Cyril Grima
email: cyril.grima@obs.ujf-grenoble.fr

François Costard
email: francois.costard@u-psud.fr

Wlodek Kofman

Bertrand Saint-Bézar

Anthony Servain

Frédérique Rémy

Jérémie Mouginot

Alain Herique

Roberto Seu

Large asymmetric polar scarps on Planum Australe, Mars: Characterization and evolution

Keywords: Mars, polar geology, Mars, polar caps, Ices, Radar observations, Tectonics

Numerous scarps with similar characteristics have been observed in the polar layered deposits of Planum Australe, Mars. They are referred to as LAPSs (for Large Asymmetric Polar Scarps) because of their typical cross-section featuring a trough between a straight slope on one side with outcrops of layered deposits and a convex slope on the other side without any outcrops. These LAPSs are restricted to the outlying region of Ultimi Lobe. Topographic data, optical images, and subsurface radar observations have been analyzed and compared to produce a complete morphologic and stratigraphic description of these scarps. In all, 167 LAPS-like features have been identified. All have similar dimensions and characteristics and appear to be deep depressions in the ice. The polar deposits have an average thickness of 1 km in this region and the LAPS depressions commonly reach half of that thickness. Subsurface data indicate that the depressions could reach bedrock at certain locations. Many surface features of the polar deposits of Mars are considered to be consequences of depositional and/or erosion processes. We propose a mechanical failure of the ice for the LAPSs origin, given the striking similarities in shape and size they show with rollover anticlines above listric faults commonly observed as a crustal extension mode on Earth. This tectonic scenario would imply a substantial outward sliding of the polar deposits in the region of Ultimi Lobe and a low basal shear stress. No information is available to determine whether such a system could be active at present. Confirmation of the "mechanical failure" hypothesis of these LAPSs on Mars is of major importance as it could be a macro-expression of fundamental differences between ice-sheet behavior under Martian and Terrestrial conditions.

(email) wlodek.kofman@obs.ujf-grenoble.fr (email) bertrand.saint-bezar@u-psud.fr (email) berkuts37@hotmail.fr (email) remy.omp@free.fr (email) jmougino@uci.edu (email) alain.herique@obs.ujf-grenoble.fr (email) roberto.seu@uniroma1.it

Introduction

The south and north polar plateaus of Mars -respectively Planum Australe and Planum Boreumare the only known examples of extraterrestrial ice-sheets comparable to those of the Earth. Many striking similarities with the Earth's ice-sheets exist. For instance, the volumes of both Planum Australe and Planum Boreum (respectively ~1.6 x10 6 km 3 and ~1.1 x10 6 km 3 ) are of the same order as that of the Greenland ice sheet (~2.6 x10 6 km 3 ), with similar average thicknesses of about 1000 to 1500 m [START_REF] Plaut | Subsurface Radar Sounding of the South Polar Layered Deposits of Mars[END_REF][START_REF] Smith | Mars Orbiter Laser Altimeter: Experiment summary after the first year of global mapping of Mars[END_REF][START_REF] Weidich | Satellite images atlas of glaciers of the world: Greenland[END_REF]. Moreover, water ice is the major ice-sheet component on both Earth and Mars, with an average impurity fraction that is likely less than 10% for Planum Australe [START_REF] Plaut | Subsurface Radar Sounding of the South Polar Layered Deposits of Mars[END_REF] and close to 5% for Planum Boreum [START_REF] Grima | North polar deposits of Mars: Extreme purity of the water ice[END_REF]. When the mass-balance is positive, the growth process of ice sheets on both planets is driven by the deposition of isochronous layers (known on Mars as polar layered deposits, PLD). Their impurity contents vary with the time of deposition [START_REF] Cutts | Models of climate cycles recorded in Martian polar layered deposits[END_REF][START_REF] Thomas | Polar deposits of Mars[END_REF]. Furthermore, the ice of the Earth' s ice sheets is crystallized in the hexagonal system (ice-Ih). Although cubic ice (Ice-Ic) could form under Martian conditions by vapor deposition on the surface, it would be metastable [START_REF] Gooding | Possible significance of cubic water-ice, H2O-Ic, in the atmospheric water cycle of Mars[END_REF]. Given the annual rise of surface temperature above the Ic/Ih irreversible transition (~150 K), the Martian ice sheets are likely exclusively composed of stable ice-Ih as on Earth. And finally, the evolution of the ice sheets on a geological time-scale is mainly driven by Milankovitch orbital cycles (obliquity, eccentricity, and precession variations) leading to planetary climate forcing [START_REF] Hays | Variations in the Earth' s Orbit: Pacemaker of the Ice Ages[END_REF][START_REF] Head | Recent ice ages on Mars[END_REF], as on Earth.

-Nevertheless, the polar environment on Mars is different than on Earth. For instance, temperatures in Martian polar regions can be as low as 150 K in both hemispheres [START_REF] Lewis | A climate database for Mars[END_REF], while the average surface pressure is 0.008 bars with seasonal variations of ± 20% [START_REF] Clifford | The State and Future of Mars Polar Science and Exploration[END_REF] and gravity is 0.38 times less than on Earth. Furthermore, the base elevations of the two Martian polar plateaus differ by around 6000 m, far more than on Earth. On Mars, the geothermal heat flow that largely determines the basal temperature of a glacier is estimated to be in the range of 15 to 45 mW m -2 (e.g. [START_REF] Mcgovern | Localized gravity/topography admittance and correlation spectra on Mars: Implications for regional and global evolution[END_REF][START_REF] Nimmo | Influence of early plate tectonics on the thermal evolution and magnetic field of Mars[END_REF][START_REF] Reese | Heat transport efficiency for stagnant lid convection with dislocation viscosity: Application to Mars and Venus[END_REF], lower than the value of ~65 mW m -2 for the Earth's continents [START_REF] Pollack | Heat flow from the earths interior -Analysis of the global data set[END_REF]. Consequently, the surface mass balance of the Earth' s ice sheets is mainly determined by melting of snow and ice, whereas condensation and sublimation are the dominant processes on Mars [START_REF] Rognon | The northern ice cap of Mars: an enigmatic evolution compared to the polar environment on Earth[END_REF]. Orbital parameters vary with a comparable timescale on Mars and Earth, but with greater amplitudes on Mars. Over the past 10 Myr, Martian obliquity has exceeded 45° and the eccentricity of the orbit has reached maximums twice that of the Earth's [START_REF] Laskar | Orbital forcing of the martian polar layered deposits[END_REF][START_REF] Ward | Long-term orbital and spin dynamics of mars[END_REF]. Taking all these similarities and dissimilarities into account, Mars can be considered as a full-scale laboratory for the study of the behavior of an Earth-like ice-sheet system over extended conditions. Consequently, the peculiarities observed on the Martian polar plateaus can help us improve our understanding of the Earth' s ice-sheet system in general and make the first steps toward meaningful comparative planetology in polar science.

With this in mind, this paper provides a geomorphologic description of polar scarps frequently observed in the region of Ultimi Lobe (UL), Planum Australe (see Fig. 1). Despite their km-size and typical asymmetric profile that will be described, equivalent formations have never been reported to date on polar glaciers on Earth. Because of their particular shape, we call these formations LAPSs (for Large Asymmetric Polar Scarps). Until now, only a few studies have reported such LAPSs on Viking images [START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF][START_REF] Thomas | Sand dune materials and polar layered deposits on Mars[END_REF], attributing them to an eolian origin. Here LAPSs are described by coupling optical images, surface topography, and radar sounding from recent datasets. This multi-instrumental approach makes it possible to establish profiles of the surface and subsurface of the LAPSs. A LAPS inventory was undertaken and 167 corresponding formations were identified across UL, providing geomorphologic elements to identify and discuss different formation processes for LAPSs. Most of the observations appear to be consistent with a mechanical failure of the ice, analogous to a "listric fault/rollover" system. This tectonic process will be discussed along with certain direct implications.

Regional context

Planum Australe overlies heavily cratered highlands that are reported to be Noachian and Hesperian in age [START_REF] Tanaka | Geologic map of the polar region of mars[END_REF]. The middle-Hesperian Dorsa Argentae Formation (DAF) is thought to be the youngest surrounding terrain of Planum Australe. The Planum Australe mound accumulated during the late Amazonian period [START_REF] Kolb | Geologic history of the polar regions of Mars based on Mars global surveyor data -II. Amazonian period[END_REF] and presently has a maximum thickness of 3700 m [START_REF] Plaut | Subsurface Radar Sounding of the South Polar Layered Deposits of Mars[END_REF]. Cratering records estimate the plateau' s surface to be as old as 10 to 100 My [START_REF] Herkenhoff | Surface Ages and Resurfacing Rates of the Polar Layered Deposits on Mars[END_REF][START_REF] Koutnik | South Polar Layered Deposits of Mars: The cratering record[END_REF].

