Irene Frizza

Hiroshi Kaminaga

Ko Ayusawa

Philippe Fraisse

Gentiane Venture

A study on the benefits of using variable stiffness feet for humanoid walking on rough terrains

This work aims to study the effects of the variable stiffness of a compliant foot on humanoid locomotion performance. Through dynamical simulations, we demonstrate that the introduction of the variable stiffness feet, changing in conjunction with the ground roughness significantly improves the walking performance on different types of rough terrain of a humanoid robot. We propose a compliant foot model with multiple viscoelastic elements in the sole. We optimize the sole stiffness for different types of uneven terrains: with rocks, tiles, and obstacles of different shapes and dimensions. We implement a variable stiffness method according to the ground roughness during the walking. Furthermore, the timing of ground scanning and optimal stiffness estimation throw the k-nearest neighbors (KNN) algorithm is described. The comparison of the results obtained with completely flat sole, compliant sole with fixed stiffness, and compliant sole with variable stiffness show the superiority of the variable stiffness feet over the two others. Finally, we present some limitations of the flexible robotic foot in the dynamic simulation.

I. INTRODUCTION

Most robotic feet have a flat and rigid structure and are therefore not deformable [START_REF] Kang | Realization of biped walking on uneven terrain by new foot mechanism capable of detecting ground surface[END_REF]- [START_REF] Hirai | The development of honda humanoid robot[END_REF]. However, maintaining balance on uneven terrain is of paramount importance for robots. To tackle this issue, in recent years, several solutions have been proposed. Some multi-joints feet [START_REF] Piazza | Toward an adaptive foot for natural walking[END_REF]- [START_REF] Narioka | Humanlike ankle-foot complex for a biped robot[END_REF] structures have been developed. Also, some structures incorporate elastic elements such as springs and dampers [START_REF] Seo | Modeling and analysis of a biomimetic foot mechanism[END_REF], [START_REF] Hashimoto | A study of function of foot's medial longitudinal arch using biped humanoid robot[END_REF], [START_REF] Kuehn | Active spine and feet with increased sensing capabilities for walking robots[END_REF]. Others contain a rubber layer under the foot sole to adapt it to the ground and absorb impacts [START_REF] Li | Flexible foot design for a humanoid robot[END_REF], [START_REF] Hirai | The development of honda humanoid robot[END_REF], [START_REF] Davis | The design of an anthropomorphic dexterous humanoid foot[END_REF].

Finding the appropriate viscoelastic properties of such robot feet is an essential process for designing new compliant feet. Finding the optimal values of the viscoelastic properties of the foot given certain rough terrains can guide the design of new robotic feet. However, increasing softness in the foot sole can reduce stability in balancing performance [START_REF] David | Study of an external passive shock-absorbing mechanism for walking robots[END_REF]- [START_REF] Catalano | Hrp-4 walks on soft feet[END_REF]. Indeed, the risk of falling due to oscillations during the standing phase is greater for structures containing multijoints and elastic properties. For this reason, the need for an adequate foot compliance model is very important [START_REF] Minakata | A study of flexible shoe system for biped robot[END_REF].

Introducing soft-behavior elements into dynamic simulation is generally a more difficult simulation problem compared to most robotics simulations which often assume that both the robot and the objects with which it operates are mechanically rigid [START_REF] Collins | A review of physics simulators for robotic applications[END_REF]. Simulators that can also support the contact dynamics between the soft and the rigid elements are 1 CNRS-AIST JRL (Joint Robotics Laboratory), UMI3218/RL, Tsukuba, Ibaraki, Japan.

2 LIRMM, Université de Montpellier, CNRS Montpellier, France. 3 The University of Tokyo, Japan.

Corresponding author: irene.frizza@aist.go.jp

Ankle Roll Joint

Torsional springs and dampers

Passive Revolute Joints Phalanges Since grounds on which robots operate may vary widely, robotic feet with variable stiffness, i.e. compliance changing according to the terrain, may offer a robust solution. Choi et al. [START_REF] Choi | Design of a variable compliant humanoid foot with a new toe mechanism[END_REF] introduced variable compliant humanoid foot design using a leaf spring and rubber balls in series. This promising approach of variable compliance feet is still very rare in the literature for humanoid robots [START_REF] Frizza | Humanoids' feet: State-of-the-art & future directions[END_REF]. For now, the studies of variable stiffness structures are almost entirely limited to the field of prostheses. Semiactive prosthetic devices that adjust their stiffness during swing phases are developed to minimize size, mass, and power consumption [START_REF] Lecomte | Variable stiffness foot design and validation[END_REF]- [START_REF] Tryggvason | Use of dynamic fea for design modification and energy analysis of a variable stiffness prosthetic foot[END_REF]. Here, the deformation of the foot allows the foot to overturn during walking, mimicking the response of a biological ankle [START_REF] Tryggvason | Use of dynamic fea for design modification and energy analysis of a variable stiffness prosthetic foot[END_REF]. In this article, for the first time, we address the problem of a humanoid robot using variable stiffness feet on uneven terrain in a dynamic simulation of walking. In Sec. II, we present the developed foot model with soft contact, its integration in a simulation environment, and the walking control law. In Sec. (III), we detail our contribution regarding the parameters selected for the simulation environment and sole stiffness optimization. Finally, in Sec. IV, we analyze the method to vary the compliance of the sole according to the ground roughness during the walking and optimal stiffness estimation. In Sec. VI, dynamic simulation experiments of walking are described. Simulation experiments are carried out with the humanoid robot HRP-4J [START_REF] Kaneko | Humanoid robot hrp-4-humanoid robotics platform with lightweight and slim body[END_REF].

II. FOOT MODELING WITH SOFT CONTACTS

A. Foot modeling and integration in Mujoco

The ability to perform dynamic simulations quickly and efficiently is necessary to test walking in multiple experimental conditions: environments, terrains, foot parameters. Introducing deformable foot behaviors into a dynamic simulation of the humanoid robot requires a reliable and consistent dynamics simulation environment. Inspired by the robotic foot SoftFoot [START_REF] Piazza | Toward an adaptive foot for natural walking[END_REF], we developed a simplified CAD model in SolidWorks to evaluate the performance mainly in the forward walking direction (showed in Fig. 1(b)).

In order to integrate it into Mujoco, we developed the hierarchical xml model of the simplified foot model. Here, we aim to replicate the three main characteristics of SoftFoot:

1) The deformable plantar fascia: the plantar arch consists of ten cuboid-shaped blocks representing the ten phalanges of the real foot. Each block replicates the same longitudinal and lateral dimensions of the real foot. The blocks are connected with revolute joints and they can rotate along the transverse axis. In order to provide visco-elasticity properties to the plantar arc of the foot, we added torsional springs and damping between the ten blocks of the plantar arc into dynamic environment (Fig. 1(a)). The visco-elasticity parameters of the foot arch are decisive to obtain the walk on uneven ground. In fact, as confirmed by the experimental results obtained in [START_REF] Catalano | Hrp-4 walks on soft feet[END_REF], a too rigid plantar fascia greatly limits the success of walking on obstacles. Therefore, it is important to properly tune the value of these parameters.

2) The completely passive system: the plantar arch is connected to the ankle by two arches: frontal arch and backward arch. The backward arc is rigidly attached to the ankle platform and it is connected with the front arch through a passive joint, allowing the rotation of the front arc along the pitch axis. The backward arc is connected with the foot sole throw a passive joint. Finally, all the revolute joints connecting the phalanges of the sole are passive, making the foot completely passive (Fig. 1(a)).

3) The close kinematic chain: the frontal arch is connected to the sole with a passive joint in order to close the kinematic chain of the foot. Indeed, there is a geometrical constraint between the frontal arch and the third phalanx of the sole. Thanks to this geometrical constraint, the two bodies are anchored to each other and they can rotate around the pitch axis.

B. Soft Contact Models

Contact between phalanges In the simulation with compliant feet, the contact between two phalanges consists of a virtual spring-damper system, with spring constant k > 0 and damper constant b > 0. The spring and damper constants are the same for all the phalanges. When the foot is in a flat position, the angles between the phalanges are null and the equilibrium position of the springs is reached (Fig. 1(a)). Springs and dampers create passive forces along pitch axis of the foot between the phalanges. As result, the sole is compliant along the walking direction.

