J P Goulmy

P Kanoute

E Rouhaud

L Toualbi

S Kruch

V Boyer

J Badreddine

D Retraint

A calibration procedure for the assessment of work hardening Part II: Application to shot peened IN718 parts

Keywords: Inconel 718, shot peening, work hardening, residual stresses, XRD

The objective of this paper is to discuss the application of the calibration methodology exposed in the previous part to shot-peened Inconel 718 specimens. Shot peening is commonly used to increase the fatigue life of critical parts such as Inconel 718 turbine discs. This surface treatment induces residual stresses, work hardening and possibly, gradients of microstructures that, in turn, affect fatigue life.

Work hardening is a quantity that represents a set of physical and mechanical phenomena related to the level of disorder reached in the microstructure of the material. Work hardening is seldom taken into account in fatigue life assessment mainly because it is not possible to characterize this quantity directly. We propose to use the calibration methodology (see part I of this paper [1]) on samples shot peened with several conditions. The three complementary experimental techniques (microhardness, XRD and EBSD) are then used to determine through correlation curves the work hardening gradients.

The methodology for characterizing the work hardening within shot peened specimens is first presented. A discussion of the applicability of the method in this context is then provided. The results obtained for the different characterization methods and microstructural effects are analyzed in two different sections. Finally, the influence of shot peening conditions on residual stresses and on work hardening is discussed, showing the interest of the proposed procedure to obtain a real picture of the mechanical state after shot peening.

Introduction

Surface enhancement methods, such as shot peening, are widely used in the aerospace or automotive industry to improve the fatigue life of critical components like turbine disk, turbine blade or connecting rod. These processes are the object of important research and development in order to optimize process conditions with the general objective of increasing the security of the structures while reducing their weight and cost [START_REF] Salah | Shot Peening Applications and Future Research in the Aerospace Industry[END_REF].

In order to accurately assess the fatigue life, the knowledge of all the mechanical fields related to the loading history applied to the material during the shot peening process is of primary importance [START_REF] Kubler | Experimental Analysis of the Shot Peening Particle Stream Using Particle Tracking and Digital Image Correlation Techniques[END_REF].

These surface treatments generate compressive residual stresses near the surface of the component that tend to delay fatigue crack initiation and prevent their propagation [START_REF] Wagner | Shot Peening: WAGNER:SHOT PEENING O-BK[END_REF], [START_REF] Mcclung | A literature survey on the stability and significance of residual stresses during fatigue[END_REF]. However, the residual stresses are not the only quantity that influences the fatigue life. The level of work hardening generated during the process can also have a significant impact [START_REF] Prevey | The Effect of Cold Work on the Thermal Stability of Residual Compression in Surface Enhanced IN718[END_REF]. Prevey observed that the evolution of the mechanical fields during thermo-mechanical cyclic loadings, in particular the relaxation of the stress state, is strongly influenced by the initial work hardening level. For the same amount of residual stresses observed locally, the level of work hardening may also significantly vary. Shot peening is thus a complex mechanical treatment that induces residual stress, strain gradients, and high plastic deformation values. To understand the impact of this process on fatigue life, it is crucial to understand the influence of work hardening and therefore be able to quantify this phenomenon.

Many authors have proposed methodologies to evaluate work hardening from calibration methods [START_REF] Prevey | The measurement of sub-surface residual stress and cold work distributions in nickel base alloys[END_REF]- [START_REF] Foss | Analysis of shotpeening and residual stress relaxation in the nickel-based superalloy RR1000[END_REF]. However, the few studies carried out in the literature to determine the work hardening induced by shot peening have shown limitations in the application of these methodologies based on the exploitation of XRD, EBSD and microhardness characterization techniques [START_REF] Prevey | The measurement of sub-surface residual stress and cold work distributions in nickel base alloys[END_REF], [START_REF] Schulze | Characteristics of Surface Layers Produced by Shot Peening[END_REF]- [START_REF] Kumar | Microstructuremechanical property correlation in shot peened and vibro-peened Ni-based superalloy[END_REF]. Shot peening is a mechanical treatment consisting of repeated high speed impacts on the surface of the treated part [14]. This generates an important strain rate gradient on the subsurface of the material, J o u r n a l P r e -p r o o f gradient that depends on time and process conditions. Numerical simulations have demonstrated that, during shot-peening, the value of the local strain rate varies from 10 -3 up to 10 4 s -1 [START_REF] Meguid | 3D FE analysis of peening of strain-rate sensitive materials using multiple impingement model[END_REF]. Thus, several questions remain open: how, in particular, these strain rate variations affect the work hardening evaluation? Moreover, microstructural variations may be observed following the application of the process [START_REF] Alexandre | Modelling the optimum grain size on the low cycle fatigue life of a Ni based superalloy in the presence of two possible crack initiation sites[END_REF]- [START_REF] Abikchi | Fatigue life and initiation mechanisms in wrought Inconel 718 DA for different microstructures[END_REF]. It could be of interest to assess the influence of the microstructure on the work hardening induced by shot peening in order to understand the possible consequences on the fatigue life. Few results are available in the literature on the quantification of work hardening after shot peening but none discuss the effects of the microstructure.

In Part I [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF], it was shown that XRD, EBSD and microhardness are sensitive to the microstructure and work hardening state. For each technique and considered microstructure, a calibration curve has been proposed. The objective of this paper is to apply these three techniques on shot peened specimens and compare their ability to evaluate shot peening induced work hardening. By mean of the calibration methodology adopted in part I, it is now possible to take into account the microstructure effect. The study of three microstructures aim to taking into account different microstructural states for a given alloy, here IN718. These microstructural changes can result in size changes of grains and/or precipitates. Therefore, three microstructures were considered in order to account for grain and precipitate size coarsening. Both "coarse grain" and "coarse grain and precipitate" microstructures were derived from the Direct Aged state, using specific heat treatments.

In this paper, we first present the material, the shot-peening conditions and the characterization procedures used on the shot-peened samples. Then, in the following parts, the results are presented and discussed. We first verify the fact that the calibration procedure is relevant to evaluate work hardening in shot peened samples. For this purpose, the shot peened samples are characterized using all three characterization techniques (XRD, microhardness and EBSD) and precautions and hypotheses for the application of the calibration method on shot peened samples are discussed. A focus is made on aspects rarely discussed in the literature: influence of shot peening on grain size and crystal orientation and effect of strain rate on work hardening profiles. In the second part, results

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obtained for the different characterization methods are analyzed. Then, the effects of the microstructures on work hardening profiles following shot peening are discussed. Finally, the influence of shot peening conditions on residual stresses and on work hardening is discussed, showing the interest of the proposed procedure to obtain a realistic picture of the mechanical state after shot peening.

Materials and methods

Material and shot peening conditions

The shot peened samples investigated in this study have been extracted from a turbine disk constituted of a Direct Aged Inconel 718 (DA). The initial microstructure has been modified such that three microstructures of this alloy have been investigated. Details concerning the heat treatments that have been applied to obtain these microstructures and their characterization in the un-shot peened state are given in the part I of the present work [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF]. The first microstructure is the reference microstructure (mean grain size around 4.2 µm and strengthening precipitates with a size of about 20 nm) obtained directly after forging and corresponding to the "as-received" material. The second microstructure is a coarse grain microstructure (mean grain size of about 35 µm and strengthening precipitates about 20 nm in size). The third microstructure includes coarse grains and coarse strengthening precipitates (mean grain size approximatively equal to 35 µm and strengthening precipitates of about 200 nm).

The samples that have been shot peened are parallelepipedic with a size of 20x15x10 mm. Before shot peening, all samples have been mechanically polished to a mirror polished surface to remove as much as possible residual stresses and work hardening induced by machining. XRD measurements have been carried out to verify that the samples were free of residual stresses (Figure 10).

The samples have been shot peened on one of the largest surfaces (see Figure 1) using a conventional shot peening nozzle and S110 steel shot. Three shot peening conditions (SP) were applied on J o u r n a l P r e -p r o o f different samples. The used shot peening conditions were chosen for two reasons. They are commonly applied on industrial components and they induce very different gradients of residual stresses and work hardening. The latter is essential for validating the proposed methodology.

The shot peening parameters are gathered in Table 2 following the NFL 06-832 AFNOR norm [14].

In the shot peening process, the number of runs of the shot peening nozzle defines the exposure time, that itself defines the coverage, corresponding to [14]. The choice was made here to adapt the number of runs of the nozzle in order to obtain the same coverage for the samples composed of the three microstructures and for a given shot peening condition. A protocol with respect to the coverage had to be defined because, for a given shot peening condition, the impacts observed on the two coarse grain microstructures are bigger than the ones observed on the DA microstructure. This is confirmed by the values of microhardness (for zero plastic strain) reported in Figure 4 part I of this article for the three microstructures. Also, it has been observed that the yield strength of coarse grain microstructures is lower than the one of the DA microstructure: R P0.2 values at 20°C: 1300 MPa for DA microstructure, 1200 MPa for coarse grain microstructure and 1120 MPa for coarse grain and coarse strengthening precipitate microstructure; this is visible in the stress-strain curves presented in Part I.

The number of runs applied to these microstructures is thus more important to ensure the same coverage as the one observed on the DA microstructure. For example, 38 runs were necessary to achieve 125% coverage with SP1 shot peening condition in the case of the DA microstructure, whereas 32 runs were necessary for microstructures with coarse grains. The number of runs applied to each microstructure are presented in Table 1. To evaluate the influence of this choice, a third shot peening condition (SP3) has also been completed, for which the same number of runs has been applied on the three microstructures. This last treatment leads to a 175% coverage for the DA microstructure and a 200% coverage for the two coarse grain microstructures.

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Experimental protocol for the shot peened samples

After shot peening, the samples have been analyzed using EBSD, XRD and microhardness. All measurements were taken at several locations to ensure the uniformity of the surface treatment. The measurement procedures were similar with the one previously described in the part I of this work [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF].

However, due to the presence of gradient in the shot peened samples, additional surface preparations have been necessary to establish the in-depth quantities of interest.

