H N Adithya

Rangaiah Kariyappa
email: r.kariyapparkari@iiap.res.in

Imada Shinsuke

Kusano Kanya

Joe Zender

Luc Damé

Giono Gabriel

Edward Deluca

Mark Weber

Solar Soft X-ray Irradiance Variability, I: Segmentation of Hinode/XRT Full-Disk Images and Comparison with GOES (1 -8 Å) X-Ray Flux

Keywords: Sun: X-ray radiation, Sun: corona, Sun: coronal features, Sun: X-ray bright points

Introduction

The radiant energy from the Sun is the primary source of energy to the Earth. This radiant energy is measured and reported as the solar irradiance. It is known that the Total Solar Irradiance (TSI) is a measure of total radiation over the entire electromagnetic spectrum and that when measured as a function of wavelength, it is called spectral irradiance. Since the radiative output of the Sun is one of the main driving forces of the terrestrial atmosphere and climate system, the study of solar energy has become of great interest and importance. Although the solar energy flux integrated over the entire spectrum is considered to be one of the major natural forces of the Earth's climate system, studying the extreme ultraviolet (EUV), ultraviolet (UV) and X-ray irradiance variability is particularly important in solar and terrestrial physics. The solar EUV, UV and X-ray fluxes play in particular a major role in the heating of the Earth's atmosphere and Solar-Terrestrial relationships. Thus it is an important issue to understand their variability and its applications in Earth's climate and space weather.

The total solar irradiance (TSI) has been monitored from several satellites for more than three decades. It has been observed and reported that the solar energy flux changes over a solar magnetic cycle. The long-term irradiance variations are due to the changing emission of bright magnetic elements e.g. [START_REF] Foukal | Magnetic modulation of solar luminosity by photospheric activity[END_REF][START_REF] Kariyappa | Contribution of chromospheric features to UV irradiance variability from spatially-resolved CA II K spectroheliograms[END_REF][START_REF] Worden | Evolution of chromospheric structures derived from Ca II K spectroheliograms: implications for solar ultraviolet irradiance variability[END_REF][START_REF] Kariyappa | CaII K imaging to understand UV irradiance variability[END_REF]Kariyappa, , 2008b;;[START_REF] Veselovsky | Global asymmetry of the Sun observed in the Extreme Ultraviolet Radiation[END_REF][START_REF] Kumara | Preliminary results on irradiance measurements from Lyra and Swap[END_REF][START_REF] Kumara | Segmentation of coronal features to understand the solar EUV and UV irradiance variability[END_REF][START_REF] Zender | Segmentation of photospheric magnetic elements corresponding to coronal features to understand the EUV and UV irradiance variability[END_REF], and the short-term irradiance variations are directly related to active regions as they evolve, fragment, and move across the solar disk [START_REF] Lean | Solar ultraviolet irradiance variations -a review[END_REF][START_REF] Giono | Origin of the solar rotation harmonics seen in the EUV and UV irradiance[END_REF][START_REF] Van Der Zwaard | Segmentation of coronal features to understand the solar EUV and UV irradiance variability III. Inclusion and Analysis of Bright points[END_REF]. The reason for the difference between the observed and modeled solar irradiance variability and the underlying physical mechanisms of solar irradiance variability are not yet fully understood. Recently some studies have been done to estimate the contribution of various coronal features to EUV and UV irradiance using spatially resolved observations [START_REF] Kumara | Segmentation of coronal features to understand the solar EUV and UV irradiance variability[END_REF][START_REF] Zender | Segmentation of photospheric magnetic elements corresponding to coronal features to understand the EUV and UV irradiance variability[END_REF]. These authors used full-disk images from PROBA2/SWAP and SDO/AIA and segmented different coronal features to understand the EUV and UV irradiance measured by the PROBA2/LYRA instrument. In addition they investigated and discussed in detail the role of magnetic field on EUV and UV irradiance variability by comparing SWAP and AIA images with SDO/HMI magnetograms. Since all features are quite dynamic in nature, and their morphological structure and magnetic configuration will continuously change, the segmentation of coronal features from the images is a difficult task.

The history of solar soft X-ray radiation (SXR) observations can be traced back to the 1960s. The measurements from sounding rockets have led to several satellite-based experiments, including SOLRAD [START_REF] Dere | The spectral dependence of solar soft X-ray flux values obtained by SOLRAD 9[END_REF], Orbiting Solar Observatory [START_REF] Hall | Solar radiation in the extreme ultraviolet and its variation with solar rotation[END_REF], SkyLab (1973SkyLab ( -1979) ) [START_REF] Vaiana | ATM observations, X-ray results[END_REF], and Solar Maximum Mission [START_REF] Acton | The soft X-ray polychromator for the Solar Maximum Mission[END_REF]. The Soft X-ray Telescope (SXT) on Yohkoh observed the full-disk of the Sun in the wavelength range from 0.2 to 3 nm from 1991 to 2001 [START_REF] Ogawara | The status of Yohkoh in orbit: an introduction to the initial scientific results[END_REF]. [START_REF] Acton | Deriving solar X ray irradiance from Yohkoh observations[END_REF] have determined the solar X-ray irradiance using an isothermal spectral model and differential emission measure (DEM). The Soft X-ray Photometer on the Student Nitric Oxide Explorer (SNOE) had several broadband channels taking daily measurements of solar X-ray irradiance with three band-passes of 2 -7 nm, 6 -19 nm, and 17 -20 nm [START_REF] Bailey | Measurements of the solar soft X-ray irradiance by the Student Nitric Oxide Explorer: first analysis and underflight calibrations[END_REF][START_REF] Bailey | Observations of the solar soft X-ray irradiance by the Student Nitric Oxide Explorer[END_REF]. The Geostationary Operational Environmental Satellite (GOES) of the National Oceanic and Atmospheric Administration (NOAA) has a series of satellites carrying X-Ray Sensor (XRS), measuring X-ray flux at 0.5 -4 Å (XRS-A) and 1 -8 Å (XRS-B) [START_REF] Viereck | AGU Fall Meeting Abstracts #SH42A-06[END_REF]). An attempt has been made to estimate a soft X-ray (SXR) background flux by subtracting out the contribution of solar flares using the soft X-ray observations from GOES (1 -8 Å). It is observed that the post-flare emission from a bright active regions appears to play a dominant role in the SXR background flux variations, whereas the quiet Sun contributions to SXR background flux are significant during solar minimum. The comparison studies of SXR irradiance values with full-disk integrated images from Yohkoh suggest that the presence of coronal holes can reduce the SXR irradiance level only in the rare case of absence of active regions [START_REF] Aschwanden | Irradiance observations of the 1 -8 Å solar soft X-ray flux from goes[END_REF].

GOES is measuring solar X-ray flux daily and its variations need to be understood from spatially resolved full-disk soft X-ray images. In particular, the segmentation of different coronal features from the X-ray images has not been done. It is important to know: (i) how the different coronal features contribute to the total solar X-ray irradiance variability, particularly the quiet Sun component (both background-BGs-and XBPs), (ii) how do they help in understanding the Sun-as-a-star total solar flux measured, (iii) how they are related to underlying photospheric magnetic features and (iv) the role of magnetic field on the X-ray irradiance variability.

In this paper, we made an attempt to address some of the problems mentioned above using, for the first time, the observations from the Hinode/XRT and GOES instruments. For this purpose, we use an algorithm developed by us in Python, which segments active regions (ARs), coronal holes (CHs), background regions (BGs), and X-ray Bright Points (XBPs) more accurately. The whole segmentation procedure has been carried out with more precise, accurate, fast, stable in both time and space, allowing for a detailed analysis of spatially resolved full-disk images and solar X-ray flux. The details of the observations, analysis, results of intensity, area, and contribution of ARs, CHs, BGs, XBPs, full-disk intensity (FDI), and GOES (1 -8 Å) flux are presented and discussed.

Observations and Analysis

We used full-disk daily images taken by the X-Ray Telescope on board the Hinode mission (Hinode/XRT). The details on the design, calibration, and measured performance of the XRT instrument up to the focal plane and the CCD camera and data handling are discussed in [START_REF] Kosugi | The Hinode (Solar-B) Mission: an overview[END_REF] and [START_REF] Golub | The X-Ray Telescope (XRT) for the Hinode mission[END_REF]. The level-2 data from February 14, 2007 to March 31, 2020 was downloaded from http://solar.physics.montana.edu/ HINODE/XRT/SCIA/synopi_images/syncmp_FITS. The xrt_prep procedure has been already applied, which subtracts a dark frame, carries out the vignetting correction, the removal of high-frequency noise, and the normalization by exposure time. In addition, the quality of the archived full-disk images has been further improved with the following three methods: (i) Images are composites of long+short exposure images to provide extended dynamic ranges and to eliminate saturations, (ii) small contamination of spots on the CCD have been corrected in the images by replacing affected pixels with the averaged intensity of the surrounding area, and (iii) the pointing information has been updated with a recently developed database that co-aligns XRT with other solar instruments [START_REF] Takeda | The Hinode/XRT full-Sun image corrections and the improved synoptic composite image archive[END_REF]. The XRT data-set used in this analysis contains composite images. It is difficult for XRT to take a single image that sees both faint and bright features, hence the composite synoptic data sets include both long and short exposure images to observe both faint and bright features. Therefore, the composite image can represent a greater dynamic range than the instrument itself is capable of producing in a single image and the composite images were suitable to use for this analysis of all the features. Images of 2007 have 2048 × 2048 pixels and after 2008 all images are 1024 × 1024 pixels in size. Out of XRT's eight diagnostic filters, we selected for a detailed analysis the images that used the Al-mesh filter, since for that set we find there are continuous observations, and because the amount of stray light is negligible throughout the studied period. An example of a segmented image obtained on March 27, 2008 with this filter is presented in Figure 1. Bad images and off centered images are manually removed for the analysis. The X-ray irradiance at Earth was observed from X-ray Sensor (XRS) onboard Geostationary Operational Environmental Satellites (GOES), which observe the Sun as a star, it can provide solar X-ray fluxes for the wavelength bands of 1 -8 Å (long channel) and 0.5 -4 Å (short channel). We choose 1 -8 Å which is closer to the XRT band. Currently there are four satellites operationally available. Since a good and continuous data set was not available for 2007 -2020 from one mission, we collected 1-minute averaged data from GOES 10, GOES 11, GOES 12, GOES 14 and GOES 15. For a comparison study with sunspot numbers (SSN), we have downloaded the daily sunspot numbers from the link: http://www.sidc.be/silso/datafiles.

We developed an algorithm in Python to extract all the features (ARs, CHs, BGs, and XBPs) from the full-disk X-ray images of the Sun. Coronal features are differentiated based on their brightness/intensity level, size/area and morphological structures. In order to identify, segment and extract the features we choose the threshold methods. Coronal holes (CHs) are extracted by intensity threshold, active region (ARs) and X-ray bright points (XBPs) are extracted by both intensity and area/size threshold method.