Contact between frontal arch and ankle

The frontal arch is free to rotate around the rotational axis of the joint between frontal and backward arc. The contact between the frontal arch and the ankle is modeled, as in the case of the phalanges of the sole, with a springdamper system. Here, the values of the stiffness and damping constants are fixed to k = 100Nm/rad and b = 80Nms/rad.

Contact between frontal arc and sole

As explained, we added a constraint between the frontal arch and sole to close the kinematic chain of the foot in the hierarchical model. The dynamic contact between the frontal arch and the foot sole is modeled as follow:

a = (1 -d)a 0 -dbv -dkr (1)
where a, v and r are respectively the joints acceleration, velocity and position. a 0 is the unforced acceleration and d the impedance. In particular, we defined d position dependent d(r) in a range between 0.9rad/Nms 2 and 0.95rad/Nms 2 . In this way, we have quite hard constraint because values of d(r) close to 0 correspond to soft constraints while values of d(r) close to 1 correspond to hard constraints. Moreover, we have chosen to use not constant value but position dependent value to take the advantage of interpolating property to avoid numerical instability. Thanks to the first term on the right-hand side, the solver interpolate between the unforced acceleration a 0 and the reference acceleration a r = -bv-kr.

The stiffness and damping values are defined as follow:

b = 2 d max t ; k = d(r) d 2 max t 2 ζ (2)
where d max = 0.95rad/Nms 2 , t is the time constant and ζ is the damping ratio. We set t = 2ms and ζ = 1. ζ = 1 corresponds to critical damping. In fact, using smaller values results in under-damped constraints, while using larger values results in excessively damped constraints.

Contact between sole and ground

The dynamic contact between the foot sole and the ground is modeled as in [START_REF] Kang | Realization of biped walking on uneven terrain by new foot mechanism capable of detecting ground surface[END_REF]. Here, d is set to 0.99 in order to reduce the slip effect during walking. Increasing the constraint impedance can be particularly effective to avoid slip behaviour between the sole and the ground. For this contact equation, t = 0.1ms and ζ = 1 are set.

C. Controller in simulation: LIPM walking stabilizer

In the dynamic simulations, we used the dynamic simulator Mujoco, the interface for simulation and robot control systems mc rtc [START_REF]mcrtc website[END_REF] and the controller already established LIPM walking stabilizer [START_REF] Caron | Stair climbing stabilization of the hrp-4 humanoid robot using whole-body admittance control[END_REF]. The robot is modelled with a linear inverted pendulum to generate the center of mass (CoM) acceleration for mc rtc. The walking controller [START_REF]Lipm walking stabilizer[END_REF] used is a position-controller with two main components for the stabilization: the DCM (Divergent Component of Motion) feedback control, which modifies the CoM acceleration based on the DCM estimation and computes the desired contact wrenches to compensate deviation from the walking pattern; and the Whole-body Admittance Control, which allows the robot to realize the desired contact wrench computing commanded velocity and acceleration of the joints. The feedback law expressed in terms of the ZMP is:

pc = pd c + ω(k P DCM + CoM K CoM )+ ω(k I DCM ) + ω(k D ˙ DCM ) -ω 2 k zmp ŻM P d (3)
where p c is the CoM position, p d c is the desired CoM position, k P , k I , k D are the proportional, derivative and integral constants of the control law, ω = g/CoM and DCM and CoM are the DCM and CoM errors. In our context, we define K I = 0 because we are using DCM bias estimator. It's purpose is to reduce the average error of DCM to zero. As the integral term of the control law uses the same quantity, we decided to set it to zero.

III. PARAMETERS OPTIMIZATION

A. Parameters selection

After creating the environment with compliant feet for the dynamic simulation, our goal is to select the most appropriate combination of parameters to improve the walking on uneven terrains and obstacles of different shapes and dimensions. In particular, the parameters we analyzed are the proportional and derivative coefficients of the LIPM walking stabilizer (k P and k D in (3)) and the stiffness coefficient of the springdamper system between the phalanges of the foot (k in (2)). The reason why we do not consider the damping constant in the parameter set is that analyzing the passive forces acting between the phalanges of the humanoid foot during walking, the contribution of the damping force is negligible compared to the contribution of the stiffness force. In fact, only a highfrequency noisy contribution is added due to the damper. For this reason, we do not include the damper coefficient (b in 2) in the tuning problem and we fix this term to 2Nms/rad. Looking at the simulation, we choose 2Nms/rad because it is a realistic value that empirically mimics the real foot behavior.

B. Bayesian optimization

We used Bayesian optimization to find the best combination of the sole stiffness coefficient and PD gains of the stabilizer of different types of ground. An a posteriori Gaussian distribution is constructed. After each computation, the value of the cost function is calculated. To find an optimal combination of the three parameters, at each optimization step the algorithm is adapted to the samples known from the previous steps. In this way, the next point to be explored is determined. This minimizes the number of steps required to find an optimal combination of parameters [START_REF]Bayesian optimization[END_REF].The bayesian optimization problem is:

max x∈A J(x) (4)
where x is the vector of the variable we are minimizing:

x = D , µ Z , σ Z , µ C , σ C .
D is the number of steps performed without falling. We define:

Z = ZM P r -ZM P m C = CoM r -CoM m (5)
where ZM P r , CoM r and ZM P m and CoM m are respectively the reference values, the measured values of the Zero Moment Point (ZMP) and of the CoM. µ Z and σ Z are respectively the mean value and standard deviation of the ZMP error Z , and µ C and σ C are respectively the mean value and standard deviation of CoM error C . J(x) is the following cost function:

J(x) = -λ D , λ Z , λ 2 Z , λ C , λ 2 C x (6)
Where the three gains λ D , λ Z and λ C used for the optimization are [10.0, 3.0, 3.0]. We have chosen these values of gains to give primary importance to the success or failure of the walking (λ D ) and secondary importance to the quality of the walk in terms of control (λ Z and λ C ). The range of definition A of the vector x is:

D ∈ Z + , [µ Z , σ Z , µ C , σ C ] ∈ R.
During the Bayesian optimization, the cost is calculated and updated every simulation time step until the robot falls or until the end of the walking path. The optimization selects the values of the parameters close to the values related to the lowest cost at previous steps. We choose the bounds of the three parameters to include as constraints in the optimization process, taking into account the following considerations.

With too large bounds for the torsional springs coefficient k, the number of simulations we need to obtain convergence to optimal stiffness increases significantly, due to a large number of values to test. In addition, the use of too large gains of the stabilizer is not feasible on the real robot.

Given the above, we choose stiffness bounds to guarantee a range from very compliant (0Nm/rad) to rigid behavior (80Nm/rad) and gains bounds to guarantee feasibility on the real robot. These values are shown in TableI.

C. Sole stiffness optimization

A total of 4400 simulations are run for the optimization of the sole stiffness and stabilizer gains, on 11 types of uneven grounds with stones, obstacles, and tiles of different sizes and shapes. The task is walking straight forward for a 3m distance trajectory (about 22 foot steps). As Table II shows, differently from the stabilizer gains where it is possible to identify a range of work, the optimal range stiffness is noticeably different depending on the type of ground. For example, for walking on large floor tiles the optimal k range is 0.02 ∼ 0.6Nm/rad, quite far from the optimal range of roof tiles 23.9 ∼ 33.3Nm/rad. For this reason, it is difficult to identify a single optimal value to designate sole compliance. This result suggests the necessity of using variable stiffness sole design for humanoid robots to walk on different types of uneven grounds.

IV. VARIABLE STIFFNESS FOOT

From the results obtained in this section III, in a complex ground not uniform in type and size of obstacles, the robot will not be able to walk with a unique value of spring coefficient in the feet soles. To account for this, the robotic foot should feature variable stiffness, with compliance changing to be able to walk in all types of terrain. We developed a variable stiffness method shown in Fig. 2. We note through the previous simulations a correspondence between the characteristics of the ground profile and the stiffness necessary to improve the performance. This suggests the study of parameters that define the shape and variation of the terrain.

A. Stiffness variation method depending on ground roughness

To vary the compliance of the sole according to the characteristics of the underlying terrain, the properties of the ground geometry were analyzed. Properties of the surface geometry are generally labeled surface roughness parameters [START_REF] Gadelmawla | Roughness parameters[END_REF]. Table III shows the ground roughness parameters analyzed in this paper and how they are calculated. In the description of the parameters, z i is the ground altitude values, n is the total number of points considered on the ground and Z i is the Fourier transform of z i . Amplitude parameters are the most important parameters to characterize surface topography. They are used to measure the vertical characteristics of the surface deviations. The parameters M , P , and K use the Fourier Analysis to study the surface irregularities, to have a consistent representation in the frequency domain.