The in-depth evolution of the full width at half maximum (FWHM) of the peaks and the residual stresses have been evaluated with XRD. The classical sin²ψ approach has been used and eleven ψ angles have been analyzed to determine line broadening depth profiles defined by the FWHM of a Pseudo Voigt profile fit. In order to be able to evaluate the quantities of interest in the depth, an electrolytic polishing, allowing material removal, was carried out with an average step of 40 µm. The material removal was controlled using a coordinate measuring machine. A correction procedure has been settled and showed that the material removal had a negligible impact on the residual stress measurements. Residual stresses and peak widths have been measured in two directions. However, no significant differences were revealed, confirming that shot peening introduces an equi-biaxial stress state in plane. The in-depth evolution of the FWHM has also been evaluated before shot peening and a constant value of the FWHM has been observed for all the untreated samples, as illustrated in Figure 10.

To obtain the microhardness profile, the samples were cut in half in the center. All cross-sections have J o u r n a l P r e -p r o o f then been prepared by mechanical polishing up to 1 μm with diamond paste followed by colloidal silica in order to achieve flat mirror polished surfaces free of damage. Ten microhardness measurements (HV 0.1 ) per depth were then performed, providing a mean hardness value and a standard deviation for each depth. Each measuring line is about 20 µm apart, leading to a profile with about ten values in the depth.

EBSD analyses were performed to evaluate if the applied shot peening conditions induce a change in grain size and crystalline orientation. For this purpose, pole figures were performed and the grain size distribution was plotted. The results are specifically discussed in section 2.3. The in-depth KAM parameter profiles were also established. To obtain this quantity, EBSD maps were analyzed by calculating the average of the KAM values at a given depth (KAMi). The KAM values obtained in the depth are thus spaced by one pixel, i.e. 0.5 µm.

Application of the calibration procedure on shot peened samples

The calibration procedure proposed in the part I of this work is applied to shot peened samples to correlate the level of work hardening with three experimental parameters: microhardness, FWHM determined from XRD measurements and KAM calculated from EBSD maps. The different steps of this analysis are presented in Figure 1.

The interest of this study is that the procedure is applied for each shot peening condition and in depth for each sample: the evolutions of the microhardness, FWHM and KAM are determined. This is illustrated in the second box of Figure 1.

In Part I, a calibration curve has been established for each experimental technique. Calibration functions have been deduced, that correlate the experimental parameter to the equivalent plastic strain. The equivalent plastic strain:

𝜀 𝑝 = √ 2 3 𝜺 𝒑 : 𝜺 𝒑 ,
where 𝜺 𝒑 is the plastic deformation tensor, has been chosen to represent the work hardening state.

The details concerning the determination of the calibration functions and the fitting of the As illustrated in the third box of Figure 1, it is then possible to use these calibration curves reversely to determine the equivalent plastic deformation in a part from the experimental measurements.

J o u r n a l P r e -p r o o f 3 Results and discussion

Relevance of the methodology for an application on shot-peened samples

In the present paper, it is proposed to evaluate the work hardening that has been generated by shot peening in IN718 parts. To use the methodology proposed in Part I, several conditions have to be verified:

1. Shot peening modifies the state of the material on the near surface of the treated part. It is thus necessary to verify that the characterization techniques are able to evaluate near surface gradients.

2. The range of plastic deformations used to establish the calibration procedure must be representative of the levels obtained during shot peening.

3. During shot-peening, a complex loading is applied to the surface of the specimen. In order to be able to quantitatively evaluate the work hardening induced by shot peening, the influence of the type of loading (monotonous, cyclic...) on the calibration procedure has to be limited.

4.

Since the calibration is dependent on the microstructure [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF], the microstructural characteristics (grain size and size of the strengthening precipitates) of the material should not change during the treatment.

5. Shot peening induces strain rate ranging from 10 -3 s -1 to 10 3 s -1 [START_REF] Boyer | A constitutive model for simulation of shot-peening and fatigue[END_REF]. Therefore, the effect of the strain rate on the work hardening profiles must be evaluated in order to define the impact of this parameter.

Each of these points are addressed in the following sections.

Characterization of near surface gradients

Figure 2 presents, for the DA microstructure, the in-depth profiles of the FWHM from XRD, KAM parameter from EBSD and microhardness HV 0.1 obtained for the "as-received" (i.e. before shot J o u r n a l P r e -p r o o f peening) and for SP2 shot peened samples. The Figure 2 shows that the measured parameters are homogeneous in the as-received material and that the three methods are able to capture an evolution of the measured quantity after shot peening. For the three techniques, a strong gradient appears near the surface to reach a plateau deeper in the material. These two phases correspond respectively to the work-hardened region and an in-depth region free of plastic deformations. In this last region, the shot-peened material exhibits values that are similar to the as-received material. Further, the KAM mapping obtained for the DA microstructure after application of SP1 and SP2

surface treatments are presented in Figure 3. As expected, these mappings show that the more severe the applied shot peening condition, the higher the KAM parameter and the greater the affected depth.

J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f

Plastic strain range

The level of plastic strain that is reached in shot peening may be very important : a plastic strain of approx. 30 to 40% can be generated at the surface of a shot peened specimen [START_REF] Prevey | The measurement of sub-surface residual stress and cold work distributions in nickel base alloys[END_REF]. The mechanical tests classically performed in a laboratory generate a limited amount of plastic strains; this is particularly the case if the method is calibrated with tensile tests alone. The range of strain used to establish the calibration may then be much smaller than the level achieved during the shot peening process. To overcome this limitation, compressive tests have been carried out in this study to explore high plastic strain levels (see section 2.2 of part I), representative of what is encountered in shot peening processes [START_REF] Boyer | A constitutive model for simulation of shot-peening and fatigue[END_REF]. This precaution is of particular importance for the EBSD characterization method since the observed saturation of the KAM parameter can lead to an underestimation of the hardening state in the material [START_REF] Soady | Evaluating surface deformation and near surface strain hardening resulting from shot peening a tempered martensitic steel and application to low cycle fatigue[END_REF], [START_REF] Sáez-Maderuelo | Plastic strain characterization in austenitic stainless steels and nickel alloys by electron backscatter diffraction[END_REF].

Influence of the loading type

The loading and deformation paths that are imposed on the material during shot peening are complex. An impact is local and induces non-proportional loading. Also, the local repetition of the impacts could be assimilated to cyclic loading such that cyclic effects are to be considered [START_REF] Zarka | On a simplified inelastic analysis of structures[END_REF]. Thus, the proposed methodology is applicable only if shot peening induces a material state that is similar to the one observed after the mechanical tests used for the calibration. Prevey investigated this point on two materials, Inconel 718 and Rene95 [START_REF] Prevey | The measurement of sub-surface residual stress and cold work distributions in nickel base alloys[END_REF]: he demonstrated that the FWHM of the XRD signal produced by a known amount of plastic strain is independent of the deformation process. In the same way, Soady et al. demonstrated that the results obtained with three characterization techniques (XRD, EBSD and microhardness) are also independent of the loading conditions (compression or tension) for a tempered martensitic steel [START_REF] Soady | Evaluating surface deformation and near surface strain hardening resulting from shot peening a tempered martensitic steel and application to low cycle fatigue[END_REF].

The shot peening process should thus not induce significant cyclic softening or hardening in the material [START_REF] Soady | Evaluating surface deformation and near surface strain hardening resulting from shot peening a tempered martensitic steel and application to low cycle fatigue[END_REF]. Such cyclic effects could effectively lead to a modification of the peak width of the XRD signal. It would then be difficult to distinguish the respective contribution of cyclic effects and work hardening on the peak width change. Cyclic softening phenomena have been observed in Inconel 718

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superalloy, but they remain relatively small at 20°C (less than 50 MPa) [START_REF] Fournier | Low cycle fatigue behavior of inconel 718 at 298 K and 823 K[END_REF].

Our work, presented in part [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF] confirms these results: the mechanical tests carried out to obtain different hardening states show that the proposed methodology remains valid for any type of loading conditions, monotonic or cyclic, tension or compression.

Influence of shot peening on grain size, crystal orientation and strengthening precipitates

A significant influence of the material microstructure on the calibration curves has been observed

(see section 4 of part I). The evaluation of work hardening in shot peened samples is therefore relevant only if the microstructural modifications induced by the treatment are limited. Because microstructural modifications have been reported in the literature for specific shot peening conditions [START_REF] Altenberger | Alternative Mechanical Surface Treatments: Microstructures, Residual Stresses & Fatigue Behavior[END_REF], it appears necessary to verify if the microstructure has been modified in the samples under consideration. For this purpose, the grain size distribution has been evaluated along with the mean grain orientation obtained with a pole figure. This quantity has been measured on the same sample and on the same area as where the misorientation EBSD maps exposed in Figure 4 were evaluated. Two characteristic zones have been defined: the zone affected by the process and the asreceived area, in the core of the material. For the DA microstructure, Figure 5.a compares the grain size distribution obtained after SP2 shot peening (the most severe treatment conditions) to the asreceived material. A qualitative comparison of these diagrams shows that no major microstructural modifications are induced by the process. J o u r n a l P r e -p r o o f SEM images have been also acquired at various depths to evaluate the influence of shot peening on the size and distribution of the strengthening precipitates. Figure 6 shows SEM images obtained at J o u r n a l P r e -p r o o f different depths (near the surface, at 10 µm, 50 µm and 200 µm below the shot peened surface) for the coarse grains and coarse strengthening precipitates microstructure. No major modifications on strengthening precipitates size have been observed for the three microstructures.

In the end, these analyses validate the fact that shot peening did not generate microstructural changes in the samples of this study.

Influence of the strain rate on the work hardening profiles obtained after shot peening

Numerical simulations of shot penning demonstrate that a strong gradient of strain rate is generated by the impacts, varying from 10 -3 s -1 to 10 4 s -1 ; the highest strain rates are obtained at the surface of the shot peened specimens and decreases rapidly in depth [START_REF] Meguid | 3D FE analysis of peening of strain-rate sensitive materials using multiple impingement model[END_REF], [START_REF] Boyer | A constitutive model for simulation of shot-peening and fatigue[END_REF]. The calibration procedures that have been proposed in the literature to evaluate work hardening have been constructed with quasistatic tests [START_REF] Prevey | The measurement of sub-surface residual stress and cold work distributions in nickel base alloys[END_REF]- [START_REF] Foss | Analysis of shotpeening and residual stress relaxation in the nickel-based superalloy RR1000[END_REF]. It seems relevant to estimate the impact of the strain rate on the evaluation of work hardening on shot peened samples. In a previous study, the level of work hardening has been correlated to FWHM data for quasi-static (10 -3 s -1 ) and high (10 3 s -1 ) strain rate loading conditions [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF].

