domain: spi chim

text:
Adrien Reynaud
email: adrien.reynaud@ifpen.fr

Mickaël Leblanc
email: mickael.leblanc@ifpen.fr

Stéphane Zinola
email: stephane.zinola@ifpen.fr

Philippe Breuil
email: philippe.breuil@mines-stetienne.fr

Jean-Paul Viricelle
email: jean-paul.viricelle@mines-stetienne.fr

Responses of a Resistive Soot Sensor to Different Mono-Disperse Soot Aerosols

Keywords: soot sensor, aerosol, nanoparticle, on-board diagnostic, electrophoresis

Since 2011, the Euro 5b European standard limits the particle number (PN) emissions in addition to the particulate mass (PM) emissions. New thermal engine equipped vehicles also have to auto-diagnose their own particulate filter (Diesel particulate filter or gasoil particulate filter) using on-board diagnostic (OBD) sensors. Accumulative resistive soot sensors seem to be good candidates for PM measurements. The aim of this study is to bring more comprehension about soot microstructures construction in order to link the response of such a sensor to particle size and concentration. The sensor sensitivity to the particle size has been studied using successively an electrostatic and an aerodynamic classification, showing the same trend.

Introduction

The soot nanoparticle emissions from thermal engines have a harmful impact on human health. Because of their small size, they deeply penetrate our breathing apparatus to reach our lungs and pulmonary alveoli. Then, they can release the carcinogenic chemical compounds they carry such as polycyclic aromatic hydrocarbons (PAH) or sulfates [START_REF] Sassi | Post-traitement des émissions polluantes des moteurs thermiques à combustion interne-Moteurs à allumage par compression[END_REF]. In 2011, the Euro 5b standard introduced a restriction regarding particle number (PN) for Diesel vehicles and it has been extended to gasoline vehicles in 2014 by the Euro 6b standard. The on-board diagnostic (OBD) of the particulate filter (DPF or GPF) is also mentioned. Currently, a pressure sensor (also called Delta P sensor) measures the pressure loss induced by the particulate matter accumulation in the particulate filter [START_REF] Walker | Controlling Particulate Emissions from Diesel Vehicles[END_REF]. However, it seems that this device is not accurate enough to detect the particulate filter leaks [START_REF] Masoudi | PM) Sensors[END_REF] which can cause a significant increase of tailpipe emissions.

Other technologies are currently the object of research works such as the radio-frequency (RF) sensors which are based on the measurement of a RF signal in the DPF cavity [START_REF] Fischerauer | Sensing the soot load in automotive diesel particulate filters by microwave methods[END_REF]. The electric charge sensors are also studied [START_REF] Ntziachristos | Exhaust Particle Sensor for OBD Application[END_REF][START_REF] Maricq | The impact of voltage and flow on the electrostatic soot sensor and the implications for its use as a diesel particulate filter monitor[END_REF] and rely on the measurement of the electrical current induced by particles pre-charged by a corona charger. The accumulative soot sensors present several advantages in this context: they are cheap and easy to integrate on the light-duty vehicles exhaust line. Among the accumulative soot sensors, we can distinguish the capacitive and the resistive soot sensors [START_REF] Kamimoto | A review of soot sensors considered for on-board diagnostics application[END_REF]. This study is focused on this last technology. In the past years, several studies showed that resistive soot sensors are a good track toward PN measurement [START_REF] Hagen | Conductometric soot sensor for automotive exhausts: Initial studies[END_REF][START_REF] Grob | Conductivity for soot sensing: Possibilities and limitations[END_REF][START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. For now, a particulate mass concentration threshold can be detected [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF][START_REF] Feulner | Conductometric Sensor for Soot Mass Flow Detection in Exhausts of Internal Combustion Engines[END_REF], but further research is required to move towards a PN threshold detection.

The sensitive area of the resistive soot sensor is made of two interdigitated platinum electrodes generating an electric field. As the soot particles are naturally electrically charged in the Diesel engine exhaust gas [START_REF] Kittelson | Electrostatic Collection of Diesel Particles[END_REF][START_REF] Jung | Measurement of Electrical Charge on Diesel Particles[END_REF], the deposit of the aerosol on the electrodes is accelerated by electrostatic phenomena [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. The sensor measures the conductance of the soot microstructures that create bridges between the electrodes, creating conductive pathways [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. The work of Grondin et al. revealed the existence of an optimal value of the polarization tension due to an equilibrium between the deposition rate by electrophoresis and the destruction of the conductive pathways by Joule effect [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. Grondin et al. also investigated the influence of the soot composition and found out that the conductance of the sensitive element of the sensor clearly decreases with an increasing organic fraction [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. Another important mechanism influencing the particle deposit is the thermophoresis and it has been used to improve the collection efficiency [START_REF] Malik | A Potential Soot Mass Determination Method from Resistivity Measurement of Thermophoretically Deposited Soot[END_REF]. Its negative impact on the deposition rate has also been investigated. A temperature gradient in the vicinity of the sensor can be artificially generated by a heater on the back of the substrate [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF] or internally caused by the heating of the deposited soot by Joule effect [START_REF] Hagen | Conductometric Soot Sensors: Internally Caused Thermophoresis as an Important Undesired Side Effect[END_REF]. Other experimental parameters have been studied such as the particle mass flow [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF][START_REF] Feulner | Conductometric Sensor for Soot Mass Flow Detection in Exhausts of Internal Combustion Engines[END_REF] or the sensor nose design [START_REF] Ochs | Particulate Matter Sensor for On Board Diagnostics (OBD) of Diesel Particulate Filters (DPF)[END_REF][START_REF] Husted | Sensing of Particulate Matter for On-Board Diagnosis of Particulate Filters[END_REF]. These studies lead to a better understanding of the soot particle deposition mechanisms.

The aim of this study is to improve our understanding of the soot deposition mechanisms on a resistive soot sensor by investigating the impact of a new parameter: the particle size. This topic was first introduced in a previous communication [START_REF] Reynaud | Soot Particle Classifications in the Context of a Resistive Sensor Study[END_REF]. Indeed, this is a way to affect both the drag force and the electrical charge distribution [START_REF] Kittelson | Electrostatic Collection of Diesel Particles[END_REF], which modify theoretically the force balance of each particle. Since a Diesel soot aerosol is composed of a wide range of particle diameters, the sensor signal may be influenced by each class of sizes. In order to perform a sensitive analysis of the particle size effect, two experimental set-ups have been used. First, the classification has been performed using a Differential Mobility Analyzer (DMA 3081, TSI, Shoreview, MN, USA). Then, an Aerodynamic Aerosol Classifier (AAC, Cambustion, Cambridge, UK) was used.

Materials and Methods

Experimental Set-up Overview

The objective of the experimental set-up is to provide to a sensor a mono-disperse aerosol from a poly-disperse soot one. It has to be mentioned that most of the set-up (particle generation, temperature monitoring, sensor monitoring) was presented in a previous work [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF]. However, the classification part of this set-up is described step-by-step in the following section.

The test bench (cf. Figure 1) was composed of a mini-CAST (Combustion Aerosol STandard, Jing Ltd., Zollikofen, Switzerland) that generated a poly-disperse soot aerosol with a concentration C 1 = 2.5 × 10 8 part./cm 3 at the temperature T 1 = 70 • C (1). The soot particles were carried to the diluter by a beveled 10 mm diameter stainless steel tube. Then, the aerosol was diluted 10 times by a clean, dry air in a VKL 10 diluter (Palas GmbH, Karlsruhe, Germany) in order to avoid any condensation and to reach temperature and concentration compatible with the inlet conditions of the DMA (T 2 = 28 • C and C 2 ∼ 10 7 part./cm 3 ) (2). The classification was operated by either the DMA or the AAC (T 3 = 28 • C and C 3 ∼ 10 4 -10 5 part./cm 3

depending on the classification technique) (3).

Sensors 2018, 18, x FOR PEER REVIEW 2 of 11 engine exhaust gas [START_REF] Kittelson | Electrostatic Collection of Diesel Particles[END_REF][START_REF] Jung | Measurement of Electrical Charge on Diesel Particles[END_REF], the deposit of the aerosol on the electrodes is accelerated by electrostatic phenomena [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. The sensor measures the conductance of the soot microstructures that create bridges between the electrodes, creating conductive pathways [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. The work of Grondin et al. revealed the existence of an optimal value of the polarization tension due to an equilibrium between the deposition rate by electrophoresis and the destruction of the conductive pathways by Joule effect [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. Grondin et al. also investigated the influence of the soot composition and found out that the conductance of the sensitive element of the sensor clearly decreases with an increasing organic fraction [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. Another important mechanism influencing the particle deposit is the thermophoresis and it has been used to improve the collection efficiency [START_REF] Malik | A Potential Soot Mass Determination Method from Resistivity Measurement of Thermophoretically Deposited Soot[END_REF]. Its negative impact on the deposition rate has also been investigated. A temperature gradient in the vicinity of the sensor can be artificially generated by a heater on the back of the substrate [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF] or internally caused by the heating of the deposited soot by Joule effect [START_REF] Hagen | Conductometric Soot Sensors: Internally Caused Thermophoresis as an Important Undesired Side Effect[END_REF]. Other experimental parameters have been studied such as the particle mass flow [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF][START_REF] Feulner | Conductometric Sensor for Soot Mass Flow Detection in Exhausts of Internal Combustion Engines[END_REF] or the sensor nose design [START_REF] Ochs | Particulate Matter Sensor for On Board Diagnostics (OBD) of Diesel Particulate Filters (DPF)[END_REF][START_REF] Husted | Sensing of Particulate Matter for On-Board Diagnosis of Particulate Filters[END_REF]. These studies lead to a better understanding of the soot particle deposition mechanisms. The aim of this study is to improve our understanding of the soot deposition mechanisms on a resistive soot sensor by investigating the impact of a new parameter: the particle size. This topic was first introduced in a previous communication [START_REF] Reynaud | Soot Particle Classifications in the Context of a Resistive Sensor Study[END_REF]. Indeed, this is a way to affect both the drag force and the electrical charge distribution [START_REF] Kittelson | Electrostatic Collection of Diesel Particles[END_REF], which modify theoretically the force balance of each particle. Since a Diesel soot aerosol is composed of a wide range of particle diameters, the sensor signal may be influenced by each class of sizes. In order to perform a sensitive analysis of the particle size effect, two experimental set-ups have been used. First, the classification has been performed using a Differential Mobility Analyzer (DMA 3081, TSI, Shoreview, MN, United States). Then, an Aerodynamic Aerosol Classifier (AAC, Cambustion, Cambridge, UK) was used.