Signatures of the former extent of the ice sheet have been observed in the DAF where esker-like ridges could be the consequence of past basal-melting [START_REF] Head | Extensive Hesperian-aged south polar ice sheet on Mars: Evidence for massive melting and retreat, and lateral flow and pending of meltwater[END_REF][START_REF] Kargel | Ancient glaciation on Mars[END_REF][START_REF] Milkovich | Meltback of Hesperian-aged ice-rich deposits near the south pole of Mars: Evidence for drainage channels and lakes[END_REF]. Present-day conditions on Mars preclude exceeding the melting temperature of water ice, except in cases of substantial geothermal anomalies [START_REF] Clifford | Polar basal melting on Mars[END_REF], not likely to be detected by any in-flight instrument, or in the case of saline water (Renno et al., -2009 and references therein). UL is an outlying region of Planum Australe, opposite DAF and extending over equivalent latitudes. It is the only part of Planum Australe below 80°S, with latitudes as low as 72°S (Fig. 1). It covers 400,000 km² (i.e. one-third of the entire plateau area).

Without this region Planum Australe would be almost perfectly symmetric instead of being offset from the pole. UL seems to have been subjected to a slight outward horizontal motion during recent geological times, revealed by two opposing signatures co-existing along its border. The first is a viscous deformation of the ice, suggested by glacial tongue-like mounds flowing outward in a series of craters dotting the edge of UL [START_REF] Byrne | History and current processes of the Martian polar layered deposits[END_REF]. Similar behavior was reported by [START_REF] Head | Mars: Evidence for geologically recent advance of the south polar cap[END_REF] in the nearby region of Promethei Lingula. The other glacial signature involves brittle processes such as slumping, landsliding and faulting that have also been reported all along the northern border of UL [START_REF] Byrne | History and current processes of the Martian polar layered deposits[END_REF][START_REF] Murray | Preliminary geological assessment of the northern edge of ultimi lobe, Mars south polar layered deposits[END_REF]. These fractures were interpreted as evidence of modest basal sliding of the whole region, seen as a brittle plate. * Location of Fig. 1 (1.5 column width).

Data

Five different data sources were used for this study:

(i) The digital elevation model of the surface acquired by the Mars Orbiter Laser Altimeter (MOLA) [START_REF] Smith | Mars Orbiter Laser Altimeter: Experiment summary after the first year of global mapping of Mars[END_REF]. We used the gridded polar map at 256 pixels/° (~230 m/pixel) with a vertical accuracy of 1 m.

(ii) Visible images from the High Resolution Imaging Science Experiment (HiRISE) with a resolution of 25 to 32 cm/pixel [START_REF] Mcewen | Mars Reconnaissance Orbiter' s High Resolution Imaging Science Experiment (HiRISE)[END_REF]. Only a few HiRISE images are available over UL.

(iii) Visible images from the High-Resolution Stereo Camera (HRSC) experiment with a resolution of 10 to 20 m/pixel [START_REF] Jaumann | The high-resolution stereo camera (HRSC) experiment on Mars Express: Instrument aspects and experiment conduct from interplanetary cruise through the nominal mission[END_REF]. Despite a more limited resolution than HiRISE, HRSC were retained because of its high coverage of the UL region.

(iv) Radar subsurface cross-sections (radargrams) of Planum Australe from SHAllow RADar (SHARAD) [START_REF] Seu | SHARAD: The MRO 2005 shallow radar[END_REF]. The radar signal (20 MHz) penetrates as deep as ~ 1500 m into water ice materials with 10 × 300 m vertical × along-track resolution. The radar waves are backscattered by dielectric gradients along the propagation path that are closely related to impurity rate variations in the ice. Thus, SHARAD is able to detect the layers forming Planum Australe. In UL, the detection signal is faint and can suddenly disappear before reappearing tens of kilometres downtrack. This could be due to a low dielectric contrast between isochronous layers. However, the regional density of SHARAD data is high and it was possible to select some radargrams to support morphologic interpretations. Initially, radargrams have a vertical timescale representing the time delay of the echoes. They have been migrated in depth by using a relative dielectric constant of 3.10 as inferred by [START_REF] Grima | North polar deposits of Mars: Extreme purity of the water ice[END_REF] for the nearly pure ice of Planum Boreum.

Off-nadir surface reflections (clutter) can generate highly delayed echoes that can be confused with subsurface structures. Adaptation of radar simulation developed by [START_REF] Nouvel | Radar signal simulation: Surface modeling with the Facet Method[END_REF] allowed us to identified the biggest clutters.

(v) Bedrock elevation as mapped by [START_REF] Plaut | Subsurface Radar Sounding of the South Polar Layered Deposits of Mars[END_REF] based on the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) data [START_REF] Picardi | Radar soundings of the subsurface of Mars[END_REF]. Compared to SHARAD, the MARSIS wavelength (1.8 to 5 MHz) provides a lower vertical resolution of ~80 m in the ice. However the signal penetrates deep enough into Planum Australe to be clearly back--scattered by the bedrock. The vertical uncertainty on the inferred basal topography is assumed to be 200 m [START_REF] Plaut | Subsurface Radar Sounding of the South Polar Layered Deposits of Mars[END_REF]. It must be emphasized that this bedrock elevation map were obtained by interpolation of a limited-density dataset: the resolution was sufficient to get the basal regional trend but not to detect local anomalies of the bedrock.

Morphological description

Spatial shape and distribution

The first recognizable feature of the LAPSs is their spatial shape. They appear to be arch-shaped, tens of kilometers long, and are distributed over the surface of the ice (Fig. 1). [START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF] described this particular signature as a scalloped or a "gull-winged" shape. This criterion enabled us to identify 167 LAPSs over Ultimi Lobe (Fig. 2). This count is not accurate for two reasons:

(i) sometimes up to three or more LAPSs are aligned and/or connected, often making it difficult to precisely define how many LAPS there actually are; (ii) the arch-shaped LAPSs can be confused with the rims of shallow-buried craters. To avoid this, we do not consider scarps that are obviously linked to a circular structure (an example is shown Fig. 5B). However, since the surface topography is quite disturbed, we cannot rule-out that some less obvious crater-linked formations were counted. The count shows that LAPSs are widespread over Ultimi Lobe. Their "horn-to-horn" width is in the range of 10 to 20 km, while some can reach 50 km. Most of them are gathered in aligned series. As already observed by [START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF], their regular size and close spacing cannot be universally explained by a crater rim origin. This strongly supports the assumption that most of the LAPSs considered in this study are not mainly linked to shallow--buried craters. As shown in Fig. 3, the concave sides of the LAPSs never face the South Pole and the dominant orientation is not towards the equator (i.e. azimuth = 0°). Neither is the orientation constant with respect to longitude. For instance, in a given area, adjacent LAPSs can be almost right-angled (e.g. white arrows in Fig. 2). The regional orientation in the polar stereographic view is also a relevant parameter. Let us define 0° and 90° as the angles for which LAPS concavities face the bottom and the right border of the map respectively. Figure 3

Cross-section

The second noteworthy feature is the asymmetric shape of LAPS cross-sections (giving them the second letter of their acronym). They exhibit a trough between a straight slope on one side, comparable to a scarp, facing a gentle convex upward slope on the other side that flattens as it rises. Figure 4 and the cross-sections in Fig. 6 illustrate this profile. The LAPS system forms extremely deep troughs in the ice with respect to the local ice thicknesses. The height of the scarps ranges from 200 m to 700 m with an average value of 400 m. The East part of UL (left side on the map) hosts the highest scarps. The resulting troughs regularly penetrate Planum Australe to half its thickness. In some cases, the trough reaches the top value of the basal elevation estimated by MARSIS. This usually occurs when the trough seems flat and parallel to the bedrock slope, as shown in Fig. 6 for cross-sections 1b-1b'and 3b-3b'. This leads to the possibility that the bedrock could in some cases become locally exposed. Some LAPSs exhibit a more complex topography on the side of the convex slope (cross-sections 4a-4a'and 4b-4b').

Their curved shape is broken by one or more flat plateaus, more or less horizontal. These LAPSs are essentially located in the heart and in the Eastern part of UL (left side of Fig. 2). Despite these very particular topographies -asymmetric and deep troughs -it is important to emphasize the relative flatness of the system. The slope of the scarps ranges from 2° to 15°, with an average value of 10°. Indeed, the mean horizontal distance between the foot and the top of the straight slope is 3.5 km. Therefore, the horizontal scale of the LAPS system is roughly an order of magnitude larger than the ice thickness. This thinness and flatness are highlighted in Fig. 6 by the inserts that display no vertical exaggeration. * Location of Fig. 4 (no particular restriction on the size).