In order to determine the average threshold values (for both intensity and size/area) for all the features, we have selected a large number of full-disk images distributed over the entire period (2007 -2020) representing the different phases of the solar activity showing both active and quiet Sun. It means that we have determined the average threshold values using a large number of images when the Sun was both active and in quiet phase.

The following steps have been followed for extraction of on-disk features for the available filter: i) Using the FITS header, center of images is found and 0.96 of solar radii disk is extracted to avoid edge/limb brightening; ii) Active regions (ARs):

-mean and median of the solar disk are calculated; -2.5× mean value set as lower limit intensity threshold; -for threshold image morphological closing operation is done with 5 × 5 kernel; -all closed contours having area more than 1000 pixels are considered as active regions.

iii) Coronal holes (CHs): -0.3× median value set as upper limit intensity threshold; -for thresholds image contours are drawn after morphological closing operation; -morphological closing is done with a 15 × 15 kernel to remove noisy grains and increase the connectivity; -all contours are considered as CHs. iv) X-ray bright points (XBPs):

-all detected ARs are masked over the solar disk (ARs are masked to avoid intense ARs' effect on XBPs detection); -once again mean value of AR-masked disk is calculated; -1.7× mean value set as lower limit intensity threshold; -for threshold image morphological closing operation is done with 5 × 5 kernel to remove small noisy grains; -contours are drawn after morphological closing operation is done; -contours having area greater than 10 pixels and less than 1000 pixels are considered as XBPs. v) Background (BGs):

-after removing all XBPs, ARs and CHs, the remaining solar disk area is considered as background regions.

We have calculated the total intensity of each feature (expressed in DN/sec or energy/area/time, -where DN is data units and represents the digitized measure of the amount of energy accumulated in a CCD pixel during an image exposure, so image exposure durations have been normalized out, and the units are directly comparable to GOES' W/m 2 ), the total area (total number of pixels) of each feature, and their contribution (%) to full-disk intensity (FDI). These values have been compared with total X-ray flux measurements from GOES. We estimated the standard deviation for the variations of intensity contributions of all the features.

Results and Discussion

Using our algorithm we segmented coronal X-ray features automatically from full-disk Hinode/XRT images observed in Al_mesh filter (Figure 1) for the period from February 14, 2007 to March 31, 2020, which covers Solar Cycle 24. We derived the corresponding total intensity, area (total number of pixels) and contribution of ARs/CHs/BGs/XBPs. We have compared the total intensity and area of ARs/CHs/BGs/XBPs to full-disk total X-ray intensity (FDI) and derived their contribution to FDI and GOES (1 -8 Å) X-ray flux for each day for the whole period. We observed from the time series plots of ARs/CHs/BGs/XBPs/FDI taken with Al_mesh (Figure 2) in comparison with GOES and sunspot numbers that the intensity of all the features is well correlated with FDI, GOES total X-ray flux and sunspot numbers. This indicates that the long-term variability of segmented features and full-disk intensity shows a correlation with GOES (1 -8 Å) solar X-ray flux and the sunspot numbers, this implies that the changes are in phase with the solar cycle. As observed from the time series plots, ARs and BG are well correlated to the total soft X-ray irradiance measured by GOES (1 -8 Å), and the intensity of XBPs and CHs seem to be moderately correlated to FDI and GOES X-ray flux. This suggests that BGs and ARs have greater impact, influence and contribution to X-ray irradiance fluctuations whereas the XBPs and CHs appear to have a smaller contribution to FDI and GOES X-ray flux variations. The important results of these are discussed in detail in the following sections.

Integrated Intensity and Area Variations of All the Features in Comparison with Full-Disk Intensity (FDI) and GOES (1 -8 Å) Flux

The time series of integrated intensity of each of the coronal features and full-disk intensity for the Hinode/XRT X-ray images observed with the Al_mesh filter, GOES (1 -8 Å) X-ray flux and sunspot numbers (SSN) are shown in Figure 2. The intensity of all the features vary with the phase of the solar cycle. It is also clear from Figure 2 that on an average over the solar cycle the integrated intensity of BGs is larger than the integrated intensity of ARs, CHs, and XBPs, but the contribution of ARs will become more dominant around solar maximum. Note that the BG intensity values will be slightly contaminated by the fuzzy regions (brighter than BG and less brighter than ARs, as seen in Figure 1) present around the ARs while segmenting the features. However, this does not change the results and conclusions. We also noticed that there is good correlation between the intensity of all the features in comparison with GOES X-ray flux and sunspot numbers indicating that they vary with solar activity.

Along with integrated intensity, we have computed the total area (total number of pixels) of ARs, CHs, BGs, and XBPs for the whole period. We have plotted the area of all the features in Figure 3 for the whole observed period and these are compared with the integrated intensity of segmented features, FDI, GOES (1 -8 Å), and SSN presented in Figure 2. From the times series shown in Figures 2 and3, we noticed that the integrated intensity values and areas of all the features exhibit the expected 27 day modulation due to the solar rotation, similar to sunspots at the photospheric level. The features' area variation (shown in Figure 3) show that the area of ARs and CHs vary with the solar activity, whereas the BGs and XBPs are anti-correlated with the phase of the solar cycle. We estimated the mean areas of all the features for the whole observed period. Most importantly, from Figure 3 and the mean areas, we can clearly notice that the BGs cover (5.82e + 5 pixels) a very large portion of the Sun's surface, whereas ARs cover (2.29e + 4 pixels) slightly smaller area compared to CHs (2.4e + 4 pixels) and larger than XBPs (1.07e + 4 pixels) in general. We observed that the filling factors of ARs/CHs/BGs/XBPs evolve throughout the solar cycle and also find from Figure 3 that the area of the features is highly variable, similar to the integrated intensity variations seen in Figure 2. This means that the total intensity of each segmented feature is affected by the intensities of the feature itself and also by the changes of its area/size. The observed intensity variation is due to the change of feature's intensity and also due to the expansion/shrinkage of of its area as it evolves. This suggests that the intensity and area of the coronal features have to be taken into account in the models to explain the irradiance variations. In Figure 4, we have plotted for the Al_mesh filter the integrated intensity of all the features (ARs, CHs, BGs and XBPs) versus the full-disk intensity (FDI), GOES X-ray flux and sunspot numbers for a period of 13 years (February 2007-March 2020, covering Solar Cycle 24). We have determined the Spearman rank correlation coefficients of the segmented features with FDI and GOES solar X-ray irradiance values and their correlation coefficients (R) are presented in Table 1. As seen from the scatter plots (Figure 4) and the results of the correlation coefficients (in Table 1) for the Al_mesh filter, the ARs and BGs are well correlated to the total integrated intensity of the full-disk (XRT) and GOES X-ray irradiance measured in 1 -8 Å. The mean correlation coefficient (R) values for ARs versus FDI and ARs versus GOES are 0.98 and 0.82, respectively. Similarly, the mean R values for BGs versus FDI and ARs versus GOES are 0.93 and 0.82, respectively. This suggests variations of ARs and BGs are of the same order of magnitude as the X-ray flux variations (although the physical units are different, the intensity is expressed in DN/s whereas the irradiance in calibrated units of W/m 2 ). Therefore BGs and ARs have a greater impact and influence on the X-ray irradiance fluctuations. The XBPs and CHs show a correlation higher than R = 0.50 (average value of R with Al_mesh from Table 1) with FDI and GOES values, indicating XBPs and CHs will moderately contribute to the total X-ray irradiance fluctuations in addition to ARs and BGs. Since the XBPs and CHs are relatively low intensity features compared to the others, the results of XBPs and CHs are very sensitive to the thresholds used and hence we have tuned the algorithm's parameters precisely to obtain more accurate results. We noticed from Figure 4 and Table 1 that FDI measured from the spatially resolved images of the whole Sun is well correlated (R = 0.81) with GOES (1 -8 Å) total X-ray flux measured Sun as a star. However, we are aware that (i) there is no perfect overlap in the responses of XRT and GOES, and (ii) our analysis leaves out the solar limb and off-limb contributions to the variability. These points suggest that further work might be able to improve the correlation between XRT-observed features and the GOES total solar X-ray flux variability. We can conclude that the spatially resolved images are important asset to use to understand the integrated intensity variability of the Sun when we observe the Sun as a star (true also in the case of solar type stars, if they have stellar X-ray irradiance measurements, where we can use Sun as a calibrator).

Number Variation of XBPs with the Solar Magnetic Activity Cycle

Using our algorithm we have separated out the XBPs component from the quiet Sun areas, since quiet Sun is the combination of both background regions (BGs) and XBPs. In the case of EUV and UV image segmentation in our previous studies [START_REF] Kumara | Segmentation of coronal features to understand the solar EUV and UV irradiance variability[END_REF][START_REF] Zender | Segmentation of photospheric magnetic elements corresponding to coronal features to understand the EUV and UV irradiance variability[END_REF] we did not separated out the bright points from the quiet Sun component, hence we had to deal with variations of the whole quiet Sun regions. In the present study, we have identified, (separated out background regions and XBPs), and extracted automatically the total number of XBPs based on their intensity levels, morphological structures and sizes (compared to other large scale features like ARs and masking them) over the full-disk images observed in Al_mesh), all this on a daily basis for the period of 13 years (February 2007-March 2020, which covers Solar Cycle 24).

In Figure 5(a), we have plotted the total number of XBPs extracted from Al_mesh filter as a function of time. In addition for a comparison study we have plotted in Figure 5(b),(c) and (d) the variation of FDI (for Al_mesh filter), GOES X-ray flux and sunspot numbers, respectively. We observed from these plots that the number of XBPs are inversely proportional to FDI, GOES X-ray flux and sunspot numbers. For more clarity we have also plotted in Figure 5(e) the number of XBPs observed in Al_mesh versus sunspot numbers (SSN) and it clearly indicates that there is an anti-correlation, with a mean correlation coefficient, R = -0.86, between the variation of XBPs numbers and the solar activity. Our investigations confirms the previous results of anti-correlation of number of XBPs with the solar cycle as derived in the quiet-Sun region from Yohkoh/SXT soft X-ray images (Hara andNakakubo-Morimoto, 2004, 2003) and also with our recent results of variation of coronal bright points derived from the analysis of EUV and UV images (SDO/AIA and PROBA2/SWAP) by [START_REF] Van Der Zwaard | Segmentation of coronal features to understand the solar EUV and UV irradiance variability III. Inclusion and Analysis of Bright points[END_REF]. Several bright point detection algorithms implemented in long-term statistical analysis have been described, using Yohkoh/SXT [START_REF] Nakakubo | Variation of X-ray bright point number over the solar activity cycle[END_REF][START_REF] Sattarov | X-ray bright points and photospheric bipoles during cycles 22 and 23[END_REF]Hara andNakakubo-Morimoto, 2003, 2004), SOHO/EIT [START_REF] Mcintosh | Nine years of EUV bright points[END_REF][START_REF] De Wit | Fast segmentation of solar extreme ultraviolet images[END_REF][START_REF] Mcintosh | On the mass and energy loading of extreme-UV bright points[END_REF], or SDO/AIA [START_REF] Alipour | Statistical properties of solar coronal bright points[END_REF][START_REF] Shahamatnia | An evolutionary computation based algorithm for calculating solar differential rotation by automatic tracking of coronal bright points[END_REF][START_REF] Arish | Extraction of active regions and coronal holes from EUV images using the unsupervised segmentation method in the Bayesian framework[END_REF][START_REF] Dorotovic | Automatic detection and tracking of coronal bright points in SDO/AIA images[END_REF] data as input. To counterweight the different intensity changes in EUV images both over the latitudinal range as well as the solar cycle changes, it has been discussed the need to apply an intensity background threshold to their BP detection algorithm [START_REF] Mcintosh | Nine years of EUV bright points[END_REF][START_REF] Mcintosh | On the mass and energy loading of extreme-UV bright points[END_REF].