Materials and Methods

Experimental Set-up Overview

The objective of the experimental set-up is to provide to a sensor a mono-disperse aerosol from a poly-disperse soot one. It has to be mentioned that most of the set-up (particle generation, temperature monitoring, sensor monitoring) was presented in a previous work [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF]. However, the classification part of this set-up is described step-by-step in the following section.

The test bench (cf. Figure 1) was composed of a mini-CAST (Combustion Aerosol STandard, Jing Ltd., Zollikofen, Switzerland) that generated a poly-disperse soot aerosol with a concentration 𝐶 = 2.5 × 10 part./cm at the temperature 𝑇 = 70 °C (1). The soot particles were carried to the diluter by a beveled 10 mm diameter stainless steel tube. Then, the aerosol was diluted 10 times by a clean, dry air in a VKL 10 diluter (Palas GmbH, Karlsruhe, Germany) in order to avoid any condensation and to reach temperature and concentration compatible with the inlet conditions of the DMA (𝑇 = 28 °C and 𝐶 ~10 part./cm ) (2). The classification was operated by either the DMA or the AAC (𝑇 = 28 °C and 𝐶 ~10 -10 part./cm depending on the classification technique) (3). Finally, the mono-disperse aerosol was driven to the 9 mm internal diameter soot sensor chamber (cf. Figure 2) -heated by a tubular furnace at the temperature of 𝑇 = 180 °C -leading to a Reynolds number 𝑅𝑒 = 55 (4). The flow was drained by a GilAir pump at 1.5 L/min. A simple heat exchanger made of an 1/4 inch external diameter and 50 cm long stainless steel tube was used Finally, the mono-disperse aerosol was driven to the 9 mm internal diameter soot sensor chamber (cf. Figure 2)-heated by a tubular furnace at the temperature of T 4 = 180 • C-leading to a Reynolds number Re = 55 (4). The flow was drained by a GilAir pump at 1.5 L/min. A simple heat exchanger made of an 1/4 inch external diameter and 50 cm long stainless steel tube was used to decrease the temperature of the heated gas to the ambient temperature. Moreover, a High Efficiency Particulate Air (HEPA) filter was used to protect the pump from potential degradation caused by the soot aerosol.

Resistive Soot Sensor

In a previous study (Ciclamen 2 French national project [START_REF] Millet | Projet CICLAMEN 2. Capteur de Particules Pour échappement Moteur; internal report[END_REF]), a resistive soot sensor has been developed by three partners (École des Mines de Saint-Étienne, IFP Énergies Nouvelles and EFI Automotive). The sensor is composed of an alumina substrate (50 mm × 5 mm × 1 mm) that supports the sensing element (3 mm× 3 mm) on one side and the heater on the other side. They are both made of platinum. The sensing element is composed of two interdigitated electrodes which are made from a platinum ink deposition at the substrate surface by screen-printing. Then, the platinum layer is engraved using a laser [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. The electrodes are 40 µm width and the interelectrode width is 20 µm (cf. Figure 3). As well as the electrodes, a heater is printed on the back of the alumina substrate in order to perform active regenerations of the soot to clean up the sensor (700 °C).

The soot sensor response is the conductance of the sensing element. A 6430 Sub-Femtoamp Remote Sourcemeter (Keithley, Solon, OH, United States) is used to apply a polarization tension between the electrodes. It helps to the construction of bridge-like structures between the electrodes (as it can be seen on Figure 3) which are responsible of the conductance increase. The construction time of the first bridge linking the electrodes is called "percolation time". Both the heater and the sensing element were monitored by a specific LabView software routine developed during a previous study [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF].

Resistive Soot Sensor

In a previous study (Ciclamen 2 French national project [START_REF] Millet | Projet CICLAMEN 2. Capteur de Particules Pour échappement Moteur; internal report[END_REF]), a resistive soot sensor has been developed by three partners (École des Mines de Saint-Étienne, IFP Énergies Nouvelles and EFI Automotive). The sensor is composed of an alumina substrate (50 mm × 5 mm × 1 mm) that supports the sensing element (3 mm× 3 mm) on one side and the heater on the other side. They are both made of platinum. The sensing element is composed of two interdigitated electrodes which are made from a platinum ink deposition at the substrate surface by screen-printing. Then, the platinum layer is engraved using a laser [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. The electrodes are 40 µm width and the inter-electrode width is 20 µm (cf. Figure 3). As well as the electrodes, a heater is printed on the back of the alumina substrate in order to perform active regenerations of the soot to clean up the sensor (700 • C).

to decrease the temperature of the heated gas to the ambient temperature. Moreover, a High Efficiency Particulate Air (HEPA) filter was used to protect the pump from potential degradation caused by the soot aerosol.

Resistive Soot Sensor

In a previous study (Ciclamen 2 French national project [START_REF] Millet | Projet CICLAMEN 2. Capteur de Particules Pour échappement Moteur; internal report[END_REF]), a resistive soot sensor has been developed by three partners (École des Mines de Saint-Étienne, IFP Énergies Nouvelles and EFI Automotive). The sensor is composed of an alumina substrate (50 mm × 5 mm × 1 mm) that supports the sensing element (3 mm× 3 mm) on one side and the heater on the other side. They are both made of platinum. The sensing element is composed of two interdigitated electrodes which are made from a platinum ink deposition at the substrate surface by screen-printing. Then, the platinum layer is engraved using a laser [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. The electrodes are 40 µm width and the interelectrode width is 20 µm (cf. Figure 3). As well as the electrodes, a heater is printed on the back of the alumina substrate in order to perform active regenerations of the soot to clean up the sensor (700 °C).

The soot sensor response is the conductance of the sensing element. A 6430 Sub-Femtoamp Remote Sourcemeter (Keithley, Solon, OH, United States) is used to apply a polarization tension between the electrodes. It helps to the construction of bridge-like structures between the electrodes (as it can be seen on Figure 3) which are responsible of the conductance increase. The construction time of the first bridge linking the electrodes is called "percolation time". Both the heater and the sensing element were monitored by a specific LabView software routine developed during a previous study [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. The soot sensor response is the conductance of the sensing element. A 6430 Sub-Femtoamp Remote Sourcemeter (Keithley, Solon, OH, United States) is used to apply a polarization tension between the electrodes. It helps to the construction of bridge-like structures between the electrodes (as it can be seen on Figure 3) which are responsible of the conductance increase. The construction time of the first bridge linking the electrodes is called "percolation time". Both the heater and the sensing element were monitored by a specific LabView software routine developed during a previous study [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF].

Soot Generation

In this work, soot particles generated by a burner were studied to model the typical soot particles emitted by a Diesel engine in order to obtain more reproducible experimental conditions, instead of using an actual engine. The mini-CAST is based on a propane-air diffusion flame quenched by a nitrogen flow. The air, nitrogen and propane flows can be modified to produce aerosols with different size distributions and chemical natures [START_REF] Moore | Mapping the Operation of the Miniature Combustion Aerosol Standard (Mini-CAST) Soot Generator[END_REF]. The criterion for choosing among the operating points was the concentration and the median mobility diameter of the generated soot particles. Based on the Grondin et al. study, it led to the operating point: Q C3H8 = 3.6 L/min, Q air,oxydation = 93 L/min, Q N 2 ,mix = 450 L/min, Q N 2 ,quench = 9 L/min. The fuel air equivalence ratio of such a combustion was 0.95, the number and mass concentration of the generated aerosol were respectively 2.5 × 10 8 part./cm 3 and 91 mg/m 3 [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. The chemical nature of the generated aerosol was investigated by Grondin et al. in a previous work [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. Several settings have been characterized but for the operating point considered in the present work, it can be highlighted that the carbon black concentration is 70 mg/m 3 , which represents 77% of the mass concentration of the aerosol. The typical soot particles emitted from Diesel engines are reported to contain a lower proportion of carbon black. The proportion of elemental carbon (mostly carbon black) is most of the time under 70% [START_REF] Shah | Emission Rates of Particulate Matter and Elemental and Organic Carbon from In-Use Diesel Engines[END_REF].