Hillocks

Another feature is regularly (but not systematically) superimposed on the typical cross-section described above i.e. a ridge that can be observed along certain scarp crests (indicated by black arrows in Fig. 6), similar to elongated hillocks. We did not find any relation between the heights of the hillocks and the height of the straight slope or any other distinctive dimensions. The hillock / straight-slope height ratio is usually < 0.2, although around 10 LAPSs, mostly concentrated in the far-east side of UL, have a ratio > 0.5. However, note that the hillocks remain extremely flat, as they are stretched horizontally over 2 to 5 km. * Location of Fig. 5 (as wide as possible).

Stratigraphy

The behavior of the internal structure of any geological formation is necessary to understand its origin. Consequently the structure of the PLD beneath the LAPS and information from subsurface radar are essential. However, as noted in section 3, the layers detected by SHARAD in the UL region correspond to faint signals. To support the radar observations, we compared them with visible images from HiRISE and HRSC. These cross-analyses made it possible to draw a partial pattern for the polar layers.

Figure 7 shows two LAPS cross-sections associated with two SHARAD radargrams. HRSC images of the surface along the groundtracks are also shown. The radargram of cross-section 5-5' clearly shows horizontal layers on the straight-slope side (right side of the picture), leading to the hypothesis that the scarp includes outcrops there. On the other side, the layers below the convex slope are curved, parallel to the surface and plunging. Unlike the case on the straight-slope side, no outcrops are expected on the convex slope. On the HRSC image associated with this crosssection, polar layer outcrops can be distinguished as gathered lines of various albedos perpendicular to the ground-track. The change in albedo is due to slope and dust content variations from one layer to another. The cross-section locates these outcrops in an area corresponding exclusively to the straight slope, confirming the radar observation. Conversely, the HRSC image does not show any such signature on the convex slope, consistent with the previous hypothesis that the layers are curved downward and do not outcrop on this side. Cross-section 6-6' shows similar behavior, although the bending of the convex-slope layers is not as clear on the radargram because of the faint detection signals. However the associated HRSC image reveals that outcrops occur only on the straight slope, implying that the convex-slope layers also plunge downward as supposed.

This asymmetric behavior of the polar layers (outcrops on the straight slope and downward bending below the convex slope) cannot be verified by radar data for the entire set of LAPSs because of the poor layer detection in the UL region as mentioned above. However, visual inspection of available images shows that outcropping on the straight slope only is a common signature of the LAPSs. Figure 8 is an example of a HiRISE image over two successive LAPSs.

The asymmetric outcrop is clear for both; however the cross-section path cuts the end of the second LAPS leading to progressive fading of the outcrops. Finally, it is not possible to comment on the structure just under the LAPS troughs given that no formation is detected by radar. This does not necessarily mean that no reflector is present. It is more likely due to the straight slope, refracting most of the back-wave power offside with respect to the instrument, making underlying features hard to detect. In addition, the recurrent clutter echoes at this location could be strong enough to hide faint signals from the subsurface. * Location of Fig. 7 (as wide as possible).

* Location of Fig. 8 (as wide as possible).

Interpretations

-

The results of the morphologic description of the LAPS surface and the stratigraphic study are summarized in Fig. 4. The different indicators support that Fig. 4 is a roughly general model for the LAPS morphology. We will now look at the possible origins of the LAPSs.

Crater filling as a possible origin of LAPSs

The arched shape of the LAPSs naturally suggests a possible link with craters. Some of the circular objects on UL are obviously buried craters that lay either on buried layered deposits (paleo-surface) or on the bedrock. Subsequent accumulation of deposits of ice on these irregularities would lead to an increasingly smooth surface topography that would nevertheless conserve a spatial signature of the initial formations with a decreasing vertical relief. In particular, the common hillocks observed along the crests could be easily interpreted in terms of crater rim prominences. However, the observed LAPSs are just semi-circular with no sign of an opposite and symmetric formation to complete the circle. Nevertheless, it should be possible to fill a particular crater or bedrock formation with successive deposits of layers in order to obtain a LAPS morphology. One possibility would be to start with a crater that has been asymmetrically eroded. Uneven deposition of ice on a crater is another mechanism that could produce a LAPS by accumulation of the deposits preferentially on one side. However, even if a LAPS production process based on crater filling can be imagined, it does not easily explain the spatial repetitions that are observed [START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF]. The repetition and similar dimensions of the LAPSs as well as their local layout in aligned formations does not support craters as the origin given that crater locations and diameters are basically random.

Erosion processes as a possible origin of LAPSs

- [START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF] performed a complete survey of optical images in order to detect surface signatures of eolian processes over the PLDs. He noticed similarities between the arched shape of LAPSs and barchan dunes formed by deposition of wind-transported sand. Since this study, the availability of an accurate digital elevation model (MOLA) allows us to show that LAPSs are instead deep depressions in the ice. This raises the possibility of LAPS formation by erosion of the ice, which could be driven either by eolian processes or sun sublimation.

The long-term efficiency of the wind loaded with abrasive particles to shape the surface of the ice is well known (e.g. [START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF][START_REF] Koutnik | Eolian controlled modification of the martian south polar layered deposits[END_REF]. Wire brush terrains, snakes, and trailing grooves are especially common eolian formations encountered on Planum Australe. They extend for hundreds of kilometers and are associated with a decameter vertical scale [START_REF] Koutnik | Eolian controlled modification of the martian south polar layered deposits[END_REF]. The objects thus formed have a vertical / horizontal ratio several orders of magnitude smaller than the typical dimensions of LAPSs. However, the relatively uniform orientation of the LAPSs is consistent with an eolian origin, except for the quasi-perpendicular LAPSs found in a single region as shown in Fig 2 .   At the North Pole, Planum Boreum exhibits numerous spiral troughs in its margin. Their asymmetric cross-sections are quite similar to those of the LAPSs. Although their precise origin

is not yet well understood, northern troughs are thought to be sustained by a process involving solar sublimation of the exposed slope together with deposition of water vapor on the pole-facing slope, leading to an asymmetric outcrop (e.g. Pelletier, 2004 and references therein). Ice flow and katabic winds could also play a key role in their evolution [START_REF] Fisher | Lineations on the "white" accumulation areas of the residual northern ice cap of Mars: Their relation to the "accublation" and ice flow hypothesis[END_REF][START_REF] Smith | Onset and migration of spiral troughs on Mars revealed by orbital radar[END_REF].

In particular, [START_REF] Smith | Onset and migration of spiral troughs on Mars revealed by orbital radar[END_REF] suggest trough migration to explain the asymmetric stratigraphy of the troughs. They suggest that katabatic winds remove material from the upwind slopes of a trough and carry it to the downwind slope. However, northern troughs have linear shapes up to hundreds of kilometers long, roughly arranged in a spiral leading out from the pole all around the cap, while LAPSs are individually restricted to a 20-km arch shape in the heart of a single region of Planum Australe. In addition, the hillocks often observed along the crests of the LAPS are not observed on the northern troughs. Thus, spatial shape and distribution considerations do not contribute to a common origin of the northern and southern scarps. Furthermore, LAPS orientations are not preferentially sunward and a number of them have their straight slope with outcrops almost parallel to longitudes, meaning that sun sublimation is not likely to have initiated them.

Tectonic scenario

On Earth, a common mode of crustal extension consists in the development of listric normal fault and their associated rollover anticline. Because of their crucial importance for commercial prospecting (hydrocarbon traps in faulted/folded strata), these structures have been studied extensively (e.g. [START_REF] Dula | Geometric-models of listri normal faults and rollover folds[END_REF][START_REF] Poblet | Fault-slip, bed-length and area variations in experimental rollover anticlines over listric normal faults: influence in extension and depth to detachment estimations[END_REF][START_REF] Shelton | Listric normal faults -an illustrated summary[END_REF].

Listric faults can be defined as curved normal faults in which the fault surface is concave upwards, its dip angle decreasing with depth. These faults occur in extension zones where there is a main detachment fracture following a curved path that connects to a horizontal décollement.

The hanging-wall may develop a syn-tectonic rollover anticline classically interpreted as a direct consequence of layer bending during hanging-wall displacement above the listric normal fault (Fig. 9). The rollover geometry is conventionally interpreted as a direct consequence of the gradual bending of the strata of the hanging-wall moving along the listric normal fault. The shift changes with a horizontal component that increases with depth to become parallel to the décollement. It is generally accepted that the latter is a level where fluid pressure is abnormally high or where materials are weak.