From an image analysis of a large sample of Kodaikanal Ca II K spectroheliograms, the intensity and area of the magnetic network elements at the center of the solar disk over two solar cycles have been estimated. From this study, it has been shown that the magnetic network elements occupy a larger area during solar minimum compared to solar maximum period (Kariyappa and Sivaraman, 1994), indicating that the network area is anti-correlated with the solar activity. Meanwhile it has been reported [START_REF] Sivaraman | Ca II K bright points and the solar cycle[END_REF] that the total number of bright points that are located in the interior of the network elements show a larger number during solar maximum compared to that of the minimum phase. This suggests that the contribution of Ca II K bright points need to be taken into account in the reconstruction of integrated Ca II K line profile and to explain the chromospheric variability. So considering the large number of XBPs present at any time in the full-disk X-ray images and their number variation for the whole observed period, the contribution of XBPs can be considered in the reconstruction of solar X-ray irradiance variability. However, the physical reason for the existence of anti-correlation between the number variation of XBPs and solar activity cycle (SSN) needs to be understood by further detailed investigations. In order to confirm the existence of anti-correlation we are in the process of identifying and counting the XBPs over a quiet Sun region, preferably away from active regions, by different approaches for the entire period of observations to overcome the contamination and contribution of active regions (detailed results of such studies will be reported elsewhere).

Contribution of Coronal Features to Total Solar X-Ray Irradiance Variations

Since the full-disk intensity (FDI) and GOES solar X-ray flux values have been measured Sun as a star, they do not reveal directly the individual contribution of different features. In order to determine the contribution of the segmented X-ray features to FD intensity and Xray irradiance variations we used an expression: contribution (%) = (100× intensity of the feature/FDI). Figure 6 represents the variation in the contribution of all the X-ray coronal features (ARs, CHs, BGs and XBPs) measured in Hinode/XRT with the Al_mesh filter to full-disk X-ray (FDI) irradiance variations. We noticed in Figure 6 2007 -2009, 2009 -2011, 2011 -2013, 2013 -2015, 2015 -2017, 2017 -2019, and 2019 -2020 March for all the features (ARs, BGs, CHs and XBPs) and FDI (DN/sec) and GOES X-ray flux (W/m 2 ). Al_mesh Filter, Contribution plot (Figure 6 inversely proportional to solar activity (large contribution during solar minimum and smaller during a solar maximum period). We also observed a similar behavior or pattern between the variations of the area (presented in Figure 3) and percentage of contributions (plotted in Figure 6) of all the coronal features, showing ARs and CHs vary in phase with the solar activity and whereas the BGs and XBPs are anti-correlated. To explain the physical reason for showing anti-correlation in the variation of the contribution of BGs and XBPs with the solar activity needs further investigations. In Table 2 we have presented the contribution values of all the features as a function of time. Figure 6 and Table 2 show that the background and active regions have a greater impact on the X-ray irradiance fluctuations and the average percentage of contributions to total solar X-ray flux from the BGs, ARs, XBPs, and CHs over the entire period of observations will be around 65%, 30%, 4% and 1%, respectively. We observe from Figure 6 and Table 2 that the BG regions are the main contributors to the total solar X-ray irradiance and covers a large area on the surface of the Sun. Similarly, ARs contribute about 30% and cover a smaller area on the Sun's surface, but they are brighter magnetic regions compared to other features. Hence the variations in both intensity and area of all the features need to be considered to explain the X-ray variability. That means the changes in both area and brightness will have more influence on the variation of total solar X-ray irradiance. The XBPs will moderately contribute (around 4%) and CHs will have negligible contribution (about 1%) to the total solar X-ray irradiance variations.

The quiet-Sun component, combination of both BGs and XBPs of the corona, is estimated to vary due to the changing size of coronal holes and quiet-Sun regions. Considering the quiet-Sun variability we can notice that the quiet-Sun intensity also varies as a function of solar cycle and from this it is clear that there is no such thing that a 'solar constant or constant background emission in soft X-ray irradiance'. This confirms the earlier findings that the intensity of the quiet-Sun varies with the solar cycle as derived from Ca II K line observations near solar disk center in the chromospheric and temperature minimum regions [START_REF] Kariyappa | Quiet-Sun variability with the solar cycle[END_REF][START_REF] Zhang | Quiet-Sun variability in a temperature minimum region[END_REF]. Hence, we conclude that the quiet-Sun is not quiet and constant, but it varies in phase with the solar activity and the changes are observed in the temperature minimum region, chromosphere, and in the corona. It is suggested that the contributions from the quiet-Sun are appreciable in the solar minimum and the soft X-ray (SXR) background flux appears to be dominated by post-flare emission from the dominant active regions and the evolution and confinement of heated plasma in flaring active regions will help to understand the variability of the full-disk integrated solar X-ray flux [START_REF] Aschwanden | Irradiance observations of the 1 -8 Å solar soft X-ray flux from goes[END_REF].

We have estimated the mean contribution of the features to total X-ray irradiance in steps of every 2 years (2007 -2009, 2009 -2011, 2011 -2013, 2013 -2015, 2015 -2017, 2017 -2019, and 2019 -2020) and the corresponding standard deviation values are also calculated for Figure 6. The reason for estimating for every 2 years is that during 2007 to 2020 (Solar Cycle 24) we find minimum, rase phase, maximum, declining phase and minimum in solar activity. The contribution of the features vary with the phase of the solar activity as discussed above. So we binned the time series into seven parts depending on the strength of the solar activity. In Table 2 we list the mean contribution for every 2 years and the values of standard deviation, and the average value of the contribution of all the features (ARs/CHs/BGs/XBPs/FDI/GOES) over the entire period of observation (2007 to 2020). It is clear from Figure 6 and Table 2 that BGs and ARs are the main contributors to total solar X-ray (SXR) flux. In addition, it suggests that the XBPs also contribute moderately to the GOES (1 -8 Å) total solar X-ray flux variations, which cannot be neglected.

In this work, for the first time, we discussed in great detail the use of segmentation in resolved soft X-ray observations to measure X-ray irradiance fluctuations. The observations from Hinode/XRT helped us to segment the images into different individual X-ray features and to estimate and study their contribution to the total solar X-ray irradiance. Our algorithm is very sensitive to the different intensity levels, an effective one, capable to automatically identify and segment the coronal features from the images. We have shown that the Xray Telescope onboard Hinode can be used to measure the solar X-ray (SXR) irradiance fluctuations. This allowed us to identify and extract different features including small-scale magnetic features like XBPs and to determine their contribution to the total solar X-ray irradiance. It is clear from Figure 6 and Table 2 that the background regions will have a large contribution to the solar X-ray irradiance, up to 65%, whereas the active regions contribute up to 30%. However, we observed that the contributions of XBPs and CHs are small (about average 4% and 1% respectively) compared to BGs and ARs. Although the XBPs are small in size, but they are intense magnetic regions and large in number at anytime over the whole solar disk. On the other hand, the XBPs will also play an important role in determining the intensity and temperature fluctuations and heating of the corona, since they are highly dynamic in nature [START_REF] Kariyappa | Intensity oscillations and heating of the coronal X-ray bright points from Hinode/XRT[END_REF][START_REF] Kariyappa | Temperature variability in X-ray bright points observed with Hinode/XRT[END_REF]. In earlier studies [START_REF] Kariyappa | Heating of the quiet solar chromosphere -part one[END_REF] it has been shown that the dynamical evolution of the bright points observed in Ca II H line will supply sufficient energy to maintain the chromospheric temperature and contribute to the heating of the quiet chromosphere and to its variability. The XBPs have also been used as tracers in estimating the coronal rotational period (Kariyappa, 2008a;[START_REF] Kariyappa | Coronal rotation from XBPs observed with Hinode/XRT[END_REF]. In general, the different bright points, such as Ca II H and K BPs, UVBPs, He I dark-points, and XBPs, observed at different wavelengths will contribute to solar irradiance variability and heating of the solar atmosphere.

The segmented coronal features observed in X-ray wavelength can be used as proxies to isolate the underlying corresponding photospheric magnetic structures for further studies. In our future work we plan to extend our algorithm's capabilities to overlay the XRT full-disk segmented images/maps on to SOHO/MDI and SDO/HMI full-disk magnetograms and to segment the photospheric magnetic features corresponding to segmented coronal Xray features. This would help us (i) to study the spatial correlation between coronal and photospheric magnetic features; (ii) to determine the total magnetic field of the segmented coronal features; (iii) to study the long-term variability of the integrated magnetic field of all the features; (iv) to quantify the role of magnetic field on the X-ray irradiance variability; (v) to understand the magnetic field fluctuations, and (vi) to construct the magnetic maps and total magnetic field of the corona. Such studies will have a wider scope for explaining the irradiance variability as all the X-ray features may be associated and related to underlying photospheric magnetic elements (since they are magnetic in origin). This research work is in progress. A similar investigations have been carried out and published recently by us [START_REF] Zender | Segmentation of photospheric magnetic elements corresponding to coronal features to understand the EUV and UV irradiance variability[END_REF] in the case of EUV and UV irradiance and magnetic field variabilities. In addition, for a future work, since the XRT instrument is producing the full-disk synoptic images with eight filters, we plan to use filter ratio method and the filters' response curves to generate the temperature maps of the full Sun using images with any two filters available, which are free from visible stray light contamination. Finally the temperature maps will be used to determine the temperature of the segmented features. This would help to understand the temperature fluctuations and long-term variations in temperature of all the segmented coronal features as a function of solar magnetic cycle. Recently we have studied the intensity variations and determined the temperature fluctuations in many XBPs observed with Hinode/XRT with two filters (Ti_poly and Al_mesh) by filter ratio method and using their response curves [START_REF] Kariyappa | Intensity oscillations and heating of the coronal X-ray bright points from Hinode/XRT[END_REF][START_REF] Kariyappa | Temperature variability in X-ray bright points observed with Hinode/XRT[END_REF].

