, as well as the existence of ester and alkyl molecules on SnO2-APTES after reaction with acyl chloride products.

Sensing measurements of different functionalized SnO2 sensors

The first part of the test under gases was to show the characteristic of the response of different sensors to ammonia gas. SnO2, SnO2-APTES, SnO2-APTES-alkyl and SnO2-APTES-ester sensors were tested under 100 ppm of ammonia balanced with 5% RH air at 25 °C. The four sensors responses (normalized conductance) are reported in Fig. 4. Fig. 4. The sensor response of SnO2 (G0=1.4×10 -5 Ω -1 ), SnO2-APTES (G0=7.9×10 -6 Ω -1 ), SnO2-APTES-alkyl (G0=1.5×10 -5 Ω -1 ), and SnO2-APTES-ester (G0=9.5×10 -6 Ω -1 ) to 100 ppm ammonia gas balanced with humid air (5%RH) at 25 °C.

SnO2 sensor

First we have to mention that the conductance of SnO2 at room temperature is measurable as reported by Ji Haeng Yu et al. [START_REF] Yu | Selective CO gas detection of CuO-and ZnO-doped SnO2 gas sensor[END_REF]. Indeed, stoichiometric SnO2 is known to be insulator at room temperature. But the used SnO2 sensitive film contains defects and the experiments were performed with 5%RH which lead to the formation of hydroxyl groups adsorbed on SnO2 surface. These two effects explain the measurable conductance base line under air balanced with 5%RH of pure SnO2. Furthermore, the conductance was still measurable even when the test was switched to dry air as the hydroxyl groups stayed adsorbed at room temperature.

The conductance of pure SnO2 decreases upon exposure to ammonia gas (Fig. 4). This type of response has been found before by Kamalpreet Khun Khun et al. [START_REF] Khun Khun | SnO2 thick films for room temperature gas sensing applications[END_REF] at temperature between 25 to 200 °C. They supposed that ammonia reacts with molecular adsorbed oxygen ion (O2 -, created as shown in Eq 1) producing nitrogen monoxide gas (NO) according to Eq 2. In presence of oxygen and at low temperature, NO can be easily transformed into NO2 which is very good oxidizing agent (Eq 3). The reaction of NO2 with SnO2 at ambient temperature causes the decrease of sensor conductance. NO2 adsorbs on SnO2 surface adsorption sites (s) and bring out electrons from the conduction band (Eq 4) [START_REF] Maeng | SnO2 Nanoslab as NO2 Sensor: Identification of the NO2 Sensing Mechanism on a SnO2 Surface[END_REF]. Thus, according to published results, the overall reaction of ammonia with SnO2 at room temperature could be written as in Eq 5. Such a mechanism is consistent with a conductance decrease upon ammonia exposure. However, actually, we have no experimental proof of such a mechanism.

Eq. (1) O2 + s + 1e -→ s-O2 - Eq. ( 2) 4NH3 + 5s-O2 -→ 4NO + 6H2O + 5s + 5e - Eq. (3) 2NO + O2 ↔ 2NO2 (equilibrium in air) Eq. (4) NO2 + s + e -→ s-NO2 - Eq. ( 5) 4NH3 + 4s + 7O2 + 4e -→ 4s-NO2 -+ 6H2O where s is an adsorption site.

SnO2-APTES sensor

When SnO2-APTES sensor was exposed to ammonia gas, no response was observed. Formation of APTES film on SnO2 prevents the water molecules to adsorb on the surface because the active sites of SnO2 are occupied by O-Si bond of APTES, and because of the hydrophobic nature of APTES film. Therefore, in the following discussion, the conventional mechanism of interaction of SnO2 with gases cannot be taken in consideration as no reactive sites are available. SnO2-APTES shows no change in conductance upon exposure to ammonia (Fig. 4). This implies that no significant interactions occur between the grafted APTES and ammonia gas. In term of polarity and other chemical properties like acidity, the amine and ammonia groups are almost the same, since amine is one of the derivatives of ammonia. Hence, such result was expected for SnO2-APTES. In addition, this behavior indicates that the SnO2 surface is well covered by APTES molecules because the negative response observed on pure SnO2 is totally inhibited.

SnO2-APTES-alkyl and SnO2-APTES-ester sensors

SnO2-APTES-alkyl and SnO2-APTES-ester exhibit increase in conductance upon exposure to 100 ppm of ammonia gas as shown in Fig. 4. However, the response of SnO2-APTES-ester is more important than for SnO2-APTES-alkyl. These responses could be related to the different polarities of the attached end groups. Indeed, Wang et al. [START_REF] Wang | Effect of Functional Groups on the Sensing Properties of Silicon Nanowires toward Volatile Compounds[END_REF][START_REF] Wang | Artificial Sensing Intelligence with Silicon Nanowires for Ultraselective Detection in the Gas Phase[END_REF] reported that the response of some functionalized sensitive films is derived from the change in electrostatic field of the attached molecular layer. Ester is a good electron withdrawing group, while alkyl is mostly considered as nonpolar. Ammonia molecule is a good nucleophilic molecule (donating), thus the interaction is between electron withdrawing (ester) and electron donating groups. In this case, dipole-dipole interaction is taking place. However, in the case of SnO2-APTES-alkyl, the interaction is of induced dipole type because ammonia is a polar molecule and alkyl end group is mostly nonpolar. It is likely that the adsorption process occurs through interaction between the nitrogen of ammonia and the end functional group of the molecular layer (alkyl and ester). It was reported before by B. Wang et al. [START_REF] Wang | Effect of Functional Groups on the Sensing Properties of Silicon Nanowires toward Volatile Compounds[END_REF], that the dipole-dipole interaction is always stronger than induced dipole interaction. This can explain the difference in the response between the SnO2-APTES-alkyl and SnO2-APTESester sensors to ammonia gas. As mentioned previously, the interaction can also result from diffusion of the gas in the molecular layer. This type of interaction is favorable only for SnO2-APTES-alkyl. It is difficult for ammonia molecules to diffuse in the molecular layer of SnO2-APTES-ester, because of the steric hindrance induced by ester end groups. This phenomenon can explain the response of SnO2-APTES-alkyl to ammonia in addition to the induced dipole interaction. These two interactions (i.e. dipole-dipole and induced-dipole) result in a modification in the dipole moment of the whole film. The variation in the molecular layer's dipole moment affects the electron mobility in SnO2 film which modifies the conductance [START_REF] Hoft | Effect of dipole moment on current-voltage characteristics of single molecules[END_REF][START_REF] Young | Effect of group and net dipole moments on electron transport in molecularly doped polymers[END_REF]. The exposure to ammonia leads to increase in electron mobility (proportional to conductance). The same behavior was founded to a selection of polar and non-polar gases but on ester, and alkyl silicon oxide functionalized substrate [START_REF] Wang | Effect of Functional Groups on the Sensing Properties of Silicon Nanowires toward Volatile Compounds[END_REF].

According to the above discussion, the response of the molecular modified sensors does not obey the conventional mechanism of direct interaction with SnO2. The response comes powerfully from the interaction of ammonia molecules with end function group of the attached layer. The response of SnO2-APTES-ester is generated from dipole-dipole interaction, while the response of SnO2-APTES-alkyl is produced from induced dipole interaction which has less significant effect.

Sensors sensitivity

Regarding the different sensors sensitivity against ammonia concentrations, Fig. 5 shows the relative responses versus ammonia concentrations of SnO2-APTES-ester in comparison with pure SnO2, SnO2-APTES and SnO2-APTES-alkyl sensors. Sensitivity is defined as the slope of the relative response curve versus ammonia concentrations, i.e., how large is the change in the sensor signal upon a certain change in ammonia concentration. Pure SnO2 and SnO2-APTES sensors have almost no sensitivity to different ammonia concentrations. In addition, SnO2-APTES-alkyl gives no significant response between 0.5 ppm and 30 ppm, but its sensitivity starts to increase from 30 ppm of ammonia. It can be noticed that SnO2-APTES-ester exhibit constant sensitivity between 0.5 ppm and 10 ppm, around 0.023 ppm -1 . After this concentration the sensor starts to become saturated and the sensitivity continuously decreases down to nearly zero at 100 ppm NH3. However, the sensitivity of SnO2-APTES-ester at concentrations higher than 30 ppm is still more significant than the sensitivity of SnO2-APTES-alkyl. The calculated LOD for ester modified SnO2 was 80 ppb. To summarize this section, SnO2-APTES-ester sensors showed good sensitivity to ammonia gas in a range of concentration compatible with breath analysis applications (sub-ppm). This sensor is studied in more details in the following part.