The morphology of the LAPSs observed on Mars is very similar to the above listric fault formation on Earth. They show a topographic asymmetry with a steeply dipping scarp facing a gentle flexure that flattens with elevation, producing a convex morphology. The radargram cross-section confirms this similarity by revealing the subsurface layer geometry. The similarity with a terrestrial example is striking (see Fig. 10). The footwall compartment clearly shows horizontal layers that appear to intersect with the escarpment, while the hanging-wall shows a flexural geometry with a gradual downward bending of layers toward the scarp. * Location of Fig. 9 (1 column width). * Location of Fig. 10 (1 column width).

The LAPS morphology on Mars thus appears to correspond to the formations produced around a listric normal fault on earth. The length of the scarp may correspond to the fault slip. The fault has a strong dip angle with respect to the outcrop, which decreases with depth to reach a level that may be the ice/bedrock interface.

Figure 9C shows a more complex evolution of the system with some subsequent faulting and tilting of individual blocks [START_REF] Davison | Listric normal-faults profiles -Calculation using bed-length balance and fault displacement[END_REF][START_REF] Hauge | A critique of techniques for modeling normal-fault and rollover geometries[END_REF][START_REF] Song | Effects of subsidiary faults on the geometric construction of listric normal fault systems[END_REF][START_REF] Williams | The geometry of listric normal faults and deformation in their hangin walls[END_REF]. It is similar to the surface characteristics observed on cross-sections 4a-4a' and 4b-4b'. The presence of this rollover geometry suggests extensive regional tectonic activity related to the slow sliding of Planum Australe on a geological time scale. In some particular situations, the semi-circular topographic scarp may correspond to some pre-existing impact crater rims associated with a circular and normal faulting system. Such structural systems favor the preferential sliding of the UL polar layered region. The implied horizontal displacement would be substantial (several kilometers, depending on the LAPS length) and mainly directed toward the quadrant of the circular histogram Fig. 3.

The rollover hypothesis can be tested by modeling the tectonic process, for instance using the area balance technique conventionally used for the restoration of a balanced cross-section [START_REF] Davison | Listric normal-faults profiles -Calculation using bed-length balance and fault displacement[END_REF][START_REF] Faure | Importance des modèles de failles dans l' équilibrage des coupes en distension[END_REF][START_REF] Gibbs | Balanced cross-section construction from seismic sections in areas of extensional teconics[END_REF]. The principle is based on the concept of plane strain or volume conservation of cross-sectional area. In Fig. 9B, it consists in measuring the area below the regional datum (area A) which is, in the case of volume conservation, equal to the volume displaced (area B). The latter is the product of the horizontal extension and the depth of the décollement (A = B = h × d). In such a model, it is assumed that the hanging-wall deforms by simple shear and that the footwall remains undeformed throughout the extension [START_REF] White | The relationships between the geometry of normal faults and that of the sedimentary layers in their hanging walls[END_REF]. The horizontal displacement (H r in Fig. 11) can be directly inferred from the crosssections (if the latter is parallel to the tectonic strike). Indeed, in the case of a listric fault, the deformation of the hanging-wall must be similar to simple shear deformation. Most of the shear angles used for earth materials are around 60° [START_REF] Dula | Geometric-models of listri normal faults and rollover folds[END_REF]. Compensation to 60° of potential gap implies that the cutoff point of the hanging-wall is projected at 60° with respect to the surface of the listric fault (Fig. 11). The horizontal displacement can be calculated as: H r = H a + h ν /tg(60°). In some case, due to slight curvature, the cutoff point may be difficult to locate. We chose to use a range of possibilities between a minimum position and a maximum position (Fig. 11). Thus, we can deduce a range for the depth of the décollement level which can be compared to the range of the ice/bedrock interface depth inferred by MARSIS. * Location of Fig. 11 (1 column width).

-Using the geometric and topographic data, we built various LAPS models. Table 1 andassociated Fig. 12 gives the results of the 8 profiles tested. For each profile, we calculated the minimum, maximum and average values of the décollement level vs depth of the bedrock interface. From Fig. 12, it appears that the depths of the décollement levels, deduced from the above models, are in good agreement with the regional depths of the bedrock interface, except for profiles 3a-3a' and 4b-4b'. For these last profiles, the depth of décollement is overestimated or underestimated by about 1 km compared to the depth of the bedrock. Of course, these disagreements could be produce by local singularities of the bedrock topography that cannot be detected by Marsis. A byproduct of this test is the horizontal displacement H r . It appears that the listric fault scenario agrees with a substantial (4-5 km on the average) horizontal displacement of the ice, corresponding approximately to 0.5 to 1% of the UL width. These values are consistent with those usually encountered for terrestrial rollover analogs. For instance, the seismic profile presented in Fig. 10 corresponds to a horizontal displacement of about 10 km [START_REF] Zhang | Mechanics of extensional wedges and geometry of normal faults[END_REF]. * Location of Table 1. * Location of Fig. 12 (1 column width).

Discussion and conclusions

We have provided a complete description of particular polar scarps observed in the ice of UL, a large outlying region of Planum Australe. Cross-analyses of topographic data, optical images, and subsurface radar observations were used to derive a morphologic and stratigraphic scheme for these scarps that we refer to as LAPS (Large asymmetric polar scarps) because of their particular cross-section with a trough between a straight and a convex upward slope, along with outcrops on the straight slope only. Although the scarps have a weak slope, they commonly reach more than half the thickness of the PLD made up of 1 to 1.5 km of stacked ice. The LAPSs are numerous and widespread over UL, but they all have similar dimensions.

Surface features on Martian PLDs are usually explained in terms of depositional and/or erosion processes. (e.g. [START_REF] Fisher | Lineations on the "white" accumulation areas of the residual northern ice cap of Mars: Their relation to the "accublation" and ice flow hypothesis[END_REF][START_REF] Howard | The role of eolian processes in forming surface features of the Martian polar layered deposits[END_REF][START_REF] Koutnik | Eolian controlled modification of the martian south polar layered deposits[END_REF][START_REF] Smith | Onset and migration of spiral troughs on Mars revealed by orbital radar[END_REF].

As an alternative process, we suggest a mechanical failure of the ice for the LAPSs origin, based on their striking similarities in shape and size with rollover anticlines above listric normal faults.

A quantitative model, based on area balance technique, has been briefly introduced to roughly test this hypothesis by computing some dimensions of a few basic geometries [START_REF] Davison | Listric normal-faults profiles -Calculation using bed-length balance and fault displacement[END_REF][START_REF] Faure | Importance des modèles de failles dans l' équilibrage des coupes en distension[END_REF][START_REF] Gibbs | Balanced cross-section construction from seismic sections in areas of extensional teconics[END_REF]). It appears that the expected depth of the décollement agrees with the bedrock depth inferred by Marsis for most of the LAPSs. It would be of interest to extend this method in the future to all the LAPSs. This system is also associated with horizontal displacement, computed to be 4-5 km on average for the LAPSs. This result agrees with what is usually observed for terrestrial rollover analogs, and would imply a substantial global outward sliding of the PLD. Such sliding has already been suggested by [START_REF] Murray | Preliminary geological assessment of the northern edge of ultimi lobe, Mars south polar layered deposits[END_REF] based on fractures observed along with undeformed layers outcropping from Planum Australe margins. Such peripheral fractures could be damping structures resulting from an outward sliding from the heart of UL. The sliding of an ice sheet is a process requiring a weak basal shear stress. Such conditions could be met in case of (i) incompetent basal sediments, (ii) basal sediments softened by melt-water, (iii) over-loaded salt acting as lubricant [START_REF] Jackson | STRUCTURAL DYNAMICS OF SALT SYSTEMS[END_REF], or (iv) a lower yield stress of Martian ice compared to Earth ice [START_REF] Banks | Forward modeling of ice topography on Mars to infer basal shear stress conditions[END_REF]. No information is available to determine whether such sliding could occur at present day or not.

Why the PLD exhibits an outward motion is also a major question resulting from the sliding hypothesis. Possible explanations include (i) an active tectonic displacement of the underlying bedrock, (ii) a simple gravitational readjustment of the accumulated ice stack, or (iii) a fluid basal-material naturally dragged up and drained outward by the weight of the glacier.

The link between the tectonic hypothesis and the hillocks often observed along the LAPS crests remains to be determined. The hillocks are possibly related to (i) a post-deposition of icy materials, in which case the LAPS topography, under the dominant wind-stream, would generate depressions and cold traps along the crests, (ii) a slight elastic rebound of the footwall after the breaking of the ice, as is commonly observed for Terrestrial tectonic phenomenon (e.g. [START_REF] Weissel | Flexural uplift of the rift flanks due to mechanical unloading of the lithosphere during extension[END_REF][START_REF] Westaway | Analysis of tilting near normal faults using calculus of variationsimplications for upper crustal stress and rheology[END_REF] and could be in our Martian case a natural consequence of listric faulting, the magnitude of which holds information on the ice rheology, (ii) a rheological response to a change in basal conditions as observed around subglacial lakes on Earth [START_REF] Remy | Ice flow physical processes derived from the ERS-1 high-resolution map of the Antarctica and Greenland ice sheets[END_REF].