This experimental campaign has been performed for the DA microstructure alone because of its interest from an industrial point of view. Moreover, this strain rate analysis has been restricted to the FWHM evaluation to limit the experimental campaign. Figure 7.a presents the calibration curves J o u r n a l P r e -p r o o f obtained for the two strain rates: for a given plastic strain, an increase in strain rate leads to an increase in the FWHM value; note that these results are in agreement with the observations made by Hoffmeister et al. [START_REF] Hoffmeister | Effects of the surface treatment on the measured diffraction peak width of Inconel 718[END_REF].

Using the calibration method (see Figure 1 in section 2.3) the work hardening is evaluated on the shot peened samples for the calibration curves corresponding to these two strain rate values. Figure 7.b presents the results obtained for the DA microstructure considering that the strain rate is constant in depth. It clearly appears that the strain rate affects the work hardening profiles obtained after shot peening, especially in the area next to the surface of the specimens. For the SP1 shot peening condition, the work hardening values reach 10% for a strain rate of 10 3 s -1 whereas they reach 25% for a strain rate of 10 -3 s -1 . For the SP2 shot peening condition, the work hardening values reach 25% for a strain rate of 10 3 s -1 while they reach 45% for a strain rate of 10 -3 s -1 .

To fully consider the influence of the strain rate gradient and its history for shot peened samples a complex analysis would be required. First, the number of tests to be carried out on the same microstructure would increase significantly since it would require a calibration curve for several strain rates. In addition, the in-depth variation of the strain rate is only accessible via a finite element simulation of the shot peening process. Furthermore, even if we restrict ourselves to the surface, the strain rate is not constant in time during the different shot impacts. Implementing such a calibration procedure could thus be of interest as a future step in the method. The analysis made here by using two extreme values of the strain rate offers the advantage of proposing limit values as a function of the depth of the work hardening level. Our results clearly show that work hardening could be underestimated if the highest strain rate is considered while considering a quasi-static loading could strongly over-estimate the work hardening.

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Work hardening after shot peening: comparison of the three characterization techniques.

Using the methodology developed in [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF], the gradient of work hardening is now evaluated in the shot peened samples within the scope of the hypotheses discussed in the previous section. As a first step, the three characterization techniques are compared with each other.

The evolution of the work hardening profiles obtained with the three characterization techniques is presented in for the DA microstructure and for the two shot peening conditions. As discussed in the previous section, the calibration has been established for two strain rates, 10 -3 s -1 and 10 3 s -1 respectively. Figure 8 clearly shows that the three techniques are able to produce a work hardening evaluation that is, as a whole, coherent with what could be expected. The three techniques produce work hardening values that are analogous for a given shot peening condition (except for microhardness in SP1; this point is discussed below). For a given depth, the work hardening evaluated by EBSD is:

-in the case of SP1, comprised between the values obtained with the calibrations made with XRD for each respective strain rate.

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-in the case of SP2, similar to the curve obtained by XRD for the 10 3 s -1 calibration and by microhardness.

As expected, work hardening decreases with depth and the affected area and work hardening levels are lower for SP1 than for SP2. Moreover, for the two shot peening conditions, EBSD and XRD methods feature a similar affected depth, corresponding to the depth that is known to be affected by shot peening for these conditions [START_REF] Prevey | The Effect of Cold Work on the Thermal Stability of Residual Compression in Surface Enhanced IN718[END_REF], [START_REF] Klotz | Surface characteristics and fatigue behavior of shot peened Inconel 718[END_REF]. This affected depth is also coherent with the results obtained with EBSD for both shot peening conditions (see Figure 3 and Figure 4). The calibration procedure demonstrates here one of its interests. Indeed, when KAM and FWHM evolutions are directly plotted as a function of depth for shot peened samples (see Figure 2), the affected work hardened depth is wider for XRD method; this is identified by the arrows: the work-hardened depth for the SP2 shot peening condition is 100 µm for the KAM parameter and 150 µm for the FWHM parameter. These results are in accordance with what has been observed without the use of a calibration procedure by Foss et al. [START_REF] Foss | Analysis of shotpeening and residual stress relaxation in the nickel-based superalloy RR1000[END_REF]. The sharp variation of the FWHM parameter for small plastic strains in the corresponding calibration curve (Figure 7) explains why the evaluation of the affected depth is, in the end, equivalent to the one evaluated with the KAM parameter.

Concerning the microhardness, it can be seen that for the shot peening conditions presented in this paper, the results are more difficult to interpret. For SP1, the evaluated work hardening is nearly zero and does not change with depth (Figure 7). The microhardness is a quantity that depends on both the work hardening and the residual stresses. In the case of shot peening, the residual stresses are important close to the surface (see Figure 10). The results obtained for SP2 suggest that the impact of residual stresses on the measurement is lower as the work hardening increases. Also, remember that the calibration method has been performed on samples that were free of residual stresses. The results obtained here confirm the difficulty, to go back in the case of shot peening to a direct quantification of work hardening without taking into account the residual stresses beforehand in the analysis of the microhardness tests; this has been already mentioned in the literature [START_REF] Li | Characterization of Residual Stresses and Accumulated Plastic Strain Induced by Shot Peening through Simulation of Instrumented Indentation[END_REF]. Ideally, to exploit this technique, it would be necessary to identify the relationship between residual stress, J o u r n a l P r e -p r o o f work hardening and microhardness using a series of finite-element simulations of micro-hardness tests with integration of the in-depth residual stress levels [START_REF] Li | Characterization of Residual Stresses and Accumulated Plastic Strain Induced by Shot Peening through Simulation of Instrumented Indentation[END_REF]. It could then be possible by an inverse method to get a consistent work hardening profile.

The results presented in this section make it possible to validate the three characterization techniques to evaluate work hardening in a shot peened part even though precautions must be taken when using microhardness due to the presence of a residual stress field. Since XRD and EBSD give similar results, the study has been limited to XRD analysis for the SP3 condition in the rest of this study.

Work hardening after shot peening: microstructure effects

The Figure 9 of the depth for different shot peening conditions. As already mentioned, two calibration curves depending on the strain rates have been built for XRD (and DA microstructure). Both work hardening evaluations are thus reported in Figure 9.a.

Note that the results presented for SP1 and SP2, for DA microstructure, are then similar to those discussed in Figure 8.

Similar affected depths can be observed in Figure 9 for the three microstructures if we focus on SP1 and SP2 shot peening conditions. However, the effects of the microstructure appear quite different on the estimated values of work hardening according to the characterization technique. Results obtained from microhardness measurements show a clear difference between the three microstructures along the entire affected depth for SP2 condition. As for SP1 treatment, the evolution seems not consistent for the two coarse grain microstructures whereas the shot peening conditions are strictly the same (Almen intensity, coverage, number of runs). As already discussed, microhardness measurement is sensitive to both residual stress and work hardening. A consistent evaluation of the work hardening in shot peened samples requires taking into account the corresponding residual stresses. They are quite different between DA and coarse grain microstructures but close between the two coarse grain microstructures, as it will be illustrated later in section 3.4. The residual stresses seem to explain the differences between DA and coarse grain microstructures but not between the two coarse microstructures. A sensitivity of the microhardness measurements to the size of the strengthening precipitates might be pointed out. The size of the precipitates is effectively increased by a factor of 10 between the two coarse grain microstructures leading to a sensible modification of the yield strength.

It can be expected that the work hardening state after shot peening is then impacted. However, it has already been mentioned in part I of the article that the indent used in this study has a rather small size (indent diagonal equal to 20 µm). The analyzed volume is approximately of the same size as the grains for the two coarse grain microstructures. This might affect the work hardening profiles obtained for the two coarse grain microstructures. In particular, the size of the precipitates could J o u r n a l P r e -p r o o f increase the dispersion when the indent is carried out on a single grain. This dispersion will be amplified after shot peening when residual stresses and work hardening are certainly not the same from one grain to another. Moreover, the residual stresses obtained from XRD correspond to an average response of the material stress state over several grains (X-ray penetration depth is estimated to be between 2 and 4 µm and the size of the analyzed volume is a few millimeters wide).

Potential differences in the stress state over one grain could not be captured. At this stage, the microhardness appears problematic to evaluate the work hardening after shot peening on the two coarse grain microstructures.

Concerning the profiles evaluated from XRD measurements, comparable values are obtained for the three SP2 shot peened microstructures, except at the surface of the samples. For a depth of less than 25-30 micrometers, differences can be observed between the three microstructures. However, the effect of strain rates (analyzed in section 3.1.5) is also particularly significant close to the surface. It is then difficult to conclude on the microstructure effect. Further investigation is required as the effect of the strain rates has not been evaluated for the coarse grain microstructures. The strain rate sensitivity may not be similar for the three generic microstructures and could explain a part of the gap noticed at the surface.

Finally, concerning the EBSD technique, comparable evolutions of the work hardening can be observed along the entire affected depth for the three microstructures. The values are also close to those obtained by XRD if the points at the extreme surface are spread out. It could be concluded that for SP1 and SP2 shot peening conditions the work hardening is quite similar for the three microstructures. This seems to be consistent with the applied shot peening conditions, which are theoretically identical for SP1 and SP2 shot peening conditions (same Almen intensity and coverage for the three microstructures). As already explained in section 2.1, the treatment parameters have been adapted according to the microstructure to achieve the same coverage. As expected, for equivalent treatment conditions, the work hardening is not very sensitive to the microstructure as for the SP1 and SP2 shot peening conditions.

J o u r n a l P r e -p r o o f

For the SP3 shot peening conditions the comparison between the three microstructures illustrated for the XRD method in Figure 9.a shows a different trend. It can then be observed that the work hardening obtained for coarse grain microstructures is higher along the entire affected depth in comparison to the DA microstructure. It is recalled that for the SP3 treatment, the same Almen intensity is applied as SP2. The shot-peening conditions are also identical in terms of number of shotpeening runs for the three microstructures but not in terms of coverage. Thus, for the SP3 condition, the coverage is around 167% for DA microstructure and 200% for coarse grain microstructures (Table 1). As observed in section 2, the yield stress is lower in the case of both coarse grain microstructures.