To obtain a measure of the roughness of each terrain, the ground altitude values are extracted from heightmaps of different types of terrains. The ground altitude of each Parameter Description S a

1 n n i=1 |z i | S q 1 n n i=1 z i 2 S sk 1 Sq 3 [ n i=1 zi 3 (i+1) ] S ku 1 Sq 4 [ n i=1 zi 4 (i+1) ] S v |min(z i )| S p |max(z i )| S z S v + S p M V ar(z(x)) = N -1 0 |Z(x)| 2 -|Z(0)| 2 P V ar( ż(x)) = N -1 0 f 0 2 n 2 |Z(x)| 2 K 1 - N -1 0 |Z(x)| 4 -|Z(0)| 4 ( N -1 0 |Z(x)| 2 -|Z(0)| 2 ) 2
heightmap frame was obtained by scanning the ground along a 3m long straight trajectory. Consequently, the ground profile of the walking trajectory was reconstructed.

The modeled foot can deform along the walking direction and, therefore, can adapt to the ground irregularities along this direction. Below a ground frequency threshold, the foot will not enter in contact along all ground profiles. For this reason, the ground profile is filtered with a low pass infinite impulse response (IIR) filter with Butterworth design, a cutoff frequency given by the inverse of the foot length, and a sampling rate of 400Hz. Subsequently, with the altitude of the filtered ground profile, the roughness parameters were calculated.

B. Estimating optimal stiffness with KNN classification

At this point, we need to understand how to connect the sole stiffness variation with the calculated roughness parameters. Through the Singular Value Decomposition (SVD) and QR decomposition, we analyzed the incidence of roughness on the sole stiffness. In particular, a matrix with all the roughness coefficients of the different terrains until powers up to the third is considered. In addition, considering the condition number of the matrix, we selected the most influential parameters in the stiffness variation: S p , P , P 2 , S 3 ku , P 3 , and S 3 p . We use k-nearest neighbors (KNN) algorithm to classify the data of roughness in order to predict correctly the sole stiffness. In our implementation 10 neighbors are used for each point. In our prediction, the points are weighted by the inverse distance to the new point. The distance metric used is a standard Euclidean distance and the algorithm used to compute the nearest neighbors is KDTree. For the model training, the data-set is composed of 360 data: 210 are relative to grounds with unique-directional variation along the walking direction, and 150 are relative to grounds with variation in all directions. The first type of ground can provide a consistent result, as the robotic foot can deform only in the walking direction. Heightmaps of these grounds are constructed from a generated random signal of 5 amplitudes between 19.5mm and 38.1mm filtered by a low-pass filter with 7 cut-off frequencies between 0.0001 and 0.1m -1 . The second type of grounds is more realistic terrains with variation also along the transversal direction. Fig. 3 shows the 5 types of grounds used for data-set training of the model: cylinder-shaped obstacles, pyramidal-shaped stones, floor tiles, cobblestones, and ellipsoidal-shaped obstacles.

Here, for each of these grounds, we used data relative to 5 different ground amplitudes between 10mm and 40mm. For both the two categories, 6 different trajectories are considered for the roughness calculation of each ground. A total of 6,000 simulations were run to construct the training data set (100 for each ground type and each ground amplitude). In the optimization, we fixed the stabilizer gains to k P = 3.4 and k D = 2.1 and we optimize only the sole stiffness parameter. Note that we use stabilizer gains that are optimal for flat feet.

The stiffness values are arbitrarily grouped in 20 clusters of range 4Nm/rad. The model predicts the cluster relative to the roughness parameters. We perform k-fold crossvalidation with 10 folds. The model has an accuracy of 0.8 with a standard deviation of 0.16. Principal components analysis shows that more than 87% of variance is explained by the first two components.

C. Range sensors and variable stiffness implementation

As explained in IV-A, the roughness of the data-set needed for the training of the KNN model is calculated thanks to the heightmap data of the grounds 2. In order to generalize our method, the goal is to walk on unknown rough terrains, in which the altitude and roughness of the ground are not known a priori. For this reason, a range sensor is placed under the first phalanx of the sole, on the tip of each foot. The position of the sensor on the tip of the foot allows for scanning of the entire area of the ground where the foot will land on the next step. During the walk, the phase in which the robot has one foot on the ground and one foot swinging is called Single Support Phase (SSP). During the simulation, the range sensor of each swing foot scans the ground when in the air in SSP. The altitude of the ground is reconstructed by the knowledge of the sensor pose relative to the fifth phalange and the phalange pose in the global frame. In the first 75% of the SSP, the altitude data are collected 4(c) 4(d). At 75% of SSP, the optimal sole stiffness is calculated through the KNN classification and it is set on the relative foot. During each step of the walking, the ground profile is reconstructed through the range sensor of the foot in the fly and the proper sole stiffness is set 4(e) 4(f). Thinking of a realistic problem, the stiffness value setting could not be instantaneous but the foot design will need time to set the stiffness. For this reason, last 25% of the SSP will be used for the stiffness setting, so that when the foot lands on the ground the stiffness has been adequately set. From the KNN classification, at 75% of the SSP, the optimal cluster is identified. To minimize the error, the mean value between the cluster boundaries is set as the sole stiffness value.

V. EXPERIMENTAL SETUP

We conducted walking simulations with the same robot and three types of sole: the original flat sole of the humanoid robot, the flexible sole with fixed stiffness set to 5Nm/rad, and the flexible sole with variable stiffness changing depending on the ground roughness. Note that the chosen fixed stiffness value was the central value of the optimal range on a wide range of terrains as result of optimization. It is also close to the estimated stiffness in the SoftFoot, already tested on the real robot when walking over obstacles [START_REF] Catalano | Hrp-4 walks on soft feet[END_REF]. For these simulations we used several grounds as described below. The test involves walking along a 3m long straight trajectory.

Types of ground Experiments were performed in two types of ground:

• Unidirectional variation ground. As explained in Sec.

IV, these types of terrain allow obtaining consistent results considering the flexibility of the foot along the same direction. For the testing, new ground signals were generated with 4 different amplitudes [START_REF] Lecomte | Variable stiffness foot design and validation[END_REF][START_REF] Caron | Stair climbing stabilization of the hrp-4 humanoid robot using whole-body admittance control[END_REF]30, and 35 mm) and 7 different frequencies (0.0002, 0.0007, 0.002, 0.007, 0.02, 0.07, and 0.2 m ). The test involves walking the 24 amplitude-frequency combinations of the terrain. The signals that generate these ground profiles are all different from the data that constitute the training data-set for the KNN method. • Bidirectional variation ground. Fig. 4 shows more realistic double-directional variation terrains used to test the walking. Here, the objective is to test the humanoid walking on the more realistic ground with rocks of different shapes and sizes, obstacles, and tiles. Adding the transverse direction of ground variation, the test becomes much more challenging considering that the foot model is deformable only along the longitudinal axis. For these experiments, additional data relating to bidirectional variation grounds are introduced into the training data-set of the model for variable stiffness.

VI. RESULTS AND DISCUSSION

Unidirectional variation ground Fig. 5 shows the comparison of the performances with the three types of sole for the combinations of ground profile amplitude (along the horizontal axis) and frequency (along the vertical axis). For each frequency-amplitude combination, the three blocks represent the variable stiffness sole, the soft sole, and the original flat sole of the robot, respectively at the top, middle, and bottom. The color represents the number of steps the robot performed without falling. As Fig. 5 shows, the humanoid with the original flat sole can not complete the path without falling in this combination of rough terrains. The introduction of the new flexible sole and then of the variable stiffness significantly improves performance and the robot can reach the total number of steps for many of the trials.

Bidirectional variation ground

The Table IV shows, in the first column, the types of grounds used to test the walk of the humanoid with the three soles: stones, roof tiles, stones bricks, small stones, and large floor tiles. Similarly to the unidirectional variation ground, the test involves walking along a 3m long straight trajectory. For each type of terrain, 4 different maximum vertical heights of the terrain are chosen, as column 2 shows. For each of the heights, we report the number of steps carried out without falling with a maximum of 22 steps to cover the 3m, for the three types of the sole. A substantial improvement in the average walking distance/number of steps is obtained due to the introduction of sole flexibility, as the comparison between columns 'Flat' and 'Soft' shows. The robot with the flat sole only walked 140/440. In particular, rigid soles show difficulty in walking since the lower ground amplitudes. Moreover, a further improvement in the number of steps performed is shown in the column 'VarStiff' with the introduction of the variable stiffness sole and the robot can execute all the path in most of the trials. More specifically, we can observe that the introduction of variable stiffness improves the performance, especially in grounds with stones and large floor tiles.

