Ahmad Abdul Karim

Thibaut Gaudin

Alexandre Meyer

Axel Buendia

Saida Bouakaz

Procedural Locomotion of Multi-Legged Characters in Dynamic Environments

Keywords: Procedural Animation, Level Of Details, Multi-Legged Characters, Dynamic Environments

We present a fully procedural method capable of generating in real-time a wide range of locomotion for multi-legged characters in a dynamic environment, without using any motion data. The system consists of several independent blocks: a Character Controller, a Gait/Tempo Manager, a 3D Path Constructor and a Footprints Planner. The four modules work cooperatively to calculate in real-time the footprints and the 3D trajectories of the feet and the pelvis. Our system can animate dozens of creatures using dedicated level of details (LOD) techniques, and is totally controllable allowing the user to design a multitude of locomotion styles through a user-friendly interface. The result is a complete lower body animation which is sufficient for most of the chosen multi-legged characters: arachnids, insects, imaginary n-legged robots, etc.

Introduction

Real or imaginary animals make frequent appearances in video games, films and virtual world simulations. In this context, animated virtual multi-legged characters like arachnids, insects, crustaceans or any imaginary n-legged robots/creatures make these virtual worlds believable and life like. The most common task that these virtual multi-legged characters perform is locomotion: the ability to move in these virtual worlds toward the points of interest. These locomotion animations are quite rich due to the variety of morphologies, gaits, body sizes/proportions and due to the complex-ity of the environment where these multi-legged characters move: overcoming objects, crossing uneven terrain, avoiding moving obstacles, etc. The challenge of modeling such motions arises from the lack of motion capture data inherent to the difficulty to obtain them and by the fact that existing animation techniques mostly address human-like characters [START_REF] Van Welbergen | Real time animation of virtual humans: A trade-off between naturalness & control[END_REF]. We propose a system (See Figure 1) that procedurally generates locomotion animations for dozens of multi-legged characters, without any motion data, in real-time using dedicated level of details (LOD) techniques, and with a reactive adaptation to a dynamic environment. We focused on generating plausible movements [START_REF] Barzel | Plausible motion simulation for computer graphics animation[END_REF] that are, at the same time, believable and fully controllable. Our technique produces a multitude of locomotion styles, generated out of many input parameters like the creature morphology, gait/tempo, the overall locomotion speed, etc. These input parameters can be edited through a user-friendly interface in real-time. We propose a foot path planning system that assign to each foot the best couple (footprint-trajectory), where the footprint is selected among several candidates, and the trajectory is the 3D path that reach this footprint. Our system generates plausible locomotion animation at different level of details (LOD): the evaluation of the possible couples (footprinttrajectory) may be adapted according to the desired LOD, typically depending on quality criteria such as the on-screen importance, visibility, etc. These footprints and trajectories may change in real-time in a reactive manner according to the dynamic of the environment and the obstacles. Finally, our system generates the pelvis trajectory according to the overall desired direction, speed, the actual context of the environment and the feet positions and feedbacks. Thus, our procedural approach provides two important features for interactive applications: reactivity and control.

Related Work

There are several techniques for generating locomotion, Multon et al. in [START_REF] Multon | Computer animation of human walking: a survey[END_REF] and Van Welbergen et al. in [START_REF] Van Welbergen | Real time animation of virtual humans: A trade-off between naturalness & control[END_REF] identified in their surveys the three major techniques: data-driven, physicsbased and procedural. Notice that few of the presented approaches were dedicated or tested on the family of animals and creatures presented in this paper, and to our knowledge, none addresses the performance issues with level of details techniques introduced in our system.