The induced work hardening (assimilated to the Von Mises equivalent plastic deformation) is therefore greater for an identical applied stress as observed on the work hardening profiles after shot peening. This observation tends to validate the interest of using the coverage parameter to ensure similar trends between different microstructures in terms of work hardening profiles as for SP2

treatment.

It can be concluded from the observations made here that the proposed calibration procedure permits to quantify precisely how the microstructure affects the level of work hardening. This result represents an opportunity in order to get a complete picture of the mechanical state of industrial parts presenting microstructural heterogeneities induced by the elaboration process.

J o u r n a l P r e -p r o o f As seen in Figure 10.a, the residual stresses determined on the as-received material are close to 0.

The slightly positive values for the DA microstructure can be related to the forging step undergone by the turbine disc from which all the samples were extracted. On the contrary the heat treatment carried out to generate the coarse grain microstructures appears to have induced a complete relaxation of the residual stresses. After shot peening, the residual stress profiles for the SP1, SP2 and SP3 conditions are consistent with results reported in the literature for similar shot peening J o u r n a l P r e -p r o o f treatments [START_REF] Prevey | The measurement of sub-surface residual stress and cold work distributions in nickel base alloys[END_REF], [START_REF] Klotz | Surface characteristics and fatigue behavior of shot peened Inconel 718[END_REF]. An increase in shot peening intensity and coverage leads to a significant increase in the maximum compressive residual stresses, located below the surface as it can be observed by comparing the SP1 and SP2 conditions. The corresponding affected depth is also greater for SP2 than for SP1 (with the notations defined in Figure 11, z σresmaxSP1 = 50 μm; z σresmaxSP2 = 80 μm), as the size of the compression zone (z σres0SP1 = 175 μm; z σres0SP2 = 225 μm). On the surface, on the other hand, the residual stresses are relatively similar (σ ressurfSP1 ≈ σ ressurfSP2 ≈ -700 MPa) for the different conditions. The comparison between SP2 and SP3 conditions on the DA microstructure seems surprising as for a lower coverage (200% for SP2 and 175% for SP3), the compressive residual stresses appear higher as well as the affected depth. Further investigation seems however necessary. The initial stress state of the DA microstructure induced by the forging process may have an influence on the residual stresses obtained after shot peening. As already mentioned, all the samples were extracted from a turbine disc and some dispersion could be expected in terms of residual stresses between different zones of the disc. This could explain the residual stress state for SP3 in comparison to SP2 condition. To ensure a consistent comparison between the shot peening conditions (and eventually between the microstructures) a preliminary step aiming to relax these initial stresses is certainly required. This result confirms nevertheless, what has already been observed in the literature [START_REF] Hatamleh | An investigation of the peening effects on the residual stresses in friction stir welded 2195 and 7075 aluminum alloy joints[END_REF]- [START_REF] Sherafatnia | Effect of initial surface treatment on shot peening residual stress field: Analytical approach with experimental verification[END_REF]: the elaboration process (forging, heat treatments,…) before shot peening has a significant influence on the residual stresses obtained after the surface treatment. on the basis of residual stresses alone, it seems preferable to apply the SP2 condition to parts subjected to thermomechanical loading, it's not so obvious when the work hardening profiles are also considered. Then, evaluating the quality of a shot peening treatment only on the knowledge of the residual stress profile could be insufficient to guarantee the service life of shot peened parts.

This work provides new opportunities in the fatigue life assessment of parts. It is then possible to quantify work hardening on a material and analyze its influence to get more accurate estimation models.

Conclusion

The calibration methodology proposed in Part I [START_REF] Goulmy | A calibration procedure for the assessment of work hardening Part I: effects of the microstructure and load type[END_REF] has been applied to different Inconel 718 microstructures that have been subjected to three different shot peening conditions.

The various limitations observed in the literature concerning the use of classical calibration procedure on shot peened samples have been discussed. It is shown that the application of the calibration procedure is relevant to assess the level of work hardening of parts after shot peening. Some precautions have however to be taken. For a given equivalent plastic strain, the FWHM is higher when the strain rate increases. A variation in strain rate applied to the material during loading modifies the measured FWHM. However, it is particularly difficult to experimentally characterize the strain rates to which the material is submitted during shot peening: it varies in time and space. To overcome this problem, limit bounds for the work hardening level after shot peening have been estimated in the present work to account for the range of strain rates involved in shot peening.

The capacity of the three characterization techniques (microhardness, EBSD, XRD) to evaluate a work hardening gradient due to shot peening was then investigated. For the shot peening conditions studied in this work, microhardness seems to be the least relevant to properly evaluate work

Acknowledgement

This work was conducted with the help of the French Technological Research Institute for Materials, Metallurgy and Processes (IRT M2P) under the CONDOR project. The authors would like to acknowledge IRT M2P and all the partners of the project led by IRT M2P. Safran is warmly thanked for its precious collaboration in this work.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to legal or ethical reasons.

J o u r n a l P r e -p r o o f

J o u r

  n a l P r e -p r o o f parameters are given in the part I of the present work.

Figure 1 .

 1 Figure 1.3 precisely describes the procedure used, by taking, as an example, the case of XRD

Figure 1 .

 1 Figure 1. Methodology for characterizing work hardening in shot-peened samples.

Figure 2 .

 2 Figure 2. Evolution for the DA microstructure of KAM, FWHM and HV 0.1 parameters as a function of depth for the as-received and shot peened samples (SP2).

Figure 3 .

 3 Figure 3. Local misorientation EBSD maps for the as-received and shot peened DA microstructure.

Figure 4

 4 Figure 4 presents local misorientation EBSD maps obtained for the three studied microstructures

Figure 4 .

 4 Figure 4. Local misorientation EBSD maps for the three studied microstructures shot peened in SP2. The grain size for the DA microstructure is 5 µm and 35 µm for the two coarse grain microstructures.

Figure 5 .

 5 b and Figure5.c further present the pole figures and grain size histograms obtained in the two zones. This confirms that the shot peening treatment applied in this study impacts neither the grain size nor the texture of the material: the pole patterns are indeed similar. The grain size distributions exhibit no major differences between the near-surface area and the core area. The same evaluations have been performed for the two coarse grain microstructures but the results are not presented here because they lead to exactly the same conclusion.

Figure 5 .

 5 Figure 5. Grain size and crystallographic texture comparison between the shot peened (SP2 condition) and asreceived materials obtained for DA microstructure.

Figure 6 .

 6 Figure 6. SEM images at different depth for the coarse grains and coarse strengthening precipitates microstructure after SP2 shot peening.

Figure 7 .

 7 Figure 7. a) Strain rate influence on the calibration curve obtained by XRD method on the DA microstructure, b) Impact of strain rate on work hardening determination in the case of SP1 shot peened DA, c) Impact of strain rate on work hardening determination in the case of SP2 shot peened DA microstructure.

Figure 8 .

 8 Figure 8. Comparison of the different characterization methods for two shot peening conditions: left SP1 and right SP2.

  compares the work hardening profiles obtained for the three microstructures of Inconel 718 superalloy (DA, coarse grain, coarse grain and coarse precipitate microstructures) The calibration curves established in part I for each microstructure and for each experimental technique (XRD, EBSD and microhardness) have been used reversely to derive the work hardening evolutions as a function J o u r n a l P r e -p r o o f

Figure 9 . 3 . 4

 934 Figure 9. Comparison of a) XRD, b) EBSD and c) Microhardness methods to evaluate the work hardening evolution in the SP1 and SP2 conditions for the three studied microstructures. Note that the scale for work hardening is the same in the seven figures

Figure 10 .

 10 Figure 10. Residual stress and work hardening profiles obtained by XRD on the three microstructures prior to shot peening (a) as well as those obtained for the three peening conditions (b) SP1, (c) SP2, (d) SP3.

Figure 11 .

 11 Figure 11. Notations used to describe the shape of shot-peening induced residual stress and work hardening profiles.

J o u r

  n a l P r e -p r o o f hardening because this method is strongly dependent on the residual stress field present in the material. On the contrary, EBSD and XRD methods are well adapted and provide complementary information. The complementarity of the XRD and EBSD techniques, which give, thanks to the calibration, similar results, is particularly interesting since it can lead to a better understanding of the deformation mechanisms occurring during manufacturing processes.Residual stress and work hardening profiles were also compared. It was shown that knowledge of the residual stress profile is insufficient to characterize the mechanical state of the material because work hardening provides additional information that needs to be taken into account when predicting the relaxation of residual stresses.Finally, the methodology has been used to characterize three microstructures with different grain and strengthening precipitates sizes. A microstructure dependency has been observed and justifies a calibration curve for each microstructure. With the work proposed here, different microstructures can now be properly compared in terms of work hardening induced by shot peening allowing an accurate assessment of the fatigue life of different industrial parts by taking into account not only the residual stress but also the amount of work hardening accumulated in the material.

Table 1 . Shot peening conditions applied to the different microstructures.

 1

	Name	Microstructure	Almen Intensity	Coverage	Number of runs of shot peening
		DA	12-13A	125%	38
	SP1	Coarse grain	12-13A	125%	32
		Coarse grain and coarse strengthening precipitates	12-13A	125%	32
		DA	22-23A	200%	24
	SP2	Coarse grain	22-23A	200%	20
		Coarse grain and coarse strengthening precipitates	22-23A	200%	20
		DA	22-23A	167%	20
	SP3	Coarse grain	22-23A	200%	20
		Coarse grain and coarse strengthening precipitates	22-23A	200%	20

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


A Loosely Coupled Vision-LiDAR Odometry using Covariance Intersection Filtering

Songming Chen1 , Vincent Frémont 1

Abstract-This paper presents a loosely-coupled sensor fusion approach, which efficiently combines complementary visual and range sensor information to estimate the vehicle ego-motion. Descriptor-based and distancebased matching strategies are respectively applied to visual and range measurements for feature tracking. Nonlinear optimization optimally estimates the relative pose across consecutive frames and an uncertainty analysis using forward and backward covariance propagation is made to model the estimation accuracy. Covariance intersection filter paves the way for us to loosely couple stereo vision and LiDAR odometry considering respective uncertainties. We evaluate our approach with KITTI dataset which shows its effectiveness to fierce rotational motion and temporary absence of visual features, achieving the average relative translation error of 0.84% for the challenging 01 sequence on the highway.