Here, the values used with the variable stiffness sole along some parts of the trajectory are high (around 40Nm/rad). Indeed, when parts of terrains require stiffness values far from 5Nm/rad, a considerable improvement is obtained due to the introduction of the variable stiffness. Conversely, with terrains like roof tiles and small stones, the advantage offered by variable stiffness is less visible. In fact, here, the fixed stiffness value is close to the optimal value.

Limitation

The introduction of the new compliant sole design causes a backward shifting of the center of mass (CoM) of the humanoid. This creates a slight imbalance of the humanoid in the backward direction during the walking. Considering that no modifications have been made to the controller, the robot stabilizer assumes flat feet parallel to the ground. Indeed, the introduction of the rotoidal joint between the foot frontal arch and the ankle allows the pitch rotation of the sole which is not considered in the actual position estimator. However, despite not being taken into account from a control point of view, the introduction of the compliant sole with variable stiffness is already contributing to better success for walking on uneven grounds.

VII. CONCLUSION AND FUTURE WORKS

We presented a compliant foot model with multiple viscoelastic elements in the sole and we optimized the sole stiffness for different types of uneven terrains. We proposed a method to use the surface parameters previously proposed for surface roughness as a vector characterizing the terrains and we used the KNN classifier to detect soil types. We have numerically proven with dynamical simulation tests that foot compliance enhances stability with respect to the rigid sole, and the variable stiffness sole further enhances the stability of humanoid walking in different environments, such as terrains with rocks, tiles, and obstacles of different sizes and shapes. The improvement results with the flexible sole were obtained even using stabilizer gains that are optimal for flat feet. Moreover, even if we introduced flexibility in the sole only along the longitudinal direction, the results have proved effective also for grounds with variation in the transverse direction.

In future works, a new robotic foot for humanoid robots will be designed. A compliant sole with the variable stiffness method connected to the ground roughness will be introduced in a physical experiment. The problem of changing the stiffness on-line during the walking will be faced introducing pneumatic actuation to the sole. Last, we will study how to improve the robot stabilization adapting the robot controller to uneven ground and flexible sole in future work.

Fig. 1 .

 1 Fig. 1. (a) Simplified model of the compliant foot. The plantar arc is composed of ten blocks representing the ten phalanxes, connected between each other by rotational joints with viscoelastic elements. The plantar arch is connected with the frontal and backward arch through rotational passive joints. (a) Deformable foot model used in dynamic simulation.

Fig. 2 .

 2 Fig.2. Proposed method to control the variable stiffness depending on the ground roughness.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Types of grounds: Cylinder Obstacles (a), Pyramidal stones (b), Cobble Stones (c), Floor Tiles (d), ellipsoidal-shaped obstacles (e)

Fig. 5 .

 5 Fig. 5. Comparison of variable stiffness, soft and flat foot performances in unique directional variation ground. The grounds are generated with signals of different amplitudes (horizontal axis) and frequencies (vertical axis). The color of the plot represents the number of steps without falling in a 3m long straight trajectory (about 22 steps). The groups of three squares represent variable stiffness, soft, and flat foot performances respectively on the top, center, and bottom.

TABLE I PARAMETERS

 I BOUNDS LIMITS USED IN THE BAYESIAN OPTIMIZATION.

	Parameter Lower bound Upper bound
	k	0 Nm/rad	80 Nm/rad
	kP	0.1	5
	kD	0.1	5

TABLE II OPTIMIZATION

 II OF STABILIZER GAINS AND SOLE STIFFNESS.

	Ground Type	Optimal gains Optimal k
		kP	kD	(Nm/rad)
	Flat floor	3.4	2.1	Entire Range
	Cylinder obst	4.8	1.8	0.2-9.5
	Stones	4.0	1.3	2.8-16.1
	Small stones	4.3	2.4	2.3-31.6
	Stones Bricks	5.0	0.5	25.4-28.9
	Pyramidal stones 4.2	1.6	0.9-48.7
	Metal iron sheets 3.6	2.7	0.7-19.9
	Large floor tiles	3.9	2.6	0.02-0.6
	Roof tiles	4.1	2.6	23.9-33.3
	Cobble stones	4.0	2.6	18.4-32.8
	Floor tiles	3.7	1.1	31.6-46.1

TABLE III ROUGHNESS

 III PARAMETERS OF THE GROUND PROFILE.[28] [29]

TABLE IV COMPARISON

 IV OF VARIABLE STIFFNESS, SOFT AND FLAT FOOT PERFORMANCES IN DOUBLE DIRECTIONAL VARIATION GROUND.

		Max ground
	Ground Type	amplitude	Number of steps (max=22)
		[mm]
			VarStiff	Soft	Flat
	Stones	55	20	6	6
		50	22	7	7
		45	22	22	7
		40	22	22	8
	Roof tiles	55	8	7	5
		50	22	8	9
		45	22	9	12
		40	22	22	16
	Stones bricks
		55	8	6	3
		50	22	5	4
		45	22	6	6
		40	22	22	6
	Small stones
		25	9	7	6
		20	13	9	7
		15	22	13	7
		10	22	22	7
	Floor tiles
		25	20	8	5
		20	22	12	6
		15	22	19	6
		10	22	22	7


Keywords: Heart failure, Young people, Ischaemia, Risk factors, Epidemiology

Aims

Heart failure (HF) in young adults is uncommon, and changes in its incidence and prognosis in recent years are poorly described.

Methods and results

The incidence and prognosis of HF in young adults (18-50 years) were characterized using nationwide medico-administrative data from the French National Hospitalization Database (period 2013-2018). A total of 1,486 877 patients hospitalized for incident HF were identified, including 70 075 (4.7%) patients aged 18-50 years (estimated incidence of 0.44‰ for this age group). During the study period, the overall incidence of HF tended to decrease in the overall population but significantly increased by ∼0.041‰ in young adults (P < 0.001). This increase was notably observed among young men (from 0.51‰ to 0.59‰, P < 0.001), particularly those aged 36-50 years. In these young men, ischaemic heart disease (IHD) was the most frequently reported cause of HF, whereas non-ischaemic HF was mainly observed in patients ≤ 35 years old. In contrast to non-ischaemic HF, the incidence of IHD increased over the study period, which suggests that IHD-related HF is progressively affecting younger patients. Concordantly, young HF patients presented with high rates of traditional IHD risk factors, including obesity, smoking, hypertension, dyslipidaemia, or diabetes. Lastly, the rates of re-hospitalization (for HF or for any cause) within two years after the first HF event and in-hospital mortality were high in all groups, indicating a poor-prognosis population.

Conclusion

Strategies for the prevention of HF risk factors should be strongly considered for patients under 50 years old.

Introduction

Heart failure (HF) is a major clinical and public health challenge that affected ∼64.3 million people worldwide in 2017. [START_REF] Roberts | Disease and Injury Incidence and Prevalence Collaborators. Global, regional, and national incidence, prevalence, and years lived with disability for 354 diseases and injuries for 195 countries and territories, 1990-2017: a systematic analysis for the global burden of disease study[END_REF] HF incidence is known to increase with age, and the median age of HF patients in Western countries exceeds 70 years. Therefore, HF is primarily considered to be a disease affecting the elderly, and most HF studies are focused on this group. However, recent studies indicated that the HF burden among the young population (defined as people < 50 years old) is increasing. [START_REF] Groenewegen | Epidemiology of heart failure[END_REF][START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF][START_REF] Jain | Trends in HF hospitalizations among young adults in the United States from 2004 to 2018[END_REF] Few specific studies have reported the incidence, aetiologies, and prognostic trends in these younger HF patients. [START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF][START_REF] Bibbins-Domingo | Racial differences in incident heart failure among young adults[END_REF][START_REF] Wong | Heart failure in younger patients: the meta-analysis global group in chronic heart failure (MAGGIC)[END_REF][START_REF] Wong | Clinical characteristics and outcomes of young and very young adults with heart failure: the CHARM programme (candesartan in heart failure assessment of reduction in mortality and morbidity)[END_REF] The rates of HF hospitalizations have remained steady in France, whereas mortality due to HF decreased by ∼3% annually during the last decade. [START_REF] Gabet | National trends in rate of patients hospitalized for heart failure and heart failure mortality in France, 2000-2012[END_REF] The same trends were observed in Sweden, Denmark and Scotland, where mortality due to HF has progressively decreased since the early 1990s. [START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF][START_REF] Christiansen | Age-Specific trends in incidence, mortality, and comorbidities of heart failure in Denmark, 1995 to 2012[END_REF][START_REF] Stewart | Trends in hospitalization for heart failure in Scotland, 1990-1996. An epidemic that has reached its peak?[END_REF] Whether these favourable trends occur in the general population of patients with HF independent of age has, to our knowledge, not been examined.