Focus on SnO2-APTES-ester sensor

Effect of humidity on the response

As known, human breath contains high amount of humidity (100 % RH at 37 °C). molecule presents high polarity, hence it can affect the response to ammonia gas. In order to check this effect, SnO2-APTES-ester was tested to ammonia gas under different amount of relative humidity ranging from 5 to 50% RH. Figure 6 shows the sensor response of SnO2-APTES-ester to 100 ppm in dry and humid air at 25 °C. Upon exposure to ammonia gas the sensor conductance increases in the four cases (dry air, 5%RH, 26%RH, and 50%RH) with fast response and recovery times. In dry air and 5%RH, the sensor shows almost the same response magnitude, 1.46 and 1.45 respectively, which decreases to 1.25 and 1.06 in 26%RH and 50%RH respectively. This means that a small quantity of relative humidity does not affect the sensor response, while at elevated amount the response starts to be less significant. A potential explanation is that the attached ester film is saturated by water molecules or adsorbed hydroxyl groups at high RH. Hence, during exposure to ammonia, there is a limitation of response due to adsorption competition between water and ammonia. For the future tests, the humidity will be kept at 5%RH. In these conditions, 5%RH at 25 °C, the response and recovery times (as defined in section 2.4) are 98 s and 130 s respectively, which is quite noticeable for a tin oxide based sensors working at room temperature.

Effect of operating temperature

In the most cases, the increase of the operating temperature of SnO2 sensors increases the sensitivity to gases. In contrast, in the case of SnO2-APTES-ester, the increase of temperature generates decrease of ammonia sensor response. As shown in Fig. 7, the response of SnO2-APTES-ester to ammonia decreases from 1.45 (25 °C) to 1.18 and 1.06 when the operating temperature is increased to 50 °C and 100 °C respectively. Therefore, the interaction of ammonia with ester attached to SnO2 is more significant at low temperature (25 °C) than at higher one (50 and 100 °C).

So, interesting conclusion is that SnO2-APTES-ester has to be operated close to room temperature (25 °C), without any power consumption for the sensitive detection of ammonia.

Effect of ammonia concentration on sensor response

The influence of ammonia concentrations on sensor response was studied at the optimum conditions defined previously, 5%RH at 25 °C. Figure 8 shows the change in conductance of SnO2-APTES-ester upon exposure to different concentrations of ammonia gas (0.2-100 ppm).

The curve at low concentrations (Fig. 8a) was used to calculate the limit of detection (LOD) of the sensor to ammonia gas which is around 80 ppb. Such a value confirms the potentiality of ammonia detection for breath analysis application.

Selectivity

Human breath contains a wide variety of volatile organic compounds which are polar or nonpolar, oxidant or reductant. It's well known that SnO2 sensor which usually operates at temperature between 350-500 °C can give a response to most of these types of gases, unfortunately without distinction (selectivity) or even with compensating effect for oxidant/reducing gases [START_REF] Pijolat | Application of membranes and filtering films for gas sensors improvements[END_REF]. Field effect transistors functionalized with ester end group have shown responses to a wide variety of volatile organic compounds like alcohols and alkanes [START_REF] Wang | Artificial Sensing Intelligence with Silicon Nanowires for Ultraselective Detection in the Gas Phase[END_REF]. This implies that these sensors also have a lack of selectivity. To check the selectivity to ammonia of the developed SnO2-APTES-ester, such sensors were tested with respect to ethanol and acetone gases.

Figure 9 shows that the SnO2-APTES-ester sensors have almost no change in conductance upon exposure to 50 ppm of acetone and ethanol at 25 °C. This means that ester modified sensor is relatively selective to ammonia, at least in regards of the two tested gases. This particular selectivity derives from the way of interactions of the grafted layer on SnO2 with ammonia gas.

As mentioned previously, interactions occur between the ester end group which is strongly electron withdrawing, and the ammonia molecule which is electron donnating. Hence, electrons are withdrawed by the attached ester end group from the ammonia molecules adsorbed on it during exposure. These interactions lead to the significant response of SnO2-APTES-ester sensor.

Other molecules like ethanol and acetone do not have this high affinity to donate the electrons to SnO2-APTES-ester, explaining minor changes in conductance upon exposure to these gases.

Conclusion

Molecularly modified SnO2 thick films were produced by screen printing and wet chemical processes. The functionalizations were carried out first by grafting of 3aminopropyltriethoxysilane followed by reaction with hexanoyl chloride or methyl adiapoyl chloride. Then, tests under gases were performed. Pure SnO2 sensor and APTES modified SnO2 didn't show any significant sensitivity to NH3 (0.5-100 ppm) at 5%RH, 25 °C, while the sensitivity to NH3 gas starts to increase from 30 ppm for alkyl modified sensor. On the contrary, ester modified sensor exhibit fast response and recovery time to NH3 gas with a limit of detection estimated to 80 ppb at 5%RH, 25 °C. In addition, this sensor shows constant sensitivity between 0.5 and 10 ppm of NH3 (0.023 ppm -1 ). Moreover, ester modified sensor is selective to NH3 gas with respect to reducing gases like ethanol and acetone. However, the relative humidity higher than 5% decreases the response. Working at room temperature, ester modified sensor may be a good candidate for breath analysis applications for the diagnosis of diseases related to ammonia gas biomarker. Such sensor could be coupled with a condenser to reduce the amount of humidity in the analyzed breath sample to 5 %RH. The sensing mechanism of ester modified SnO2 and the selectivity in regards of various volatile organic compounds have to be investigated in further work.

Fig. 1 .

 1 Fig. 1. Photograph of the two SnO2 sensor sides deposited by screen printing.

Fig. 2 .

 2 Fig. 2. Schematic illustration of SnO2-APTES, SnO2-APTES-alkyl, and SnO2-APTES-ester synthesis steps. Hexanoyl chloride represent the molecule C5H8ClOR with R: CH3 and methyl adipoyl chloride is the molecule C5H8ClOR with R: COOCH3.

Fig. 3 .

 3 Fig. 3. ATR-FTIR spectra of SnO2 (red curve), SnO2-APTES (green curve), SnO2-APTES-alkyl (blue curve), and SnO2-APTES-ester (black curve) films.

Fig. 5 .

 5 Fig. 5. Relative response of pure SnO2, SnO2-APTES, SnO2-APTES-alkyl, and SnO2-APTESester sensors versus ammonia concentrations balanced with 5% RH air at 25 °C.

Fig. 6 .

 6 Fig. 6. The sensor response curves of SnO2-APTES-ester to 100 ppm of ammonia gas balanced with dry air, 5%RH, 26%RH, and 50%RH at 25 °C.

Fig. 7 .

 7 Fig. 7. The sensor response of SnO2-APTES-ester to 100 ppm ammonia gas balanced with humid air (5%RH) at 25 °C, 50 °C, and 100 °C.

Figure

  Figure 8b also shows the stability of the sensor with time. The stability of metal oxide is a challenge mostly for room temperature gas sensors. Present results show quite good stability of the baseline. The drift is around 0.98% over 16 hours.

Fig. 8 .

 8 Fig. 8. The change in conductance of SnO2-APTES-ester upon exposure to different concentrations of ammonia gas in humid air (5%RH) at 25 °C, a) [NH3] ranging from 0.2 to 5 ppm and b) [NH3] ranging from 5 to 100 ppm.

Fig. 9 .

 9 Fig. 9. The sensor response of SnO2-APTES-ester upon exposure to 50 ppm of ammonia, ethanol, and acetone gases in humid air (5%RH) at 25 °C.


Latifah Nurahmi
email: latifah.nurahmi@irccyn.ec-nantes.fr

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Philippe Wenger
email: philippe.wenger@irccyn.ec-nantes.fr

Josef Schadlbauer
email: josef.schadlbauer@uibk.ac.at

Manfred Husty
email: manfred.husty@uibk.ac.at

Reconfiguration analysis of a 4-RUU parallel manipulator

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

To the best of the authors knowledge, the notion of operation mode was initially introduced by Zlatanov et al. in [START_REF] Zlatanov | Constraint Singularities as C-Space Singularities[END_REF] to explain the behaviour of the three degree-of-freedom (3-dof ) DYMO robot which can undergo a variety of transformation when passing through singular configurations. In [START_REF] Kong | Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method[END_REF], the author analysed the types of operation modes and the transition configurations of the 3-RER 1 Parallel Manipulator (PM) based upon the Euler parameter quaternions. Walter et al. in [START_REF] Walter | A Complete Kinematic Analysis of the SNU 3-UPU Parallel Robot[END_REF] used the Study's kinematic mapping to show that the 3-UPU PM built at the Seoul National University (SNU) has nine different operation modes. Later in [START_REF] Walter | Kinematic Analysis of the TSAI 3-UPU Parallel Manipulator Using Algebraic Methods[END_REF], the authors revealed five different operation modes of the 3-UPU PM proposed by Tsai in 1996 [START_REF] Tsai | Kinematics of a 3-DOF Platform with Three Extensible Limbs[END_REF]. By using same approach, Schadlbauer et al. in [START_REF] Schadlbauer | The 3-RPS Parallel Manipulator from an Algebraic Viewpoint[END_REF] found two distinct operation modes of the 3-RPS PM proposed by Hunt in 1983 [START_REF] Hunt | Structural Kinematics of In-Parallel-Actuated Robot-Arms[END_REF]. Later in [START_REF] Schadlbauer | Operation Modes in Lower-Mobility Parallel Manipulators[END_REF], the authors characterized the motion type in both operation modes by using the axodes. The self-motions of this manipulator were classified in [START_REF] Schadlbauer | Self-Motions of 3-RPS Manipulators[END_REF]. Another PM of the 3-RPS family is the 3-RPS Cube PM and was proposed by Huang et al. in 1995 [10]. Nurahmi et al. in [START_REF] Nurahmi | Kinematic Analysis of the 3-RPS Cube Parallel Manipulator[END_REF][START_REF] Nurahmi | Motion Capability of the 3-RPS Cube Parallel Manipulator[END_REF] found that this manipulator has only one operation mode in which the 3-dof general motion and 1-dof Vertical Darboux Motion occur inside the same operation mode.