I. INTRODUCTION

Accurate state estimation and a good knowledge of the surrounding environment are crucial for autonomous driving. Many autonomous vehicles use range-based Li-DAR and/or vision-based stereo cameras to perform the task of ego-motion estimation. The most frequently used sensors (Camera and LiDAR) have their own strengths and weaknesses under different working conditions. Laser scanners, for example, are good at measuring distance but are quite sensitive to fog and rain. Cameras are more commonly applied to extract the visual cues of the scene, but cannot work in low illumination conditions. Since range and visual sensors appear to be complementary, their combination allows to compensate respective shortcomings. The main challenge for longterm state estimation is error accumulation, especially in environmentally degenerate scenarios. The fusion of range and visual sensors could restrict the local uncertainties and allow to confine the odometry drift.

ORB-SLAM [START_REF] Mur-Artal | Orbslam: a versatile and accurate monocular slam system[END_REF] is regraded as a typical representative of vision-based SLAM. Oriented FAST and Rotated BRIEF (ORB) features are extracted and matched for real-time state estimation, and meanwhile the bag-ofwords (BOW) dictionary is queried for loop closure and drift cancellation. This method can accurately localize the mobile platform and create a sparse feature map of its surroundings with limited computation resources. ORB-SLAM2 is proposed in [START_REF] Mur-Artal | Orb-slam2: An open-source slam system for monocular, stereo, and rgb-d cameras[END_REF] with stereo observations based back-end which solves the scale ambiguities for trajectory estimation.

Scan-matching is a fundamental process to estimate platform motion and to create a 3D map with LiDAR. A popular approach for LiDAR based localization is LOAM [START_REF] Zhang | Loam: Lidar odometry and mapping in real-time[END_REF]. It conducts Iterative Closest Point (ICP) scan-matching for 3D point clouds registration, which is followed by a global scan-to-map alignment in order to reduce local errors. Feature alignment problem can be solved using the well known Levenberg-Marquardt optimizer.

Despite the success and popularity of ORB2 SLAM [START_REF] Mur-Artal | Orb-slam2: An open-source slam system for monocular, stereo, and rgb-d cameras[END_REF] and LOAM [START_REF] Zhang | Loam: Lidar odometry and mapping in real-time[END_REF], they are in fact deterministic algorithms. They do not effectively handle the sources of uncertainty. As a result, they provide overconfident state estimation results across frames. The main contribution of this paper is to propose a loosely coupled sensor fusion approach for vehicle localization with range and visual sensors. Measurement uncertainties for visual and range sensors are properly defined for state estimation. Backward covariance propagation [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF] is utilized to transform the covariance from measurement domain to estimation domain. At the same time, forward covariance propagation is leveraged to transform the uncertainty from manifold space to Euclidean space. The covariance intersection filtering enables adaptive fusion of the two sensors given their respective uncertainties.

The remainder of this paper is divided into the following sections. Section II presents the related sensor fusion approaches to improve the perception and localization performance. The loosely coupled vision-LiDAR odometry approach is proposed in Section III and tested with the KITTI odometry benchmark in Section IV. In Section V, a concise conclusion is given which is followed by the future work plan.

II. RELATED WORK

A. Enhanced Visual State Estimation

The state estimation result reached by visual-SLAM algorithms can be further enhanced via integrating Li-DAR measurements. In [START_REF] Graeter | Limo: Lidarmonocular visual odometry[END_REF], depth information from LiDAR measurements was utilized for visual feature tracking after LiDAR points being projected onto image frames. At the same time, visual semantic information was used for removing outliers and increasing the weights of static landmarks. Instead of using visual feature points, in [START_REF] Shin | Direct visual slam using sparse depth for camera-lidar system[END_REF] a SLAM system using visual photometric information was proposed. Its performance was enhanced with the involvement of sparse LiDAR point cloud for depth acquisition. However, as pixel resolution was much greater than LiDAR point cloud one, many pixels were not assigned the depth value, thus extra interpolation was needed to make up the missing values.

B. Enhanced LiDAR State Estimation

In many cases, LiDAR scan-matching is used for local motion estimation and visual hint is utilized for loop closure validation. The accuracy of LiDAR based localization was improved in [START_REF] Liang | Visual laser-slam in large-scale indoor environments[END_REF], with visual feature aided loop detection to reduce the accumulated drift. In [START_REF] Zhu | Loop detection and correction of 3d laser-based slam with visual information[END_REF], the visual keyframes were utilized to assist the laser-based slam to perform local and global bundle adjustments. Furthermore, the LiDAR scan-to-scan matching can be improved using the initial guess from visual estimation as demonstrated in [START_REF] Pandey | Visually bootstrapped generalized icp[END_REF].

C. Concurrent Visual-LiDAR State Estimation

There are also many works which coupled both Li-DAR and visual state estimation process together. Zhang et al. [START_REF] Zhang | Visual-lidar odometry and mapping: Low-drift, robust, and fast[END_REF] designed V-LOAM pipeline which used high frequency vision based odometry as the motion prior and corrected with high precision, low frequency lidar scan matching estimation afterwards. The framework in [START_REF] Seo | A tight coupling of vision-lidar measurements for an effective odometry[END_REF] did not rely on visual estimation as the motion initial guess for lidar odometry. They took in both visual and LiDAR measurements, stacking and minimizing both modalities' residuals during the optimization phase. However, as mentioned in Section I, they did not consider the uncertainty during state estimation process, which may cause the overconfident estimation prone to certain sensor modality.

III. PRESENTATION OF THE METHOD

In Fig. 1, it can be seen that the proposed sensor fusion framework starts with a descriptor-based visual feature tracking module to estimate vehicle ego-motion. Meanwhile, LiDAR distance-based scan-to-scan matching runs in parallel for state estimation. Backward covariance propagation transforms the uncertainty from measurement space to estimation space, which helps to obtain the uncertainty of frame-to-frame state estimation for both sensor modalities. Covariance intersection filtering ensures that the uncertainty of state does not expand after the sensor fusion, which combines two frameto-frame poses elegantly. Then, the robustified pose is used for LiDAR point cloud registration. Scan-to-map matching afterwards further reduces local drift caused by frame-to-frame estimation. The updated odometry is the final output which is published at 10Hz frequency.

To denote coordinate systems, this paper's convention is to use uppercase letter to indicate different coordinate frames. In visualization and sensor fusion steps, vehicle pose is expressed with 3D translation and RPY Euler angle rotation. However, in order to avoid singularity problem, we optimize on their manifold which is detailed in Section III-A.1. In the following, coordinate systems being used are explained.

• Camera sensor coordinate system {C t } at timestamp t is defined at the camera optical center. The x-axis, y-axis and z-axis point rightward, downward and forward respectively as the camera configuration in [START_REF] Geiger | Vision meets robotics: The kitti dataset[END_REF]. • LiDAR sensor coordinate system {L t } at timestamp t is defined at the LiDAR scanner center. The xaxis, y-axis and z-axis point forward, leftward, and upward respectively as the LiDAR configuration in [START_REF] Geiger | Vision meets robotics: The kitti dataset[END_REF]. • World coordinate system {W } is defined as {C 0 } which is the initial frame of the camera coordinate system, and lidar-camera extrinsics T C L is assumed to be known beforehand. C t :

C t xi =   Pr l K, Θ C t-1 C t , C t-1 x i Pr r K, Θ C t-1 C t , C t-1 x i   (1)
where C t xi = ûi,l , vi,l , ûi,r , vi,r T is the prediction in the current frame and C t-

1 x i = u i,l , v i,l , u i,r , v i,r
T is its correspondence in the previous frame. In general, the optimal relative camera transformation can be estimated by minimizing the weighted squared error of measurements and predictions.

Θ * C t-1 C t = argmin Θ C t-1 C t F(x, Θ C t-1 C t ) = argmin Θ C t-1 C t N c ∑ i=1 C t x i -C t xi 2 Σ (2) where • 2
Σ is the Mahalanobis distance with Σ -1

x i as the information matrix for the i th measurement. To handle estimation parameters that do not belong to Euclidean spaces, the common strategy is to transfer the error minimization to its corresponding manifold. In our case, iterative optimization update for estimated parameters is made using Lie algebraic perturbation model. Operator is generalization of the normal addition operator, which is defined as δ ε Θ exp(δ ε) Θ, then Ji ( Θ) can be written as

Ji ( Θ) = ∂ e i (δ ε Θ) ∂ δ ε δ ε→0 (3)
As a result, we can apply the famous Levenberg-Marquardt method without considering additional constraint such as rotation matrix orthogonality.

2) LiDAR odometry: The same way in LiDAR odometry, edge and planar LiDAR points are tracked to recover the LiDAR pose. For each LiDAR scan point, local curvature c is computed to evaluate its smoothness considering the surrounding area. Let S be a group of points in the vicinity of x i in the same scan layer.

c = 1 |S| • L t x i ∑ j∈S, j =i L t x i -L t x j (4)
Edge and planar points are defined based on c values. The edge line constructed by two edge points at previous frame ( L t-1 x j , L t-1 x l ) ∈ L t-1 E forms the correspondence of L t x i . L t-1 x j and L t-1 x j are selected according to nearest neighbour criteria and they belong to The planar patch represented by three points at previous frame ( L t-1 x j , L t-1 x l , L t-1 x m ) ∈ H t-1 forms the correspondence of L t x i . We assume that the closest neighbor of L t x i is denoted as L t-1 x j . L t-1 x l , L t-1 x m are second and third nearest neighbors of L t x i , one belonging to the same scan layer of L t-1 x j , and the other in the consecutive scan layer of L t-1 x j . With the corresponding relationship of the feature points in hand, according to Fig. 2, we are able to calculate the distance from a feature point to its correspondence.

d E = L t x i -L t-1 x j × L t x i -L t-1 x l L t-1 x j -L t-1 x l (5)
d H = L t x i -L t-1 x j L t-1 x j -L t-1 x l × L t-1 x j -L t-1 x m L t-1 x j -L t-1 x l × L t-1 x j -L t-1 x m (6)
The optimal LiDAR relative pose can be obtained by minimizing the weighted sum squared distances of edge and planar points to their correspondences. Σ x -1

Ei and Σ x -1 Hi stand for the information matrix of E i th edge and H i th planar measurement points and we take the same optimization strategy as in Section III-A.1.