Figure 1

 1 Figure 1 A sample of the segmented image observed with the Al_mesh filter, showing all the features: ARs (yellow), CHs (violet), BGs (rest) and XBPs (red) obtained with Hinode/XRT on March 27, 2008.

Figure 2

 2 Figure 2 Total intensity of solar X-ray features: (a) ARs, (b) CHs, (c) BGs, (d) XBPs, (e) full-disk intensity (FDI) (observed by Hinode/XRT with the Al_mesh filter), (f) GOES (1 -8 Å) X-ray flux, and (g) sunspot numbers (SSN) for a period starting from February 14, 2007 to March 31, 2020.

Figure 3

 3 Figure 3 Total area (total number of pixels) variation of solar X-ray features: (a) ARs, (b) CHs, (c) BGs, and (d) XBPs (observed by Hinode/XRT with the Al_mesh filter) for a period starting from February 14, 2007 to March 31, 2020.

Figure 4

 4 Figure 4 Using images taken with the Al_mesh filter, total intensity of all the features versus FDI and GOES flux: (a) ARs versus FDI, (b) CHs versus FDI, (c) BGs versus FDI, (d) XBPs versus FDI and (e) ARs versus GOES, (f) CHs versus GOES, (g) BGs versus GOES, (h) XBPs versus GOES and (i) FDI versus GOES. The correlation coefficient values (R) are marked on the top left corner of each plot.

Figure 4

 4 Figure 4 (Continued)

Figure 5

 5 Figure 5 (a) Variation of the total number of XBPs using the Al_mesh filter, (b) FDI with the Al_mesh filter, (c) variation of GOES X-ray flux, (d) Sunspot number variation, and scatter plot: (e) number of XBPs with the Al_mesh versus SSN, for a period starting from February 14, 2007 to March 31, 2020.

Figure 6

 6 Figure 6 Variation of Contribution of ARs, CHs, BGs and XBPs to FDI with the Al_mesh filter: (a) ARs contribution variation, (b) CHs contribution variation, (c) BGs, and (d) XBPs contribution variations to FDI, for a period starting from February 14, 2007 to March 31, 2020.

  (a) and Figure 6(b) that the contribution of ARs and CHs changes with the phase of the solar cycle, whereas the BGs and XBPs contribution values, presented in Figure 6(c) and Figure 6(d), respectively, are

Table 1

 1 Spearman rank correlation coefficients between GOES (1 -8 Å) and full-disk intensity (FDI) with coronal features (ARs, BGs, CHs and XBPs) as observed with Hinode/XRT.

	Correlation coefficients (R)
	Al_mesh filter (Figure 4)
	FDI vs GOES	0.81
	CH vs GOES	0.56
	XBPs vs GOES	0.42
	AR vs GOES	0.82
	BG vs GOES	0.70
	CH vs FDI	0.63
	XBPs vs FDI	0.51
	AR vs FDI	0.98
	BG vs FDI	0.93

Table 2

 2 Mean contribution (%) and corresponding standard deviation (Sigma) values determined in steps of 2 years for the periods: February

  )

		CHs	CHs		XBPs	XBPs	ARs	ARs
		Mean	Sigma	Mean	Sigma	Mean	Sigma
	2007(Feb)-09	0.73	0.67		4.80	1.68	13.84	14.18
	2009-11	0.37	0.49		4.11	1.06	25.50	15.61
	2011-13	0.62	0.49		2.94	1.06	50.90	10.37
	2013-15	1.34	0.67		1.94	0.94	53.77	08.13
	2015-17	1.12	0.72		3.02	1.09	42.46	12.19
	2017-19	0.37	0.39		3.80	1.35	17.15	14.38
	2019-20(March)	0.17	0.20		5.43	1.07	08.68	07.85
	Average	0.67	0.52		3.75	1.18	30.32	11.82
	Al_mesh Filter, Contribution plot (Figure 6)
		BGs	BGs	FDI		FDI	GOES	GOES
		Mean	Sigma	Mean		Sigma	Mean	Sigma
	2007(Feb)-09 2009-11 2011-13 2013-15 2015-17 2017-19 2019-20(March) Average	80.63 70.42 46.55 41.94 53.40 77.67 86.32 65.28	13.04 14.18 9.78 7.74 11.54 13.39 7.10 10.97	4.49E + 7 7.54E + 7 1.89E + 8 2.34E + 8 1.48E + 8 0.57E + 8 0.45E + 8 1.13E + 8	1 .45E + 7 2 .91E + 7 8 .49E + 7 1 0 .24E + 7 8 .00E + 7 2 .06E + 7 0 .87E + 7 4 .86E + 7	7 .38E -9 8 .70E -8 6 .76E -7 9 .28E -7 4 .79E -7 0 .56E -7 0 .65E -7 3 .28E -7	1 .62E -8 1 .94E -7 1 .31E -6 1 .05E -6 0 .72E -6 1 .35E -7 0 .48E -7 5 .00E -7

Acknowledgements Hinode is a Japanese mission developed and launched by ISAS/JAXA, in collaboration with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. The Hinode science team at ISAS /JAXA has conducted the Scientific operation of the Hinode mission. This team mainly consists of scientists from different institutes in the partner countries. JAXA and NAOJ (Japan), STFC (UK), NASA (USA), ESA, and NSC (Norway) have provided the support for the post-launch operation. The Hinode team had contributed all their efforts in the design, build, and operation of the mission. RK wishes to express his sincere thanks to all members of ISEE for the support provided under Joint International Research Program. A part of these results were presented by RK at the PSTEP-4 International Symposium held at ISEE/Nagoya University from January 28 -30, 2020 and his visit to this symposium has been supported by MEXT/JSPS KAKENHI Grant Number JP15H05816. The authors are thankful to the referee for the valuable comments and suggestions that improved the manuscript considerably.

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Sébastien Kleff

Justin Carpentier

Nicolas Mansard

Ludovic Righetti

On the Derivation of the Contact Dynamics in Arbitrary Frames: Application to Polishing with Talos

Contact dynamics relies on the simultaneous satisfaction of constraints at the robot body level and at the contact level. At both levels, various formulations can be chosen that all must lead to the same results, given the same hypothesis, hence the little importance of their details. Yet when using it in an optimal control problem, a particular formulation is often imposed by the task to be performed by the robot. In this paper, we detail the formulation of the contact quantities (force, movement) in an arbitrary frame imposed by the task. In that case, we will show that we are typically not interested in working in the local frame (attached to the robot contact point), nor in the world frame, but in a user-defined frame centered at the contact location with a fixed orientation in the world. The derivations can then be used for 6D, 3D or normal (pure-sliding) contact. We implemented the corresponding derivatives on top of the contact dynamics of the rigid-body dynamics library Pinocchio in the optimal control solver Crocoddyl. We show that a unique formulation is able to handle several operational orientations, by achieving several surfacing tasks in model predictive control with the robot Talos.

I. INTRODUCTION

The advent of fast nonlinear model predictive control (MPC) on torque-controlled robots is enabled by dedicated numerical optimal control solvers that exploit the structure of the dynamics, such as DDP [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF], [START_REF] Mastalli | Crocoddyl: An efficient and versatile framework for multi-contact optimal control[END_REF]. Despite the successful deployment of these controllers on hardware [START_REF] Meduri | Biconmp: A nonlinear model predictive control framework for whole body motion planning[END_REF]- [START_REF] Marti-Saumell | Full-body torque-level non-linear model predictive control for aerial manipulation[END_REF], contact tasks are still challenging to realize and subject to active research. In this context, designing such tasks suppose the knowledge of desired contact forces and accelerations in a "convenient" frame -what convenient means is up to the user. However, analytical derivatives of multi-body dynamics algorithms are often expressed in local frames by construction, i.e. in frames attached at all instants to moving parts of the robot (following the efficiency of local formulation [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF]). Yet from a user point of view, this representation is not the most intuitive and complicates the design of contact tasks. Indeed, think for instance of locomotion task where we would like to avoid slippage: the contact force could be regularized around a vertical force in the cost objective. As a matter of fact, doing so requires to express the residual of the contact force and its derivatives in a frame that is aligned with the world frame (i.e. with gravity) yet centered at all times with the robot foot. This kind of task is the motivation of our paper:

expressing derivatives in such a custom frame, although done using basic spatial algebra tools, is not trivial.

In the literature, several contact models have been used in MPC. The rigid contact dynamics implemented in Pinocchio [START_REF] Carpentier | The Pinocchio C++ library: A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives[END_REF] is utilized for instance in [START_REF] Kleff | Introducing Force Feedback in Model Predictive Control[END_REF]- [START_REF] Galliker | Bipedal locomotion with nonlinear model predictive control: Online gait generation using whole-body dynamics[END_REF]. In our previous work [START_REF] Kleff | Introducing Force Feedback in Model Predictive Control[END_REF] the target force was expressed in a local frame whose alignment with the surface normal was part of the task. In [START_REF] Sleiman | A unified mpc framework for whole-body dynamic locomotion and manipulation[END_REF], [START_REF] Galliker | Bipedal locomotion with nonlinear model predictive control: Online gait generation using whole-body dynamics[END_REF] contact force constraints are expressed in a surface-aligned frame but the contact forces are optimized as control variables and not explicitly tracked. Soft contact models are also exploited, e.g. in [START_REF] Gold | Model Predictive Interaction Control for Industrial Robots[END_REF] where a visco-elastic force objective expressed in the surface coordinates, or in [START_REF] Erez | An integrated system for real-time model predictive control of humanoid robots[END_REF] where iLQG is used with Mujoco [START_REF] Todorov | Mujoco: A physics engine for model-based control[END_REF]. But Mujoco performs the Recursive Newton-Euler Algorithm only in local or world coordinates. Contact invariant approaches allowing to optimize control policies through contact have also been proposed in [START_REF] Neunert | Whole-body nonlinear model predictive control through contacts for quadrupeds[END_REF], [START_REF] Mordatch | Discovery of complex behaviors through contact-invariant optimization[END_REF], [START_REF] Cleac'h | Fast contact-implicit model-predictive control[END_REF] but it is unclear in which coordinate frame they are expressed. Overall there seem to be little focus on this topic among the MPC community, and the most popular contact models are usually based on local formulations. MPC requires the explicit computation of the derivatives, hence an effort of the community to provide differentiable contact simulators [START_REF] Todorov | Mujoco: A physics engine for model-based control[END_REF], [START_REF] Werling | Fast and feature-complete differentiable physics for articulated rigid bodies with contact[END_REF]- [START_REF] Le Lidec | Differentiable simulation for physical system identification[END_REF]. By basing our contact formulation on a differentiable contact solver [START_REF] Mastalli | Crocoddyl: An efficient and versatile framework for multi-contact optimal control[END_REF], [START_REF] Budhiraja | Differential dynamic programming for multi-phase rigid contact dynamics[END_REF], we pursue the same objective and provide all necessary derivations to obtain the derivatives of the contact dynamics.