HF is the final stage of a wide range of cardiac impairments, and myocardial ischaemia is the prevailing cause in older patients. However, specific non-ischaemic HF aetiologies (including congenital heart disease, myocarditis, different types of cardiomyopathies, or toxic-related lesions of the myocardium) are more frequently observed among younger patients. [START_REF] Ziaeian | Epidemiology and aetiology of heart failure[END_REF] Data from the Swedish hospital discharge Registry and from the Framingham Heart Study suggest that the increasing burden of HF earlier in life may be explained by an increase in the prevalence of metabolic (i.e. obesity and diabetes) risk factors for HF combined with a higher attributable weight of these risk factors among younger people. [START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF][START_REF] Tromp | Age dependent associations of risk factors with heart failure: pooled population based cohort study[END_REF] These factors are well-known contributors to ischaemic HF that develops in older patients. Whether the incidence of HF cases can be attributed to a change in the incidence of ischaemic or nonischaemic causes of HF in younger patients has yet to be deciphered.

In this study, we used data from the French National Hospitalization Database (Programme de Médicalisation des Systèmes d'Information, PMSI) to investigate recent trends in HF incidence among patients under 50 years old. We also investigated gender-and age-specific trends in HF incidence and underlying aetiologies in this population.

Methods

Study design and data source

We performed a retrospective cohort study using data from patients with HF extracted from the Programme de Médicalisation des Systèmes d'Information (PMSI) French national database. The PMSI medical and administrative database contains standardized discharge reports for all patients discharged from all public or private hospitals in France, and it covers a population of ∼67 million French people. Each discharge report provides administrative and demographic data, discharge diagnoses, procedures, and comorbidities. Diagnoses are encoded as primary or secondary using the French version of the International Classification of Diseases, 10th revision (ICD-10). Discharge reports are mandatory and serve as the basis for hospital funding. The PMSI database includes a unique anonymous identifier for each patient; this enables all stays by the same patient to be linked, even if the stays were in different establishments. [START_REF] Girard | Epidemiology of pulmonary tuberculosis in France. Can the hospital discharge database be a reliable source of information?[END_REF]

Patient selection

Patients included in this study were incident cases of HF who were admitted to hospital between 2013 and 2018 in France. These patients were identified in the PMSI database by at least one stay coded with an I50 (ICD-10) diagnosis corresponding to Heart Failure and no stay coded with this code in 2011 and 2012. This timeframe was selected as optimal to identify incident HF as we found that a second HF stay occurred in the first 2 years after the first HF events for the very large majority (∼85%) of patients (data not shown). We subsequently grouped the patients aged 18-50 years according to gender and age group (18-30, 31-35, 36-40, 41-45, and 46-50 years old) for further analysis. This study complied with rules of the French Data Protection Committee (Commission Nationale Informatique et Liberté, CNIL, MR-005, registration number: F20220208162131).

Extracted variables

We extracted age and gender data at the time of the first hospitalization stay for HF and all hospitalization stay's dates and diagnoses and their outcome (death) from 2011 until 2020, as nationwide hospitalization data were available only after this period (see Supplementary material online, Tables S1 andS2).

Statistical methods

To calculate incidences of HF for each year from 2013 to 2018, the number of new cases of HF was divided by the French population estimate of the corresponding age group. The French population of 2013 was selected as the reference. Annual incidences by age were calculated as the averages over the study period of incidences per age. P-values were obtained from χ² test statistics for binary variables. Cochran-Armitage trend test was used to assess for changes in incidence over time. The binomial law was used to estimate confidence intervals when possible. Additional statistical methods are reported in Supplementary material online.

For each age group and gender, we then calculated the incidence of patients presenting with the main aetiologies. Patients were classified into one of three aetiological subgroups: ischaemic heart disease (IHD), myocarditiscardiomyopathy, and other causes (other-not stated) according to ICD-10 codes (see Supplementary material online, Table S1). All patients with an ICD-10 code between I20 and I25 within the 2 years prior to or at the time of hospitalization for the first HF event were placed in the IHD group. For patients without an ICD-10 code between I20 and I25, a patient was placed in the myocarditis-cardiomyopathy group if he/she had an ICD-10 code between I40 and I43 within the 2 years prior to or at the time of hospitalization for the first HF event. Finally, remaining patients without these ICD-10 codes were classified in the other group. Incidences of each subgroup were calculated for each age group and gender along with their 95% confidence interval assuming a binomial distribution. A survival analysis with reversed time was performed to explore time to first prior hospitalization using Kaplan-Meier estimates according to aetiology, age group and gender subgroups.

Finally, to investigate the prognosis of patients with HF, we calculated the observed proportion of re-hospitalization for HF for any cause within the 2 years after the first HF event and of hospital death within 2 years or at the time of first HF event by age group, gender, and aetiological subgroup.

Results

General trends

Between 2013 and 2018, we identified 1 486 877 patients hospitalized for incident HF among the 67 million French inhabitants covered in the PMSI database. Among these patients with incident HF, 467 031 (31.4%) were patients aged 18-70 years and 70 075 (4.7%) were patients aged 18-50 years (Table 1). About one-third of HF patients aged 18-70 years were women.

By the end of the study period, the incidence of HF was estimated to be 3.65‰ in the whole French population and 0.44‰ in the young (18-50 years old) French population. In the whole French population, the overall incidence of HF decreased over the 2013-18 period from 3.99‰ to 3.65‰ (P < 0.001, Figure 1A), whereas we observed a significant increase of ∼0.041‰ in the incidence of HF in the 18-50-year-old group (P < 0.001) (Figure 1B). This evolution was explained by a significant increase in the incidence of HF among young men (from 0.51‰ to 0.59‰, P < 0.001), whereas the incidence of HF among young women remained stable over the study period (∼0.28‰). The incidence exponentially increased by age, from ∼0.07‰ at 18 years old to 1.31‰ at 50 years old (Figure 1C). Figure 1D shows that the evolution of incidence of HF according to age was similar for men and women but remained higher in the young male population compared to the young female population at every investigated age. At 50 years old, the incidence of HF was more than two times higher for men than for women (1.84‰ vs. 0.80‰ respectively, P < 0.001).

We further described the evolution of HF incidence in men and women aged 18-50 years (Figure 1E). We observed a positive change in HF incidence over the study period after 30 years old for men and women, with a maximal change reached in the 36-40 year old age group (+22% for men and +17% for women). In age groups 41-45 and 46-50 years, changes observed in men were much higher (+16% and +14%, respectively) than those in women (+3% and +1%, respectively) (Figure 1E).

Underlinying heart failure causes and trends in prevalence over the study period

Figure 2A shows the incidence of HF hospitalizations with a concomitant diagnosis or a diagnosis within the prior 2 years of either myocarditis-cardiomyopathy, IHD, or other causes according to gender and age group. Myocarditis-cardiomyopathy and IHD diagnoses were found in 23% and 20%, respectively, of patients aged 18-50 years, whereas these incidences were 11% and 30% for patients aged 51-70 years. We also found that the incidence of IHD-related HF was positively associated with age, whereas the incidence of patients with a myocarditiscardiomyopathy diagnosis significantly decreased with age in both genders. Myocarditis-cardiomyopathy diagnostic codes were more frequently found in the youngest HF patients and decreased across all age groups from 32% to 11% in men and from 22% to 11% in women.

Reciprocally, the incidence of IHD-related HF patients was minimal for the youngest HF patients (10% and 6% in men and women, respectively) but was more likely than myocarditis-cardiomyopathy at a younger age for men than for women (36-40 years old for men vs. 41-45 years old for women). The incidences of patients diagnosed with IHD-related HF in the 46-50-year-old age group (38% and 23% in men and women, respectively) were similar to those observed in the 51-70-year-old age group (37% and 24% in men and women, respectively), suggesting that a plateau was reached by age 46 for both genders. The large and heterogeneous group of patients in the other group (i.e. patients without a concomitant diagnosis of either myocarditis-cardiomyopathy of IHD within the prior 2 years or at the time of hospitalization) showed no qualitative differences between age groups, but a difference between genders was detected, with rates higher in women than in men (P < 0.001). Corresponding data to Figure 2A are provided in Supplementary material online, Table S3.