Θ * L t-1 L t = argmin Θ L t-1 L t N E ∑ E i =1 d E i Σ x -1 Ei d E i + N H ∑ H i =1 d H i Σ x -1 Hi d H i (7)

B. Uncertainty analysis

Robust state estimation should be able to provide the uncertainty information associated with the vehicle pose estimates. The sensor fusion phase is driven by the uncertainties in the estimation domain. Thus, we analyse the uncertainties coming from visual and range sensors via forward and backward covariance propagation.

1) Visual sensor uncertainty:

Although the optimal relative pose can be obtained by minimizing Eq. 2, its accuracy also depends on the precision of the corresponding feature points, more specifically, the level of the image pyramid they belong to. The image pyramid [START_REF] Adelson | Pyramid methods in image processing[END_REF] is a series of image collections whose resolution gradually decreases in the shape of a pyramid. The image pyramid can be sequentially matched to ensure scale invariance during feature tracking. In our case, the image pyramid has 8 levels with the same scale factor 1.2 between two consecutive levels. We assume that all points considered in the optimization procedure are wellmatched pixel features with only zero mean Gaussian noise N(0, Σ V

x i ), with σ x u i,l(r) = σ x v i,l(r) = 1.2 level-1 as standard deviation for i th measurement. Jacobian matrix Ji (Θ * ) is defined in Eq. 3, and it converts the uncertainty from measurement space to estimation space. Since we optimize on manifold, let ε = log(Θ * ) a 6d vector in the Lie algebra space, Jacobian matrix J m2e = ∂ e ε ∂ ε is indispensable to propagate the covariance from manifold space to Euclidean space for data visualization and fusion. Then we can obtain the uncertainty of frameto-frame relative pose Σ V Θ * through Eq. 8 and the result is shown in Fig. 3.

Σ V Θ * = J V m2e ( N c ∑ i=1 ( JV i (Θ * )Σ V x i -1 JV i (Θ * ))) -1 J V m2e (8)
2) Range sensor uncertainty: our case, a Velodyne HDL-64E is used which provides a (0 • ∼ 360 • ) azimuth field of view (θ ) and (-24.9 • ∼ 2 • ) elevation field of view (φ ). According to official velodyne data sheet, range accuracy can reach up to 2 cm which is quite small compared with its range limit 120 m. Hence, each measurement is treated equally and we can set measurement uncertainty Σ L

x i as identity matrix for each point. Based on such assumption, we obtain the uncertainty of scan-to-scan relative pose Σ L Θ * through Eq. 9 and the result is shown in Fig. 4.

Σ L Θ * = J L m2e ( N E +N H ∑ i=1 ( JL i (Θ * )Σ L x i -1 JL i (Θ * ))) -1 J L m2e (9)
3) Covariance Intersection Filtering: Covariance intersection [START_REF] Julier | Using covariance intersection for slam[END_REF] is a variant of Gaussian process sensor fusion which can combine two estimates under unknown correlations. The covariance intersection combination formulas are given by x i Ji (Θ * ) will lead to unreliable uncertainty estimation marked as black dash lines in Fig. 3. Covariance intersection can ensure that the resulting estimate is conservative, which efficiently filters out the unstable estimation. In order to simplify the fusion parameterization, only planar translation t x and t z and yaw angle r y are fused and updated. The fused pose will better register the lidar map point and we adopt the multi-level voxel scan-to-map matching as in [START_REF] Seo | A tight coupling of vision-lidar measurements for an effective odometry[END_REF] to reduce the frame-toframe estimation drift.

Σ f used Θ = ω Σ L Θ -1 + (1 -ω) Σ V Θ -1 -1 Θ f used = Σ f used Θ ω Σ L Θ -1 Θ L + (1 -ω) Σ V Θ -1 Θ V (10)

IV. EXPERIMENTAL RESULTS

The KITTI dataset [START_REF] Geiger | Vision meets robotics: The kitti dataset[END_REF] contains stereo sequences and Velodyne HDL-64E LiDAR point clouds captured in urban and highway environments. We use the metric of average relative translation error t rel proposed in [START_REF] Geiger | Vision meets robotics: The kitti dataset[END_REF] for evaluation purpose. To have a fair comparison, the ORB SLAM2 loop closure module is deactivated. Evaluation results are obtained on a laptop with an Intel i7-9750H CPU and 32GB of RAM. We choose three typical sequences 01 (Highway), 02 (Urban+Country) and 07 (Urban) from KITTI dataset to make analysis and detailed quantitative result is shown in Tab. I1 .

Our Loosely-Coupled Vision-LiDAR Odometry(LC-VLO) outperforms state-of-the art approaches for challenging trajectory in sequence 01. When driving on a highway scenario, few distinctive visual features are available, see Fig. 5, which makes descriptor-based feature tracking erroneous and thus causes poor pose estimation for visual sensor. Our proposed approach ensures a consistent odometry estimation even moving at high speed. In sequence 02, our loosely coupled odometry is not as good as the ORB-SLAM2 due to lack of horizontal lines or planes to constraint the drift along the vertical axis. However, it does efficiently prevent large divergence occurrence like in A-LOAM2 method. The large divergence mainly results from A-LOAM's inappropriate distance-based matching strategy. Far edge points are more likely to be mismatched when encountering large rotational motion. It happens at the middle of sequence 02 where A-LOAM method loses the tracking of features and fails to confine the estimation error. Uncertainty analysis is able to detect the potential deficiencies in the early scan-to-scan step and mitigate feature misalignment problem. Our LC-VLO is superior to ORB-SLAM2 and A-LOAM for sequence 7, which shows that multi-level voxel scanto-map matching procedure is indispensable to reduce frame-to-frame estimation drift. Overall, the proposed LC-VLO adaptively fuses vision and LiDAR estimation, which is able to improve estimation performance in individual sensor degenerate cases, especially for the challenging KITTI sequence 01 and 02. be easily replaced according to personalized demands, which makes our approach very flexible. As our current approach does not consider loop closure, we will focus on exploiting visual semantic hints for robust feature tracking and place recognition to further ameliorate the

Fig. 1 .

 1 Fig. 1. Overview of the proposed sensor fusion scheme

A. State estimation 1 )

 1 Visual odometry: In feature-based stereo vision odometry, key points with local descriptors are matched to deduce the camera motion with scale metrics. Provided camera intrinsics K, stereo feature points belonging to the previous frame are triangulated in the first step. And then transformed triangulated points are reprojected via the perspective projection operation Pr l (•), Pr r (•) onto the left and right images respectively considering the 6 dof state estimation variable Θ C t-1

Fig. 2 .

 2 Fig. 2. Scheme of edge and planar LiDAR points correspondence projection

Fig. 3 .

 3 Fig. 3. Visual sensor pose estimation uncertainty along t x and t z for KITTI sequence 01

Fig. 4 .

 4 Fig. 4. Range sensor pose estimation uncertainty along t x and t z for KITTI sequence 01

Fig. 5 .Fig. 6 .

 56 Fig. 5. Few distinctive ORB visual features for tracking on the highway scenario, KITTI sequence 01

TABLE I COMPARISON

 I OF ACCURACY ( IN PERCENTAGE ).

	Sequence	Metric t rel	ORB-SLAM2	A-LOAM	LC-VLO
	00		0.88%	0.77%	0.74%
	01		1.40%	2.27%	0.84%
	02		0.79%	4.91%	1.50%
	03		0.77%	1.24%	0.87%
	04		0.45%	1.23%	1.08%
	05		0.61%	0.70%	0.43%
	06		0.73%	0.62%	0.58%
	07		0.90%	0.63%	0.50%
	08		-%	-%	-%
	09		0.90%	1.09%	1.01%
	10		0.59%	1.69%	0.52%
	localization accuracy in the future.

S. Chen, V. Frémont are with the Laboratoire des sciences du numérique de Nantes (LS2N), UMR 6004, at École Centrale de Nantes, 44321 Nantes, France, songming.chen@ec-nantes.fr, vincent.fremont@ec-nantes.fr

The sequence 08 is not evaluated due to ground truth flaw with manual inspection

Advanced implementation of LOAM, https://github.com/HKUST-Aerial-Robotics/A-LOAM

* This work was supported by China Scholarship Council


David Pérez-Morales

Vincent Fremont

Information-Theoretic Sensor-Based Predictive Control for Autonomous Vehicle Navigation: A Proof of Concept

This paper explores the feasibility of an Information-Theoretic Sensor-Based Predictive Control (IT-SBPC) approach for autonomous navigation in presence of pedestrians. Our technique relies solely in sensor data expressed relative to the vehicle and therefore no localization is inherently required. By combining the advantages of the informationtheoretic framework and sensor-based formalism, the proposed technique drives the vehicle safely and smoothly towards the desired goal. Several real-time simulated scenarios, showing that the car is able to reach the goal with centimeter-level accuracy, validate the effectiveness of our approach.

I. INTRODUCTION

Vehicles and Intelligent Transportation Systems have followed a quick mutation thanks to the growing use of electric technology and the emergence of autonomous navigation techniques. The hype created around self-driving vehicles even lead parts of the industry to overestimate their arrival [START_REF] Khalid | Ford CEO says the company 'overestimated' self-driving cars[END_REF] as commercial products. In spite of the efforts from the industry and academia to reach fully autonomously driving capabilities (level 5), there 's not yet a system with such capacities [START_REF] Ackerman | Toyota's Gill Pratt on Self-Driving Cars and the Reality of Full Autonomy[END_REF]. According to different studies, pedestrian populating environments present the major challenges due to the ethical, social and legal considerations that arise [START_REF] Adouane | Toward fully autonomous vehicle navigation: From behavioral to hybrid multi-controller architectures[END_REF], [START_REF] Bellet | From semi to fully autonomous vehicles: New emerging risks and ethico-legal challenges for human-machine interactions[END_REF]. Moreover, crowded environments present challenges even for experienced drivers. Therefore, being able to navigate safely and effectively under such conditions would represent a major milestone towards fully autonomous driving.