Finally, the particle size distribution was measured both with an electrostatic classifier and an aerodynamic classifier as mentioned before, and shown on Figure 4. It can be seen that the median mobility diameter and the median aerodynamic diameter were different (respectively 70 nm and roughly 60 nm) as well as the standard deviation. This can be explained by the relation between the aerodynamic diameter d a and the electric mobility diameter d m which is not linear because of the Cunningham slip correction factor C c . Indeed, the relation between the mobility diameter, the volume equivalent diameter and the aerodynamic diameter is given by McMurry et al. [START_REF] Mcmurry | The Relationship between Mass and Mobility for Atmospheric Particles: A New Technique for Measuring Particle Density[END_REF]:

d 2 a C c (d a ) = 1 χ ρ p ρ 0 d 2 ve C c (d ve ) = ρ e f f ρ 0 d 2 m C c (d m ), ( 1
)
where χ is the shape factor used to correct the drag of non-spherical particles, ρ p the density of the particle, ρ 0 an arbitrary density (1000 kg/m 3 ) and ρ e f f the effective density, defined here as the ratio:

ρ p d 3 ve d 3 m [24].
Sensors 2018, 18, x FOR PEER REVIEW 4 of 11

Soot Generation

In this work, soot particles generated by a burner were studied to model the typical soot particles emitted by a Diesel engine in order to obtain more reproducible experimental conditions, instead of using an actual engine. The mini-CAST is based on a propane-air diffusion flame quenched by a nitrogen flow. The air, nitrogen and propane flows can be modified to produce aerosols with different size distributions and chemical natures [START_REF] Moore | Mapping the Operation of the Miniature Combustion Aerosol Standard (Mini-CAST) Soot Generator[END_REF]. The criterion for choosing among the operating points was the concentration and the median mobility diameter of the generated soot particles. Based on the Grondin et Finally, the particle size distribution was measured both with an electrostatic classifier and an aerodynamic classifier as mentioned before, and shown on Figure 4. It can be seen that the median mobility diameter and the median aerodynamic diameter were different (respectively 70 nm and roughly 60 nm) as well as the standard deviation. This can be explained by the relation between the aerodynamic diameter 𝑑 and the electric mobility diameter 𝑑 which is not linear because of the Cunningham slip correction factor 𝐶 . Indeed, the relation between the mobility diameter, the volume equivalent diameter and the aerodynamic diameter is given by McMurry et al. [START_REF] Mcmurry | The Relationship between Mass and Mobility for Atmospheric Particles: A New Technique for Measuring Particle Density[END_REF]:

𝑑 𝐶 (𝑑 ) = 1 𝜒 𝜌 𝜌 𝑑 𝐶 (𝑑 ) = 𝜌 𝜌 𝑑 𝐶 (𝑑 ), (1)
where 𝜒 is the shape factor used to correct the drag of non-spherical particles, 𝜌 the density of the particle, 𝜌 an arbitrary density (1000 kg/m 3 ) and 𝜌 the effective density, defined here as the ratio: 𝜌

Electrostatic Classification by a DMA

Inside the DMA, soot particles are carried between two concentric electrodes by an air flow (sheath flow Q sheath = 15 L/min in this work). The particles are then submitted to both the Coulomb force and the drag force induced by the sheath flow. By changing the electric field or the sheath flow, the electrical mobility diameter of the outlet mono-disperse aerosol can be modified. After their neutralization by a RX-neutralizer (TSI 3088) it is assumed that the number N n of elementary charges e (= 4.8 × 10 -10 e.s.u.) carried by any particle follows the Boltzmann equilibrium distribution [START_REF] Liu | Electrical neutralization of aerosols[END_REF]:

N n = N 0 × exp - n 2 e 2 d m k B T , ( 2
)
where N 0 is the number of neutral particles, d m the mobility diameter of the particles, k B the Boltzmann constant and T the temperature of the particles. Consequently, a low number of particles are actually carrying electric charges. Thus, according to equation ( 2), a small amount only of classified particles are transmitted to the outlet of the DMA i.e., approximately 20% for the median mobility diameter of d m = 70 nm [START_REF] Hinds | Aerosol Technology. Properties, Behaviour, and Measurement of Airborne Particles[END_REF] (cf. Figure 5). Furthermore, the aerosol thus generated is essentially composed of single charge nanoparticles.

Sensors 2018, 18, x FOR PEER REVIEW 5 of 11

Electrostatic Classification by a DMA

Inside the DMA, soot particles are carried between two concentric electrodes by an air flow (sheath flow 𝑄 = 15 l/min in this work). The particles are then submitted to both the Coulomb force and the drag force induced by the sheath flow. By changing the electric field or the sheath flow, the electrical mobility diameter of the outlet mono-disperse aerosol can be modified. After their neutralization by a RX-neutralizer (TSI 3088) it is assumed that the number 𝑁 of elementary charges 𝑒 ( = 4.8 × 10 e. s. u. ) carried by any particle follows the Boltzmann equilibrium distribution [START_REF] Liu | Electrical neutralization of aerosols[END_REF]:

𝑁 = 𝑁 × 𝑒𝑥𝑝 - ² ² , ( 2
)
where 𝑁 is the number of neutral particles, 𝑑 the mobility diameter of the particles, 𝑘 the Boltzmann constant and 𝑇 the temperature of the particles. Consequently, a low number of particles are actually carrying electric charges. Thus, according to equation ( 2), a small amount only of classified particles are transmitted to the outlet of the DMA i.e., approximately 20% for the median mobility diameter of 𝑑 = 70 nm [START_REF] Hinds | Aerosol Technology. Properties, Behaviour, and Measurement of Airborne Particles[END_REF] (cf. Figure 5). Furthermore, the aerosol thus generated is essentially composed of single charge nanoparticles. The blue area represents the concentration computed by the SMPS algorithm, based on the Boltzmann equilibrium. The effective aerosol (i.e., the particles sent to the sensor chamber) is the green area.

Aerodynamic Classification by an AAC

The aerodynamic classification was performed by an AAC (Cambustion). Because of their injection between two rotating cylinders, soot particles are submitted to both an aerodynamic flow (sheath flow) and their own centrifugal force. Thus, particles are classified according to their aerodynamic diameter 𝑑 which is the diameter of a sphere with standard density that settles at the same terminal velocity as the particle of interest. Since this method does not classify them according to their charge, a higher number of particles is transmitted to the outlet mono-disperse aerosol (cf. Figure 6). Moreover, we could suppose that the global charge of the aerosol has an impact on its sensitivity to electrophoresis, leading to different deposit phenomena, probably more realistic than the previous technique. The blue area represents the concentration computed by the SMPS algorithm, based on the Boltzmann equilibrium. The effective aerosol (i.e., the particles sent to the sensor chamber) is the green area.

Aerodynamic Classification by an AAC

The aerodynamic classification was performed by an AAC (Cambustion). Because of their injection between two rotating cylinders, soot particles are submitted to both an aerodynamic flow (sheath flow) and their own centrifugal force. Thus, particles are classified according to their aerodynamic diameter d a which is the diameter of a sphere with standard density that settles at the same terminal velocity as the particle of interest. Since this method does not classify them according to their charge, a higher number of particles is transmitted to the outlet mono-disperse aerosol (cf. Figure 6). Moreover, we could suppose that the global charge of the aerosol has an impact on its sensitivity to electrophoresis, leading to different deposit phenomena, probably more realistic than the previous technique.

Sensors 2018, 18, x FOR PEER REVIEW 5 of 11

Electrostatic Classification by a DMA

Inside the DMA, soot particles are carried between two concentric electrodes by an air flow (sheath flow 𝑄 = 15 l/min in this work). The particles are then submitted to both the Coulomb force and the drag force induced by the sheath flow. By changing the electric field or the sheath flow, the electrical mobility diameter of the outlet mono-disperse aerosol can be modified. After their neutralization by a RX-neutralizer (TSI 3088) it is assumed that the number 𝑁 of elementary charges 𝑒 ( = 4.8 × 10 e. s. u. ) carried by any particle follows the Boltzmann equilibrium distribution [START_REF] Liu | Electrical neutralization of aerosols[END_REF]:

𝑁 = 𝑁 × 𝑒𝑥𝑝 - ² ² , ( 2
)
where 𝑁 is the number of neutral particles, 𝑑 the mobility diameter of the particles, 𝑘 the Boltzmann constant and 𝑇 the temperature of the particles. Consequently, a low number of particles are actually carrying electric charges. Thus, according to equation ( 2), a small amount only of classified particles are transmitted to the outlet of the DMA i.e., approximately 20% for the median mobility diameter of 𝑑 = 70 nm [START_REF] Hinds | Aerosol Technology. Properties, Behaviour, and Measurement of Airborne Particles[END_REF] (cf. Figure 5). Furthermore, the aerosol thus generated is essentially composed of single charge nanoparticles. The blue area represents the concentration computed by the SMPS algorithm, based on the Boltzmann equilibrium. The effective aerosol (i.e., the particles sent to the sensor chamber) is the green area.