Accordingly, a general methodology for the type synthesis of reconfigurable mechanisms has been proposed and several new reconfigurable mechanisms have been generated. In [START_REF] Kong | Type Synthesis of Parallel Mechanisms with Multiple Operation Modes[END_REF][START_REF] Kong | Type Synthesis of Variable Degrees-of-Freedom Parallel Manipulators with Both Planar and 3T1R Operation Modes[END_REF], the authors proposed a general method based upon the screw theory to synthesize a PM that can perform two operation modes. In [START_REF] He | Kinematic Analysis of a Single-Loop Reconfigurable 7R Mechanism with Multiple Operation Modes[END_REF], a novel 1-dof single-loop reconfigurable 7-R mechanism with multiple operation modes based upon the Sarrus mechanism, was proposed. The following year, the reconfiguration analysis of this mechanism based on the kinematic mapping and the algebraic geometry method was presented in [START_REF] Kong | Type Synthesis and Reconfiguration Analysis of a Class of Variable-DOF Single-Loop Mechanisms[END_REF].

By using the theory of the displacement groups, the lower-mobility PM with multiple operation modes and different number of dof were presented in [START_REF] Fanghella | Parallel Robots that Change Their Group of Motion[END_REF]. Refaat et al. in [START_REF] Refaat | Two-Mode Overconstrained Three DOFs Rotational-Translational Linear-Motor-Based Parallel-Kinematics Mechanism for Machine Tool Applications[END_REF] introduced a family of 3-dof PM that can exhibit two 1T1R modes by using Lie-group theory. In [START_REF] Gogu | Maximally Regular T2R1-Type Parallel Manipulators with Bifurcated Spatial Motion[END_REF], Gogu introduced several PM with two 2T1R modes. In [START_REF] Gan | Mobility Change in Two Types of Metamorphic Parallel Mechanisms[END_REF], a new joint was presented and added in the manipulator architecture hence it allows the moving platform to change the motion types. By adding a rTPS limb which has two phases, a new metamorphic parallel mechanism is introduced in [START_REF] Gan | Unified Kinematics and Singularity Analysis of a Metamorphic Parallel Mechanism with Bifurcated Motion[END_REF]. The link-coincidence-based geometric-constraint method is proposed in [START_REF] Dai | Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds[END_REF] to obtain reconfigurable mechanisms originated from carton folds and packaging dated back to 1996. At the same year, Wohlhart in [START_REF] Wohlhart | Kinematotropic linkages[END_REF] showed mechanisms that changed mobility through singularities.

In [START_REF] Li | Parallel Mechanisms with Bifurcation of Schoenflies Motion[END_REF], Li and Hervé investigated several PM with two distinct Schönflies modes. The Schönflies motion contains three independent translations and one pure rotation about an axis of fixed direction, namely 3T1R. The authors continued in [START_REF] Lee | Isoconstrained Parallel Generators of Schoenflies Motion[END_REF] to present the systematic approach to synthesize the iso-constrained parallel Schönflies motion generators with two identical 5-dof limbs.

The type synthesis of the 3T1R PM with four identical limb structures was performed in [START_REF] Kong | Type Synthesis of Parallel Mechanisms[END_REF], which leads to a kinematic architecture with four revolute actuators, namely the 4-RUU PM. In [START_REF] Masouleh | Solving the Forward Kinematic Problem of 4-DOF Parallel Mechanisms (3T1R) with Identical Limb Structures and Revolute Actuators Using the Linear Implicitization Algorithm[END_REF], eight solutions of the direct kinematics were enumerated by using the linear implicitization algorithm. Amine et al. in [START_REF] Amine | Singularity Analysis of the 4-RUU Parallel Manipulator Using Grassmann-Cayley Algebra[END_REF][START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures[END_REF] investigated the singularity conditions of the 4-RUU PM by using the Grassmann-Cayley Algebra and the Grassmann Geometry. It is shown that the 4-RUU PM is an over-constrained manipulator and it shares some common properties among the constraint wrenches.

By using an algebraic description of the manipulator and the Study's kinematic mapping based upon [START_REF] Husty | Algebraic Methods in Mechanism Analysis and Synthesis[END_REF], a characterization of the operation modes of the 4-RUU PM are discussed in more details in this paper. Due to the unique topology of the RUU limb that comprises two links with one revolute actuator attached to the base, the actuated joint angle always appears in every constraint equation. This kinematic issue does not allow to compute a primary decomposition because the constraint equation changes for every joint inputs. As a consequence, the 4-RUU PM is decomposed into two iso-constrained 2-RUU PM. The constraint equations of each 2-RUU PM are initially derived and the primary decomposition is computed. It turns out that the 2-RUU PM has three 4-dof operation modes. By combining the results of primary decomposition from both 2-RUU PM, the operation modes of the 4-RUU PM can be characterized. It reveals that the 4-RUU PM has two 4-dof operation modes and one 2-dof operation mode. The singularities are examined by deriving the determinant of the Jacobian matrix of the constraint equations with respect to the Study parameters. It is shown that the manipulators are able to change from one operation mode to another operation mode by passing through the configurations that belong to both modes. The singularity conditions are mapped onto the joint space. Eventually, the changes of operation modes are illustrated.

This paper is organized as follows: A detailed description of the manipulator architecture is given in Section 2. The constraint equations of the manipulators are expressed in Section 3. These constraint equations are used to identify the operation modes in Section 4. In Section 5, the singularity conditions and self-motions are presented. Eventually, the operation modes changing of the 4-RUU PM is discussed in Section 6.

Nurahmi, Caro, Wenger, Schadlbauer, and Husty, submitted to Mech. and Mach. Theory 4 The 4-RUU PM shown in Fig. 1, is composed of a square base, a square moving platform, and four identical limbs. The origin O of the fixed frame Σ 0 and the origin P of the moving frame Σ 1 are located at the center of the square base and the square moving platform, respectively. Each limb is composed of five R-joints such that the second and the third ones, as well as the forth and the fifth ones, are built with intersecting and perpendicular axes. Thus they are assimilated to U-joint. The printed model of this manipulator is shown in Fig. 2. The first R-joint is attached to the base and is actuated. Its rotation angle is defined by θ 1i (i = 1, ..., 4). The axes of the first and the second joints are directed along Z-axis. The axis of the fifth joint is directed along z-axis. The second axis and the fifth axis are denoted by v i and n i (i = 1, ..., 4), respectively. The axes of the third and the forth joints are parallel. The axis of the third joint is denoted by s i (i = 1, ..., 4) and it changes instantaneously as a function of θ 2i (i = 1..4) as shown in Fig. 3, hence: The first R-joint of the i-th limb is located at point A i with distance a from the origin O of the fixed frame Σ 0 . The first U-joint is denoted by point B i with distance l from point A i . Link A i B i always moves in a plane normal to v i . Hence the coordinates of points A i and B i expressed in the fixed frame Σ 0 are:

PSfrag A 1 A 2 A 3 A 4 C 1 C 2 C 3 C 4 B 1 v 1 n 1 s 1 s 1 O P r X Y Z x y z Σ 0 Σ 1
s i = ( 0, cos(θ 2i ), sin(θ 2i ), 0 ) T , i = 1, ..., 4 (1)
r 0 A 1 = ( 1, a, 0, 0 ) T r 0 B 1 = ( 1, l cos(θ 11 ) + a, l sin(θ 11 ), 0 ) T r 0 A 2 = ( 1, 0, a, 0 ) T r 0 B 2 = ( 1, l cos(θ 12 ), l sin(θ 12 ) + a, 0 ) T r 0 A 3 = ( 1, -a, 0, 0 ) T r 0 B 3 = ( 1, l cos(θ 13 ) -a, l sin(θ 13 ), 0 ) T r 0 A 4 = ( 1, 0, -a, 0 ) T r 0 B 4 = ( 1, l cos(θ 14 ), l sin(θ 14 ) -a, 0 ) T (2)
The moving platform is connected to the limbs by four U-joints, of which the intersection point of the R-joint axes is denoted by C i . The length of the moving platform from the origin P of the moving frame Σ 1 to point C i is defined by b. The length of link B i C i is defined by r. The coordinates of point C i expressed in the moving frame Σ 1 are:

r 1 C 1 = ( 1, b, 0, 0 ) T r 1 C 3 = ( 1, -b, 0, 0 ) T r 1 C 2 = ( 1, 0, b, 0 ) T r 1 C 4 = ( 1, 0, -b, 0 ) T (3)
As a consequence, there are four design parameters a, b, l, and r; and four joint variables θ 11 , θ 12 , θ 13 , and θ 14 that determine the motions of the 4-RUU PM.