We identified a significant increase in HF incidence in young men aged 41-50 years over the 2013-18 study period (Figure 1E). A significant increase in the incidence of IHD-related HF (from 40% to 43%, P < 0.001, Figure 2B) occurred, whereas incidences of a concomitant diagnosis of myocarditis-cardiomyopathy or other-not stated diagnoses decreased or were stable over the study period (Figure 2B; Supplementary material online, Table S4).

Figures 2C-F are Kaplan-Meier estimations of having an IHD or a myocarditis-cardiomyopathy diagnostic code, respectively, before the first HF event for men and women according to age group. Overall, these time-reversed graphs showed that HF cause was reported at the time of HF hospitalization for the majority of patients aged < 50 years, both for IHD and for myocarditis-cardiomyopathy. In contrast, the incidence of patients with a known IHD prior the first HF event was much more important in patients aged 51-70 years (Figure 2C and D), but this pattern was not observed for myocarditis-cardiomyopathy (Figure 2E andF). Overall, these data show that the ischaemic origin of HF became highly prevalent in men > 40 years old and in women > 45 years old but not in younger patients with very premature HF.

We next looked at the prevalence of diagnostic codes related to cardiovascular risk factors, lifestyle habits, and other cardiac and non-cardiac disorders in these patients. We found that the overall burden of comorbidity in this young HF population was substantial. Table 2 shows that the population of young HF patients had more major HF risk factors (and often multiple factors) than anticipated in such a young population in France. [START_REF] Cherfan | Unhealthy behaviors and risk of uncontrolled hypertension among treated individuals-the CONSTANCES population-based study[END_REF][START_REF] Blacher | Prevalence and management of hypercholesterolemia in France, the Esteban observational study[END_REF][START_REF] Vallee | Patterns of hypertension management in France in 2015: the ESTEBAN survey[END_REF][START_REF] Chevreul | The burden and treatment of diabetes in France[END_REF] Indeed, the classical atherosclerotic risk factors (i.e. smoking, hypertension, dyslipidaemia, diabetes, or obesity) were substantially high and significantly more frequent in patients with IHD than in patients with myocarditis-cardiomyopathy or others diagnosis. Similarly, unhealthy lifestyles habits (smoking, alcohol or drug use) and cancer were frequently found in this population, regardless of gender. We further looked to temporal trends in HF risk factors (see Supplementary material online, Figure S1) and found a significant increase in the rates of smoking habits and obesity over the study period in patients presenting incident HF, whereas the rates of other risk factors were stable or decreasing.

Prognosis of young patients with heart failure

Figure 3A-C illustrates the probability of re-hospitalization for HF (Figure 3A) or for any causes (Figure 3B) and the probability of death for any causes (Figure 3C) within 2 years after the first HF hospitalization. Corresponding estimates are provided in Supplementary material online, Table S5. For patients aged 18-50 years, proportions of re-hospitalization for HF were ∼24% and 30% for patients with IHD-related HF and myocarditis-cardiomyopathy diagnoses, respectively, and 31% and 35%, respectively, for patients aged 51-70 years. It remained stable over age group for IHD-related HF and myocarditis-cardiomyopathy, whereas it increased from 15% to 23% over age group for patients with a diagnosis of other-not stated. The proportion of re-hospitalization for any cause was ∼75% and remained stable over age groups and gender. The proportion of deaths in hospital for any cause within 2 years after the first HF hospitalization was high, at ∼10% for patients aged 18-50 years, and it was similar between patients with IHD and myocarditiscardiomyopathy. Patients with other causes displayed higher all-cause mortality rates, indicating that these patients were likely to have additional extra-cardiac conditions.

Discussion

In this nationwide study conducted in France using data from 2013 to 2018, we found that HF was rare in patients aged < 50 years (4.7%), but HF incidence progressively increased in young men and particularly in those aged 36-50 years. In these patients, IHD was the most frequently reported cause of HF, and its incidence increased over the study period. This result suggests that IHD-related HF is increasing and is progressively affecting younger patients. Concordantly, we found that young HF patients frequently presented with multiple comorbidities, including high rates of traditional risk factors for IHD. The proportion of re-hospitalization (for HF or for any cause) within 2 years after the first HF event was high for the entire study population (∼25% for HF and 75% for any causes) regardless of the primary aetiology and age group, indicating a population at poor prognosis. In-hospital mortality rates also were high (roughly 10%). The strength of our study is the large patient cohort that included the entire French population hospitalized for HF from 2011 to 18 (1 486 877 patients regardless of age). Patients from 18 to 70 years old were followed over a median of 73 months before and 58 months after first HF event.

Previous data from Sweden, the USA, and Australia have shown that HF admissions levelled off and eventually declined during the last two decades. [START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF][START_REF] Jain | Trends in HF hospitalizations among young adults in the United States from 2004 to 2018[END_REF][START_REF] Weber | Changing age-specific trends in incidence, comorbidities and mortality of hospitalised heart failure in Western Australia between 2001 and 2016[END_REF][START_REF] Andersson | Epidemiology of cardiovascular disease in young individuals[END_REF] This is in line with the overall trend observed in our study in France, indicating a worldwide tendency. [START_REF] Groenewegen | Epidemiology of heart failure[END_REF] However, we found that epidemiological trends in France depended on age, with increasing HF incidence in the young male population. This confirms and expands the progressive increase in HF incidence in young patients previously reported in Sweden. [START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF] By analysing the associated diagnostic codes, we found that this increase in young men was likely linked to an increase of ischaemic HF. We found that IHD-related HF was one of the most prevalent causes of HF in men > 40 years old and women > 45 years old but not in younger patients with very premature HF, who were more likely to present with cardiomyopathies. The higher likelihood of observing IHD-related HF in young men is supported by observations from others countries. In the USA, more than 200,000 30-54-year-old patients were hospitalized every year with a diagnosis of acute myocardial infarction, [START_REF] Andersson | Epidemiology of cardiovascular disease in young individuals[END_REF] and men had the highest incidence rate across all age groups. [START_REF] Virani | Heart disease and stroke statistics-2020 update: a report from the American Heart Association[END_REF] Similar results were reported in the UK, where the prevalence of acute myocardial infarction reach 0.5% in men and 0.18% in women aged 35-44 years. [START_REF] Egred | Myocardial infarction in young adults[END_REF] The progressive increase of IHD at younger age can be linked to the overall increase of inactivity, atherothrombosis risk factors, and related comorbidities. [START_REF] Park | The health status of young adults in the United States[END_REF] Smoking, hypertension, dyslipidaemia, diabetes, and obesity are classical risk factors leading to atherosclerosis and coronary artery disease and were frequently observed in our patients (notably in those with IHD-related HF) as compared to the known incidences in healthy French people of the same age. In addition, even if we cannot ensure a direct causality link with the increase of IHD, we found a significant increase over the study period in the rates of smokers and obesity suggesting that preventive efforts should be particularly strengthened regarding these two major risk factors. The CONSTANCES study, a general prospective population-based cohort with 200 000 French adults aged 18-69 years at inception, reported that 12.5% and 10.3% of French women and men, respectively, aged 18-49 years presented with obesity (BMI > 30 kg.m -2 ), whereas 22.9% were obese in our population. [START_REF] Cherfan | Unhealthy behaviors and risk of uncontrolled hypertension among treated individuals-the CONSTANCES population-based study[END_REF] According to the French national ESTABAN study, the expected rate of arterial hypertension among the young men (18-34, 53-44, and 45-54 years old) should be 11.7%, 17%, and 36.6% respectively. [START_REF] Vallee | Patterns of hypertension management in France in 2015: the ESTEBAN survey[END_REF] Moreover, the prevalence of arterial hypertension is significantly lower among young women (18-34, 53-44, and 45-54 years old), at 1.5%, 9.1%, and 21.1%, respectively. [START_REF] Vallee | Patterns of hypertension management in France in 2015: the ESTEBAN survey[END_REF] The expected rate of dyslipidaemia (e.g. hypercholesterolaemia) among the French population (47.3 ± 14.6 years old) should be 23.3% (27.8% for men and 19.0% for women) according the ESTABAN study, [START_REF] Blacher | Prevalence and management of hypercholesterolemia in France, the Esteban observational study[END_REF] but it exceeded 30% in young HF patients with IHD. Finally, the number of diabetic patients (type 1 or 2) among our population was much higher than anticipated by the French Health Insurance agency (Système national d'information inter-régimes de l'assurance maladie, Sniiram), as only 2.5% of the young French population present with diabetes. [START_REF] Chevreul | The burden and treatment of diabetes in France[END_REF][START_REF] Mandereau-Bruno | Prévalence du diabète traité pharmacologiquement (tous types) en France en 2015. Disparités territoriales et socio-économiques[END_REF] In our study, young French patients with IHD-related HF also presented with significantly higher rates of risk factors. A recent study reported important age differences in the effect of traditional risk factors on the development of future HF. Specifically, risk factors for HF carried a stronger relative risk among young people compared to older people. [START_REF] Tromp | Age dependent associations of risk factors with heart failure: pooled population based cohort study[END_REF] These results have to be interpreted in line with the dramatic increase in the prevalence of metabolic diseases with cardiac consequences in young people. [START_REF] Park | The health status of young adults in the United States[END_REF] Our data likely suggest a progressive increase in the prevalence of IHD-related HF in younger patients.