Aerodynamic Classification by an AAC

The aerodynamic classification was performed by an AAC (Cambustion). Because of their injection between two rotating cylinders, soot particles are submitted to both an aerodynamic flow (sheath flow) and their own centrifugal force. Thus, particles are classified according to their aerodynamic diameter 𝑑 which is the diameter of a sphere with standard density that settles at the same terminal velocity as the particle of interest. Since this method does not classify them according to their charge, a higher number of particles is transmitted to the outlet mono-disperse aerosol (cf. Figure 6). Moreover, we could suppose that the global charge of the aerosol has an impact on its sensitivity to electrophoresis, leading to different deposit phenomena, probably more realistic than the previous technique. The resolution R s of the AAC is defined as the set point divided by the full width half maximum of the instrument's transfer function in the size domain [START_REF]Aerodynamic Aerosol Classifier AAC User Manual, Version 1.10; User Interface Software v1.100[END_REF]. It can be expressed as a function of ∆ which represents the width of the size distribution function at mid concentration (cf. Equation ( 3)):

R s = d a ∆ , (3)
In this work, the value of ∆ was kept approximatively constant. The settings are summed up in Table 1.

Experimental Protocol

Before the signal acquisition, one hour was spent heating up the soot sensor chamber, flowing clean and heated air through it. Moreover, the soot generator (mini-CAST) required at least 50 min to reach a steady state operation. The size distribution function was then measured either by the DMA or the AAC and the CPC in order to check the number concentration and the operating of every device of the set-up. Finally, the soot sensor was cleaned, being wiped with a clean ethanol soaked-cloth. After being positioned with the sensing element facing the aerosol flow, the sensor was regenerated at 700 • C for 3 min.

Results and Discussion

Response to an Electrostatically Classified Aerosol

In order to perform a sensitivity analysis over the mobility diameter, four different diameters were studied: 60 nm, 70 nm, 80 nm and 100 nm. 60 nm, 70 nm and 80 nm were chosen to have a similar concentration (cf. Figure 4). Several sensor signal acquisitions are reported in Figure 7. Moreover, several polarization tensions were studied (30 V, 50 V, 70 V and 100 V). However, the measurements did not let us draw a conclusion about any correlation between the polarization tension and the percolation time. This was consistent with a previous study [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF], concluding that for the present operating point of the soot generator (mini-CAST Jing Ltd.) the polarization tension does not affect the response time of the sensor between 40 V and 90 V.

When compared with the response to high concentration aerosols (10 8 part./cm 3 ) [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF], the sensor signals were composed of rare and random conductance steps (cf. Figure 7), due to the soot bridge constructions. Indeed, in the case of typical responses to high concentration aerosols, until 6 events per second were detected for the mini-CAST operating point considered in this study, with a polarization tension equal to 60 V [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. The randomness is explained by the low PN concentration (from 9.0 × 10 3 part./cm 3 to 1.3 × 10 4 part./cm 3 depending on the particle size) which made these events infrequent, leading to a low reproducibility. Concerning the height of the steps, it varied from 0.4 µS (60 nm) to 0.02 µS (100 nm). This could be explained by several phenomena such as different morphologies or chemical natures of the soot bridges. Moreover, a continuous decrease of the conductance was observed after every step. A hypothesis is that an erosion mechanism contributes to decrease the radius of the bridges, decreasing slowly the conductance. Under this assumption, the energy necessary to activate the erosion is provided by the electrical current crossing the bridge-like structure, producing heat by Joule effect. A similar phenomenon has been observed by Sediako et al. [START_REF] Sediako | Real-time observation of soot aggregate oxidation in an Environmental Transmission Electron Microscope[END_REF]: by heating a single soot particle at low temperature (550-800 °C), they measured the real-time morphological changes within an Environmental Transmission Electron Microscope (ETEM).

According to this hypothesis, it is expected that a similar phenomenon may be observed for real Diesel soot. Indeed, Sharma et al. [START_REF] Sharma | Experimental Study of Carbon Black and Diesel Engine Soot Oxidation Kinetics Using Thermogravimetric Analysis[END_REF] report that the activation energy for Diesel soot is close to the average value for several carbon black samples. However, at the moment, the electrical and thermal conductivities of the soot are not accurate enough to evaluate the temperature of the bridge and then to conclude about the nature of the erosion mechanisms.

Considering the randomness of the signals and their low reproducibility, the percolation time was the only parameter that was studied in detail here. A correlation had been identified between the percolation time and the mobility diameter (Figure 8). However, the 100 nm aerosols were not considered for this correlation due to the lower PN concentration (roughly half the concentration of the other points because of the Gaussian shaped size distribution function). Indeed, previous studies have shown that decreasing concentrations leads to increasing percolation times [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. Nevertheless, the percolation time did not follow the Gaussian shape of the size distribution function, meaning that the trend induced by the size of the particles was predominant over the concentration effect in this range. Moreover, a continuous decrease of the conductance was observed after every step. A hypothesis is that an erosion mechanism contributes to decrease the radius of the bridges, decreasing slowly the conductance. Under this assumption, the energy necessary to activate the erosion is provided by the electrical current crossing the bridge-like structure, producing heat by Joule effect. A similar phenomenon has been observed by Sediako et al. [START_REF] Sediako | Real-time observation of soot aggregate oxidation in an Environmental Transmission Electron Microscope[END_REF]: by heating a single soot particle at low temperature (550-800 • C), they measured the real-time morphological changes within an Environmental Transmission Electron Microscope (ETEM).

According to this hypothesis, it is expected that a similar phenomenon may be observed for real Diesel soot. Indeed, Sharma et al. [START_REF] Sharma | Experimental Study of Carbon Black and Diesel Engine Soot Oxidation Kinetics Using Thermogravimetric Analysis[END_REF] report that the activation energy for Diesel soot is close to the average value for several carbon black samples. However, at the moment, the electrical and thermal conductivities of the soot are not accurate enough to evaluate the temperature of the bridge and then to conclude about the nature of the erosion mechanisms.

Considering the randomness of the signals and their low reproducibility, the percolation time was the only parameter that was studied in detail here. A correlation had been identified between the percolation time and the mobility diameter (Figure 8). However, the 100 nm aerosols were not considered for this correlation due to the lower PN concentration (roughly half the concentration of the other points because of the Gaussian shaped size distribution function). Indeed, previous studies have shown that decreasing concentrations leads to increasing percolation times [START_REF] Brunel | Smart Soot Sensor for Particulate Filter OBD[END_REF]. Nevertheless, the percolation time did not follow the Gaussian shape of the size distribution function, meaning that the trend induced by the size of the particles was predominant over the concentration effect in this range.

Response to an Aerodynamically Classified Aerosol

In the previous section, the low concentration of the mono-disperse aerosols was identified as a randomness factor, having a negative impact on the reproducibility of the results. In order to improve the experiments, the aerosol was classified using an AAC, leading to more concentrated aerosols ( 1.3 × 10 part./cm vs. 7.2 × 10 part./cm ) without electric charge segregation. Two different mono-disperse aerosols were studied: 70 nm and 50 nm of median aerodynamic diameter. These specific diameters were chosen because of the close number concentration (cf. Table 1). In each case, two measurements were done to evaluate the reproducibility of the sensor response. Moreover, the polarization tension was 60 V in each case.

In Figure 9, it can be seen that the sensor signals had a significantly different shape, closer to the typical responses observed in previous studies performed at high concentrations and for polydisperse aerosols [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF]. Nevertheless, a zoom on the signal close to the percolation time showed that the conductance was still made of random steps with a higher frequency (cf. Figure 9b). Several negative steps were observed and could be attributed to bridges blow off [START_REF] Fragkiadoulakis | Modeling a resistive soot sensor by particle deposition mechanisms[END_REF]. What is more, the conductance decreases after each step seemed to be lower or nonexistent and the height of the steps (approximately 0.01 µS) was significantly lower compared to the response to the DMA classified aerosols. Also, the reproducibility of the signal was improved.

Response to an Aerodynamically Classified Aerosol

In the previous section, the low concentration of the mono-disperse aerosols was identified as a randomness factor, having a negative impact on the reproducibility of the results. In order to improve the experiments, the aerosol was classified using an AAC, leading to more concentrated aerosols (1.3 × 10 4 part./cm 3 vs. 7.2 × 10 5 part./cm 3 ) without electric charge segregation. Two different mono-disperse aerosols were studied: 70 nm and 50 nm of median aerodynamic diameter. These specific diameters were chosen because of the close number concentration (cf. Table 1). In each case, two measurements were done to evaluate the reproducibility of the sensor response. Moreover, the polarization tension was 60 V in each case.

In Figure 9, it can be seen that the sensor signals had a significantly different shape, closer to the typical responses observed in previous studies performed at high concentrations and for poly-disperse aerosols [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF]. Nevertheless, a zoom on the signal close to the percolation time showed that the conductance was still made of random steps with a higher frequency (cf. Figure 9b). Several negative steps were observed and could be attributed to bridges blow off [START_REF] Fragkiadoulakis | Modeling a resistive soot sensor by particle deposition mechanisms[END_REF]. What is more, the conductance decreases after each step seemed to be lower or nonexistent and the height of the steps (approximately 0.01 µS) was significantly lower compared to the response to the DMA classified aerosols. Also, the reproducibility of the signal was improved.

Response to an Aerodynamically Classified Aerosol

In the previous section, the low concentration of the mono-disperse aerosols was identified as a randomness factor, having a negative impact on the reproducibility of the results. In order to improve the experiments, the aerosol was classified using an AAC, leading to more concentrated aerosols ( 1.3 × 10 part./cm vs. 7.2 × 10 part./cm ) without electric charge segregation. Two different mono-disperse aerosols were studied: 70 nm and 50 nm of median aerodynamic diameter. These specific diameters were chosen because of the close number concentration (cf. Table 1). In each case, two measurements were done to evaluate the reproducibility of the sensor response. Moreover, the polarization tension was 60 V in each case.