In this paper, we propose to derive the formulation of the rigid contact model, which includes the constrained joint acceleration and contact forces, in arbitrary frames of reference. We emphasize that this is a technical difficulty that needs to be overcome by most MPC practitioners. We present here a formal and generic way of dealing with rigid contact models of any dimensions in arbitrary frames. In particular, we focus on the generic spatial (6D) formulation from which all other formulations can be deduced. The calculations are not found explicitly in the literature, but from our experience they are tedious and error-prone. We hope that it can serve other researchers from the MPC community to design contact tasks. We also provide an open-source implementation of these derivatives on top of the Pinocchio [START_REF] Carpentier | The Pinocchio C++ library: A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives[END_REF] and Crocoddyl [START_REF] Mastalli | Crocoddyl: An efficient and versatile framework for multi-contact optimal control[END_REF] libraries and we validate experimentally these developments in a nonlinear MPC framework on the torque-controlled humanoid robot Talos.

II. BACKGROUND

In this section we recall the rigid contact model [START_REF] Udwadia | A new perspective on constrained motion[END_REF], the constrained forward dynamics [START_REF] Budhiraja | Differential dynamic programming for multi-phase rigid contact dynamics[END_REF] and the derivation of their analytical derivatives [START_REF] Carpentier | Analytical derivatives of rigid body dynamics algorithms[END_REF]. Then we introduce the reference frame of interest in which we want to express contact tasks, as well as the spatial algebra tools used in the paper.

A. Constrained forward dynamics

A rigid contact is a pure kinematic constraint. The Lagrangian dynamics of a robot subject to rigid contacts is derived from Gauss' least action principle [START_REF] Udwadia | A new perspective on constrained motion[END_REF] stating that constrained accelerations are the closest to free accelerations

min a q 1 2 a q -M (q) -1 (τ q -b(q, v q )) 2 M (q) (1a) s.t. f (q) = 0 (1b)
where q ∈ R n q is the vector of joint positions,

f ∶ R n q → R m is the contact constraint, a q ∈ R n v is the vector of constrained joint accelerations, v q ∈ R n v is the vector of joint velocities, M (q) ∈ S n v
+ is the joint space inertia matrix, b(q, v q ) ∈ R n v is the vector of centrifugal, Coriolis and gravity forces, τ q ∈ R n v is the vector of joint torques.

Differentiating twice (1b) leads to the quadratic program

min a q 1 2 a q -M (q) -1 (τ q -b(q, v q )) 2 M (q) (2a) s.t. J(q)a q + Jc (q)v q α 0 (q, v q ) = α * (2b)
where J(.) ∶ R n q → R m×n v is the Jacobian of the contact constraint, α(q, v q ) = J(q)a q + α 0 (q, v q ) is the contact acceleration, α 0 (q, v q ) ∈ R m is the contact acceleration drift, and α * ∈ R m is the desired contact acceleration, typically a numerical stabilization term. This program can be solved efficiently using proximal methods [START_REF] Carpentier | Proximal and sparse resolution of constrained dynamic equations[END_REF]. Dropping the dependencies in q, v q , the KKT conditions of (2) read

[ M J T J 0 ] K [ a q -λ ] y = [ τ q -b α * -α 0 ] σ (3)
The Lagrange multipliers λ ∈ R m correspond to the contact forces. The constrained forward dynamics is the solution map of the system (3)

FD ∶ { R n q × R n v × R n v → R n v × R m (q, v q , τ q ) ↦ y = K -1 σ (4)
Given position, velocities and torques, it computes the constrained joint acceleration and contact forces. We are interested in computing

∂ FD ∂q , ∂ FD ∂v q , ∂ FD ∂τ q
in a particular reference frame introduced later on. We also define the inverse dynamics function

ID ∶ { R n q × R 2n v × R m → R n v (q, v q , a q , λ) ↦ τ q = M a q + b -J T λ (5)

B. Differentiating the constrained dynamics

Let us now recall how the derivatives of FD can be expressed in terms of the derivatives of ID and of the contact acceleration [START_REF] Carpentier | Analytical derivatives of rigid body dynamics algorithms[END_REF]. Without loss of generality, we assume α * = 0. Let z be any of the variables q, v q , τ q . The derivatives of (4) are given by the following proposition Proposition 1:

∂ FD ∂z = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -K -1 [ ∂ ∂z ID(q, v q , a q , λ) ∂ α ∂z ] if z ≡ q or v q -K -1 [ -I n v 0 ] if z ≡ τ q (6)
Proof: We simply differentiate y = FD(q, v q , τ q ). Let ′ denote the differentiation operator w.r.t. z.

y ′ = -K -1 K ′ K -1 σ + K -1 σ ′ (7)
y ′ = -K -1 (K ′ y -σ ′ ) (8)
y ′ = -K -1 ⎛ ⎝ K ′ [ a q -λ ] -[ τ ′ q -b ′ -α ′ 0 ] ⎞ ⎠ (9)
y ′ = -K -1 ⎛ ⎝ [ M ′ J ′T J ′ 0 ] [ a q -λ ] -[ τ ′ q -b -α ′ 0 ] ⎞ ⎠ (10)
y ′ = -K -1 [ M ′ a q -J ′T λ + b ′ -τ ′ q J ′ a q + α ′ 0 ] (11)
When z ≡ τ q , the lower part vanishes and the upper part is reduced to -I n v . When τ q ≡ q or v q , we recognize the derivatives of ( 5) and (2b) w.r.t. q and v q .

C. Spatial algebra tools

We remind here important definitions of spatial algebra 1 .

A placement is an element X of the special Euclidean group SE(3). Given two frames A, B, the transformation of motions from B to A is characterized by

A X B A X B ≜ [ A R B 0 - A R B A p × B A R B ] (12)
where A R B ∈ SO(3), A p B ∈ R 3 are the rotation and translation from frame B to frame A respectively, and × transforms a vector in R 3 into its associated skew-symmetric matrix. Forces are transformed with

A X * B ≜ ( A X B ) -T (13)
Spatial velocities and accelerations are motions in

M ≃ R 6 denoted by ν ∈ R 6 , α ∈ R 6 ν = [ ν ω ν v ] , α = [ α ω α v ] (14)
where ν ω , ν v ∈ R 3 represent the angular and linear velocities, α ω , α v ∈ R 3 are the angular and linear components of the spatial acceleration. The spatial velocity (resp. acceleration) of frame B expressed in frame A is denoted by The force acting at the center of frame B expressed in A by A λ B . The subscript B will be omitted from spatial motions and forces when the frame of interest is not ambiguous. The spatial cross product operator and its adjoint are defined as

A ν B (resp. A α B ). The spatial force in F ≃ R 6 is denoted by λ λ = [ λ η λ f ] (15)
ν× ≜ [ ν × ω 0 ν × v ν × ω ] ( 16
)
ν× * ≜ -ν× T ( 17
)
We recall now the time derivative of a rigid transformation Proposition 2: Let A, B be arbitrary frames.

∂ ∂t ( B X A ) = B (ν A -ν B ) × B X A (18)
The proof can be found in [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF]. This proposition will be used to derive the derivatives of interest in the next section.

D. Reference frames convention

We define the following reference frames and their short names (see Figure 1). On purpose, we follow the conventions established in the software Pinocchio [START_REF] Carpentier | The Pinocchio C++ library: A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives[END_REF] where all the equations recalled in this section are efficiently implemented.

• WORLD (O) : inertial frame, fixed w.r.t. the observer • LOCAL (L) : attached to a moving part of interest, typically to a joint of the robot (a.k.a. "body" frame) • CENTERED (♡): centered at the local frame, but with axes oriented like WORLD at all times The ♡ reference frame is the one in which we would like to express contact tasks. It is related to frames O, L as follow

O X ♡ (q) = [ I 0 -p × I ] , ♡ X L (q) = [ R 0 0 R ] ( 19
)
where R = O R L (q) is the rotation from L to O and p = O p L (q) is the translation from L to O.

III. DERIVATIVES OF THE CONSTRAINED DYNAMICS IN THE ♡ FRAME

Note that Proposition 1 is agnostic to the reference frame. In this section, we show how the analytical derivatives of inverse dynamics and contact acceleration used in (6) can be derived in the ♡ frame by using their counterparts known in L [START_REF] Carpentier | Analytical derivatives of rigid body dynamics algorithms[END_REF]. Our analysis focuses on the generic case of a single

6D contact (m = 6)
2 such that the m-dimensional vectors α, λ introduced in II-A correspond to spatial quantities α, λ.

A. Derivative of the spatial acceleration in ♡

The following proposition provides the derivatives of ♡ α (spatial acceleration of L expressed in ♡) in terms of the derivatives of L α (spatial acceleration of L expressed in L).

Proposition 3:

∂ ♡ α ∂τ q = 0 (20) ∂ ♡ α ∂v q = ♡ X L ∂ L α ∂v q (21) ∂ ♡ α ∂q = ♡ X L ∂ L α ∂q -[ ♡ α × ω R L J ω ♡ α × v R L J ω ] ( 22
)
where L J ω represents the angular part (i.e. top 3 rows) of the local contact Jacobian.

Proof: The derivatives w.r.t. τ q is trivially 0 since α doesn't depend on τ q . For q, v q we differentiate

♡ α = ♡ X L (q) L α (23) ♡ α ′ = ♡ X L (q) L α ′ + ♡ X L (q) ′L α (24)
♡ X L (q) does not depend on v q so the second equality follows. For q, the second term involves a tensor-vector product that can be evaluated directly using Proposition 2:

∂ ∂t ( ♡ X L ) = ♡ (ν -ν ♡ ) × ♡ X L (25)
where ν, ν ♡ are the spatial velocities of frames L and ♡ respectively. Moreover, by definition of the frame

♡ ♡ ν = [ O ν ω R L ν v ] , ♡ ν ♡ = [ 0 R L ν v ] (26)
Injecting Eq. ( 26) in ( 25) and using the anti-commutativity of the spatial cross product

∂ ∂t ( ♡ X L ) L α = [ O ν ω 0 ] × ♡ X L L α (27) = - ♡ α × [ O ν ω 0 ] (28) = [ - ♡ α × ω R L J ω - ♡ α × v R L J ω ] v q (29)
We used O ν ω = R L ν ω and the Jacobian definition L ν ω = L J ω v q . Observing that v q = ∂ q ∂t , the result follows.