Constraint equations

In this section, the constraint equations are derived whose solutions illustrate the possible poses of the moving platform (coordinate frame Σ 1 ) with respect to Σ 0 . To obtain the coordinates of points C i and vectors n i expressed in Σ 0 , the Study parametrization of a spatial Euclidean transformation matrix M ∈ SE(3) based on [START_REF] Husty | Algebraic Methods in Mechanism Analysis and Synthesis[END_REF] is used.

M = x 2 0 + x 2 1 + x 2 2 + x 2 3 0 T 3×1 M T M R (4)
where M T and M R represent the translational and rotational parts of the transformation matrix M, respectively, and are expressed as follows:

M T =   2(-x 0 y 1 + x 1 y 0 -x 2 y 3 + x 3 y 2 ) 2(-x 0 y 2 + x 1 y 3 + x 2 y 0 -x 3 y 1 ) 2(-x 0 y 3 -x 1 y 2 + x 2 y 1 + x 3 y 0 )   M R =   x 2 0 + x 2 1 -x 2 2 -x 2 3 2(x 1 x 2 -x 0 x 3 ) 2(x 1 x 3 + x 0 x 2 ) 2(x 1 x 2 + x 0 x 3 ) x 2 0 -x 2 1 + x 2 2 -x 2 3 2(x 2 x 3 -x 0 x 1 ) 2(x 1 x 3 -x 0 x 2 ) 2(x 2 x 3 + x 0 x 1 ) x 2 0 -x 2 1 -x 2 2 + x 2 3   (5)
Nurahmi, Caro, Wenger, Schadlbauer, and Husty, submitted to Mech. and Mach. Theory 7

The parameters x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 , which appear in matrix M, are called Study parameters. These parameters make it possible to parametrize SE(3) with dual quaternions. The Study's kinematic mapping maps each spatial Euclidean displacement of SE(3) via transformation matrix M onto a projective point X [x 0 : x 1 : x 2 : x 3 : y 0 : y 1 : y 2 : y 3 ] in the 6-dimensional Study quadric S ∈ P 7 [START_REF] Schadlbauer | The 3-RPS Parallel Manipulator from an Algebraic Viewpoint[END_REF], such that: SE(3) → X ∈ P 7 (x 0 : x 1 : x 2 : x 3 : y 0 : y 1 : y 2 : y 3 ) T = (0 : 0 : 0 : 0 : 0 : 0 : 0 : 0) T

Every projective point X will represent a spatial Euclidean displacement, if it fulfils the following equation and inequality:

x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0, x 2 0 + x 2 1 + x 2 2 + x 2 3 = 0 (7)
Those two conditions will be used in the following computations to simplify the algebraic expressions. First of all, the tangent half-angle substitutions are performed to rewrite the trigonometric functions of θ 1i and θ 2i (i = 1, ..., 4) in terms of rational functions of a new variable t ij . However, the tangent half-angle substitutions will increase the degree of variable and make the computation quite heavy.

cos(θ

ij ) = 1 -t 2 ij 1 + t 2 ij , sin(θ ij ) = 2t 2 ij 1 + t 2 ij , i = 1, 2, j = 1, ..., 4 (8)
where

t ij = tan( θ ij 2
). The coordinates of points C i and vectors n i expressed in the fixed frame Σ 0 are obtained by:

r 0 C i = M r 1 C i , n 0 i = M n 1 i , i = 1, ..., 4 (9)
The coordinates of all points are given in terms of the Study parameters and the design parameters. The constraint equations can be obtained by examining the design of RUU limb. The link connecting points B i and C i is coplanar to the vectors v i and n 0 i . Accordingly, the scalar triple product of vectors (r 0 C ir 0 B i ), v i and n 0 i vanishes, namely:

(r 0 C i -r 0 B i ) T . (v i × n 0 i ) = 0 , i = 1, ..., 4 (10)
After computing the corresponding scalar triple product and removing the common denominators, the following constraint equations come out:

g 1 : (at 2 11 -bt 2 11 -lt 2 11 + a -b + l)x 0 x 1 + 2lt 11 x 0 x 2 -(2t 2 11 + 2)x 0 y 0 + 2lt 11 x 3 x 1 + (-at 2 11 -bt 2 11 + lt 2 11 -a -b -l)x 3 x 2 + (-2t 2 11 -2)y 3 x 3 = 0 (11a) g 2 : (l -lt 2 12 )x 0 x 1 + (at 2 12 -bt 2 12 + 2lt 12 + a -b)x 0 x 2 -(2t 2 12 + 2)x 0 y 0 + (at 2 12 + bt 2 12 +2lt 12 + a + b)x 3 x 1 + (lt 2 12 -l)x 3 x 2 -(2t 2 12 + 2)y 3 x 3 = 0 (11b) g 3 : (at 2 13 -bt 2 13 + lt 2 13 + a -b -l)x 0 x 1 -2lt 13 x 0 x 2 + (2t 2 13 + 2)x 0 y 0 -2lt 13 x 1 x 3 + (-at 2 13 -bt 2 13 -lt 2 13 -a -b + l)x 2 x 3 + (2t 2 13 + 2)x 3 y 3 = 0 (11c) g 4 : (lt 2 14 -l)x 0 x 1 + (at 2 14 -bt 2 14 -2lt 14 + a -b)x 0 x 2 + (2t 2 14 + 2)x 0 y 0 + (at 2 14 + bt 2 14 -2lt 14 + a + b)x 1 x 3 + (-lt 2 14 + l)x 2 x 3 + (2t 2 14 + 2)x 3 y 3 = 0 (11d)
To derive the constraint equations corresponding to the link length r of link B i C i , the distance equation can be formulated as: (r 0

C i -r 0 B i ) 2 = r 2 .
As a consequence, the following four equations are obtained:

g 5 : (a 2 t 2 11 -2abt 2 11 -2alt 2 11 + b 2 t 2 11 + 2blt 2 11 + l 2 t 2 11 -r 2 t 2 11 + a 2 -2ab + 2al + b 2 -2bl + l 2 - r 2 )x 2 0 -8blt 11 x 0 x 3 + (4at 2 11 -4bt 2 11 -4lt 2 11 + 4a -4b + 4l)x 0 y 1 + 8lt 11 x 0 y 2 + (a 2 t 2 11 -2a bt 2 11 -2alt 2 11 + b 2 t 2 11 + 2blt 2 11 + l 2 t 2 11 -r 2 t 2 11 + a 2 -2ab + 2al + b 2 -2bl + l 2 -r 2 )x 2 1 -8 blt 11 x 1 x 2 + (-4at 2 11 + 4bt 2 11 + 4lt 2 11 -4a + 4b -4l)x 1 y 0 -8lt 11 x 1 y 3 + (a 2 t 2 11 + 2abt 2 11 -2 alt 2 11 + b 2 t 2 11 -2blt 2 11 + l 2 t 2 11 -r 2 t 2 11 + a 2 + 2ab + 2al + b 2 + 2bl + l 2 -r 2 )x 2 2 -8lt 11 x 2 y 0 +(4at 2 11 + 4bt 2 11 -4lt 2 11 + 4a + 4b + 4l)x 2 y 3 + (a 2 t 2 11 + 2abt 2 11 -2alt 2 11 + b 2 t 2 11 -2blt 2 11 + l 2 t 2 11 -r 2 t 2 11 + a 2 + 2ab + 2al + b 2 + 2bl + l 2 -r 2 )x 2 3 + 8lt 11 x 3 y 1 + (-4at 2 11 -4bt 2 11 + 4l t 2 11 -4a -4b -4l)x 3 y 2 + (4t 2 11 + 4)y 2 0 + (4t 2 11 + 4)y 2 1 + (4t 2 11 + 4)y 2 2 + (4t 2 11 + 4)y 2 3 = 0 (12a) g 6 : (a 2 t 2 g 7 : (a 2 t 2 13 -2abt 2 13 + 2alt 2 13 + b 2 t 2 13 -2blt 2 13 + l 2 t 2 13 -r 2 t 2 13 + a 2 -2ab -2al + b 2 + 2bl + l 2 - r 2 )x 2 0 + 8blt 13 x 0 x 3 + (-4at 2 13 + 4bt 2 13 -4lt 2 13 -4a + 4b + 4l)x 0 y 1 + 8lt 13 x 0 y 2 + (a 2 t 2 13 -2 abt 2 13 + 2alt 2 13 + b 2 t 2 13 -2blt 2 13 + l 2 t 2 13 -r 2 t 2 13 + a 2 -2ab -2al + b 2 + 2bl + l 2 -r 2 )x 2 1 + 8 blt 13 x 1 x 2 + (4at 2 13 -4bt 2 13 + 4lt 2 13 + 4a -4b -4l)x 1 y 0 -8lt 13 x 1 y 3 + (a 2 t 2 13 + 2abt 2 13 + 2al t 2 13 + b 2 t 2 13 + 2blt 2 13 + l 2 t 2 13 -r 2 t 2 13 + a 2 + 2ab -2al + b 2 -2bl + l 2 -r 2 )x 2 2 -8lt 13 x 2 y 0 + ( -4at 2 13 -4bt 2 13 -4lt 2 13 -4a -4b + 4l)x 2 y 3 + (a 2 t 2 13 + 2abt 2 13 + 2alt 2 13 + b 2 t 2 13 + 2blt 2 13 + l 2 t 2 13 -r 2 t 2 13 + a 2 + 2ab -2al + b 2 -2bl + l 2 -r 2 )x 2 3 + 8lt 13 x 3 y 1 + (4at
+ a 2 + 2ab + b 2 + l 2 -r 2 )x 2 1 + (-4blt 2 14 + 4 bl)x 1 x 2 + (4lt 2 14 -4l)x 1 y 0 + (4at 2 14 + 4bt 2 14 -8lt 14 + 4a + 4b)x 1 y 3 + (a 2 t 2 14 -2abt 2 14 + b 2 t 2 14 +l 2 t 2 14 -r 2 t 2 14 -4alt 14 + 4blt 14 + a 2 -2ab + b 2 + l 2 -r 2 )x 2 2 + (4at 2 14 -4bt 2 14 -8lt 14 + 4a -4b)x 2 y 0 + (-4lt 2 14 + 4l)x 2 y 3 + (a 2 t 2 14 + 2abt 2 14 + b 2 t 2 14 + l 2 t 2 14 -r 2 t 2 14 -4alt 14 -4blt 14 + a 2 + 2ab + b 2 + l 2 -r 2 )x 2
3 + (-4at 2 14 -4bt 2 14 + 8lt 14 -4a -4b)x 3 y 1 + (4lt 2 14 -4l)x 3 y 2 + (4t 2 14 + 4)y 2 0 + (4t 2 14 + 4)y 2 1 + (4t 2 14 + 4)y 2 2 + (4t 2 14 + 4)y 2 3 = 0 (12d) To derive the constraint equations corresponding to the axes s i of each limb, the scalar product of vector --→ B i C i and vector s i should vanish, as : (r 0 C ir 0 B i ) T s i = 0. Hence, the following constraint equations are obtained: 7) is added since all solutions have to be within the Study quadric, i.e.: g 13 : x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0. To exclude the exceptional generator (x 0 = x 1 = x 2 = x 3 = 0), we add the following normalization equation:

g 9 : (-at
g 14 : x 2 0 + x 2 1 + x 2 2 + x 2 3 -1 = 0.
It assures that there is no point of the exceptional generators appears as a solution.

Operation modes

The 4-RUU PM is an over-constrained mechanism [START_REF] Amine | Singularity Analysis of the 4-RUU Parallel Manipulator Using Grassmann-Cayley Algebra[END_REF][START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures[END_REF], therefore it can be decomposed into two iso-constrained 2-RUU PM as shown in Fig. 4. The printed model of the 4-RUU PM presented in Fig. 2, can also be decomposed into 2-RUU PM as shown in Fig. 5. The first mechanism consists of the 1st and the 3rd limbs, hence it is named the 2-RUU (I) PM. The second mechanism consists of the 2nd and the 4th limbs, hence it is named the 2-RUU (II) PM. The moving platforms of both mechanisms move independently. When the moving frames of both mechanisms are coincident (accordingly P I ≡ P II ), we obtain the 4-RUU PM. As a consequence, the operation modes of the 4-RUU PM are determined by the linear combination of the results of primary decomposition of the 2-RUU (I) and the 2-RUU (II) PM, as presented in the following. The operation modes and the self-motions of the 2-RUU PM are presented in more detail in [START_REF] Nurahmi | Operation Modes and Self-motions of a 2-RUU Parallel Manipulator[END_REF].

A 1 A 2 A 3 A 4 C 1 C 2 C 3 C 4 X Y Z x x

The 2-RUU (I) PM

In the 2-RUU (I) PM, the first and the second R-joints in each limb are actuated. The design parameters are assigned as a = 2, b = 1, l = 1, r = 2 (in units that need nod be specified). The set of eight constraint equations is written as a polynomial ideal with variables {x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 } over the coefficient ring C[t 11 , t 13 , t 21 , t 23 ], defined as: I (I) = g 1 , g 3 , g 5 , g 7 , g 9 , g 11 , g 13 , g 14 . At this point, the following ideal is examined:

J (I) = g 1 , g 3 , g 13 .
The primary decomposition is computed to verify if the ideal J (I) is the intersection of several smaller ideals. Indeed, the ideal J (I) is decomposed into three components as: J (I) = 3 k=1 J k(I) , with the results of primary decomposition:

J 1(I) = x 0 , x 3 , x 1 y 1 + x 2 y 2 J 2(I) = x 1 , x 2 , x 0 y 0 + x 3 y 3 J 3(I) = (
Accordingly, the 2-RUU (I) PM under study has three operation modes. The computation of the Hilbert dimension of ideal J k(I) with t 11 , t 13 , t 21 , t 23 treated as variables shows that: dim(J k(I) ) = 4 (k = 1, ..., 3). To complete the analysis, the remaining equations are added by writing:

K k(I) : J k(I) ∪ g 5 , g 7 , g 9 , g 11 , g 14 , k = 1, ..., 3 (15)
It follows that the 2-RUU (I) PM has three 4-dof operation modes. This type of manipulator is called invariable-dof PM in [START_REF] Kong | Type Synthesis of Variable Degrees-of-Freedom Parallel Manipulators with Both Planar and 3T1R Operation Modes[END_REF]. Each system K k(I) corresponds to a specific operation mode that will be discussed in the following.

System K 1(I) : 1st Schönflies mode

In this operation mode, the moving platform is reversed about an axis parallel to the XY -plane of Σ 0 by 180 degrees from the "identity condition". The identity condition is when the moving frame and the fixed frame are coincident, i.e. Σ 1 ≡ Σ 0 and the transformation matrix is an identity matrix. The condition x 0 = 0, x 3 = 0, x 1 y 1 + x 2 y 2 = 0 are valid for all poses and are substituted into the transformation matrix M, such that:

M 1(I) =       1 0 0 2(x 1 y 0 -x 2 y 3 ) x 2 1 -x 2 2 2x 1 x 2 0 2(x 1 y 3 + x 2 y 0 ) 2x 1 x 2 -x 2 1 + x 2 2 0 - 2y 2 x 1 0 0 -1       (16)
From the transformation matrix M 1(I) , it can be seen that the 2-RUU (I) PM has 3-dof translational motions, which are parametrized by y 0 , y 2 , y 3 and 1-dof rotational motion, which is parametrized by x 1 , x 2 in connection with x 2 1 + x 2 2 -1 = 0 [START_REF] Schadlbauer | Operation Modes in Lower-Mobility Parallel Manipulators[END_REF]. The z-axis of frame Σ 1 attached to the moving platform is always pointing downward in this operation mode and the moving platform remains parallel to the base.