In Figure 9, it can be seen that the sensor signals had a significantly different shape, closer to the typical responses observed in previous studies performed at high concentrations and for polydisperse aerosols [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF]. Nevertheless, a zoom on the signal close to the percolation time showed that the conductance was still made of random steps with a higher frequency (cf. Figure 9b). Several negative steps were observed and could be attributed to bridges blow off [START_REF] Fragkiadoulakis | Modeling a resistive soot sensor by particle deposition mechanisms[END_REF]. What is more, the conductance decreases after each step seemed to be lower or nonexistent and the height of the steps (approximately 0.01 µS) was significantly lower compared to the response to the DMA classified aerosols. Also, the reproducibility of the signal was improved. Before coming to conclusions, it is important to mention that we can dismiss the impact of the concentration because in this case, 70 nm aerosols had a lower concentration than 50 nm aerosols. Indeed, on the one hand, a lower concentration tends to increase the percolation time. On the other hand, it was previously observed that a higher diameter tends to decrease the percolation time (cf. Figure 8). Consequently, as the percolation time was lower for the 70 nm aerosols, it should be true a fortiori if their concentrations were strictly equal.

Discussions

It was observed that the percolation time increased when the particle size decreased. The same trend was observed using two different classification techniques: a DMA and an AAC. This means that the sensor is sensitive to the particle size.

Several phenomena can explain the impact of the particle size on the sensor response. The first hypothesis is a geometric criterion saying that a higher number of small nanoparticles is required to link two 20 µm spaced electrodes. This leads to a lower bridge construction frequency that is to say a lower slope of the signal dG dt , which was indeed observed experimentally in Figure 9. Moreover, the size of the particles also affects their motion in the neighborhood of the electrodes. In their numerical study, Teike et al., wrote the force balance of a particle close to the electrode as [START_REF] Teike | Multiscale Lattice-Boltzmann Approach for Electrophoretic Particle Deposition[END_REF]:

m p ∂ → v p ∂t = → F e + → F d + → F b , (4)
where m p is the mass of the particle and → v p its velocity. The forces can be expressed by:

               → F e = -N
where → E is the electric field generated by the polarization of the electrodes, d ve the geometric equivalent diameter of the particle, C c the Cunningham slip correction coefficient. η f , ν f , T f and ρ f are respectively the dynamic viscosity, the kinematic viscosity, the temperature and the density of the gas. ∆t is the relaxation time of the particle. The randomness of the Brownian motion is modeled by the vector → ζ , which is a zero-mean, unit-variance-independent Gaussian random number.

As it can be seen in the Equation ( 5), the drag force → F d can be approximated by the Stokes' law modified by a shape factor χ and depends directly on the particle diameter. The Brownian force also depends directly on the particle diameter. The Coulomb force, however, depends indirectly on the particle diameter because the number of elementary charges carried by a soot particle follows the Boltzmann equilibrium (cf. equation ( 2)) which also depends on the size of the particle. Among those three forces, all of them depends on the size of the particle, leading to the conclusion that the trajectory in the vicinity of the sensor varies according to the size of the particle. We can make the hypothesis that the deposit position of the bigger particles is more suitable for the construction of the bridge-like structures rather than smaller particles.

However, the sensitivity to particle size can be explained by other factors, such as the chemical nature of soot. Indeed, the chemical nature of the soot may also depend on their size, and it is known that the electrical conductivity of the bridge-like structures depends on their chemical nature [START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF].

Conclusions

The aim of this study was to investigate the sensitivity of a resistive soot sensor response to the particle size. To do this, a mono-disperse aerosol was first generated by a DMA. The physical principle of the DMA led to low concentration aerosols (<10 5 part./cm 3 ), inducing a low reproducibility of the sensor signal. The AAC, due to its physical principle, was able to generate more concentrated aerosols, which was suitable in the context of this study. Indeed, an increased reproducibility was observed.

The trend observed using each classification processes were found to be identical: smaller particles increased the percolation time. Thus, it was shown that the resistive soot sensor was sensitive to the particle size, even at low concentration. The resistive soot sensor is a complex system, thus there are numerous physical and chemical phenomena leading to the trend highlighted in this work. It can be interpreted as a geometric phenomenon implying that a higher number of small particles is required to link the electrodes, rather than big ones.

Finally, the size of the particles impacts several forces, modifying their trajectories. Present results allow to suppose that the bigger particles are deposited in the more suitable zones to build bridges. A numerical analysis of the electrophoresis could bring information about the deposition process of the particulate matter and about the impact of the particle size over the sensor signal.

Figure 1 .

 1 Figure 1. Schematic view of the experimental bench. A CPC 3022A, TSI (Condensation Particle Counter) has been used to measure the distribution size function of the soot aerosol.

Figure 1 .

 1 Figure 1. Schematic view of the experimental bench. A CPC 3022A, TSI (Condensation Particle Counter) has been used to measure the distribution size function of the soot aerosol.

Sensors 2018 ,

 2018 18, x FOR PEER REVIEW 3 of 11to decrease the temperature of the heated gas to the ambient temperature. Moreover, a High Efficiency Particulate Air (HEPA) filter was used to protect the pump from potential degradation caused by the soot aerosol.

Figure 2 .

 2 Figure 2. Illustration of the soot sensor chamber.

Figure 3 .

 3 Figure 3. (a) Illustration of the resistive soot sensor; (b) Typical scanning electron microscopy of the interdigitated electrodes [15]. It shows the soot bridge-like microstructures linking the electrodes after the percolation time is reached.

Figure 2 .

 2 Figure 2. Illustration of the soot sensor chamber.

Figure 2 .

 2 Figure 2. Illustration of the soot sensor chamber.

Figure 3 .

 3 Figure 3. (a) Illustration of the resistive soot sensor; (b) Typical scanning electron microscopy of the interdigitated electrodes [15]. It shows the soot bridge-like microstructures linking the electrodes after the percolation time is reached.

Figure 3 .

 3 Figure 3. (a) Illustration of the resistive soot sensor; (b) Typical scanning electron microscopy of the interdigitated electrodes [15]. It shows the soot bridge-like microstructures linking the electrodes after the percolation time is reached.

Figure 4 .

 4 Figure 4. (a) Mobility diameter size distribution of a poly-disperse aerosol generated by the mini-CAST. The measurements are performed using a Scanning Mobility Particle Sizer (SMPS); (b) Aerodynamic diameter size distribution of the same aerosol measured downstream of an AAC with a CPC.

Figure 4 .

 4 Figure 4. (a) Mobility diameter size distribution of a poly-disperse aerosol generated by the mini-CAST. The measurements are performed using a Scanning Mobility Particle Sizer (SMPS); (b) Aerodynamic diameter size distribution of the same aerosol measured downstream of an AAC with a CPC.

Figure 5 .

 5 Figure 5. Illustration of the working principle of an aerosol classification by electrostatic technique. The blue area represents the concentration computed by the SMPS algorithm, based on the Boltzmann equilibrium. The effective aerosol (i.e., the particles sent to the sensor chamber) is the green area.

Figure 6 .

 6 Figure 6. Illustration of the working principle of an aerosol classification by aerodynamic technique.

Figure 5 .

 5 Figure 5. Illustration of the working principle of an aerosol classification by electrostatic technique.The blue area represents the concentration computed by the SMPS algorithm, based on the Boltzmann equilibrium. The effective aerosol (i.e., the particles sent to the sensor chamber) is the green area.

Figure 5 .

 5 Figure 5. Illustration of the working principle of an aerosol classification by electrostatic technique. The blue area represents the concentration computed by the SMPS algorithm, based on the Boltzmann equilibrium. The effective aerosol (i.e., the particles sent to the sensor chamber) is the green area.

Figure 6 .

 6 Figure 6. Illustration of the working principle of an aerosol classification by aerodynamic technique.Figure 6. Illustration of the working principle of an aerosol classification by aerodynamic technique.

Figure 6 .

 6 Figure 6. Illustration of the working principle of an aerosol classification by aerodynamic technique.Figure 6. Illustration of the working principle of an aerosol classification by aerodynamic technique.

Figure 7 .

 7 Figure 7. Sample of the soot sensor responses to a DMA-classified aerosol for different mobility diameters and different polarization tensions (red: 30 V; blue: 50 V; green: 70 V; black: 100 V). (a) 𝑑 = 60 nm ; (b) 𝑑 = 70 𝑛𝑚 ; (c) 𝑑 = 80 nm ; (d) 𝑑 = 100 nm.

Figure 7 .

 7 Figure 7. Sample of the soot sensor responses to a DMA-classified aerosol for different mobility diameters and different polarization tensions (red: 30 V; blue: 50 V; green: 70 V; black: 100 V). (a) d m = 60 nm; (b) d m = 70 nm; (c) d m = 80 nm; (d) d m = 100 nm.

11 .Figure 8 .

 118 Figure 8. Overview of the percolation time for different mobility diameters and concentrations. The red line is the correlation between the averaged percolation times and the mobility diameters, without taking into account the 100 nm diameter measurements ( 𝑡 = -455 × 𝑑 + 39737, 𝑅 = 0.9463).

Figure 9 .