A deeper study of possible tectonic activity in UL should be undertaken because of its implications on the ice rheology and PLD basal conditions. In particular, it could be a macroexpression of fundamental differences between ice-sheet behavior under Martian and Terrestrial conditions, given that the viscosity of ice on Earth would not be expected to generate such faulting. [START_REF] Plaut | Subsurface Radar Sounding of the South Polar Layered Deposits of Mars[END_REF], taking into account the ±200 m uncertainty. Note the typical asymmetric cross-section, perpendicular to the spatial arch shape, with a straight slope facing a convex slope. The general behavior of the polar layers deduced from the stratigraphic study was added to create a typical model of a LAPS cross-section. The outcrops on the straight slope are outlined, while the downward bending of the subsurface layers is visible below the convex slope. The question mark beneath the LAPS is a reminder that no radar data regarding the subsurface structure are available at the corresponding location (see section 4.2 for details).

Hillocks are also present.

  shows this regional orientation in a circular histogram. The whole set of LAPSs appears to be essentially directed towards the same quadrant between 0° and 90°. * Location of Fig. 2 (no particular restriction on the size).* Location of Fig.3(no particular restriction on the size).

Figure captions Fig. 1 .

 captions1 Figure captions

Fig. 2 .

 2 Fig. 2. Arch-shaped features are indicated in bold red. The arches marked by X' s are LAPSs that

Fig. 3 .

 3 Fig. 3. (Top) Histogram of the LAPS azimuths. The Y-axis is the number of occurrences. A

Fig. 4 .

 4 Fig. 4. 3D view of a LAPS (located at the lower right corner of Fig. 5A). Illumination is from the

Fig. 5 .

 5 Fig. 5. Five enlargements (A-E) for the corresponding regions indicated in Fig.1. The background

Fig. 6 .

 6 Fig. 6. Cross-sections of some LAPSs. The cross-section locations are indicated in Fig. 5. Black

Fig. 7 .

 7 Fig. 7. Cross-sections (5-5' and 6-6') of two LAPSs, associated with HRSC pictures (DLR/FU

Fig. 8 .

 8 Fig. 8. Cross-section (7-7') of two successive LAPSs, associated with the HiRISE picture

Fig. 9 .

 9 Fig. 9. Sketches illustrating the typical geometric evolution of hanging-wall deformation in a

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Seismic profile of a rollover fold associated with a listric fault in the Bohai Gulf of

Fig. 12 .

 12 Fig. 12. Graphic illustration of the minimum, maximum and average depth of the décollement

Figure 6 Figure

 6 Figure 6

  Figure 12

Table 1 .

 1 Depth of the décollement level determined for 8 LAPS profiles.

	Profile	Min. value (km) H a h v H r	Max. value (km) H a h v H r	Measured area (km 2 )	Depth of décollement (km) Min. Max. slip slip Mean Min. Max. Mean Depth of bedrock (km)
	1aa' 2.31 0.26 2.46 3.23 0.26 3.38	3.67	1.49 1.09 1.29 0.90 1.30 1.10
	1bb' 2.98 0.28 3.14 3.70 0.27 3.86	1.87	0.60 0.48 0.54 0.20 0.60 0.40
	2aa' 3.00 0.43 3.25 4.10 0.45 4.36	4.37	1.35 1.00 1.17 0.70 1.11 0.91
	2bb' 3.20 0.35 3.40 3.80 0.34 4.00	2.75	0.81 0.69 0.75 0.70 1.11 0.91
	3aa' 2.10 0.19 2.21 3.70 0.22 3.83	2.29	1.04 0.60 0.82 1.08 1.50 1.29
	3bb' 5.70 0.40 5.93 8.80 0.40 9.03	5.82	0.98 0.64 0.81 0.36 0.76 0.56
	4aa' 4.40 0.58 4.74 5.90 0.57 6.23	5.33	1.12 0.86 0.99 0.65 1.05 0.85
	4bb' 3.80 0.55 4.12 6.90 0.55 7.22	8.11	1.97 1.12 1.55 0.67 1.04 0.86

* Location of Fig.6(as wide as possible).

Acknowledgements

The Shallow Radar (SHARAD) was provided by the Italian Space Agency and operated by the InfoCom Department, University of Rome ''La Sapienza''. Thales Alenia Space Italia is the prime contractor for SHARAD and is in charge of in-flight instruments and the SHARAD Operations Center. The Mars Reconnaissance Orbiter mission is managed by the Jet Propulsion Laboratory, California Institute of Technology, for the NASA Science Mission Directorate, Washington, DC. Lockheed Martin Space Systems, Denver, Colorado, is the prime contractor of the orbiter. The authors are grateful to the European space agency (ESA) and the French space agency (CNES) for supporting this work. We thank Pierre Beck, Harvey Harder and two anonymous reviewers for their careful reviews that greatly improved this paper.


Lamri Nehaoua

Hichem Arioui

Said Mammar

Motorcycle Riding Simulator: How to Estimate Robustly the Rider's Action?

This paper deals with a motorcycle riding simulator and addresses two key issues: 1) reconstruction of rider's action, considered as the main input to the simulation system and 2) design an appropriate force feedback, on the handlebar, emulating a tire-road contact.

To answer the first challenge, a Walcott-Zak based sliding-mode observer is designed for the rider torque estimation. The reconstructed torque is used as the main control of the virtual motorcycle dynamic model, in order to actuate the simulator's platform. The steering system is modeled as a haptic display subjected to a couple of action-reaction torques: rider and tire-road dynamics.

Besides, a torque feedback is implemented to compensate the lack of the real tire-road contact.

The control approach is based on a robust tracking problem of a reference steering angle by using H ∞ optimization technique.

I. INTRODUCTION

Motorcycle riders are considered as the most vulnerable and risky road users. Indeed, the incidence of motorcycle fatalities is still increasing, during last decades. In spite of the preventive measurements, as well as, recently repressive installations (radar enforcement of speed limit) and laws, the number of motorcyclists killed did not drop whereas considerable progresses in term of security (passive and active one) system were observed for the others car categories.

Passive and active security systems, such as ABS, ESP, safety belts and airbags, represent a major advance in term of driver safety. Most new cars are endowed with these safety systems that begin to emerge on two-wheeled vehicles. If it is clear that these security systems have a

proven efficiency, no information or studies exist about their real exploitation by the road users (riders and drivers). Consequently, the well control of assistance systems necessarily passes by users' knowledge and training. Moreover, number of accidents is a direct consequence of the lack of knowledge about risks incurred in some riding situations. Thus, it appears leading to develop a training/simulation tool.

Nowadays, driving simulators seem to be a perfect tool to answer crucial challenges such as: training of new drivers, risk awareness, road safety, etc. Their use becomes fundamental thanks to the possibility of carrying out representative experiments under reconfigurable conditions and scenarios (weather, infrastructure: urban/highway, etc.) Nevertheless, designing a simulator requires to make compromises between the quality of the reproduced motions and the overall cost of the mechatronics architecture. If designing a vehicle simulator is a very active research field since several years [START_REF] Chen | NADS at the University of Iowa: A tool for driving safety research[END_REF], [START_REF] Nehaoua | Design and control of a small-clearance driving simulator[END_REF], [START_REF] Arioui | From design to experiments of a 2-dof vehicle driving simulator[END_REF], the simulation of motorcycle riding remains stammering with few prototypes in the world [START_REF] Miyamaru | Development of motorcycle riding simulator[END_REF], [START_REF] Cossalter | Development and validation of a motorcycle riding simulator[END_REF], [START_REF] Arioui | Mechatronics, design, and modeling of a motorcycle riding simulator[END_REF]. The question of the immersion (the quality of the reproduced motions) is even more complex for a two-wheeled vehicle as well as of the balance (lean motion).

Generally, a driving simulator is built-up around two main components: a mechanical architecture and simulation software. This later consists of several computer units including the visual and the traffic environment generator, the control of mechanical platform and the virtual vehicle represented by its dynamic model. The vehicle dynamic model allows the computation of the vehicle states according to the simulator's user actions to update the visual environment and to actuate the mechanical platform. Consequently, the vehicle dynamic model is one of the most important components of a driving simulator.