System K 2(I) : 2nd Schönflies mode

In this operation mode, the condition x 1 = 0, x 2 = 0, x 0 y 0 + x 3 y 3 = 0 are valid for all poses. The transformation matrix in this operation mode is written as:

M 2(I) =       1 0 0 0 -2(x 0 y 1 -x 3 y 2 ) x 2 0 -x 2 3 -2x 0 x 3 0 -2(x 0 y 2 + x 3 y 1 ) 2x 0 x 3 x 2 0 -x 2 3 0 - 2y 3 x 0 0 0 1       (17)
From the transformation matrix M 2(I) , it can be seen that the 2-RUU (I) PM has 3-dof translational motions, which are parametrized by y 1 , y 2 , y 3 and 1-dof rotational motion, which is parametrized by x 0 , x 3 in connection with x 2 0 + x 2 3 -1 = 0 [START_REF] Schadlbauer | Operation Modes in Lower-Mobility Parallel Manipulators[END_REF]. In this operation mode, the z-axis of frame Σ 1 attached the moving platform is always pointing upward and the moving platform remains parallel to the base.

The systems K 1(I) and K 2(I) have the same motion type, i.e. Schönflies motion, however they do not have configurations in common. It occurs since the orientation of the moving platform is not the same from one operation mode to the other. The z-axis of frame Σ 1 attached to the moving platform in system K 1(I) is always pointing downward (the moving platform is always titled by 180 degrees), while in the system K 2(I) , the z-axis of frame Σ 1 attached to the moving platform is always pointing upward.

System K 3(I) : Third mode

In this operation mode, the moving platform is no longer parallel to the base. The variables x 3 , y 0 , y 1 can be solved linearly from the ideal J 3(I) and are shown in Eq. [START_REF] Refaat | Two-Mode Overconstrained Three DOFs Rotational-Translational Linear-Motor-Based Parallel-Kinematics Mechanism for Machine Tool Applications[END_REF]. Since solving the inverse kinematics of t 11 , t 13 are quite computationally expensive, the joint variables t 11 , t 13 are considered to be the independent parameters of this mode. Then the parameters y 2 , y 3 can be solved in terms of (x 0 , x 1 , x 2 , t 11 , t 13 ). Substituting back the parameters y 2 , y 3 into Eq. ( 18), then the Study parameters x 3 , y 0 , y 1 , y 2 , y 3 are now parametrized by (x 0 , x 1 , x 2 , t 11 , t 13 ). Accordingly, the 2-RUU (I) PM will perform two translational motions, which are parametrized by variables t 11 , t 13 and two rotational motions, which are parametrized by variables x 0 , x 1 , x 2 in connection with the normalization equation g 14 .

x 3(I) = (t ). As a consequence, in this operation mode, the links B i C i (i = 1, 3) from both limbs are always parallel to the same plane and the axes s i (i = 1, 3) from both limbs are always parallel too.

The 2-RUU (II) PM

In the 2-RUU (II) PM, the first and the second R-joints in each limb are also actuated. The design parameters are assigned with the same values as a = 2, b = 1, l = 1, r = 2 (in units that need not be specified). The set of eight constraint equations is written as a polynomial ideal with variables {x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 } over the coefficient ring C[t 12 , t 14 , t 22 , t 24 ], defined as: I (II) = g 2 , g 4 , g 6 , g 8 , g 10 , g 12 , g 13 , g 14 . At this point, the following ideal is examined:

J (II) = g 2 , g 4 , g 13 .
The primary decomposition is computed and it turns out that the ideal J (II) is decomposed into three components as: J (II) = 3 k=1 J k(II) , with the results of primary decomposition:

J 1(II) = x 0 , x 3 , x 1 y 1 + x 2 y 2 J 2(II) = x 1 , x 2 , x 0 y 0 + x 3 y 3 J 3(II) = (3t
Accordingly, the 2-RUU (II) PM under study has three operation modes. The computation of the Hilbert dimension of ideal J k(II) with t 12 , t 14 , t 22 , t 24 treated as variables shows that: dim(J k(II) ) = 4 (k = 1, ..., 3). To complete the analysis, the remaining equations are added by writing:

K k(II) : J k(II) ∪ g 6 , g 8 , g 10 , g 12 , g 14 , k = 1, ..., 3 (20)
It follows that the 2-RUU (II) PM has three 4-dof operation modes too. The system K 1(II) is identical with the system K 1(I) (as explained in Section 4.1.1), in which the moving platform is titled about an axis of parallel to XY -plane of Σ 0 by 180 degrees and it can exhibit the Schönflies motion with pure rotation about Z-axis. The system K 2(II) is identical with the system K 2(I) (as explained in Section 4.1.2), where the moving platform can exhibit 3-dof independent translations and one pure rotation about Z-axis. In the system K 3(II) , the moving platform is no longer parallel to the base. The variables x 3 , y 0 , y 1 can be solved linearly from the ideal J 3(II) and are shown in Eq. [START_REF] Gan | Unified Kinematics and Singularity Analysis of a Metamorphic Parallel Mechanism with Bifurcated Motion[END_REF]. Since solving the inverse kinematics of t 12 , t 14 are quite computationally expensive, the joint variables t 12 , t 14 are considered to be the independent parameters in this third mode. Then the parameters y 2 , y 3 can be solved in terms of (x 0 , x 1 , x 2 , t 12 , t 14 ). Substituting back the parameters y 2 , y 3 into Eq. ( 21), the Study parameters x 3 , y 0 , y 1 , y 2 , y 3 are now obtained and parametrized by (x 0 , x 1 , x 2 , t 12 , t 14 ). Hence the moving platform of the 2-RUU (II) PM will perform two translational motions which are parametrized by t 12 , t 14 and two rotational motions which are parametrized by x 0 , x 1 , x 2 in connection with the normalization equation g 14 .

Under the system K 3(II) , the joint angles t 22 and t 24 can be computed from the equations g 10 , g 12 . It reveals that no matter the value of the first actuated joint (t 12 , t 14 ) in each limb, these equations (g 10 , g 12 ) vanish for two real solutions, namely (1.)

t 22 = - 1 t 24 (θ 22 = π + θ 24 )
and (2.) t 22 = t 24 (θ 22 = θ 24 ). It means that in this operation mode, the links B i C i (i = 2, 4) from both limbs are always parallel to the same plane and the axes s i (i = 2, 4) from both limbs are always parallel too.

Noticeably, the third mode of the 4-RUU PM is a 2-dof operation mode since two input joint angles are sufficient to define the pose of the manipulator. This operation mode was referred to coupled motion in [START_REF] Amine | Singularity Analysis of the 4-RUU Parallel Manipulator Using Grassmann-Cayley Algebra[END_REF][START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures[END_REF]. Since the system K 3 is a lower dimension operation mode, namely 2-dof , this type of manipulator is called variable-dof PM in [START_REF] Kong | Type Synthesis of Variable Degrees-of-Freedom Parallel Manipulators with Both Planar and 3T1R Operation Modes[END_REF].

X Y Z O A 1 A 2 A 3 A 4

Singularities and Self-motions

The 4-RUU PM reaches a singularity condition when the determinant of its Jacobian matrix vanishes. The Jacobian matrix is the matrix of all first order partial derivative of the constraint equations with respect to the Study parameters. Since the 4-RUU PM has more than one operation mode, the singular configurations can be classified into two different types, i.e. the singularity configurations that belong to a single operation mode and the singularity configurations that belong to more than one operation mode. The common configurations than belong to more than one operation mode allow the 4-RUU PM to switch from one operation mode to another operation mode, which will be discussed in Section 6.

The singular poses are examined by taking the Jacobian matrix from each system of polynomial and computing its determinant. From practical point of view, the singularity surface is desirable also in the joint space. Hence the expression of the Jacobian determinant is added The second factor is y 3 = 0, when the moving platform is coplanar to the base, the 4-RUU PM is always in a singular configuration. Finally, the last factor of S 2 : det(J 2 ) = 0 is analysed. Due to the heavy elimination process, the actuated joint angles are assigned as t 11 = 0, t 12 = 1, and t 13 = 0. The elimination gives a univariate polynomial of degree 18 in t 14 as: 99544625t 18 14 -1042686200t 17 14 + 4293155895t 16 14 -9293913184t 15 14 + 10513736564t 14 14 -175591 6864t 13 14 -14239053636t 12 14 + 24856530336t 11 14 -20314694418t 10 14 + 4683758224t 9 14 + 92888105 78t 8 14 -13708185120t 7 14 + 10456187332t 6 14 -5370369152t 5 14 + 1960220428t 4 14 -507121440t 3 14 +89099433t 2 14 -9580248t 14 + 476847 = 0 (31) One singularity configuration in the 2nd Schönflies mode can be obtained by solving Eq. (31), for example t 14 = 1. Then the direct kinematics of at least one singularity pose can be computed with θ 11 = 0 • , θ 12 = 90 • , θ 13 = 0 • , θ 14 = 90 • and it is shown in Fig. 8.