 9 Figure 9. Soot sensor responses of two aerodynamically classified aerosols. Red curves: 50 nm aerosol; blue curves: 70 nm aerosol. The different responses are differentiated by their line width. The polarization tension is 𝑉 = 60 V for each experiment. (a) Overview; (b) zoom on the percolation time.

Figure 8 .

 8 Figure 8. Overview of the percolation time for different mobility diameters and concentrations. The red line is the correlation between the averaged percolation times and the mobility diameters, without taking into account the 100 nm diameter measurements (t perc = -455 × d m + 39737, R 2 = 0.9463).

Sensors 2018 , 11 .Figure 8 .

 2018118 Figure 8. Overview of the percolation time for different mobility diameters and concentrations. The red line is the correlation between the averaged percolation times and the mobility diameters, without taking into account the 100 nm diameter measurements ( 𝑡 = -455 × 𝑑 + 39737, 𝑅 = 0.9463).

Figure 9 .

 9 Figure 9. Soot sensor responses of two aerodynamically classified aerosols. Red curves: 50 nm aerosol; blue curves: 70 nm aerosol. The different responses are differentiated by their line width. The polarization tension is 𝑉 = 60 V for each experiment. (a) Overview; (b) zoom on the percolation time.

Figure 9 .

 9 Figure 9. Soot sensor responses of two aerodynamically classified aerosols. Red curves: 50 nm aerosol; blue curves: 70 nm aerosol. The different responses are differentiated by their line width. The polarization tension is V pol = 60 V for each experiment. (a) Overview; (b) zoom on the percolation time.

  such a combustion was 0.95, the number and mass concentration of the generated aerosol were respectively 2.5 × 10 part./cm and 91 mg/𝑚[START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. The chemical nature of the generated aerosol was investigated by Grondin et al. in a previous work[START_REF] Grondin | Développement d'un capteur de suie pour application automobile: Etude des paramètres clés affectant sa réponse[END_REF]. Several settings have been characterized but for the operating point considered in the present work, it can be highlighted that the carbon black concentration is 70 mg/m , which represents 77% of the mass concentration of the aerosol. The typical soot particles emitted from Diesel engines are reported to contain a lower proportion of carbon black. The proportion of elemental carbon (mostly carbon black) is most of the time under 70%[START_REF] Shah | Emission Rates of Particulate Matter and Elemental and Organic Carbon from In-Use Diesel Engines[END_REF].

			al. study, it led to the operating point: 𝑄	=
	3.6 L/min , 𝑄 ,	= 93 L/min , 𝑄 ,	= 450 L/min , 𝑄 ,	= 9 L/min . The fuel air
	equivalence ratio of

Table 1 .

 1 Overview of the AAC parameter R s for different aerodynamic diameters.

	d a [nm]	R s [-]	∆ [nm]	C [part./cm 3 ]
	50	3.6	13.9	5.6 × 10 5
	70	5.2	13.5	4.2 × 10 5

Acknowledgments:

The authors would like to acknowledge the companies Cambustion and Intertek for lending of the Aerodynamic Aerosol Classifier (AAC).

Author Contributions: Supervision, S.Z., M.L., P.B. and J.-P.V.; investigation, A.R.; writing-review and editing, S.Z., M.L., P.B. and J.-P.V.; writing-original draft, A.R.

Funding: This research received no external funding.

Conflicts of Interest:

The authors declare no conflict of interest.


Thomas Robert
email: thomas.robert@ifsttar.fr

Julien Causse

Gérard Monnier

Estimation of external contact loads using an inverse dynamics and optimization approach: general method and application to sit-to-stand maneuvers

Keywords: Inverse Dynamics, Under-determinate problem, External contact loads, Joint loads, Optimization, Sit to stand

Estimation of external contact loads using an inverse dynamics and optimization approach: general method and application to sit-to-stand maneuvers.

The proposed method is tested on human sit-to-stand maneuvers performed holding an handle with one hand, i.e. asymmetrical movements with multiples external contacts. Three sets of measured and unmeasured contact load components and three cost functions are considered and simulated results are compared to experimental data. For the population and movement studied, better results are obtained for a least square sharing between actuated degrees of freedom of the relative motor torques (motor torques normalized by the maximal torque production capacity). Moreover, the number of unknown ECL components does not signicantly inuence the results. In particular, measuring only the vertical 1. Introduction Human motion analysis focuses more and more on dynamic variables. In particular, analysis of the external contact loads (ECL) allows characterizing the nature and the role of interactions between a subject and its environment. For example, [START_REF] Chateauroux | Car egress analysis of younger and older drivers for motion simulation[END_REF] classied hand/vehicle contacts during # car ingress/egress motions based on their role on the motion: exploration of the environment, balance maintenance and participation to the displacement.

Knowledge of the ECL is also necessary to analyze the dynamics of a motion, typically the internal loads such as the net joint torques (NJT) or muscle forces. These internal loads are not directly measurable with non invasive techniques.

They are usually estimated using inverse dynamic approaches, i.e. based on the knowledge of the system's kinematics, its inertia properties and the ECL acting on it (e.g. [START_REF] Dariush | A well-posed, embedded constraint representation of joint moments from kinesiological measurements[END_REF][START_REF] Doriot | A three-dimensional kinematic and dynamic study of the lower limb during the stance phase of gait using an homogeneous matrix approach[END_REF]Dumas et al., 2007b;[START_REF] Silva | Kinematic data consistency in the inverse dynamic analysis of biomechanical systems[END_REF][START_REF] Winter | Biomechanics and Motor Control of Human Movement[END_REF].

Classically ECL are directly recorded using load sensors. However, direct # measurement may be technically complex, especially for ambulatory or out of the lab experiments. Besides, use of force sensors tends to reduce the ecological aspect of an experiment. A classical illustration would be the force plate targeting problem in gait analyses [START_REF] Challis | The variability in running gait caused by force plate targeting[END_REF][START_REF] Oggero | Probability of valid gait data acquisition using currently available force plates[END_REF]. Another example would be ergonomic experiments performed in complex environments, such as car ingress-egress [START_REF] Causse | Dynamic analysis of car ingress/egress movement: an experimental protocol and preliminary results[END_REF][START_REF] Kim | Development of discomfort evaluation method for car ingress motion[END_REF] where the diculty to equip every mock-up parts lead experimenters to limit the possible contact areas for the subject, and eventually limit the variety and naturalness of movements.

It is therefore necessary to use modeling techniques to indirectly estimate # the ECL from the known kinematics of body segments (measured or simulated).

One of the method could be to use a detailed model of the contacts, in particular for the foot-ground contact during locomotion [START_REF] Fluit | A simple controller for the prediction of three-dimensional gait[END_REF][START_REF] Pandy | Computer modeling and simulation of human movement[END_REF].

However such models are dicult to adjust, potentially expensive in computation and sensitive to the modeling hypotheses [START_REF] Dorn | Estimates of muscle function in human gait depend on how foot-ground contact is modelled[END_REF]. An alternative ! is to use an inverse approach, i.e. to estimate the ECL acting on a system that lead to its observed motion [START_REF] Pillet | Evaluation of force plate-less estimation of the trajectory of the centre of pressure during gait. comparison of two anthropometric models[END_REF][START_REF] Sanders | Accuracy of derived ground reaction force curves for a rigid link human body model[END_REF]. However, one of the limitations of this approach is that it becomes under-determined if there are more unknown ECL components than independent governing equations. This usually occurs when there are multiple unmeasured contacts loads !# with the environment. Thus, setting an assumption on the load sharing becomes necessary.

In the past, only few studies have proposed to deal with these types of underdetermined problems. Most of them focused on bipedal phase of the gait and represented the load sharing with mathematical functions [START_REF] Davis | Decomposition of superimposed ground reaction forces into left and right force proles[END_REF][START_REF] Hardt | A ve body -three dimensional dynamic analysis of walking[END_REF][START_REF] Koopman | An inverse dynamics model for the analysis, reconstruction and prediction of bipedal walking[END_REF][START_REF] Ren | Whole body inverse dynamics over a complete gait cycle based only on measured kinematics[END_REF]. While these methods provided satisfying results, they were very specic to the given task and the population. A more generic approach for solving the load sharing problem consists by minimizing a cost function was presented by [START_REF] Vaughan | Closed loop problems in biomechanics. part iian optimization approach[END_REF]. Instead of focusing only on the ECL problem, [START_REF] Vaughan | Closed loop problems in biomechanics. part iian optimization approach[END_REF] chose to explicitly introduce the unknown net joint loads (NJL) into the problem. It allowed to estimate NJL and ECL in the same procedure and also to use NJL into the cost function. Based on this approach, they obtained interesting results for dierent kind of movements. However, the study suered from several limitation: there results were strongly dependent on the initial # guess of the solution, the methods was only tested in 2D and in cases with no more than two unknown ECL wrenches, and the results were reported for only one cost function.

This indicates that there is still a lack of methodologies for estimating the ECL, particularly in under-determined cases. This study proposes a generic ## method which utilizes a combination of inverse dynamics and optimization. It is intended to be relatively simple, computationally cost eective and robust.

The proposed method is tested on human sit-to-stand maneuvers performed holding an handle with one hand, i.e. asymmetrical movements with multiples external contacts. Inuence of the number of unknown ECL components and $ of dierent cost functions are evaluated.

Modeling

This section introduces the generic problem to be solved in this study.

Description of system considered

The system considered is a whole body human model made of n s rigid seg-$# ments linked by n j joints representing a total of n dof degrees of freedom (DoF).