The incidence of HF in young patients can also be influenced by socio-economic factors that were not investigated in our study. Recent studies have reported that HF patients in low-middle-income countries were younger than patients in high-income countries. [START_REF] Groenewegen | Epidemiology of heart failure[END_REF][START_REF] Agbor | An overview of heart failure in low-and middle-income countries[END_REF] A recent study in the U.S. showed that nearly half of young patients with HF reside in zip codes in the lowest quartile of national household income. [START_REF] Jain | Trends in HF hospitalizations among young adults in the United States from 2004 to 2018[END_REF] Additional analyses will be now be needed to better define the relations between HF incidence, HF risk factors and socio-economic factors in young vs. older patients.

We also found that young HF patients had a poor prognosis that was comparable to that observed in older patients, with a high rate of rehospitalizations in the 2 years following HF diagnosis. HF is known to be associated with the highest 30-day readmission rate (around 20%-25%) regardless of the primary aetiology. [START_REF] Dharmarajan | Diagnoses and timing of 30-day readmissions after hospitalization for heart failure, acute myocardial infarction, or pneumonia[END_REF][START_REF] Patil | Readmissions among patients admitted with acute decompensated heart failure based on income quartiles[END_REF][START_REF] Jencks | Rehospitalizations among patients in the medicare fee-for-service program[END_REF] Gender also seems to have an impact on re-hospitalization among the elderly. In one study, women older than 60 years had a 2.5 times greater risk for rehospitalization than men. [START_REF] Howie-Esquivel | Effect of gender, ethnicity, pulmonary disease, and symptom stability on rehospitalization in patients with heart failure[END_REF] However, in our study the observed proportion of re-hospitalization was similar between men and women younger than 50 years. Additionally, some studies found that patients with HF presented with significant comorbidities and reported that 45% of rehospitalizations were non-cardiovascular related, [START_REF] Muzzarelli | Predictors of early readmission or death in elderly patients with heart failure[END_REF][START_REF] Aranda | Current trends in heart failure readmission rates: analysis of medicare data[END_REF] which is in line with our results. Patients who were readmitted had more peripheral vascular disease, diabetes, and stroke when compared with HF patients who were not readmitted after their index hospitalization. [START_REF] Aranda | Current trends in heart failure readmission rates: analysis of medicare data[END_REF] These studies indicate that comorbid conditions may be significant predictors of repeat hospitalization of HF patients.

Regarding proportion of death within 2 years after the first HF event, 2-year mortality data among young adults with HF are scarce. In line with our results, 1-year case fatality rates after a first HF hospitalization in a New South Wales, Australia, study were 13% in the 45-49-year-old age group. [START_REF] Robertson | The health services burden of heart failure: an analysis using linked population health data-sets[END_REF] The Swedish National Registry reported rates of 11%-12% for those aged 18-54 years, which is high but lower than that of older patients. [START_REF] Barasa | Heart failure in young adults: 20-year trends in hospitalization, aetiology, and case fatality in Sweden[END_REF] More recently, a Canadian study reported a 5-year mortality rate of 28% among patients aged 20-44 years. [START_REF] Wong | Heart failure in young adults is associated with high mortality: a contemporary populationlevel analysis[END_REF] We found concordant results in our study, and all of these reports highlight the important role of surveillance in the immediate post-discharge period. Practitioners should be warned of high re-hospitalization and mortality risks in young HF patients, and strict monitoring should be recommended.

Our study illustrates the need for better identification and prevention of HF in the young population. Our results suggest that many cases are related to modifiable risk factors, including smoking, obesity, dyslipidaemia, diabetes, and hypertension, as observed in young patients with first myocardial infarction. [START_REF] Yandrapalli | Modifiable risk factors in young adults with first myocardial infarction[END_REF] Additional analyses on the temporal trends of HF risk factors further identify growing rates of smoking habits and obesity, suggesting a growing contribution of these two factors, even if the causality link would remain to be demonstrated. Currently, screening for these traditional risk factors is not proposed for young patients in France (and in other countries). Our data indicate that the associated cause leading to HF was not identified before the first HF hospitalization for many young patients. Therefore, efforts to detect, prevent, and treat cardiovascular risk factors in young patients (beginning at 36 years old in men and 40 years old in women) should be recommended to efficiently limit incident HF. Reciprocally, some young patients had IHD or myocarditis-cardiomyopathy diagnostic codes before the first HF event. Our study highlights the need for a specific monitoring of these young patients in order to decrease the transition to HF.

Our study had some limitations. First, it was performed using medico-administrative databases, and the use of discharge diagnosis codes could result in underestimation of relevant data. More particularly, diagnoses that do not require significant medical care might not be coded. Therefore, we cannot exclude that associated diagnoses and risk factors are underestimated, while HF requiring significant medical treatment is not likely to be underestimated. This will result in higher rates for associated diagnoses and risk factors than the observed ones that were already high is such a population of young adults. Second, we used hospital admission rates and not true incidence. However, the French medical care organization ensures that all young adults having a first episode of HF will be referred to a reference hospital and the PMSI database is likely exhaustive for HF diagnoses. Third, the coding practices might have change over time, but if this occurred, the resulting bias would not be influenced by ages or genders, thus limiting the risk. Fourth, the database does not provide indications on medications, HF classification, or left ventricular ejection fraction. However,

Figure 1 (

 1 Figure 1 (A) Incidence of heart failure in the French population (all age groups) from 2013 to 2018. (B) Standardized incidence of heart failure in the 18-50-year-old French population. (C) Incidence of heart failure according to age from 2013 to 2018 in the young French population (18-50 years) assessed in this study. (D) Incidence of heart failure according to age from 2013 to 2018 in men and women. (E) Relative evolution of incidence of heart failure between 2013 and 2018 in the young population according to age group and gender.

Figure 2 (

 2 Figure 2 (A) Concomitant diagnosis [myocarditis-cardiomyopathy (red), ischaemic heart disease (green), or other-not stated (grey)] either within the 2 years prior to or at the time of hospitalization for heart failure for men and women according to age group. (B) Relative evolution of incidence of myocarditis-cardiomyopathy (red), IHD (green), or other-not stated (grey) over the 2013-18 study period in the young male population aged 41-50 years. (C/D) Kaplan-Meier estimations of having a prior ischaemic heart disease preceding the first heart failure hospitalization according to age group in men (C) and women (D). (E/F) Kaplan-Meier estimations of having a prior myocarditis/cardiomyopathy preceding the first heart failure hospitalization according to age group in men (E) and women (F). P < 0.001 for all comparisons.

Figure 3

 3 Figure 3 According to age group, gender, and the primary aetiology (categories myocarditis-cardiomyopathy, ischaemic heart disease, and other-not stated): (A) Observed proportion of re-hospitalization for heart failure within 2 years after the first heart failure event. (B) Observed proportion of any re-hospitalization within 2 years after the first heart failure event. (C) Observed proportion of death within 2 years after the first heart failure event.