In order to deal with pedestrian populating environments, Model Predictive Path Integral (MPPI) Control techniques appear as an interesting path to explore due to the prediction step inherently considered. MPPI control algorithms and its derivatives have been extensively investigated for autonomous navigation tasks [START_REF] Williams | Aggressive driving with model predictive path integral control[END_REF], [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF], [START_REF] Kusumoto | Informed Information Theoretic Model Predictive Control[END_REF]. Some of the key features of this type of techniques are that it is a derivativefree method optimal control method that can naturally deal with non-linear, non-convex and non-smooth system dynamics and cost functions. Moreover, it is straightforward to impose constraints on both the state and control inputs of the system.

Another interesting alternative for autonomous navigation tasks is the use of sensor-based control techniques. In particular, they have been proven to be valid for navigation [START_REF] De Lima | Sensor-Based Control with Digital Maps Association for Global Navigation: A Real Application for Autonomous Vehicles[END_REF], dynamic obstacle avoidance [START_REF] Kang | Dynamic obstacles avoidance based on image-based dynamic window approach for human-vehicle interaction[END_REF] and for parking applications in static [START_REF] Perez-Morales | Laser-Based Control Law for Autonomous Parallel and Perpendicular Parking[END_REF] and dynamic environments [START_REF]Multi-Sensor-based Predictive Control for Autonomous Parking in Presence of Pedestrians[END_REF]. A key benefit of these techniques is their great robustness against modeling and calibration errors, particularly when then pose of the robot doesn't need to be reconstructed. Moreover, because they rely only in locally perceived information, they do not suffer from potential localization issues.

In this work we aim to provide a proof of concept that combines the advantages of the Information-Theoretic Model Predictive Control (ITMPC) framework [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF] and the sensorbased formalism [START_REF] Chaumette | Visual servo control, part I : Basic Approaches[END_REF] by proposing an Information-Theoretic Sensor-Based Predictive Control approach capable of navigating towards a given goal while avoiding collision with (potentially dynamic) surrounding obstacles. Given that this work is a proof of concept, a single obstacle is considered in the environment. Moreover, dynamic obstacles are assumed to be pedestrians.

In the next section the kinematic model of the vehicle, the multi-sensor formalism and the interaction model that allows to describe the navigation task and collision avoidance constraints are presented. Afterwards, the control strategy is presented in Section III. The simulation setup is described in Section IV and the obtained results are presented in Section V. Finally, some conclusions are given in Section VI.

II. MODELING AND NOTATION A. Car-like robot model and notation

Considering the well-known kinematic model of a car with rear-wheel driving [START_REF] De Luca | Feedback control of a nonholonomic car-like robot[END_REF], the vehicle's twist is defined by the following column vector (elements separated by a semicolon):

v m = [v xm ; θm ], (1)
where v xm and θm are, respectively the longitudinal (along x m ) and rotational velocities expressed in the moving base frame F m . Additionally, one can link the steering angle δ to θm using the following equation:

θm = v xm tan δ l wb . ( 2
)
where l wb is the distance between the front and rear wheel axles. Therefore, it is possible to consider as the actual control input of the robotized vehicle the following expression:

v r = [v xm ; δ] (3)
Nonetheless, in order to minimize the chattering produced by the stochastic nature of the proposed approach, the time derivative of (3) is considered as the input of the transition model introduced in the next section: As such, it is possible to retrieve (3) from (4) with a simple numerical integration:

u = [ vxm ; δ] (4)
v r = v r + t s u (5)
where t s is the sampling period.

It should be noted that, thanks to the multi-sensor-based formalism considered (introduced in the next subsection), our closed-loop control law does not need to have any knowledge about the Cartesian pose of the vehicle.

The vehicle used for simulation, represented by its bounding rectangle in Fig. 1a, is a Renault ZOE (Fig. 1b). Its relevant dimensional parameters are presented in Table I.

B. Multi-sensor modeling

For the sake of clarity, the considered multi-sensor modeling (detailed in [START_REF] Kermorgant | Dealing with constraints in sensorbased robot control[END_REF]) is recalled in this subsection.

Let us consider a robotic system equipped with k sensors (Fig. 2a) that provide data about the robot pose in its environment. Each sensor S i gives a signal (sensor feature)

s i of dimension d i with k i=1 d i = d.
In a static environment, the sensor feature derivative can be expressed as follows:

ṡi = Li vi = Li i Tm vm ( 6
)
where Li is the interaction matrix [START_REF] Chaumette | Visual servo control, part I : Basic Approaches[END_REF] of s i (dim( Li ) = d i × 6) and i Tm is the 3D screw transformation matrix that expresses the sensor twist vi (which is expressed in its corresponding frame F i ) with respect to the robot twist vm , expressed in the mobile frame F m which is used as control frame.

Denoting s = (s 1 ; . . . ; s k ) the d-dimensional signal of the multi-sensor system, the signal variation over time can be linked to the moving vehicle twist:

ṡ = Ls vm (7)
with Ls = L Tm , where L and Tm are obtained by concatenating either diagonally or vertically, respectively, matrices Li and

i Tm ∀ i ∈ [1 . . . k].
Planar world assumption: Assuming that the vehicle to which the sensors are rigidly attached evolves in a plane and that the sensors and vehicle have vertical parallel z axes, all the twists are reduced to [v xi ; v yi ; θi ] hence the reduced forms (denoted by the accent ˇ) of the various matrices and vectors (denoted by the accent ˘) are considered. Ľi is of dimension

d i × 3, vm = [v xm ; v ym ; θm ] and i
Ťm is defined as:

i Ťm =   cos( m θ i ) sin( m θ i ) x i sin( m θ i ) -y i cos( m θ i ) -sin( m θ i ) cos( m θ i ) x i cos( m θ i ) + y i sin( m θ i ) 0 0 1   (8)
where m t i = [x i ; y i ] and m θ i are, respectively, the position and orientation of S i (frame F i ) with respect to F m expressed in F m . Furthermore, since in the considered model the control frame F m is attached to the vehicle's rear axle with origin at the point M (Fig. 1a), the robot twist vm can be further reduced to [START_REF] Khalid | Ford CEO says the company 'overestimated' self-driving cars[END_REF]. Thus, it is possible to write:

ṡ = L s v m (9)
where L s is composed of the first and third columns of Ľs . Moreover, if the appropriate considerations are taken in the interaction model, sensor-based control strategies are capable of dealing with moving features [START_REF] Shahriari | Robotic visual servoing of moving targets[END_REF]. [START_REF] Kusumoto | Informed Information Theoretic Model Predictive Control[END_REF] can be modified to account for the velocity of moving sensor features as follows:

ṡ = Ls (v m -vmo ) ( 10
)
where vmo is the twist of the moving feature expressed in the vehicle frame. Furthermore, if at some point a moving feature becomes static (i.e. vmo is null), [START_REF] Perez-Morales | Laser-Based Control Law for Autonomous Parallel and Perpendicular Parking[END_REF] would simply become ṡ = Ls vm , expression that can be easily reduced to [START_REF] Kang | Dynamic obstacles avoidance based on image-based dynamic window approach for human-vehicle interaction[END_REF] with no further impact on the modeling. For this work we assume that vmo is known.

C. Interaction model

For the interaction model it is assumed that the system is capable of perceiving its desired goal and (potentially dynamic) surrounding obstacles as points in, respectively, polar (ρ, θ) and Cartesian (x, y) coordinates. Therefore, the features in polar coordinates and the associated interaction matrix are denoted as [START_REF] Chaumette | Visual servo control, part II: advanced approaches[END_REF]:

s ρθ = [ρ; θ], (11) Ľρθ =
cos(θ)sin(θ) 0

sin(θ) ρ -cos(θ) ρ -1 . (12)
Similarly, the features in Cartesian coordinates and the associated interaction matrix are denoted as [START_REF] Chaumette | Visual servo control, part I : Basic Approaches[END_REF]:

s xy = [x; y], (13)
Ľxy = -1 0 y 0 -1 -x . ( 14
)
Without entering in matters of social interaction, the moving obstacle is assumed to be a pedestrian. It is modeled as a point in Cartesian coordinates with a given orientation (Fig. 2b).

III. INFORMATION-THEORETIC SENSOR-BASED PREDICTIVE CONTROL

The ITMPC [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF] strategy, which our work is based upon, is a stochastic Model Predictive Control method that can be applied to non-linear (with respect to the control input or the state) systems and with non-convex cost objectives. The key idea of the approach is to evaluate the cost of thousands of trajectories (rollouts) sampled from the system dynamics using a Monte-Carlo simulation in real time.

Let us consider a discrete time dynamical system of the form

s t+1 = f (s t , w t ) (15)
where s ∈ R d is the state of the system at time t, w t ∈ R n is the input to the system at time t and f is the statetransition function of the system. It is assumed that we have no direct control over the input variable w t and instead w t is a random vector generated by a white-noise proces with density function w t ∼ N (u t , Σ) in which we have a direct control over the mean u t . This assumption translates to the low-level controller achieving the set-point with some error that satisfies a Gaussian distribution. Given a finite-time horizon t ∈ {0, 1, 2, . . . , T -1}, the stochastic optimal control aims to find a control input sequence ũ = {u 0 , u 1 , . . . , u T -1 } ∈ R n×T that minimizes the expectations E [S(s t , w)] with respect to all the simulated rollouts generated using [START_REF] Shahriari | Robotic visual servoing of moving targets[END_REF]. The optimal control problem can be formulated as:

J = min ũ∈U E [S(s t , w)] = min ũ∈U E φ(s T ) + T -1 t=0 (q(s t ) + γu T t Σ -1 (u t -w t )) (16
) where U is the set of admissible command sequences, S(s t , w) is the state-dependent cost to go of a rollout while φ(s T ) and q(s t ) are, respectively, the terminal and the statedependent running costs, and γ = λ(1α) where λ is the inverse temperature and 0 <= α <= 1 is a parameter that balances the requirements of low energy (α = 0, ũ is pushed to zero) and control smoothness (α = 1, ũ is kept near the current planned control sequence). As noted in [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF], this formulation is closely related to the well-known policy gradient theorem typically used in reinforcement learning. φ, q, Σ, γ, α, λ: Cost functions/parameters; SGF: Savitzky-Golay convolutional filter; 1 while task not completed do 2 s 0 ← GetState(); Our Information-Theoretic Sensor-Based Predictive Control strategy is detailed in Algorithm 1. It starts by getting the current state of the system and then producing K trajectory samples. Each rollout is generated by simulating forward the dynamics using a randomly generated control sequence. A cost based on the state and inputs is computed for each rollout. Having computed the cost of each trajectory (which is initialized to zero), the control sequence is updated based on the minimum sampled cost and the probability-weighted average over all the perturbation sequences. Afterwards, the updated control sequence is passed through a Savitzky-Golay Filter (SGF). A simple numerical integration (5) is used to obtain the control input to be sent to the vehicle. Finally, the control sequence of is slid down in order to warm-start the next control loop iteration.