X Y Z O A 1 A 2 A 3 A 4
The determinant of Jacobian det(J 2 ) also vanishes in two particular conditions, namely when all the actuated joint angles have the same values and when the first links of each limb are pointing inward toward the origin O of the fixed frame Σ 0 . In the first condition, when all the actuated joint angles have the same values, the moving platform gains 1-dof self-motion. During the motion, the first links of each limb stay fixed and the moving platform can perform a rotational motion. Let us consider the actuated joint angles being t 11 = 0, t 12 = 0, t 13 = 0, t 14 = 0 and the 1-dof self-motion is parametrized by x 3 , as shown in Fig. 9. 5.3 Self-motions in Third mode (K 3 )

X Y Z O A 1 A 2 A 3 A 4
X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4
Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4 Figure 10: First translation of self-motion in K 2 . X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4
Before computing the self-motion of the 4-RUU PM in the third mode K 3 , the singularity conditions of the 2-RUU (I) and 2-RUU (II) PM in K 3(I) and K 3(II) , respectively, are first discussed. The determinants of the Jacobian matrices are computed in each system K 3(I) and K 3(II) . The determinants of these Jacobian matrices consist of eleven factors that are defined as:

S 3(I) : det(J
X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4
Figure 13: Self-motion when θ 11 = θ 14 and θ 12 = θ 13 in K 3 .

X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4 X Y Z A 1 A 2 A 3 A 4

Operation mode changing

There exist common configurations where the mechanism, i.e. the 4-RUU PM, can switch from one operation mode to another operation mode. These configurations are well known as transition configurations. Transition configuration analysis is an important issue in the design process and control of the parallel manipulators with multiple operation modes [START_REF] Kong | Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method[END_REF]. However, the 1st Schönflies mode and the 2nd Schönflies mode do not have configurations in common, since the variables x 0 , x 1 , x 2 , x 3 can never vanish simultaneously. It means that the 4-RUU PM cannot switch from the 1st Schönflies mode to the 2nd Schönflies mode directly.

To change from the 1st Schönflies mode to the 2nd Schönflies mode, the 4-RUU PM should pass through the third mode, namely system K 3 . There exist some configurations in which the manipulator can switch from the 1st Schönflies mode to the third mode or vice versa, and these configurations belong to both operation modes. Noticeably, these configurations are also singular configurations since they lie in the intersection of two operation modes.

In the following, the conditions on the actuated joint angles for the 4-RUU PM to change from one operation mode to another are presented. Each pair of ideals {K i ∪ K j } is analysed and the Study parameters are eliminated to find common solutions.

6.1 1st Schönflies mode (K 1 ) ←→ Third mode (K 3 )

To switch from the 1st Schönflies mode (K 1 ) to the third mode (K 3 ) or vice versa, one should find the configurations of the 4-RUU PM that fulfil the condition of both operation modes, namely (J 1 ∪ J 3 ). Then all Study parameters are eliminated to find an equation in terms of the actuated joint angles t 11 , t 12 , t 13 , t 14 , written as: 9t ... = 0 (34) In this transition configurations, the moving platform is twisted about an axis parallel to XY -plane of Σ 0 by 180 degrees and the actuated joint angles fulfil Eq. (34). The three conditions of the self-motions (1.) t 11 = t 12 and t 13 = t 14 , (2.) t 11 = -1/t 13 and t 12 = -1/t 14 , and (3.) t 11 = t 14 and t 12 = t 13 given in Section 5, are contained in Eq. (34). It shows that the moving platform is in a transition configuration of the 1st Schönflies mode K 1 and the third mode K 3 that amounts to a self-motion.

2nd Schönflies mode (K 2 ) ←→ Third mode (K 3 )

To switch from the 2nd Schönflies mode (K 2 ) to the third mode (K 3 ) or vice versa, one should find the configurations of the 4-RUU PM that fulfil the condition of both operation modes, namely (J 2 ∪ J 3 ). Then all Study parameters are eliminated to find an equation in terms of the actuated joint angles t 11 , t 12 , t 13 , t 14 , written as: 9t The moving platform of the 4-RUU PM is in a transition configuration between K 2 and K 3 when the moving platform is parallel to the base and the actuated joint angles fulfil Eq. (35). The three conditions of the self-motions (1.) t 11 = t 12 and t 13 = t 14 , (2.) t 11 = -1/t 13 and t 12 = -1/t 14 , and (3.) t 11 = t 14 and t 12 = t 13 given in Section 5, are contained in Eq. ( 35). It means that the moving platform is in a transition configuration of the 2nd Schönflies mode K 2 and the third mode K 3 that amounts to a self-motion. As a consequence, the transition between systems K 1 and K 2 occurs through the third system K 3 , which is lower dimension operation mode and amounts to a self-motion. The transition from K 2 to K 1 through the third mode K 3 with the condition of the actuated joint angles t 11 = t 12 and t 13 = t 14 , is shown in Fig. 15(a)-15(f).

Conclusions

In this paper, the method of algebraic geometry was applied to characterize the type of operation modes of the 4-RUU PM. The 4-RUU PM is initially decomposed into two 2-RUU PM. The constraint equations corresponding to two 2-RUU PM are derived and the primary decomposition is computed. It reveals that the 2-RUU PM have three 4-dof operation modes. However, when they are assembled to be the 4-RUU PM, its operation modes are composed of two 4-dof Schönflies modes and one 2-dof operation mode.

The singularity conditions were computed and represented in the joint space, i.e., the actuated joint angles (t 11 , t 12 , t 13 , t 14 ). It turns out that every configuration in the 4-dof third modes of both 2-RUU PM, is always in singularity and it amounts to a self-motion. However, every configuration in the 2-dof third mode of the 4-RUU PM is not always in singularity, i.e., self-motion. The self-motion in this operation mode occurs if the actuated joint angles fulfil some particular conditions, namely (1.) t 11 = t 12 and t 13 = t 14 , (2.) t 11 = -1/t 13 and t 12 = -1/t 14 , and (3.) t 11 = t 14 and t 12 = t 13 . The 4-RUU PM is able to switch from the 1st Schönflies mode to the 2nd Schönflies mode by passing through the third mode, which contains self-motion.

Figure 1 :

 1 Figure 1: The 4-RUU PM.

Figure 2 :

 2 Figure 2: Printed model of 4-RUU PM.

2 A 3 A 4 B 1 B 2 B 3 B 4 Figure 3 :

 23412343 Figure 3: Parametrization of the first two joint angles in each leg from top view.

Figure 4 :

 4 Figure 4: The 4-RUU PM decomposed into two 2-RUU PM.

Figure 5 :

 5 Figure 5: Printed model of 4-RUU PM decomposed into two 2-RUU PM.

Figure 6 :

 6 Figure 6: The third mode K 3 .

Figure 7 :

 7 Figure 7: Singularity pose in the 1st Schönflies mode K 1 .

Figure 8 :

 8 Figure 8: Singularity pose in the 2nd Schönflies mode K 2 .

Figure 9 :

 9 Figure 9: Self-motion when θ 11 = θ 12 = θ 13 = θ 14 = 0 in K 2 .

X

Figure 11 :

 11 Figure 11: Second translation of self-motion in K 2 .

Figure 14 :

 14 Figure 14: Self-motion when θ 11 = π + θ 13 and θ 12 = π + θ 14 in K 3 .

Figure 15 :

 15 Figure 15: Transition from the 2nd Schönflies mode to the 1st Schönflies mode via the third mode with θ 11 = θ 12 and θ 13 = θ 14 .

  2 11 t 2 13 x 1 -t 2 11 t 13 x 2 + t 11 t 2 13 x 2 + 2t 2 13 x 1 + t 11 x 2 -t 13 x 2 + x 1 )x 0 (3t 2 11 t 2 13 x 2 + t 2 11 t 13 x 1 -t 11 t 2 13 x 1 + 2t 2 11 x 2 + 4t 2 13 x 2 -t 11 x 1 + t 13 x 1 + 3x 2 ) + t 2 11 t 13 x 1 -t 11 t 2 13 x 1 + 2t 2 11 x 2 + 4t 2 13 x 2 -t 11 x 1 + t 13 x 1 + 3x 2 13 x 2 y 3 -t 11 t 2 13 x 2 1 -2t 11 t 2 13 x 2 2 + t 11 t 2 13 x 2y 3 + 2t 2 13 x 1 y 3 -t 11 x 2 2 + t 11 x 2 y 3 -t 13 x 2 1 -2t 13 x 2 2 -t 13 x 2 y 3 -x 1 x 2 + x 1 y 3 ) + t 2 11 t 13 x 1 -t 11 t 2 13 x 1 + 2t 2 11 x 2 + 4t 2 13 x 2 -t 11 x 1 + t 13 x 1 + 3x 2 )x 1 -t 11 x 0 x 2 2 + t 11 x 1 x 2 y 2 -t 13 x 0 x 2 1 -2t 13 x 0 x 2 2 -t 13 x 1 x 2 y 2 -x 0 x 1 x 2 -3x 2 2 y 2 )(18)Under this operation mode, the joint angles t 21 and t 23 can be computed from the equations g 9 , g 11 . It turns out that no matter the value of the first actuated joints (t 11 , t 13 ) in each limb, these equations vanish for two real solutions, namely (1.) t 21 = -1 t 23

	y 0(I) = -	1 13 x 2 (t 2 11 t 2 3t 2 11 t 2 13 x 1 x 2 +
	t 2 11 t 2 13 x 1 y 3 -t 2 11 t 13 x 2 2 -t 2 11 t y 1(I) = (3t 2 11 t 2 13 x 2 (t 2 1 11 t 2 13 x 0
	x 1 x 2 -3t 2 11 t 2 13 x 2 2 y 2 -t 2 11 t 13 x 0 x 2 2 -t 2 11 t 13 x 1 x 2 y 2 -t 11 t 2 13 x 0 x 2 1 -2t 11 t 2 13 x 0 x 2 2 + t 11 t 2 13 x 1
	x 2 y 2 -2t 2 11 x 2 2 y 2 -4t 2 13 x 2 2 y 2