Table 1

 1 . ....... ...... ....... ...... ...... ....... ...... ....... ...... ....... ...... ....... ...... ....... ...... ...... ....... ...... ....... ...... ....... ...... ....... ...... ...... ....... ...... ....... Patients hospitalized for HF in France between 2013 and 2018

	All (n)	Men (%)	Women (%)

HF, heart failure.

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Funding

This work was supported by the University Research Federation against Heart Failure (FHU2019, PREVENT_Heart Failure), AP-HP (APHP.centre, CRIU funding), and INSERM.

Data availability

This study was performed using the PMSI (Programme de Médicalisation des Systèmes d'Information) French national database and restrictions apply to the availability of these data, which were used under license for this study.

most of young patients with HF present with aetiologies associated with reduced ejection fraction [START_REF] Fayol | Aetiological classification and prognosis in patients with heart failure with preserved ejection fraction[END_REF] which has well-established management guidelines. Fifth, in order to identify patients with a new HF diagnosis, we excluded patients with a HF diagnosis in the 2 years before the first HF event; however, patients could have had an older HF diagnosis. However, we show that this limitation was minor based on the high tendency of re-hospitalization for HF in the early stage after first hospitalization for HF.

In conclusion, the increase of HF incidence in young adults (< 50 years old) documented in our study differs from the general trend noted in older patients. Among the young population, men seemed to be more vulnerable to premature HF than women. We found that myocarditis-cardiomyopathy and IHD were major causes of HF admission, with inverse probability trends according to age group. In the young population, and especially in young men, ischaemic HF was the predominant form of HF, and our data suggest that it is progressively increasing. The young adults hospitalized for premature HF also presented with high rates of major modifiable risk factors for ischaemic HF, including obesity, dyslipidaemia, smoking, hypertension, and diabetes. The observed proportion of re-hospitalization for HF or from any cause within 2 years after the first HF event is alarming. Although the cause of the increased incidence of HF among the young population is not fully known, it may reflect true epidemiological changes linked to increased prevalence of cardio-metabolic risk factors in young people. This study highlights the need for a specific care pathway for HF patients under 50 years old. Strategies for the prevention of HF risk factors should be strongly considered for patients under 50 years old.

Supplementary data

Supplementary data is available at European Heart Journal online.

Conflict of interest:

The APHP, which employs Pr. J.-S.H., has received research grants from Bioserenity, Pliant Thx, Sanofi, Servier, and Novo Nordisk. He has also received speaker, advisory board, or consultancy fees from Alnylam, Amgen, Astra Zeneca, Bayer, Bioserenity, Boerhinger Ingelheim, MSD, Novartis, Novo Nordisk, and Vifor Pharma, all unrelated to the present work. Other authors declare no competing financial interests.


Characterization of OMEGA/MEx CO 2 non-LTE limb observations on the dayside of Mars

Introduction

The upper atmosphere of a terrestrial planet is a region difficult to sound, both by in-situ and remote sounding [START_REF] Mueller-Wodarg | Planetary Upper Atmospheres[END_REF]. This atmospheric region is characterized by non-local thermodynamic equilibrium (non-LTE) that occurs when collisions between atmospheric species are not enough efficient in transferring energy. The CO 2 non-LTE emission at 4.3 µm originates in the upper layers of the atmosphere and is a feature common to the three terrestrial planets with an atmosphere (Venus, Earth, and Mars). It provides a useful tool to gain insight into the atmospheric processes at these altitudes [START_REF] Lopez-Puertas | Non-LTE radiative transfer in the atmosphere[END_REF]. Non-LTE fluorescent emissions were first observed in the Earth's upper atmosphere in CO 2 bands at 15 and 4.3 µm [START_REF] Curtis | Thermal Radiation in the Upper Atmosphere[END_REF] and were later observed on several planets in different spectral bands. Ground-based observations of CO 2 laser bands at 10 µm in the atmospheres of Venus and Mars [START_REF] Deming | Observations of the 10-micron natural laser emission from the mesospheres of Mars and Venus[END_REF] were interpreted as non-LTE emissions by several atmospheric models developed in the 1980s [START_REF] Deming | Modeling of the 10micron natural laser emission from the mesospheres of Mars and Venus[END_REF]. On Jupiter, Saturn and Titan non-LTE emissions were identified in the CH 4 band at 3.3 µm [START_REF] Drossart | Fluorescence in the 3μm bands of methane on Jupiter and Saturn from ISO/SWS observations[END_REF]. More recently, CO 2 non-LTE emission at 4.3 µm was detected in the upper atmosphere of Mars and Venus by the PFS (Planetary Fourier Spectrometer) and OMEGA (Visible and Infrared Mapping Spectrometer) experiments on board the European spacecraft Mars Express [START_REF] Formisano | Observations of non-LTE emission at 4-5 microns with the planetary Fourier spectrometer abord the Mars Express mission[END_REF][START_REF] Drossart | Limb observations of infrared fluorescence of CO 2 from OMEGA/Mars Express, Second workshop on Mars atmosphere modelling and observations[END_REF][START_REF] López-Valverde | Analysis of CO 2 non-LTE emissions at 4.3 μm in the Martian atmosphere as observed by PFS/Mars Express and SWS/ISO[END_REF] and by VIRTIS (Visible and Infrared Thermal Imaging Spectrometer) on board the European Venus Express [START_REF] Gilli | Limb observations of CO 2 and CO non-LTE emissions in the Venus atmosphere by VIRTIS/Venus Express[END_REF]. These observations led to the development of a more comprehensive non-LTE model for the upper atmosphere [START_REF] López-Valverde | Analysis of CO 2 non-LTE emissions at 4.3 μm in the Martian atmosphere as observed by PFS/Mars Express and SWS/ISO[END_REF][START_REF] López-Valverde | Modeling the atmospheric limb emission of CO 2 at 4.3 μm in the terrestrial planets[END_REF]. According to these models, during daytime the solar radiation in several near-IR bands from 1 to 5 µm produce enhanced state populations of many CO 2 vibrational levels which cascade down to lower states emitting photons in diverse 4.3 µm bands. These emissions produce what is observed.

Observations

The OMEGA/MEx experiment, combining imaging and spectroscopy in the near infrared, is acquiring a very large dataset of dayside limb observations of the upper atmosphere of Mars in the wavelength range 0.38-5.1 μm [START_REF] Bibring | OMEGA: Observatoire pour la Minéralogie, l'EAU, les Glaces et l'Activité[END_REF] and with an instantaneous field of view (IFOV) of 1 mrad/pixel, corresponding to a vertical spatial resolution of few kilometers. Since January 2004, more than 170 dayside limb observations have been acquired at various locations, seasons and illuminations (Fig. 1).

Results

The variability of the non-LTE emission with latitude, altitude, solar illumination (SZA) and season was analyzed and compared to predictions derived by non-LTE model. Fig. 2 shows the SZA-altitude cross section of radiance at 4.3 µm. Both the altitude and the intensity of the peak emission change with SZA. The peak emission is slightly affected up to ~40º SZA. At larger SZA values both the intensity and the altitude of the peak decline. These results are in agreement with the theoretical expectations for a solar fluorescent emission [START_REF] López-Valverde | Modeling the atmospheric limb emission of CO 2 at 4.3 μm in the terrestrial planets[END_REF]. We are currently studying the behaviour of the instrument during limb sounding by determining the exact size and shape of the instantaneous field of view, the pointing error and the actual noise level, in order to detect potential bias or systematic errors. Once the characterization of the dataset available is achieved, OMEGA limb observations of non-LTE emissions will be used to estimate the temperature and the density variability of the upper mesosphere/lower thermosphere. Density variations are not well known on Mars at these altitudes, the only information being obtained from aerobraking observations at a fixed local time [13] and more recently by the Mars Express ultraviolet spectrometer SPICAM [START_REF] Forget | The density and temperatures of the upper martian atmosphere measured by stellar occultations with Mars Express SPICAM[END_REF]. However, most data acquired by SPICAM was obtained at nighttime, thus not allowing studying the diurnal cycle in detail. OMEGA dayside observations will add new information in this direction. Also, we plan to exploit the good vertical resolution of OMEGA to learn about non-LTE processes in the upper atmosphere and their variation with altitude, from the study of the different CO 2 bands contribution profiles.

Figure 1 :

 1 Figure 1: distribution of dayside limb spectra binned 2º in Ls (solar longitude) and latitude. The colour scale indicates the local hour.

Figure 2 :

 2 Figure 2: map of OMEGA radiances at 4.30 μm versus SZA and altitude. 10 OMEGA cubes were used for this plot.