3 S k ← InitTrajectoryCost(), S k ∈ R K ; 4 for k ← 0 to K -1 do 5 s ← s 0 ; 6 Sample ε k = { k 0 , k T -1 } ∈ N (0, Σ); 7 for t ← 0 to T -1 do 8 if k ≥ (1 -α)K then 9 w t = u t + k t ; 10 else 11 w t = k t ; 12 s ← f (s, w t ); 13 S k + = q(s) + γu T t Σ -1 (u t -w t ); 14 S k + = φ(s); 15 β ← min k [S k ]; 16 η ← K-1 k=0 exp -1 λ (S k -β) ; 17 for k ← 0 to K -1 do 18 w k ← 1 η exp -1 λ (S k -β) ;
The key difference with the work presented in [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF] is the use of a numerical integration step in the transition model (Algorithm 2) and to obtain the final control input v r to be sent to the vehicle. As acknowledged in [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF] nature of the Information-Theoretic Model Predictive Control strategy can lead to significant chattering. We found that even after applying a SGF to the control sequence, the non-smooth behavior of the control was still strongly present, especially when the vehicle starts relatively far away from the goal. By using the aforementioned numerical integration step (5) we managed to considerably reduce this effect.

IV. SIMULATION SETUP

In order to evaluate the performance of our proposed IT-SBPC approach we make use of an in-house developed fast prototyping environment using the same software architecture as the one inside the real vehicle. In addition to behaving nearly identically (form a software architecture point of view) to the real car, this fast prototyping environment simulates as well the dynamics of the vehicle leading to rather realistic simulations. Moreover, to simulate the pedestrian we make use of a modified version [START_REF] Prédhumeau | Modeling and Simulating Pedestrian Social Group Behavior with Heterogeneous Social Relationships[END_REF] of the open source crowd simulator pedsim ros [START_REF] Okal | pedsim ros GitHub repository[END_REF]. The pedestrian generated by pedsim ros is aware of the vehicle and thus its behavior/motions are influence by it.

Considering that the task is to reach a goal while avoiding collision, the state-dependent cost function is defined as:

q(s) = e T Qe + v T m Rv m + 10 4 C ( 17
)
where e = s ρθ -s * ρθ , with s * ρθ being the desired value (i.e. the goal to be reached) of s ρθ . Q and R are weighted matrices that regulate the influence of, respectively, the different elements of the task sensor features and of the vehicle's twist. These matrices are defined as:

Q = diag(0.55, 1.0) (18a) R = diag(2.5, 30.0) (18b)
Finally, to avoid collision with the obstacle perceived as s xy , C = w(x)w(y), where w(s) is a generic smooth weighting function (Fig. 3) that depends on the current value of a given sensor feature. The activation boundaries s s ± , s ± of w(s) are defined such that they describe two bounding rectangles for the vehicle 1.0 m and 4.0 m longer and 0.7 m and 3.7 m wider than l ve and w ve respectively. These bounding rectangles are denoted by the red and orange ones in Figs. 4a-7a. The size of the smaller bounding rectangle has a direct impact on the safety distance to the obstalce while the size of the larger one affects the distance at which the vehicle starts to react to the obstacle. The sizes chosen were found empirically aiming to balance safety and comfort requirements while avoiding being overly convservative. The minimum w -and maximum w + values of w(s) are, respectively 0 and 1. The rest of the controlrelated parameters are given in Table II.

It should be noted that, contrary to the typical real-time implementations of MPPI-like techniques which make use of GPUs [START_REF] Williams | Model Predictive Path Integral Control: From Theory to Parallel Computation[END_REF], [START_REF]Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving[END_REF], [START_REF] Mohamed | Model Predictive Path Integral Control Framework for Partially Observable Navigation: A Quadrotor Case Study[END_REF], our implementation runs on a single core of an AMD Ryzen 9 5950X.

Finally, all the simulations presented in the next section start with the goal placed at 51 m straight ahead of the car and s * ρθ = [1; 0]. The goal placement was chosen in order to let the car smoothly reach its maximum allowed velocity before having to perform any avoidance maneuver. s = [s ρθ ; s xy ] is assumed to be already expressed in the control frame F m .

V. RESULTS

A. Static obstacle

We start the assessment of our approach with the easiest case: A static obstacle placed between the starting and desired positions, specifically at 25 m straight ahead of the car. As it can be seen in Fig. 4, the vehicle successfully and smoothly (Fig. 4d) reaches the goal while avoiding collision with the static obstacle by slightly turning to the right to then later turn left to keep moving towards the goal. The linear velocity v xm (Fig. 4b) is generally smooth. The steering angle δ (Fig. 4c) starts slightly unstable but it gets more stable as the linear velocity approaches the maximum value and it remains stable even when avoiding the obstacle. The final error is e = [-0.0162; -0.0057].

B. Pedestrian with parallel motion, same orientation

We now move to a case where a simulated pedestrian is present in the environment. Its motion is parallel to what the ideal motion of the vehicle would be if no obstacles were to be present. At the moment when the vehicle has to avoid the pedestrian, both agents have mostly the same orientation. Similarly to the previous case, the vehicle successfully reaches the goal while avoiding collision (Fig. 5), this time by slightly turning to the left to then later turn right

C. Pedestrian with parallel motion, opposite orientation

We now present a different case considering the same simulated environment as in the previous one. This time the vehicle and the pedestrian have mostly opposite orientations at the moment of the collision avoidance maneuver. Again, the vehicle successfully reaches the goal while avoiding collision (Fig. 6). The evolution of the control and task features signals is very similar to the previous case. The final error is e = [-0.0193; -0.0019].

D. Pedestrian with perpendicular motion

Finally we consider a case where the pedestrian and vehicle motions' are mostly perpendicular. In this case as well the car successfully reaches the goal while avoiding collision (Fig. 7). This time our approach predicts that, given the pedestrian's motion, the best option to avoid collision is to turn slightly to the left, even if that means that at some point the car steers towards the pedestrian (Fig. 7a) because, by the time the car would reach the line traced by the pedestrian, it would have already moved downwards and thus no collision occurs. The general comments on the evolution of the control and task features signals from the previous cases apply as well for this one. The final error is e = [-0.0160; -0.0083].

VI. CONCLUSIONS

In this work we have presented an approach that combines information-theoretic and sensor-based control techniques in order to navigate towards a given goal while avoiding collision with obstacles. On the one hand, the sensorbased formalism allows to define the navigation task and collision avoidance constraints without inherently requiring any knowledge of the car's nor pedestrian's position in a reference fixed frame. On the other hand, the informationtheoretic framework allows to naturally handle non-linear and non-smooth dynamics by forward simulating thousands of trajectories in order to find an optimal control sequence.

The obtained results are rather encouraging as the vehicle consistently manages to reach the goal without colliding with the obstacle in many different scenarios. Moreover, the car consistently achieves rather small final ||e|| ≤ 0.021 values.

Thanks to the relatively lightweight computations required by the sensor-based formalism, our implementation is able to run on a single CPU core of an AMD Ryzen 9 5950X. Nonetheless, in order to handle multiple obstacles future implementations would require some level of parallelization, be it on CPU or GPU. Future work will focus on this issue.

Another interesting research path is to investigate different cost functions to deal with arbitrarily shaped obstacles and social interaction with pedestrians and other vehicles in order to deal with the navigation challenges (safety, comfort, efficiency) that multiple surrounding agents (pedestrians, vehicles, etc) would cause.

Fig. 1 .Fig. 2 .

 12 Fig. 1. (a) Kinematic model diagram for a car-like rear-wheel driving robot. (b) Robotized Renault ZOE used as a basis for the simulation model

Algorithm 1 :

 1 Information-Theoretic Control Data: f : Transition mmodel; K: Number of rollouts; T : Number of time steps; ũ = {u 0 , u 1 , . . . , u T -1 }: Initial control sequence;

19 for t ← 0 to T -1 do 20 ũ ← ũ + K- 1 k=0 w k ε k 21 ũ

 1920121 ← SGF(ũ); 22 v r ← NumericalIntegration(v r , u 0 ); 23 SendToActuators(v r ); 24 for t ← 1 to T -1 do 25 u t-1 = u t ; 26 u T -1 = Init(u T -1 );

Fig. 3 .

 3 Fig. 3. Generic smooth weighting function w(s)

Fig. 5 .Fig. 6 .

 56 Fig. 4. Static obstacle

Fig. 7 .

 7 Fig. 7. Pedestrian with perpendicular motion

  , the stochastic Algorithm 2: Transition Model f Data: s, w: State and input; v r : Current control input; v max , δ max : Maximum control input values; t s : Sampling period; 1 v r ← NumericalIntegration(v r , w); 2 v r ← ClampMaxValues(v r ); 3 v m ← BuildTwist(v r ); 4 s ρθ ← s ρθ + L s ρθ t s v m ; 5 s xy ← s xy + Ľsxy t s (v mvmo );

David Pérez-Morales and

Vincent Fremont are with LS2N, Laboratoire des Sciences du Numérique de Nantes, École Centrale de Nantes, 1 rue de la Noë, 44321 Nantes, France 1 David.PerezMorales@ls2n.fr 2 Vincent.Fremont@ls2n.fr

ACKNOWLEDGMENT

This research is part of the HIANIC project, funded by the French National Research Agency (ANR-17-CE22-0010).