To simulate a dynamic model, the vehicle steering angle is commonly used in automotive analysis and control. This fact makes possible to simulate the vehicle dynamics without any knowledge of the driver torque applied on the steering-wheel. Indeed, the driver's torque and the corresponding steering angle can be modeled by a minimal phase transfer function even by a simple proportional factor [START_REF] Sentouh | A human-centred approach of steering control modelling[END_REF] because of the vehicle chassis and the steering system dynamics are completely decoupled. However, this assumption cannot be transcribed for two-wheeled vehicles for many reasons such: counter-steer phenomena, direct link between steering system and the front wheel and the effect of the roll motion. Hence, the motorcycle is primarily controlled by the applied rider's steering torque. Consequently, if the torque sensor is not available (cost August 26, 2012 DRAFT reason), an estimation procedure must be carried out to reconstruct this input. To the author's knowledge, contributions of this work have never been addressed before.

The problem of designing observers for linear systems with unknown inputs has been extensively studied in the literature [START_REF] Darouach | On the novel approach to the design of unknown input observers[END_REF], [START_REF] Luenberger | An introduction to observers[END_REF]. Among them, the sliding mode observers have been used for their robustness, insensitivity to the matched unknown inputs and finite/exact time convergence [START_REF] Kalsi | sliding mode observers for systems with unknown inputs: a hign-gain approach[END_REF]. To construct such observer, two necessary and sufficient conditions should be satisfied: 1) the plant model must be a minimal phase system and 2) the so-called observer matching condition must be satisfied. However, the second condition is not guaranteed for a large range of mechanical systems. In this way, a great effort is invested to deal with the observer matching condition limitation and sliding mode observers have been extended to a large wide of physical systems [START_REF] Floquet | A canonical form for the design of unknown input sliding mode observers[END_REF]. In [START_REF] Floquet | on sliding mode observers for systems with unknown inputs[END_REF], additional output signals are generated from the measured outputs by using sliding mode differentiator, and the resulting augmented system satisfies well the observer matching condition.

In this paper, we discuss the possibility of using unknown input sliding mode observer for the estimation of the rider's torque applied on the simulator's handlebar. Since the system does not satisfy the observer matching condition, additional outputs are generated using high order sliding mode exact differentiator.

This paper is organized as follows: section II is dedicated to the simulator and the torque feedback system description. Also, in this section the identification process of the steering system used as the new interface between the rider and the virtual vehicle is presented. Section III highlights the design procedure of the rider's torque estimation. Section (IV) deals with the torque feedback control approach. Finally, results, conclusion and appendixes complete the paper.

II. RIDING SIMULATOR AND TORQUE FEEDBACK SYSTEM

A. Motorcycle riding simulator

Within the framework of the French SIMACOM research project (2005-2009), a riding motorcycle simulator is the result of a collaboration between the INRETS 1 and IBISC 2 -UEVE 3 laboratories in France. • as a training tool for new riders with different scenarios: normal traffic environment, dangerous riding situations (avoidance, emergency braking, nearly failing or slipping situations, etc.)

• to carry out riders behaviors studies in such situations.

First investigations have led to 5 degrees of freedom (DoF) mechanical platform including a double haptic feedback on the handlebar and three basic motions which consist of pitch, roll and yaw rotations.

Figure [START_REF] Chen | NADS at the University of Iowa: A tool for driving safety research[END_REF] presents a picture of the achieved riding simulator. More details concerning the mechatronics aspects can be found in [START_REF] Nehaoua | Design and modeling of a new motorcycle riding simulator[END_REF], [START_REF] Arioui | Mechatronics, design, and modeling of a motorcycle riding simulator[END_REF].

August 26, 2012 DRAFT

B. Two-wheeled vehicle dynamics

In order to actuate the simulator's platform a vehicle dynamic model must be introduced. This model aims to predict the behavior of a virtual motorcycle according to rider's actions on the platform.

Many works have been addressed the modeling of single track vehicles. The first study, reported by Whipple, goes back to 1899 and relates to the bicycle's stability and balance [START_REF] Whipple | Stability of the motion of a bicycle[END_REF].

In 1971, a reference study was undertaken by Sharp [START_REF] Sharp | The stability and control of motorcycles[END_REF]. It examined the motorcycle's stability and its sensitivity to the variations of the geometrical and tire's parameters. More recently, the team of Cossalter at the University of Padova (Italy) has been invested for a several years on the development and the experimental validation of dynamic models intended for competition motorbike. The FastBike project presents a nonlinear model of the two-wheeled vehicle described as a set of 6 bodies, allowing the simulation of 11 DoF [START_REF] Cossalter | A motorcycle multibody model for real time simulation based on the natural coordinates approach[END_REF].

G r G f v x v y F yf F yr δ, τ r φ ψ τ d -τ b ǫ ǫ Fig. 2

. Geometrical representation of the Sharp's motorcycle model

In general, two dynamical characteristics are considered: in-plane and out-of-plane modes.

In-plane mode includes longitudinal, vertical and pitch motions whereas the out-of-plane mode represents lateral, yaw, roll and steer motions.

In the present paper, the out-of-plane dynamics is described by Sharp's 1971 model. The motorcycle is represented as a set of two rigid bodies linked through a steering mechanism August 26, 2012 DRAFT (Figure 2). The first body G f is the front solid composed by the front wheel, the fork and the handlebar display. The second body G r is the rear solid including the rear wheel, the engine, the tank and the seat. Here, the rider is supposed to be rigidly attached to the rear body frame and his relative motion is not taken into account. Equations of motion are expressed by the following state-space representation:

ẋv = A v x v + B v τ r (1)
where

x v = [v y , ψ, φ, δ, ϕ, δ, F yr , F yf ]
T is the state vector and τ r is the rider torque applied on the handlebar. Matrices A v and B v are variables and function of the longitudinal velocity v x , which is governed by :

vx = 1 m e τ d -τ b r D + m ψv y + F wx (2)
Remark 2.1: See the appendix for states and variables definitions.

C. Steering system description and modeling

For this platform, the motorcycle front body dynamics, including the wheel and the fork system, is substituted by a DC motor. The new resulting configuration, handlebar and motor, is considered as a new "connection" between the rider and the "virtual" motorcycle (i.e. dynamic model). Then, the DC motor ensures efforts feedback resulting from the road-tire interaction to the rider. At the same time, this device performs a significant role in estimating the action of the rider.

August 26, 2012 DRAFT The implemented new display includes several parts (Figure 3), described as:

• an original handlebar of a Yamaha 125 cm 3 motorcycle.

• the handlebar is attached to the DC motor axis through a double pulley-belt to increase the produced torque.

• an optical encoder for the measurement of the steer angle. This sensor is mounted in parallel axes through a pulley-belt system with respect to the DC motor shaft axis to improve the resolution of the rotation measure.

Since the electrical dynamics of the DC motor is very fast comparing to the mechanical one, it will be not considered below. Then, the equation describing the mechanical behavior of the DC motor is given by:

J m θm + β v,m θm = τ m -τ cm (3)
where, θ m is the rotor axis rotation, τ m is the delivered motor torque and τ cm is the load August 26, 2012 DRAFT torque. Moreover, the dynamics of the handlebar system can be expressed by:

J h θh + β v,h θh + β s sign( θh ) = τ r + N τ cm (4)
Where θ h is the handlebar steer angle and τ r is the rider's torque applied on the handlebar.

Knowing that θ m = N θ h and τ m = k t i, it results that:

J eq θh + β eq θh + β s sign( θh ) = τ r + N k t i (5)
with:

J eq = J h + N 2 J m β eq = β v,h + N 2 β v,m
Equation ( 5) describes the dynamic behavior of the new rider-motorcycle interface. J m , β v,m , k t and N are known parameters.

Remark 2.2:

When modeling a DC motor, the electrical dynamics is considered as follows:

L di dt + R i = u -k e θh (6)
where L and R are the motor's armature induction and resistance, k e is the motor's voltage constant. Nevertheless, since the motor's armature induction is very low, the current dynamics di/dt is neglected, and it is supposed that the motor is driven by the current signal i. So, the transformation of current signal to an equivalent voltage signal is given by:

u = Ri + k e θh (7)

D. Parametric Identification

Identification of the simulator's handlebar parameters is of crucial importance. In fact, misidentification of inertial parameters leads to a lack in state observation and control laws with degraded performances. In this section, we aim to estimate the handlebar's inertia J h and friction parameters β v,h and β s . In the considered identification process, the DC motor is driven by a

voltage signal in open-loop (without state feedback) and the rider torque is set to zero (τ r =0). In August 26, 2012 DRAFT this case, only the motor current i and position angle θ h are measured while the angular velocity θh is obtained by differentiation.

First, equation ( 5) must be rewritten in a linear form with respect to parameters to be estimated such as:

Φp = N k t i (8)
where p = [J eq , β eq , β s ] T is the vector containing all parameters to be identified and Φ =

[ θh , θh , sign( θh )].

Several methods were developed in the literature for the parametric identification [START_REF] Landau | Identification et commande des systèmes[END_REF], [START_REF] Aström | Adaptive control[END_REF].