(θ 21 = π + θ 23 ) and (2.) t 21 = t 23 (θ 21 = θ 23

  3(I) ) = (t 21 t 23 + 1)(-t 23 + t 21 )x 0 (t 2 13 + 1) 3 (t 2 11 + 1) 3 ... = 0 S 3(II) : det(J 3(II) ) = (t 22 t 24 + 1)(-t 24 + t 22 )x 0 (t 2 It can be seen that the first two factors of S 3(I) and S 3(II) in Eq. (32) are the necessary conditions for the 2-RUU (I) and 2-RUU (II) PM to be in the systems K 3(I) and K 3(II) , respectively (as explained in Sections 4.1 and 4.2). They are (1.) t 21 = -1 t 23 (θ 21 = π+θ 23 ) and (2.) t 21 = t 23 To investigate the self-motion in the third mode K 3 of the 4-RUU PM, let us recall the first every configuration in the third mode K 3 of the 4-RUU PM is not always in self-motion. The self-motion of the third mode K 3 occur if and only if the actuated joint angles (t 11 , t 12 , t 13 , t 14 ) fulfil particular conditions.

	14 + 1) 3 (t 2 12 + 1) 3 ... = 0	(32)

(θ 21 = θ 23 ) for the 2-RUU (I) PM, and (1.) t 22 = -1 t 24 (θ 22 = π+θ 24 ) and (2.) t 22 = t 24 (θ 22 = θ 24 )

for the 2-RUU (II) PM. It means that each configuration in the systems K 3(I) and K 3(II) amounts to a self-motion.

R, P, E, U, S denote revolute, prismatic, planar, universal and spherical joints, respectively.

The 4-RUU PM

In the 4-RUU PM, the first R-joint in each limb is actuated. The design parameters are assigned with the same values as a = 2, b = 1, l = 1, r = 2. The set of ten constraint equations is written as a polynomial ideal with variables {x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 } over the coefficient ring C[t 11 , t 12 , t 13 , t 14 ], defined as: I = g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 13 , g 14 . At this point, the following ideal is examined: J = g 1 , g 2 , g 3 , g 4 , g 13 . Since the 4-RUU PM can be assembled by the 2-RUU (I) and 2-RUU (II) PM, the ideal J can be written as the linear combination of the results of primary decomposition from the 2-RUU (I) and 2-RUU (II) PM. It is noteworthy that the first and second components of the 2-RUU (I) and 2-RUU (II) PM are identical, so that J 1(I) = J 1(II) and J 2(I) = J 2(II) .

with:

As a consequence, the 4-RUU PM has four operation modes. To complete the analysis, the remaining equations are added by writing:

The systems K 1 and K 2 are 4-dof operation modes, which correspond to the 1st Schönflies mode and the 2nd Schönflies mode, as explained in Sections 4.1.1 and 4.1.2, respectively. However, the characterization of the system K 3 needs to be discussed further, as presented hereafter.

System K 3 : Third mode

The third mode of the 4-RUU PM is characterized by the system K 3 . In this mode, the primary decomposition leads to the ideal J 3 and all polynomial equations in this ideal should vanish simultaneously. Hence the variables x 3 , y 0 , y 1 , y 2 , y 3 can be obtained in cascade from the ideal J 3 , such that:

)

Not all polynomial in the ideal J 3 vanishes and it remains two polynomial equations, as follows:

Nurahmi, Caro, Wenger, Schadlbauer, and Husty, submitted to Mech. and Mach. Theory 18

x 2 1 -... = 0 (26) As two iso-constrained 2-RUU PM are assembled to be the 4-RUU PM by combining the results of primary decomposition into J 3 , one of the 2-RUU PM is dependent to another. Accordingly, one of the 2-RUU PM can be selected to represent the third mode of the 4-RUU PM. The new ideals are defined corresponding to the two 2-RUU parallel manipulators as follows:

Both ideals in Eq. ( 27) are solved separately to show that they lead to the same results. The variables x 3 , y 0 , y 1 , y 2 , y 3 obtained in Eq. ( 25) are then substituted into ideals L I , L II in Eq. [START_REF] Masouleh | Solving the Forward Kinematic Problem of 4-DOF Parallel Mechanisms (3T1R) with Identical Limb Structures and Revolute Actuators Using the Linear Implicitization Algorithm[END_REF]. The variable x 0 can be solved from g 14 and the equations g 5 , g 6 , g 7 , g 8 split into two factors. The first factors of these equations have the same mathematical expression and lead to the 1-dof self-motion, which will be discussed further in Section 5.3.

The second factors are analysed thereafter. The variable x 1 is solved from each ideal and yields two equations in terms of (x 2 , t 29). Back substitution into Eqs. ( 25)-( 28), all Study parameters can be solved and one of the manipulator poses can be obtained as shown in Fig. 6 with θ 11 = -84

Singularities in 1st Schönflies mode (K 1 )

The determinant of the Jacobian matrix is computed in the system K 1 , which consists of five constraint equations over five variables (x 1 , x 2 , y 0 , y 2 , y 3 ). Hence the 5 × 5 Jacobian matrix can be obtained. The factorization of the determinant of this Jacobian matrix S 1 : det(J 1 ) = 0 yields three factors. The inspection of the first factor shows the singularity configurations that lie in the intersection with the system K 2 . However, this factor is neglected since the systems K 1 and K 2 do not have configurations in common, i.e. x 0 , x 1 , x 2 , x 3 can never vanish simultaneously.

The second factor is y 2 = 0, when the moving platform is coplanar to the base, the 4-RUU PM is always in a singular configuration. Eventually, the third factor of S 1 : det(J 1 ) = 0 is analysed. This factor is added to the system K 1 and the remaining five Study parameters are eliminated. Due to the heavy elimination process, the actuated joint angles are assigned as t 11 = -2, t 12 = -1, and t 13 = 1/2. The elimination yields a univariate polynomial of degree 14 in t 14 as:

Singularities and Self-motions in 2nd Schönflies mode (K 2 )

The determinant of the Jacobian matrix is computed in the system K 2 , which consists of five constraint equations over five variables. Therefore the 5 × 5 Jacobian matrix can be obtained. The determinant of this Jacobian matrix S 2 : det(J 2 ) = 0 consists of three factors too. The investigation of the first factor gives the condition in which the mechanism is in the intersection between the systems K 1 and K 2 . As explained in Section 5.1, this factor is removed. The variable x 1 can be solved and it yields two equations in terms of the actuated joint angles only. The first equation is similar to Eq. ( 28) and the second equation takes the form: 3t = 0 (33) The variable x 2 is an independent parameter. As a consequence, when all the corresponding actuated joints are actuated, there is an additional 1-dof rotational motion exhibited by the moving platform. This motion is a self-motion and it is parametrized by the variable x 2 . Two equations in Eq. ( 28) and Eq. (33) are solved to find the relations among the actuated joint angles in the self-motion, namely: Since two 2-RUU PM are assembled perpendicularly as in the 4-RUU PM, only one example between the self-motion of solutions 1 and 2 is shown. The example of self-motion of solution 2 is shown in Fig. 13 with θ 11 = θ 14 = 90 • and θ 12 = θ 13 = 0 • . Figure 14 shows the example of self-motion of solution 3 with θ 11 = 90 • , θ 12 = 0 • , θ 13 = -90 • , θ 14 = 180 • . Every configuration in the third modes of the 2-RUU (I) and 2-RUU (II) PM amounts to a selfmotion. However, when the 2-RUU (I) and 2-RUU (II) PM are assembled to obtain the 4-RUU PM,