B. Derivative of ID in ♡

The derivatives of (5) in ♡ are given in terms of the local ID derivatives by the following proposition. Proposition 4:

∂ ∂τ q ( ♡ ID) = 0 (30) ∂ ∂v q ( ♡ ID) = ∂ ∂v q ( L ID) (31) ∂ ∂q ( ♡ ID) = ∂ ∂q ( L ID) - L J T [ L λ × η L J ω L λ × f L J ω ] (32)
Proof: By definition, ID in ♡ and its derivative are given by

♡ ID(q, v q , a q , ♡ λ) = M a q + b - ♡ J T ♡ λ (33) ♡ ID ′ (q, v q , a q , ♡ λ) = M ′ a q + b ′ -( ♡ J T ) ′ ♡ λ (34)
where a q , ♡ λ are considered constant during the differentiation. The differential w.r.t. τ q is trivially 0. For q, v q , we observe that the only terms depending on the reference frame are the contact wrench λ and the contact Jacobian. Using that ♡ J = ♡ X L L J, we have

( ♡ J T ) ′ ♡ λ = ( L J T ♡ X T L ) ′ ♡ λ (35) = L (J T ) ′ ♡ X T L ♡ λ + L J T ( ♡ X T L ) ′ ♡ λ (36)
The first term is computed naturally with the local derivatives of ID. Since ♡ X T L in the second term doesn't depend on v q , the second equality follows. For q, we observe that ♡ X T L = L X ♡ . Let us use Proposition 2 for the tensor-vector product

∂ ∂t ( ♡ X T L ) = ∂ ∂t ( L X ♡ ) (37) = L (ν ♡ -ν) × L X ♡ (38)
The spatial velocities of ♡ and L expressed in L are

L ν ♡ = [ 0 L ν v ] , L ν = [ L ν ω L ν v ] (39)
Plugging expressions (39) in (38)

∂ ∂t ( ♡ X T L ) ♡ λ = [ - L ν ω 0 ] × L X ♡ ♡ λ (40) Let us notice that L X ♡ = L X * ♡ since L X ♡ is orthogonal. So L X ♡ ♡ λ = L X * ♡ ♡ λ = L λ and ∂ ∂t ( ♡ X T L ) ♡ λ = [ - L ν ω 0 ] × L λ (41) = L λ × [ L ν ω 0 ] (42) = [ L λ × η L J ω L λ × f L J ω ] v q (43)
Finally, the result follows from observing that v q = ∂ q ∂t .

C. Projecting onto lower dimensional constraints

The case of an m-dimensional constraint for m < 6 can be deduced by projections from the 6D case. It can be useful, as illustrated in our experiments, to express tasks where the motion is constrained only in some directions. Let us define the set I ⊂ {1, ..., 6} of m dimensions. The constrained forward dynamics read

[ a q -λ I ] = K -1 [ τ q -b α * ,I -α 0,I ] (44)
were the subscript λ I means the m-dimensional sub-vector obtained by masking indices in λ that are not in I. Following this notation, the constrained forward dynamics function is

FD I (q, v q , τ q ) ≜ P I (FD(q, v q , τ q )) (45)
where

P I ∶ R n v × R m → R n v × R m represents the projection
onto dimensions i ∈ I, i.e. P I (a q , λ) ≜ (a q , λ I ).

IV. EXPERIMENTAL RESULTS

This section demonstrates the relevance of our method on a polishing task with the torque-controlled humanoid Talos.

A. Experimental setup

The upper body (torso + right arm) is controlled in MPC while the lower body is maintained in a fixed posture using position control, so we have n q = n v = 6. The DDP implementation is used to solve the optimal control problem is available in Crocoddyl 3 and the rigid-body dynamics computations are performed in Pinocchio 4 . The analytical derivatives of the constrained forward dynamics in the ♡ frame which were derived in Section III, are implemented in the open-source library Sobec5 . The task is exert with the right wrist a constant normal force on a flat rigid surface while drawing a circle. This task is demonstrated on both horizontal and vertical surfaces. For this task, the desired contact force is naturally expressed in the ♡ frame, as the local frame attached to the wrist of the robot is not necessarily aligned with the surface normal direction.

B. MPC formulation

We use the force feedback MPC formulated in [START_REF] Kleff | Introducing Force Feedback in Model Predictive Control[END_REF], which solves the following Optimal Control Problem (OCP) min w(.),z(.)

∫ T 0 L(z(t), w(t), λ(t), t)dt + L T (z(T )) (46) s.t. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ż(t), λ(t) = F (z(t), w(t)) z(0) = z 0 z(t) ∈ Z, w(t) ∈ W
where z ≜ (q, v q , τ q ) is the state, w is the control (a.k.a computed) torque, z 0 is the initial state, L, L T the running and terminal costs, Z, W represent state and control constraints. The dynamics F gathers the constrained forward dynamics st -order low-pass filter on the torques with angular cut-off frequency ω 0 , i.e. ż(t), λ(t) are such that q(t) = v q (t) (47) vq (t), λ(t) = FD I (q(t), v q (t), τ q (t)) (48)

τq (t) = ω 0 (w(t) -τ q (t)) (49)
The contact model used in this case is 1D (m = 1), i.e. the robot is free to move in the plane orthogonal to the surface normal. The definition of the index set I depends on the experiment: when the task is to maintain contact with a horizontal plane, then I = {3} (constraint acting along the z-dimension in ♡), whereas for the vertical plane contact, I = {2} (constraint acting along the y-dimension in ♡). The cost function describing the sanding task has the form 4b. The contact forces are estimated from joint torques sensors, joint position encoder measurements and joint velocities computed through finite differences using the forward dynamics. We can see that the robot is able to track the forces expressed in the ♡ frame. It is important to note that without the mathematical developments presented in this paper, it would have been impossible to specify the task directly in this frame. We would have had to make the wrist orientation as part of the main task, i.e. to hard-code an operational frame that must remain oriented like the world frame at all times. This imposes an additional constraint on the robot's motion and makes the task more difficult to realize. By expressing the contact task directly in the appropriate frame, the controller is free to trade-off wrist orientation against force tracking.

L(z, w, t) = c z ∥A z (z -z 0 )∥ 2 + c lim z ∥B z (z)∥ 2 + c λ ∥A λ ( ♡ λ - ♡ λ)∥ 2 + c R ∥A R (R(q) ⊖ R)∥

D. Extension and implementation

The model covered in this paper is only considering the case where the contact occurs between a robot body and the world. It is implemented for the 3 proposed frames LOCAL, CENTERED and WORLD in Sobec 5 . It can be generalized to the universal case where the contact occurs between two frames C 1 and C 2 that can both be either attached to a moving body or fixed in the world (not limiting to one contact being fixed). We then recover the LOCAL formulation when C 1 is attached to the robot and C 2 is fixed, and the CENTERED formulation when C 2 is attached to the robot and C 1 is fixed. This extension has been chosen for the implementation of the constraint dynamics soon released in Pinocchio v3.

The extension of the model proposed here to this universal description is handled in an extended version of this paper6 .

V. CONCLUSION

In this paper we derived derivatives of the contact acceleration and forces to perform contact tasks in MPC in an arbitrary frame. The main outcome of this study is that nonlinear skew terms need to be added to the rotated local derivatives. We showed experimentally that our calculations enable to exert forces in arbitrary directions. We hope this paper will help MPC practitioners to design contact tasks without having to go through these tedious calculations. It is also worth emphasizing that the centered frame can be in fact any frame deemed relevant for the task by the user (only the rotation matrix needs to be changed, not the maths). As future work we intend to include these derivatives more systematically in our software stack and propagate their use to locomotion and whole-body multi-contact problems. Another interesting lead can be to optimize the computation of the projected constraint derivatives.

Fig. 1 :

 1 Fig. 1: Representation of the 3 reference frames

  Fig.2: Sanding task on a horizontal table (top) and on a vertical board (bottom). Analytical derivatives of the constrained forward dynamics in the ♡ frame enables to easily design and realize a contact task using nonlinear MPC.

2 +c 2 + c w ∥A w w∥ 2

 222 p ∥A p (p(q) -p(t))∥ R are the end-effector frame 3D position and rotation respectively, A z , A w , A p , A λ , A R are activation weight matrices, c z , c lim z , c w , c lim w , c λ , c p , c R are scalar costs weights, ⊖ represents the difference in SO(3), B z , B w are weighted quadratic barriers. The reference force to exert on the contact surface is ♡ λ, expressed in the centered frame and the trajectory to track on the surface plane is p(t).

Fig. 3 :

 3 Fig. 3: Screenshot of the horizontal (top) and vertical (bottom) sanding tasks.

  (a) End-effector position in the x, y-plane of the centered ♡ frame for the horizontal surface sanding (b) End-effector position in the x, y-plane of the centered ♡ frame for the vertical surface sanding

Fig. 4 :

 4 Fig. 4: End-effector trajectories in the surface plane for horizontal (top) and vertical (bottom) sanding tasks

For a more detailed treatment the reader can refer to[START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF], Section

2.9.

Adding more contacts is not an issue -they simply need to be stacked properly in the KKT system and in the derivatives. The case of lower dimensional constraints is discussed in Section III-C.

https://github.com/loco-3d/crocoddyl

https://github.com/stack-of-tasks/pinocchio

https://github.com/MeMory-of-MOtion/sobec

https://hal.archives-ouvertes.fr/hal-03758989

ACKNOWLEDGMENT

The authors would like to thank Nahuel Villa, Côme Perrot and Pierre Fernbach for helping in the experiments and Guilhem Saurel for his amazing support in software development.

This work was in part supported by the National Science Foundation (grants 1825993, 1932187, 1925079 and 2026479), Meta Platforms Inc., and ANR Dynamo-grade (ANR-21-LCV3-0002)


Pierre-Emmanuel Arduin
email: pierre-emmanuel.arduin@dauphine.psl.eu

Káthia Marc ¸al De Oliveira

Christophe Kolski
email: christophe.kolski@uphf.fr

Trust me and Click! A Pilot Study of Cognitive Walkthrough for Phishing Emails

Keywords: Cognitive Walkthrough, Phishing, Online Deception, Usability Inspection

The growing importance of new markets on the Internet in the 1990s led to the development of new guidelines and interface design principles ensuring online consumer trust. These guidelines have been rapidly spoofed by attackers who integrated interface design principles maliciously to deceive users with phishing emails. In this paper we propose to adapt an inspection method, Cognitive Walkthrough, to understand how users walk through the processing of a phishing email. 23 experienced evaluators used the proposed method, Cognitive Walkthrough for phishing emails, in a pilot study and gave a feedback on it. The method allows to analyze the areas of a phishing email eliciting users to trust, to act, and to distrust. These areas are then commented allowing to understand the interface design principles exploited maliciously by attackers when they are saying to users: Trust me and click!