The kinematics and inertia properties of the segments are considered to be known.

Dynamics and constraints of the system

The dynamic wrenches for each segments, due to the inertial and gravita-% tional eects, can be estimated from their kinematics and inertia properties and aggregated to obtain the dynamic wrench of the overall system. This wrench has to be balanced by the known and unknown ECL, resulting in a system of 6 linear equations with n u unknowns. Alternatively, we chose to write the Newton-Euler equations for each segments, introducing the internal loads acting %# between the segments. It led to a set of 6n s linear equations (6 per segments) with n u + 6n j unknowns (corresponding to the unknown ECL components and the NJL):

a eq x = b eq (1)
were x = [S U ] t is a single vector grouping all the unknowns, S is the vectors containing the 6n j NJL (3 forces and 3 torques per joint), U is the vectors of the & n u unknown ECL components, a eq is a matrix relating the joints and unknown ECL components with the corresponding aected segments, and b eq is the eect of the dynamic wrench, known ECL and loads due to the gravity for all the segments (see Appendix for the details).

The physical limitations of the studied system, such as unilaterality of the &# contacts or maximum motor torques that could be developed, are also considered in this study. In order to make the resolution of the problem faster and more robust( see Section 2.3), we chose to express these limitations as linear equations of the unknown vector x. Mathematical description of these constraints are given in Appendix. '

The description of the system dynamics and limitations boils down to the linear system:

   a eq x = b eq a c x ≤ b c (2)

Resolution

This study focuses on under-determined problems for which there are fewer governing equations than the unknowns (6n s < 6n j +n u ). In order to select only '# one solution among all the possibles, additional expectations for this solution are introduced via a function to be minimized. For numerical purposes (developed in the discussion) we propose: 1) to write this function in a specic form (squared euclidean norm of a linear function of x), 2) to add an additional term to it the squared euclidean norm of x with a very low weighting factor and 3) to set the equality constraints as penalties. Eventually, it led to the following optimization problem:

min x α ∥sx∥ 2 + β ∥x∥ 2 + γ ∥a eq x -b eq ∥ 2 , s.t. a c x ≤ b c (3)
where ∥ ∥ represents the euclidean norm, s is a matrix dening a linear transformation of x, and α, β and γ are weighting factors.

This problem can be rewritten as a quadratic programming problem: #

min x 1 2 x T Hx + g T x, s.t. a c x ≤ b c (4)
with H = 2αs T s + βI + γa T eq a eq a positive denite matrix (see Discussion) and g T = -2γb T eq a eq .

Case Study

Task and motions studied

Experimental data were collected on six healthy young male volunteers (stature = 1752±61 mm, weight = 66±8 kg and age = 26±3 yo), having no musculoskeletal disorder.

Subjects were initially seated on a at rigid seat with feet on the ground and the right hand grasping a horizontal handle. The handle was centered in the sagittal plane of the subject, located at the level of the subject's eyes when # seated and at a distance corresponding to the shoulder-wrist length. They were asked to rise from the seat to a standing posture using the handle (see Figure 1). Three dierent seat heights were tested: 50% (H 50 ), 75% (H 75 ) or 100% (H 100 ) of the knee height. The task was repeated two times for H 50 and H 100 and six times for H 75 .

Experimental Data Collection and Processing

A Motion Analysis R ⃝ system, sampling at 100Hz, was used to capture the trajectories of 51 reective markers placed on the subject. All ECL between the subject and its environment were recorded using three force plates (under each foot and under the seat) and a 6-axes load sensor placed between the handle # and the frame.

Kinematics was reconstructed using a global optimization procedure [START_REF] Wang | Validation of a model-based motion reconstruction method developed in the realman project[END_REF] and a 16 rigid bodies -33 degrees of freedom kinematic model (see Figure 2). The segment inertia parameters were estimated from regression equations (Dumas et al., 2007a). NJL were computed using a recursive ! approach [START_REF] Doriot | A three-dimensional kinematic and dynamic study of the lower limb during the stance phase of gait using an homogeneous matrix approach[END_REF] and two calculation strategies (bottom-up and top-down) converging at the lumbar joint [START_REF] Robert | Validation of net joint loads calculated by inverse dynamics in case of complex movements: application to balance recovery movements[END_REF]. These NJL computed using all the contact loads information will later be used as reference to estimate the quality of the optimization (see Section 3.5).

Modeling Hypotheses !#

The methodology proposed in section 2 is applied using the measured kinematics and dierent subsets of the measured ECL.

Three dierent sets of known (i.e. which are measured) and unknown ECL were considered (see Figure 2): 1) S 1 -both seat and hand contact wrenches are known; 2) S 2 -only seat contact wrench is known; 3) S 3 -only the vertical force " of seat contact wrench is known and other components are assumed absent.

When unknown, hand to handle interaction was represented by a full wrench applied at the projection of the third metacarpophalangeal joint on the handle.

The contact between the foot and the ground were modeled as 3 punctual unilateral contacts with tangential force limited by friction (static coecient µ=0.5) "# and positioned under the 1st and the 5th metatarsal heads and the heel. These 3 punctual contact wrenches were further aggregated to represent the foot to ground interaction with a single contact wrench. Three types of constraints were applied: unilaterality and dry friction con-# straints for punctual contacts, and maximal motor torques that could be developed by the subject (see Table 1). Mathematical description of these constraints are given in Appendix.

Table 1 about here.

Resolution of the Optimization Problem

## Three dierent expectations on the load sharing were tested: minimization in a least square sense of 1) the unknown external contact forces (ECF); 2) the net joint torques (NJT), 3) the motor torques normalized by their maximum allowable value (MT % see Table 1). It resulted in three cost functions C i 's of the form: $

C i (x) = α i ∥s i x∥ 2 + β ∥x∥ 2 + γ ∥a eq x -b eq ∥ 2 (5)
with s i matrices dening the linear transformation of x the unknown vector of ECL and NJL into the vector of ECF, NJT or MT % respectively (details in Appendix). Values of α i were adjusted based on the average value of s i (see Table 2), while β and γ were set to 10 -5 and 10 5 respectively.

These quadratic programming problems were solved using the Matlab 4

⃝ $#
quadprog solver. The algorithm was automatically initialized for the rst frame of the motion (initial point set to a vectors of ones (MathWorks, 2012)). For the remaining frames it was initialized using the results of the previous frame.

Analyses

Three cost functions (C 1 -C 3 ), three contact congurations (S 1 -S 3 ) and % three experimental seat heights (H 50 , H 75 and H 100 ) were considered. Seven combinations of these variables were tested (see Table 2). For each of them, the NJL and the unknown ECL were estimated for every maneuver performed by the six subjects. Forces and torques were normalized by body weight (BW)

and body weight times body height (BW.BH) respectively [START_REF] Hof | Scaling gait data to body size[END_REF]. Peak %# values of the norm of the simulated ECL and NJT were extracted and referred as P sim hereafter. In addition, time history proles of the norm of the simulated data were compared to their reference curves. RMS dierences normalized by the amplitude of the reference curve were computed and referred as R % . The reference data were: 1) the measured experimental data for the ECL; 2) the & results of the inverse dynamics performed using the experimentally measured ECL (see section 3.2) for the NJT.

ANOVAs and Tukey's post-hoc tests were used to analyze the eects of 1) the cost function on R % in S 2 and H 75 (three rst rows of Table 2), 2) the contact conguration on R % using C 3 and in H 75 (rows three to six of Table 2), &# and 3) the seat height on P sim using C 3 and in S 3 (last three rows of Table 2).

Results

The procedure successfully converged for all the tested motions and congurations. An evaluation was performed to observe if equality constraints set as penalties were satised. It was found that the residual values of ∥a eq .x -b eq ∥ ' (see Equation ( 5))remained negligible (less than 10 -4 ) in all cases, except for some motions in contact the conguration S 1 , where it reached up to 3 N.m around the seat-o for torque constraints along the transverse axis. Figure 3 displays an example of reference data (measured ECL and NJT obtained through inverse dynamics with the measured ECL) for a representative '# subject in conguration S 2 and H 75 . These results are comparable in term of proles and amplitude to the classical data available in the literature for sitto-stand maneuver [START_REF] Bahrami | Biomechanical analysis of sit-to-stand transfer in healthy and paraplegic subjects[END_REF][START_REF] O'meara | The eects of unilateral grab rail assistance on the sit-to-stand performance of older aged adults[END_REF]. In this Figure 3, simulated results obtained in the same situation for dierent cost functions are also superimposed to the reference data. Both the shape of the simulated curves and their amplitudes are overall in good agreement with the reference ones.

An example of simulated and measured center of pressure trajectories under each foot is shown in Figure 4. The loaded area are comparable. In this particular case the experimental trajectory goes outside the modeled base of support # (BoS) for some frames of the motion, whereas the simulated trajectory remains inside. A similar analysis was performed to evaluate the eect of the contact cong-uration using the cost function C 3 in seat height H 75 . An ANOVA with factors Contact and Contact conguration performed on the R % for the ECF showed that neither the Contact conguration nor the interaction has a signicant eect (p = .157 and p = .321, respectively). Similar results are obtained for the NJT.

Two ANOVAs with factors Contact and Seat height performed on the P sim # for the ECF and for the NJT conrmed that the Seat height has a signicant eect on the P sim . Post-Hoc tests showed that the higher the seat the smaller the P sim (H 100 < H 75 < H 50 ).