We choose the adaptive gradient method because of its simplicity in off-line or on-line implementation [START_REF] Craig | Adaptive Control of Mechanical Manipulators[END_REF]. This method consists of optimizing a quadratic cost function

C f = 1 2 (τ ref -τ ) 2
, where τ ref = N k t i and τ = Φp. Adaptation law is governed by:

ṗ = -KΦ T (N k t i -Φp) (9)
where K is the adaptation matrix coefficients adjusted to ensure rapid convergence and also tied to the different excitation trajectories (slow or rapid reference trajectories). Finally, the different parameters are obtained by integrating ṗ (equation 9). 4Hz] and a bipolar rectangular signal, are used to actuate the handlebar torque feedback system.

The resulting handlebar position θ h is then differentiated to get θh , the motor current is measured and with equation ( 7), the voltage signal u is reconstructed. In this section, we present the reconstruction of the rider's action based on sliding mode observer approach ( [START_REF] Walcott | Combined observer-controller synthesis for uncertain dynamical systems with applications[END_REF]).

The state-space representation of the simulator's handlebar system is derived from equation [START_REF] Cossalter | Development and validation of a motorcycle riding simulator[END_REF] and can be expressed by:

ẋ = Ax + Bu + D(τ r + τ p ) y = θ h = Cx (10)
where, x = [ θh , θ h ] T is the state vector, u = k t N i is the control input, τ r is the unknown driver's torque to be estimated and τ p = -β s sign( θh ) is a known external bounded perturbation induced by the dry friction of the handlebar axis. In equation [START_REF] Kalsi | sliding mode observers for systems with unknown inputs: a hign-gain approach[END_REF], D is of full column rank and C is of full row rank. The unknown input τ r , is obviously bounded, hence, τ r ≤ ρ where ρ > 0.

Application of the sliding mode observer [START_REF] Walcott | Combined observer-controller synthesis for uncertain dynamical systems with applications[END_REF] to the system (10) yields to: August 26, 2012 DRAFT

ẋ = Ax + Bu + L(y -ŷ) -DE(ŷ, y, η) (11)
with ŷ = C x is the estimated output and:

E(ŷ, y, η) = η F (y -ŷ) F (y -ŷ) 2 + ǫ ( 12
)
is a nonlinear injection term where ǫ is a small parameter used to reduce chattering and η is a design parameter.

The term L(y -ŷ) is used to improve the convergence of the observer. The existence of the observer gain L is ensured by the observability/detectability of (A, C) where L is defined such that the matrix (A -LC) is Hurwitz and has a prescribed eigenvalues in the open left-half plane.

In this case, for any Q symmetric definite positive, there is a unique solution P symmetric definite positive, such that:

(A -LC) T P + P(A -LC) + Q = 0 (13)
and P satisfies, for some matrix F, the following condition:

PD = C T F T (14)
The previous observer is insensitive to the external perturbation if and only if the well-known matching condition rank(D) = rank(CD) is satisfied. However, for system [START_REF] Kalsi | sliding mode observers for systems with unknown inputs: a hign-gain approach[END_REF], it is easy to check that rank(D) = rank(CD), which means that the observer matching condition is not fullfiled. Nevertheless, according to [START_REF] Floquet | on sliding mode observers for systems with unknown inputs[END_REF], the sliding mode observer method can be extended for systems where the matching condition is not satisfied by generating additional output signals.

For the present system, this means that we need the information of the motor's position rate (Figure 6). Consequently, the sliding mode observer ( 11) is re-defined as following:

ẋ = Ax + Bu + L(y a -ŷa ) -DE(ŷ a , y a , η) (15)
where y a = [θ h , θh ] T and ŷa = C a x. The generated additional output θh is obtained by differentiating the measured position using a robust super-twisting exact differentiator [START_REF] Levant | Robust exact differentation via sliding mode technique[END_REF]. As reported by Walcott and Zak, the error's dynamics of the observer is asymptotically stable for η ≥ ρ. This means that after a finite time convergence of the differentiator, we have y a = C a x.

Thus, the observer's error can be defined by:

ė = (A -LC a )e + Dτ r + DE(ŷ a , y a , η) (16)
and after a finite time, the state estimation error e = x -x converge to the sliding surface {S : FC a e = 0}. According to equations 14 and 15, the unknown input is estimated by:

D T PD {τ r + E(ŷ a , y a , η)} = 0 → τ r = -E(ŷ a , y a , η) (17)
Remark 3.1: The main advantage of Walcott-Zak observer is its simplicity of use and well suited implementation especially for low order systems.

Remark 3.2:

The main disadvantage of Walcott-Zak observer is the difficulty of finding a solution P which satisfies simultaneously equations ( 13) and ( 14). For a low order system, the problem of finding the matrix P can be done by hand calculation. Otherwise, a method proposed in [START_REF] Hui | Observer design for systems with unknown inputs[END_REF] can be applied.

Remark 3.3: The driver's torque is, by definition, a bounded input. Here, we consider a torque's peak value about 70N.m. Furthermore, a smooth approximation of sign(.) by tanh(.)

is assumed [START_REF] Yao | Robust adaptive nonlinear control with guaranteed transient performance[END_REF].

August 26, 2012 DRAFT

IV. TORQUE FEEDBACK CONTROL

The tire-road interaction and the gyroscopic effect generate a torque along the steering system.

So, when driving a real motorcycle, a rider action is applied on the vehicle handlebar to counter the generated torque and hence drive the motorcycle toward an equilibrium trajectory.

On a riding simulator, the lack of tire-road interaction and gyroscopic effect makes the simulation of a given maneuver impossible. The visual environment information alone is not sufficient for a simulator's user to drive a virtual motorcycle, so, it is essential to implement a torque feedback (figure 7). In this paper, the torque feedback is implemented as a robust tracking problem in order to follow a reference steering angle δ generated by the motorcycle dynamic model. As shown in the figure 7, when a driver torque τ r is applied on the simulator's handlebar, it is immediately reconstructed by the unknown input observer. Afterward, the estimated torque τr is used as control input to the virtual motorcycle dynamic model. The robust controller is designed to bring the simulator's handlebar position θ h as closely as to its reference value δ despite the driver action. Consequently, one can deduce that this control scheme is equivalent to a rider's torque rejection and from the motorcycle equilibrium it can be deduced that the torque delivered by the motor is the same as that generated by tire-road interaction, gyroscopic effects and other dynamic phenomenon. August 26, 2012 DRAFT This control approach is more efficient since it avoids recompiling the resulted torque (requiring rate and acceleration information) and also avoiding algebraic loops which may cause instabilities. Furthermore, high performances (in term of tracking) are not required since the objective is to restitute admissible torque and the rider adjusts his trajectory, according to the visual rendering, by applying more or less effort.

Reconsider the state-space representation of the simulator's handlebar system described by equation ( 10) as following:

ẋ = Ax + Bi + Dw y = θ h = Cx ( 18
)
where w = τ r + τ p . In transfer function representation, the system output is given by:

y = θ h = G h i + G w w (19)
Under parameter uncertainties, the previous output are rewritten as following:

θ h = G h,∆ i + G w,∆ w (20)
where, G h,∆ , G w,∆ are the perturbed transfer functions. In the following, a feedback loopshaping design procedure is adopted (Figure 8). In this procedure, the parameter uncertainties are represented as a perturbation on the normalized left coprime factor of the system model [START_REF] Glover | Robust stabilization of normalized coprime factor plant descriptions with H∞ bounded uncertainty[END_REF], [START_REF] Mcfarlane | A loop-shaping design procedure using H∞ synthesis[END_REF]. Consequently, the perturbed plant G h,∆ is given by:

G h,∆ = (M -∆ M ) -1 (N h + ∆ N h ) (21)
G w,∆ = (M -∆ M ) -1 N w where [M, N h ] ∈ H ∞ are the left coprime factors, ∆ M , ∆ N h ∈ H ∞ are unknown modeling
perturbations. By replacing [START_REF] Levant | Robust exact differentation via sliding mode technique[END_REF] in [START_REF] Walcott | Combined observer-controller synthesis for uncertain dynamical systems with applications[END_REF], we obtain:

M θ h = N h i + N w w + φ (22) August 26, 2012 DRAFT ¡ ¢ £ ¤ ¥ £ ¦ § ¨© ¢
Fig. 8. Closed-loop system for the normalized coprime factor model in which φ = ∆ N h i + ∆ M θ h . The aim here is to design a controller i = Kθ h that stabilizes the closed-loop system against the external perturbation w and unstructured parameter uncertainties φ. This is achieved by minimizing H ∞ norm of the transfer T [φ,w] T →[θ,i] T such that:

T [φ,w] T →[θ,i] T < γ (23)
where T [φ,w]→[θ,i] is given by:

T [φ,w]→[θ,i] =   I K   (I -G h K) -1 M -1 G w (24)
Finally, finding K which minimizes γ such that T [φ,w]→[θ,i] ∞ < γ is equivalent to the problem of minimizing the four blocs H ∞ one [START_REF] Green | Linear Robust Control[END_REF] given by:

T [φ,w]→[θ,i] =   I K   (I -G s K) -1 I G s G w (25)
where, G s = W 2 G h W 1 is the shaped version of G h . By using Hifoo toolbox [START_REF] Gumussoy | Multiobjective robust control with hifoo 2.0[END_REF], an optimal H ∞ reduced-order controller with performance level γ * = 2.5587 is obtained as the following:

K(s) = -2.355s -613.7 s + 1445 (26)
August 26, 2012 DRAFT

V. SIMULATION RESULTS AND EXPERIMENTAL TESTS

This part aims to point-out the effectiveness of the overall architecture of figure [START_REF] Sentouh | A human-centred approach of steering control modelling[END_REF], including Walcott-Zak unknown input observer, exact differentiator and the robust torque feedback. We discuss the reconstruction of the applied rider's torque results and the disturbance rejection for the steering angle tracking.