I. INTRODUCTION

The "businessization" of the Internet [START_REF] Imase | As an element of the regional innovation cluster, the citizen/non-profit sector fulfills the "seedbed function" of the new industry[END_REF] in the early 1990s led to new markets and implied new challenges for Human System Interactions. Mainframe computers interacting with experts in the 1970s became personal computers interacting with employees in the 1980s. These became connected to the Internet in the 1990s and interacting with citizens of the world through to the World Wide Web. Researchers developed then models and guidelines ensuring online consumer trust [START_REF] Ba | Evidence of the effect of trust building technology in electronic markets: Price premiums and buyer behavior[END_REF], [START_REF] Egger | Affective design of e-commerce user interfaces: How to maximise perceived trustworthiness[END_REF]. Originally developed for legitimate companies purposes, these guidelines have been rapidly spoofed by phishers to deceive individuals [START_REF] Dhamija | Why phishing works[END_REF].

Past studies highlighted that users from different socioeconomic groups or different countries have different privacyrelated experiences and perceptions [START_REF] Saleh | What about my privacy, habibi?[END_REF]. An experiment made by Butavicius et al. [START_REF] Butavicius | When believing in technology leads to poor cyber security: Development of a trust in technical controls scale[END_REF] concluded that firewalls and antiviruses are more trusted than spam filters and social media privacy settings. In 2021, more than 300 billion emails are sent everyday in the world [START_REF] Radicati-Group | Email statistics report[END_REF] and when facing an email, decisionmaking relies on simple cues making most phishing emails being peripherally processed [START_REF] Vishwanath | Why do people get phished? testing individual differences in phishing vulnerability within an integrated, information processing model[END_REF]. The pilot study presented in this paper focuses on the first step of attacks beginning with phishing emails as an entry point. We argue that emails may be regarded as Human System Interactions phenomena, where user actions may be requested by the system.

Considering an increasing malicious adoption of interface design principles by attackers, we propose to adapt Cognitive Walkthrough [START_REF] Lewis | Testing a walkthrough methodology for theory-based design of walk-up-and-use interfaces[END_REF], an inspection method historically developed for usability inspection, to the case of phishing emails. An originality of this work is to focus on the evaluation of a kind of maliciousness quality, i.e. the extend to which the attacker has integrated interface design principles in a phishing email to deceive the user.

In the second section of this paper, we present the background theory and assumptions: usability inspection and Cognitive Walkthrough, phishing and online deception. In the third section, we present how Cognitive Walkthrough can be adapted to phishing emails. In the fourth section, we present a pilot study that involved 23 participants: the experiment procedure and the participants, the preliminary results, and a discussion on the limits and ethical implications of this research. The overall purpose of this paper is to raise awareness on how usability principles may be misused by attackers targeting legitimate users, i.e. maliciousness quality. Our proposition leads to understand the interface design principles exploited by attackers when they are saying to users: Trust me and click!

II. BACKGROUND THEORY AND ASSUMPTIONS

In this section, we first present usability inspection and Cognitive Walkthrough. Second, we draw from the literature to present phishing and how users may be deceived through online deception.

A. Usability inspection and Cognitive Walkthrough

Past research highlighted early the idea that software evaluation was not well accepted within organizations [START_REF] Nielsen | Guerrilla HCI: Using discount usability engineering to penetrate the intimidation barrier[END_REF], often considered as too expensive, time-consuming, and difficult to implement. In the mid-1990s, inspection methods appeared as a potential way to persuade professionals to adopt evaluation: they are convenient, affordable, and manageable. Traditionally, these methods intervene early in a user interface design to identify, qualify and quantify usability problems [START_REF] Virzi | Usability inspection methods[END_REF]. In this paper, we propose a paradigm switch by adapting and studying how a method such as Cognitive Walkthrough (CW) could be applied a posteriori to inspect maliciousness quality, i.e. how usability principles have been maliciously used by attackers to deceive users.

CW is an inspection method inspired by Norman's Action Theory [START_REF] Norman | Cognitive engineering[END_REF] with the idea of learning through exploration, allowing the detection of usability defects and the estimation of their degree of seriousness. It relies on: (i) the ease of performing a task with a minimum of knowledge of the system, and (ii) the ease of learning through exploration of the user interface. During a preparation phase, the evaluator specifies series of tasks to be assessed and each task is broken down into action sequences. Then, during an evaluation phase, the evaluator inspects each action by walking through the interface; he/she answers to questions, identifies problems, and reports them in problem description sheets.

Actually, even if when receiving an email a user do not need to walk through an interface, we argue that emails in general and phishing emails in particular may be regarded as Human System Interactions phenomena: user actions are requested such as clicking on a link or downloading an attachment for example. This constitutes an originality of the study presented in this paper. Indeed, the scientific community proposed several variants and evolutions of CW [START_REF] Mahatody | State of the art on the cognitive walkthrough method, its variants and evolutions[END_REF] and, when looking at a variation such as CW for the Web, we discovered a method for detecting and fixing errors occuring when browsing a website [START_REF] Blackmon | Cognitive walkthrough for the web[END_REF]. First, the user parses the page into a range of subregions and generates a brief description of each subregion (attention phase). Second, he/she generates descriptions of all graphic widgets in the target subregion and acts on the closest to his or her goals [START_REF] Kitajima | A comprehensionbased model of web navigation and its application to web usability analysis[END_REF].

Dhamija et al. [START_REF] Dhamija | Why phishing works[END_REF] performed a CW on 200 sample attacks. The 22 participants browsed legitimate and phishing websites with no information on their trustworthiness, had to say if the site was legitimate or not, state their confidence in their evaluation, and explain their reasoning. The authors highlighted three main dimensions in attackers' strategies: (i) lack of knowledge, (ii) visual deception, and (iii) lack of attention. This approach is rather different from the one of the work presented in this paper: not only we focus on emails as user interfaces requiring user actions, but also the participants are not acting as common users receiving emails but they have the role of evaluators, as if they were inspecting usability. Collected data will lead to understand usability problems and we argue that such data could also lead to a better understanding of the interface design principles used maliciously by attackers to deceive users.

B. Phishing and online deception

Matheson and Zanna [START_REF] Matheson | Persuasion as a function of selfawareness in computer-mediated communication[END_REF] already proposed in the late 1980s a computer-mediated persuasion study. Participants had to complete decision-making problems in two situations: (i) faceto-face, reading persuasive communication printed on paper, (ii) computer-mediated, reading persuasive communication on a screen. The results showed then an increased feeling of privacy in the computer-mediated situation compared to the face-to-face situation.

Phishing, i.e. the practice of directing users to fraudulent websites [START_REF] Dhamija | Why phishing works[END_REF], is well discussed in the literature through psychometric studies [START_REF] Butavicius | When believing in technology leads to poor cyber security: Development of a trust in technical controls scale[END_REF], context studies [START_REF] Ayaburi | Understanding phishing susceptibility: An integrated model of cue-utilization and habits[END_REF], or training studies [START_REF] Singh | Training to detect phishing emails: Effects of the frequency of experienced phishing emails[END_REF]. Yu et al. [START_REF] Yu | Mouse behavior as an index of phishing awareness[END_REF] presented recently that slow mouse movements are good predictors of phishing threat detection. Nevertheless, the literature seems to lack of an adapted usability inspection method, such as Cognitive Walkthrough (CW, see Section II-A), to evaluate a kind of maliciousness quality, i.e. the extend to which the attacker has integrated interface design principles in a phishing email to deceive the user.

Indeed, online interactions as well as a constant multitasking led individuals to rely more on mental shortcuts such as cognitive biases [START_REF] Mcalaney | Cybersecurity as a social phenomenon[END_REF]. Such biases may be used as online weapons of influence [START_REF] Muscanell | Weapons of influence misused: A social influence analysis of why people fall prey to internet scams[END_REF] when a user interface is designed to integrate persuasion principles [START_REF] Cialdini | Influence: The psychology of persuasion[END_REF] and leads to effective phishing [START_REF] Ferreira | An analysis of social engineering principles in effective phishing[END_REF]. Users often suffer from engaging in an effortful and careful way of thinking [START_REF] Fiske | Social cognition: From brains to culture[END_REF], particularly when interactions are online [START_REF] Guadagno | Have we all just become "robo-sapiens"? reflections on social influence processes in the internet age[END_REF]. In the early 2000s authors as Cialdini et al. [START_REF] Cialdini | Training in ethical influence[END_REF] warned on the potentiality of persuasion principles used maliciously on the Internet, asking for what they called "ethics of influence". Authors such as Xiao et al. [START_REF] Xiao | Online persuasion mechanisms and processes-a research agenda[END_REF] go deeper recently with a call to understand comprehensively online persuasion mechanisms and processes. As the reader may guess, this is not what we are going to propose in this paper, focusing on the evaluation of interface design principles integrated maliciously in a phishing email to deceive users.

If Zuckerman et al. [START_REF] Zuckerman | Learning to detect deception[END_REF] proposed in the early 1980s studies to detect deception through videotape experiments, recent research considers phishing susceptibility as a likelihood of being phished [START_REF] Hong | Keeping up with the joneses: Assessing phishing susceptibility in an email task[END_REF] and opposes it to phishing resistance [START_REF] Butavicius | When believing in technology leads to poor cyber security: Development of a trust in technical controls scale[END_REF]. Some authors also consider trust and distrust [START_REF] Arduin | Trusting security when sharing knowledge?[END_REF], even if others such as Moody et al. [START_REF] Moody | Which phish get caught? an exploratory study of individuals' susceptibility to phishing[END_REF] argue that neither the disposition to trust nor to distrust influence phishing susceptibility prediction. Authors as Nicho et al. [START_REF] Nicho | Evaluating user vulnerabilities vs phisher skills in spear phishing[END_REF] proposed to summarize the literature on variables affecting phishing susceptibility and we note that very few concern is given to usability inspection.

We propose in this paper to go further in studying how attackers adopted interface design principles to deceive users. By adapting an inspection method, CW, to phishing emails, we are studying how users walk through the analysis of the content, how they decide to trust, to follow a call-to-action, and to distrust a phishing email.

III. COGNITIVE WALKTHROUGH FOR PHISHING EMAILS

We argue that phishing emails are entry points of phishing attacks and that the presence of user-action calls within these emails led us to consider them as Human System Interactions phenomena.

The considered task of the walkthrough is the processing of an email by a user when he/she receives it. Such a task is broken down into actions such as the identification of trusteliciting cues and reasoning explanations, the identification of calls to potentially risky actions and consequent damages, and the identification of distrust-eliciting design cues and reasoning explanations.