Discussion

Numerical Aspects

! This paper presents a general method to estimate unmeasured external contact loads acting on a system whose kinematics and inertial properties are known. This method was intended to be relatively simple and robust. For the case study considered the optimization problem converged systematically, and one could remark the smoothness of the time prole for the simulated vari-!# ables, although the problem is solved frame by frame. It is notably imputable to the numerical properties of the optimization problem. We chose to write the problem as a quadratic program (quadratic cost function and linear constraints). For this type of problem, if the H matrix of the cost function (see equation 5) is positive denite (always non negative, nil only for x = 0) and if " the problem has a solution, then it exists a unique minimum that is the global minimum. The solver will thus systematically and quickly converge toward the desired solution.

The cost functions used in this study are the sum of three terms (see Equation ( 5)). The rst and last terms are strictly non negative functions that may "# be equal to zero for non zero value of x. Adding the squared norm of x, even with a very small weight, ensures that the cost function can only be nil only when x = 0, i.e. the cost functions are positive denite. In other words, the rst and last terms of the cost function did not necessarily nullify the space of solution. Typically the foot contact model used in this study (3 contact points) # induced a redundancy in the horizontal plane: an innity of individual contact forces combinations summed up to the same overall contact force for the foot, and thus the same motor torques and the same value of the optimization criterion. The role of the second part of the criterion, slightly weighted, is then to nullify the space of solution and as such to improve the stability and con-## vergence eciency of the optimization algorithm. Because of its small relative weight, it did not signicantly changed the results in terms of the resultant of ECL or NJL.

This study focused on under-determined problems. However, in some situations, the system of equation ( 2) may become over-constrained. It typically $ happens in case of temporary inconsistencies between the modeling hypotheses and the imposed experimental data (estimated accelerations or BSIP). In this study we chose to set the equality constraints as penalties in the cost function.

It led to the optimization problem (3), only constrained by the set of inequalities representing the system's limitations. These later are independent of the $# experimental data and, besides modeling incoherences, unlikely to nullify the space of solution. Consequently the problem (3) always has at least a solution.

Although this solution only satises as well as possible the equations of motion, it may be acceptable for small residual values of the penalties. In this study over-constrained situations may happen around the seat-o in the contact con-% guration S 1 . As displayed in Figure 4, the foot contact wrenches may be too tightly constrained for some motion, leading to small discrepancies between the modeled BoS and the experimental CoP trajectory. These discrepancies could be compensated by the hand contact wrench when this later is not imposed, but led to an over-constrained problem when the hand contact wrench is imposed %# (contact conguration S 1 ). The observed residual values remained relatively small, and the obtained solutions were thus acceptable. However, a track for improvement would be to nely dened the shape of the BoS, by adding contact points for example.

Case study: cost function and applicability &

The proposed methodology was applied to sit-to-stand motions with an handle. Dierent load sharing hypotheses were tested. Cost function C 3 provide the better results (the smaller R % ). It principally implies a least square sharing between actuated DoF of the relative motor torques (motor torques normalized by the maximal torque production capacity). It therefore prevents to overload &# those DoF which are having small capacity. This eect is best seen on the right upper limb, where the use of C 3 tend to limit the loads along the right upper limb DoF, with smaller force production capacities relative to those of the lower limbs. This criterion is quite similar to the minimization of the individual muscle stress (indivudual muscle forces normalized by their physiological cross ' sectional area) classically used for solving the under-determinated muscle force sharing problem (e.g. [START_REF] Erdemir | Model-based estimation of muscle forces exerted during movements[END_REF], for a review). Other criteria could have been considered, in particular considering the p-norm of s i x to the power p with p ̸ = 2 and the min/max criterion [START_REF] Rasmussen | Muscle recruitment by the min/max criterion a comparative numerical study[END_REF]. It could also be interesting to extend this problem to include individual muscle forces in the '# criterion. It would not change the nature of the problem as, by neglecting the contraction dynamics, the muscle forces can be expressed as a linear function transformation of the motor torques via a level arm matrix.

Three dierent contact congurations were considered in this study. Statistical analysis showed that this parameter did not signicantly inuence the ! quality of the results (dierence between reference and simulated data). In particular, reducing the number of unknowns did not improve the R % (S 1 vs S 2 ).

Results obtained in S 3 (using only the vertical force component of the seat contact wrench) are important in terms of application. They suggest that, for the studied movement and population, measuring only the vertical force under the !# seat would allow a relatively correct estimation of all ECL and NJT: not only the values of R % were small (about 10% for the feet ECL and 20 % for the NJT), but the inuence of an experimental parameters (the Seat Height) was also correctly predicted. In this case, peak values of NJT and ECL increased when the seat height decreased, as observed from the experimental data [START_REF] Robert | Dynamics of sit-to-stand motions: eect of seat height, handle use and asymmetrical motions[END_REF]. ! This could be of particular importance for ergonomic studies where usually the goal is to understand the inuence of dierent experimental parameters on the motion dynamics and which often suer from instrumentation diculties.

Limitations

The methodology was only evaluated for one type of motion and a small !# sample of single population. Load sharing criterion may evolve between motion and population, and may include other parameters such as the comfort or balance. This should be further studied.

Moreover, the contact congurations considered in this study remained constant during the motion. Although the kinematics and measured contact loads ! may suciently drive the problem, discontinuities in solution may appear when considering intermittent contacts. One could thus consider describing the problem using parametric curves and to solve it as a whole instead of frame by frame.

Conict of interest statement

Figure 2

 2 Figure 2 about here.

Figure 5

 5 Figure 5 displays the normalized RMS dierences between the simulated and reference data (R % ) averaged accross trials and subjects for each contact and joints and the three cost functions tested (contact conguration S 2 and seat height H 75 ). Very high values for the right wrist NJT are due to very small amplitudes of the reference curves. Therefore, they are discarded in further analyses. Overall, R % are smaller for C 3 and higher for C 1 . This is conrmed by an ANOVA with factors Contact and Cost function performed on the R % values of the ECF. It showed a signicant eect (p<0.05) of the Cost function, the #

Figure 1 Figure 1 :

 11 Figure 1

Figure 2 :

 2 Figure 2

Figure 3 :

 3 Figure 3

Figure 4 :

 4 Figure 4

  Figure 5

Figure 5 :

 5 Figure5: R % (normalized RMS dierences between simulated and reference data -see section 3.5) for dierent cost functions and dierent joint and contacts, averaged accross subjects and trials, in contact conguration S 2 and seat height H 75 . Vertical bars represents ± one standard error of the mean (SEM). Results above 120% are given in the gure. The high values for the right wrist net joint torques are due to small amplitudes of the experimental curves.

Table 1 Table 1 :

 11 Maximum torques for the dierent degrees of freedom of the considered model

	Joint	Motion	Maximum torque Source
			value (N.m)
	Hip Joint	Flexion	185	Chan et al. (2006)
		Extension	190	Chan et al. (2006)
		Abduction	190	Delp (1990)
		Adduction	190	Delp (1990)
		Internal Rotation 60	Delp (1990)
		External Rotation 60	Delp (1990)
	Knee Joint	Flexion	100	Chan et al. (2006)
		Extension	168	Chan et al. (2006)
		Internal Rotation 20	Expertize
		External Rotation 20	Expertize
	Ankle Joint	Dorsiexion	126	Chan et al. (2006)
		Plantarexion	126	Chan et al. (2006)
		Inversion	20	Expertize
		Eversion	20	Expertize
	Shoulder Joint Flexion	92	Chan et al. (2006)
		Extension	67	Chan et al. (2006)
		Abduction	71	Chan et al. (2006)
		Adduction	67	Chan et al. (2006)
		Internal Rotation 52	Chan et al. (2006)
		External Rotation 33	Chan et al. (2006)
	Elbow Joint	Flexion	77	Chan et al. (2006)
		Extension	46	Chan et al. (2006)
		Pronation	15	Garner and Pandy (2000)
		Supination	15	Garner and Pandy (2000)
	Wrist Joint	Flexion	185	Garner and Pandy (2000)
		Extension	190	Garner and Pandy (2000)
		Abduction	190	Garner and Pandy (2000)
		Adduction	190	Garner and Pandy (2000)
	Lumbar Joint Flexion	143	Chan et al. (2006)
		Extension	234	Chan et al. (2006)
		Lateral Flexion	159	Chan et al. (2006)
	Neck Joint	Flexion	100	Chan et al. (2006)
		Extension	100	Chan et al. (2006)

Table 2 Table 2 :

 22 The dierent combination of parameters tested, and the corresponding weights of the cost function.

	#	Cost	Contact	Seat	α i	β	γ	Tested Eect
		Function	Conguration	Height
	1 2 3	C 1 C 2 C 3	S 2 S 2 S 2	H 75 H 75 H 75	10 -1 10 -5 10 5 1 10 -5 10 5 500 10 -5 10 5	Cost Function
	4 3 5	C 3 C 3 C 3	S 1 S 2 S 3	H 75 H 75 H 75	500 500 500	10 -5 10 5 10 -5 10 5 10 -5 10 5	Contact Conguration
	6	C 3	S 3	H 50	500	10 -5 10 5
	5	C 3	S 3	H 75	500	10 -5 10 5	Seat Height
	7	C 3	S 3	H 100	500	10 -5 10 5