A. Simulation

We consider here a square torque signal as depicted in (figure 10). This signal will be referred as the reference steering torque. In the first simulation, the dry friction effect is neglected. Moreover, the simulation is carried out for the nominal system and for an uncertainty of 50% on motor's parameters (inertia and viscous damping). Figure [START_REF] Floquet | A canonical form for the design of unknown input sliding mode observers[END_REF] illustrates the reference steering torque versus the estimated one by using the Walcott-Zak observer. One can conclude that the observer ensures an exact convergence, in a finite time, under noiseless signals existence for the nominal system and an asymptotic convergence for the perturbed plant. Besides the unknown input estimation, the present observer can also estimate all system states (position and velocity). Figure [START_REF] Floquet | on sliding mode observers for systems with unknown inputs[END_REF] represents the simulated steering position and its rate using the handlebar model of equation [START_REF] Kalsi | sliding mode observers for systems with unknown inputs: a hign-gain approach[END_REF] and the estimated ones. It can be shown that for state estimation, the used observer is robust against parameter uncertainties and ensures exact convergence in finite time but for noiseless signals.

Finally, figure [START_REF] Nehaoua | Design and modeling of a new motorcycle riding simulator[END_REF] shows both performances of the robust controller and the robust differentiator. The proposed controller allows a robust tracking of a reference handlebar position δ even in the presence of parameter uncertainties. The same remark can be stated for the differentiator, used to generate additional outputs in order to satisfy matching condition. Hence, a compensation of τ p is necessary to extract the torque information which will be used by the motorcycle model. Without compensation, the effect of dry friction may be undesirable especially under low motorcycle speed where the rider's torque to be applied is reduced.

Remark 5.1: It should be mentioned that the robust differentiator convergence is ensured for noiseless input signal. In the case of stochastic noisy signals, a filter must be introduced. Nevertheless, in experimentation, the measured position noise is very limited since this measurement is obtained from a digital encoder and acquired using CAN bus interface.

Remark 5.2: In experimentation, an undesirable rider's hand motion can be occurred even if the rider has no attention to manipulate the simulator's handlebar. This motion can generate a chattering induced by the sign( θh ) function. To avoid this situation, a dead zone of a small width is introduced on the handlebar's velocity.

B. Experimental tests

In this section, we present experiments carried out on the motorcycle riding simulator. According to the visual environment, the rider applies a torque to follow a defined trajectory. Only the handlebar position θ h and the motor current i are measured. In figures [START_REF] Sharp | The stability and control of motorcycles[END_REF][START_REF] Cossalter | A motorcycle multibody model for real time simulation based on the natural coordinates approach[END_REF], one can notice an exact estimation of the system states and an asymptotic convergence of the driver torque with compensation. It is also shown that the robust controller satisfies requirements on the position reference tracking. In figure [START_REF] Landau | Identification et commande des systèmes[END_REF], the position rate as obtained by the robust differentiator is reported. Finally, the effectiveness of the proposed torque feedback method is shown in figure [START_REF] Aström | Adaptive control[END_REF] where the estimated rider's torque is compared to motor delivered torque computed from the measured current (τ m = k t N i). The effectiveness of the proposed torque feedback method is shown in figure [START_REF] Aström | Adaptive control[END_REF] where the estimated rider's torque is compared to motor delivered torque computed from the measured current (τ m = k t N i). The objective of this paper is precisely the reconstruction of unknown inputs, whose rider's torque. To this end, we implement a sliding-mode unknown input observer of Walcott-Zak type. This choice is motivated by the fact that this observer is quite simpler to use and achieves comparable results to other approaches in terms of robustness and convergence.

In this way, a new rider-simulator interface is designed. A DC motor is implemented on the handlebar torque feedback. The different mechanical parameters are identified and a robust approach, based on an optimal H ∞ reduced-order controller, is developed for the feedback torque restitution.

Since, the handlebar plant does not satisfy the matching observer condition, additional outputs are generated by using a robust exact differentiator. It may be noted that, the use of a CAN bus is very useful to limit the influence of noises especially the current measurement one. The performance of the overall architecture is simulated and tested. For our current application (riding simulation), obtained results are very satisfactory especially when the dry friction is compensated.

σ f                    E =                 

Fig. 1 .

 1 Fig. 1. SIMACOM's Riding Simulator Platform (INRETS-UEVE)

Fig. 3 .

 3 Fig. 3. Steering system of the riding simulator

Fig. 4 .

 4 Fig. 4. Applied control voltage (a chirp sine 0.1-4Hz)

Fig. 5 .

 5 Fig. 5. Validation with a bipolar rectangular control voltage

Fig. 6 .

 6 Fig. 6. Diagram of the unknown input sliding mode observer

Fig. 7 .

 7 Fig. 7. Bloc diagram of the virtual motorcycle and the handlebar system

Fig. 9 .

 9 Fig. 9. Open-loop architecture of the riding simulator

Fig. 10 .

 10 Fig. 10. Applied Rider's torque τr

Fig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. Rider's torque estimation for the nominal and perturbed systems

Fig. 14 .

 14 Fig. 14. Compensation of dry friction: (left) impact on rider's torque estimation, (right) impact on reference position generation

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Performances of the robust controller and state estimation

Fig. 17 .Fig. 18 .

 1718 Fig.[START_REF] Landau | Identification et commande des systèmes[END_REF]. Angular velocity θh : differentiated using exact robust differentiator and estimated using Walcott-Zak observer

2 D - 1 τr

 21 wheel spin inertia and dynamic radius me equivalent mass, me = m + J R r rider's torque τ d , τ b driving and braking forces Fwx wind gust Fyr, F yf rear and front side slip force Av state matrix Av = M -1 E Bv input matrix Bv = M -1 [0, 0, 0, 1, 0, 0, 0] T M motorcycle mass matrix E motorcycle generalized effort vector Handlebar system θm, θ h motor and handlebar rotations Jm, βv,m inertia and damping of motor's rotor J h , β v,h , βs inertia, damping and dry friction of handlebar N pulley transmission ratio τm, i, kt induced torque, current and torque factor of motor first and second derivative of a variable x w.r.t time x estimate of a variable x x T transpose of vector or matrix x

1 0

 1 b 22 b 23 b 24 0 0 -b -l 0 b 32 0 b 34 b 35 b 36 0 0 0 b 42 b 43 C δ b 45 b 46 0 η Parameters aij (mass matrix M is symmetric positive definite matrix)a11 = m f + mr a12 = m f k a13 = mrh + m f j a14 = m f e a22 = m f k 2 + Irz + I f z cos 2 ǫ + I f x sin 2 ǫ a23 = m f kj -Crxz + (I f z -I f x ) cos ǫ sin ǫ a24 = m f ek + I f z cos ǫ a33 = mrh 2 + m f j 2 + Irx + I f x cos 2 ǫ + I f z sin 2 ǫ a34 = m f ej + I f z sin ǫ a44 = m f e 2 + I f z Parameters b12 = (m f + mr)vx b22 = m f kvx b23 = -(i f y /R f + iRy/Rr)vx b24 = -i f y /R f sin ǫvx b32 = (m f j + mrh + i f y /R f + iRy/Rr)vx b34 = i f y /R f cos ǫvx b35 = -(m f j + mrh)g b36 = Z f ηm f eg b42 = (m f e + i f y /R f sin ǫ)vx b43 = -i f y /R f cos ǫvx b45 = Z f ηm f eg b46 = (Z f ηm f eg) sin ǫ