First, the evaluator identifies in the email a set of trusteliciting areas for the user: T . The factors eliciting trust for each area t ∈ T are then noted by the evaluator in open text comments fields. Second, the evaluator identifies in the email a set of areas calling the user to act and potentially posing a The areas T , A, and D selected by several evaluators will allow to produce heatmaps [START_REF] Bojko | Informative or misleading? heatmaps deconstructed[END_REF] showing the most trust-eliciting, calling a user action, and distrust-eliciting areas in the email, whereas an analysis of the open text comments will allow to highlight the most occurring explanations on reasons eliciting trust, on possible risks following a user action, and on reasons eliciting distrust. Some of the areas calling the user to act (A) may be malicious and lead him/her to constitute an insider threat [START_REF] Arduin | Insider Threats[END_REF] Indeed, we consider that a problem occurs with phishing emails when an area calling the user to act is also eliciting his/her trust. This is the case when the intersection between the sets A (the areas calling the user to act) and T (the trust-eliciting areas) is not empty: A ∩ T ̸ = ∅. The areas eliciting user distrust provide also useful comparative data. Fig. 1 presents the actions completed by the evaluator during an inspection with CW for phishing emails.

IV. PILOT STUDY

In this section, we first present the experiment procedure and the 23 evaluators. Second, we discuss the preliminary results and third, the limits and ethical implications of this research.

A. Experiment procedure and participants

This pilot study has been conducted with 23 evaluators (11 male, 12 female, 15 PhD, 6 MSc, 2 BSc). In the following we refer to these evaluators involved in the study as the participants. The average age of the participants is 42.1 years. 30.4% of the participants reported having inspected ten or more user interfaces (52.2% between three and ten, 47.8% none). Thus participants show different levels of expertise in terms of inspection. In terms of phishing, they were 52.2% to recognize having already been a victim of phishing (39.1% never). 78.3% declared themselves as Computer Scientists and 21.7% as Managers. Data collection was managed with the online reaction time experiment software Qualtrics [START_REF] Barnhoorn | Qrtengine: An easy solution for running online reaction time experiments using qualtrics[END_REF].

The experiment procedure was as follows: a brief text informed the participants about the purpose and the context of the study: they were not users receiving phishing emails but evaluators acting as if they were inspecting usability. Participants were informed that a variation of CW for phishing emails will be used and they had then to give their informed consent to take part.

In this pilot study, two emails were sequentially shown to the participants with no information about their trustfulness. They inspected each email one at a time and faced, for each, three pages with questions structured following the method presented in Section III and by completing the actions presented in Fig. 1. Then, the participants were asked questions about their impressions on the inspection procedure and their profile (age, gender, academic level, area of expertise, number of usability inspections already conducted).

At the end of the experiment, the research team manually determined for each email the set A∩T showing the existence of areas calling the user to act also trusted by him/her.

B. Preliminary results of the pilot study

Through this experiment, we collected two categories of data: (i) clicked areas in the emails and (ii) open text comments. Participants had no time constraint and the average response time was 26.06 minutes (min. = 00:04:35, max. = 01:23:44). The participant with the higher response time was a cybersecurity expert who admitted having deeply analyzed the emails. The second higher response time was 01:06:20 (no explanation given).

1) Heatmaps of the clicked areas: For the 23 participants and the 2 emails, 3 images with clicked areas were collected (trust-eliciting, user-action-call, distrust-eliciting), namely: 23×2×3 = 138 images of emails with clicked areas. These 138 images were aggregated by the Qualtrics software to 2 × 3 = 6 heatmaps: 2 trust-eliciting, 2 user-action-call, and 2 distrust-eliciting areas. Fig. 2 presents the heatmaps showing the trust-eliciting areas (T ), the areas calling a user action (A), and the distrust-eliciting areas (D) for the two considered emails.

The participants identified an average of 3.6 areas as trusteliciting (min. = 0, max. = 10), 2.3 areas as calling a user action (min. = 1, max. = 6) and 3.3 areas as distrust-eliciting (min. = 1, max. = 10), by taking the two emails together. 166 areas were identified as trust-eliciting, 105 as calling a user action, and 154 as distrust-eliciting by the 23 participants for the two emails.

In this pilot study, the average number as well as the total number of trust-eliciting areas are higher than for distrusteliciting areas, even if the participants were aware of the nature of the inspection: CW for phishing emails. Indeed, distrusteliciting areas, as shown in Fig. 2.D, highlight a focus of participants on the body of the text when seeking for distrusteliciting cues, but less numerous than trust-eliciting areas. These results also show that the intersections between the sets A and T for the two emails are not empty (see Figs. 2.A and 2.T), meaning that the attackers have successfully integrated interface design principles to call a user action while eliciting him/her to trust such a call.

2) Analysis of the open text comments: As presented in Sections III and IV-A, the participants should explain why they selected these areas. In this pilot study, up to 10 areas may be clicked by the participants and thus up to 10 open text fields (one explanation per click) for each question may be completed or left empty. On a potential total of 23 × 2 × 3×10 = 1, 380 responses, we collected actually 351 responses corresponding to the areas clicked by the participants. These 351 responses were manually tagged by the research team and 303 tags were affected, some responses having no tag: e.g. "it leaves me in doubt. . . " or "see previous answer". On these 303 tags, 131 were assigned for trust-eliciting explanations, 58 for potential risks of user actions, and 114 for distrust-eliciting explanations.

Trust-eliciting areas are explained by the presence of a usual and good-looking logo [START_REF] Saleh | What about my privacy, habibi?[END_REF], the presence of a signature or human contact [START_REF] Guadagno | Have we all just become "robo-sapiens"? reflections on social influence processes in the internet age[END_REF], a suitable formatting [START_REF] Ferreira | An analysis of social engineering principles in effective phishing[END_REF] and phrasing [START_REF] Dhamija | Why phishing works[END_REF], as well as a verifiable address [START_REF] Dhamija | Why phishing works[END_REF]. The possible risks of a user action identified by the participants are a hidden redirection [START_REF] Mcalaney | Cybersecurity as a social phenomenon[END_REF] and personal data theft [START_REF] Lewis | Testing a walkthrough methodology for theory-based design of walk-up-and-use interfaces[END_REF]. Finally, distrusteliciting areas are explained by spelling mistakes [START_REF] Moody | Which phish get caught? an exploratory study of individuals' susceptibility to phishing[END_REF], an irrelevant content [START_REF] Hong | Keeping up with the joneses: Assessing phishing susceptibility in an email task[END_REF], a threatening message [START_REF] Fiske | Social cognition: From brains to culture[END_REF], the presence of a suspicious link [START_REF] Crossler | Future directions for behavioral information security research[END_REF], as well as a poor phrasing [START_REF] Cialdini | Training in ethical influence[END_REF], a generic message [START_REF] Cialdini | Influence: The psychology of persuasion[END_REF] or a poor formatting [START_REF] Butavicius | When believing in technology leads to poor cyber security: Development of a trust in technical controls scale[END_REF].

C. Limits and ethical implications of this research

After the inspection, 78.3% of the participants described the procedure as "simple to realize". We noted that 43.5% would not have wanted to be more guided, whereas 39.1% would have preferred. Nevertheless 73.9% agreed that being more guided would have influenced their responses. We noted a 65.2% agreement that the proposed inspection method allowed them to transcribe how a user processes a phishing email and a 39.1% indifference with the feeling of missing something (34.8% disagree, 26.1% agree).

The general comments of the 23 evaluators involved in this pilot study showed an interest for the proposed adaptation of CW for phishing emails. Some of them pointed out limits such as the context of the experiment: "I would have liked to receive both emails on my address to experience it just as in real life". Others pointed out the technical constraints of the tool: "I would have preferred to select areas and not just click on them", "not seeing the URLs associated with the links does not really allow to analyze the emails" . Indicating how the user contextualizes the email to his/her topicality is also presented as a lack for some participants: "More than specific elements, the links between these elements usually help me to know if the email is fraudulent", "I would check if the signatory actually works for the company sending me the email" . As the reader may guess, the research team is now working on how to improve the method from these comments on the pilot study presented in this paper.

The ethical implications of this research have to be considered. Indeed, the risk of a malicious use of our results or the proposed inspection method should not be neglected. Attackers may use this research as a way to develop a "how to" guide and the risk of dual use [START_REF] Rath | Evolution of different dual-use concepts in international and national law and its implications on research ethics and governance[END_REF] is important. Phishing emails are particularly interesting because their efficiency increases not only with the quality of their maliciouness, but also when organizational countermeasures show their limits in mitigating sloppiness and ignorance [START_REF] Crossler | Future directions for behavioral information security research[END_REF]. Consequently, identifying the interface design principles used maliciously by attackers would be a way to understand how users walk through these cues in particular and the processing of an email in general.

The importance of awareness and formation remains then crucial as a way to improve the processing of emails. We argue that education, higher education, and management could be effective vehicles to raise awareness among users on how they process emails. Such awareness will lead users to better understand how they may be deceived by an attacker through the system. A simple email may appear so informative, so passive, and consequently so harmless. Nevertheless, we have to be aware and make users aware that HTML code and hypertext browsing allow attackers not only to deceive us with a simple email, but also to call and to lead us to act.

V. CONCLUSION

In this paper we propose a new procedure relying on CW to inspect the maliciousness quality of phishing emails, i.e. the extend to which the attacker has integrated interface design principles to deceive the user.

With the presented pilot study involving 23 evaluators, we highlighted trust-eliciting areas, areas calling for a potentially risky user action, and distrust-eliciting areas. We observed that some of the trust-eliciting areas were also calling for a potentially risky user action, showing maliciousness quality. We raised a call for further investigation notably on links between the identified areas, the profile of participants using CW for phishing emails, and the final trust-and-click user action. We then analyzed the open-text explanations filled by the participants to highlight key trends on: why users trust? what do they risk? and why do they distrust? We finally proposed to open and keep open controversy as the considered research material and results may be maliciously used as a "how to" guide by attackers.

The idea of a "design for successful guessing" proposed by Lewis et al. [START_REF] Lewis | Testing a walkthrough methodology for theory-based design of walk-up-and-use interfaces[END_REF] was to facilitate the problem solving mechanisms with the design of an interface easy to learn by exploration. With this pilot study, we focused on the first step of attacks beginning with a phishing email. This work is currently continuing, as the research team thoroughly analyzes the data per participant to highlight trends on explanations and reasons eliciting trust, user actions and distrust. Moreover, analyzing overlapped areas and quantitative measures as well as crossing data from the open text comments and the heatmaps are other research directions. like any other cognitive process, guessing may be deceived by attackers who integrated maliciously interface design principles, as if they were saying to users: Trust me and click!

Fig. 1 .

 1 Fig. 1. Actions completed by the evaluator during the inspection of a phishing email.

Fig. 2 .

 2 Fig. 2. Trust-eliciting (T ), calling a user action (A), and distrust-eliciting (D) areas for the two phishing emails considered in this pilot study (N = 23). The trademarks were hidden after the experiment.

ACKNOWLEDGMENT

The authors would like to thank the 23 evaluators who performed this pilot study and provided valuable insight into this work.