Data-driven techniques use input data to produce the animation (like Motion Capture Data). These techniques, although the most natural looking, are fixed for a specific morphology and a specific context. Physics-based techniques are more generic and adapt better to the environment. They do so by simulating the actual physical forces/torques that act on the articulated body. But they are computationally expensive due to the number of equations to solve, which limits the number of characters simulated at the same time. Several works were proposed to overcome the problems of the previous techniques or to mix them, like: Adapting the motion data to different morphologies (Motion Retargeting) using inverse kinematics in [START_REF] Gleicher | Retargetting motion to new characters[END_REF][START_REF] Multon | From motion capture to real-time character animation[END_REF] or mesh based techniques in [START_REF] Edmond | Spatial relationship preserving character motion adaptation[END_REF]. Using inverted pendulum model (IPM) [START_REF] Kajita | The 3d linear inverted pendulum mode: A simple modeling biped walking pattern generation[END_REF] to avoid complex kinematics and dynamics calculations on the original multi-segments skeleton. In [START_REF] Tsai | Real-time physics-based 3d biped character animation using an inverted pendulum model[END_REF] they animate the upper body of the character using Motion Capture data while the lower body is animated using an IPM. In [START_REF] Kwon | Control systems for human running using an inverted pendulum model and a reference motion capture sequence[END_REF] they optimize their IPM controller using Motion Capture data. In [START_REF] Coros | Locomotion skills for simulated quadrupeds[END_REF] they animate a quadruped in real-time using forward dynamics, using forces data extracted out of real quadruped Motion Capture data. Finally, several papers introduce a specific offline optimization pass, to accommodate for predefined environmental topologies and changes [START_REF] Wu | Terrain-adaptive bipedal locomotion control[END_REF][START_REF] Wang | Optimizing walking controllers for uncertain inputs and environments[END_REF]. The lack of motion data and the difficulties to obtain them for the family of animals and creatures that we simulate has led us to avoid data-driven techniques. In our system we concentrated more on generating plausible movements [START_REF] Barzel | Plausible motion simulation for computer graphics animation[END_REF], making it more adapted (performance wise) to real-time applications than the physics-based ones.

Procedural techniques use mathematical formulas and algorithms to generate the locomotion without any real motion data. They do so using empirical and biomechanics concepts. These techniques offer high level of controllability over data-driven ones: practically no morphological constraints, good adaptability to the environment, etc. These techniques take less computation time per character, making it possible of animating several characters at the same time in real-time. However, they may suffer from the lack of naturality in the movement, a disadvantage that we try to overcome in this paper by integrating basic biomechanics principles. We took inspiration from the work of Boulic et al. in [START_REF] Boulic | A global human walking model with real-time kinematic personification[END_REF] on a human walking model and the work of Singh et al. [START_REF] Singh | Footstep navigation for dynamic crowds[END_REF] that show how avoidance in dynamic crowds can be improved using biomechanics-based footstep planning.

In parallel to the locomotion animation, a path and footprints planning are needed in order to navigate through the environment. In [START_REF] Gyu | Planning biped locomotion using motion capture data and probabilistic roadmaps[END_REF] they search valid footprints using probabilistic navigation graphs that take into account posture transition. Then, these footprints are substituted with corresponding motion clips. Lamarche in [START_REF] Lamarche | Topoplan: A topological path planner for real time human navigation under floor & ceiling constraints[END_REF] generates navigation graphs and ceiling information and uses these information to calculate the best footprints around the optimized trajectory, and to generate the final motion using motion data, allowing for 3D navigation in complex environments. Path planning techniques are also largely studied in robotics. In [START_REF] Kuffner | Footstep planning among obstacles for biped robots[END_REF][START_REF] Kuffner | Online footstep planning for humanoid robots[END_REF][START_REF] Raibert | Bigdog, the rough-terrain quaduped robot[END_REF] (H7 Humanoid Robot™ and Boston BigDog™) their planner produce the footprints using feedforward loops, correcting itself after each step for adaptation. In [START_REF] Chestnutt | Footstep planning for the honda asimo humanoid[END_REF] (Honda ASIMO™) they plans a sequence of footstep positions to navigate toward a goal location while avoiding static and dynamic obstacles, their method uses a time-limited planning horizon. Comparing to our purpose of computer animation, robotics have to deal with the complexity related to the mobile robot mechanical constraints and with the real world physics that we do not fully have.

Most of previously cited techniques are Foot placement driven locomotion, where the feet drive the locomotion and the overall trajectory of the center of mass. In [START_REF] Van Basten | The step space:example-based footprintdriven motion synthesis[END_REF] they generate locomotion by overlaying Motion Capture data on predefined foot placement, while Torkos et al. in [START_REF] Torkos | Footprintbased quadruped motion synthesis[END_REF] generate the locomotion of a quadruped using external imposed footprints, and real-time physics. These techniques suffer from the problem that the lower body (the legs) usually do not control the locomotion. On the contrary, the upper body imposes a logical trajectory that the lower body tries to follow [START_REF] Berthoz | La simplexité[END_REF]. Our system respects this concept: the pelvis adapt its movement based on the feet feedback, but at the same time it imposes the overall trajectory.

When considering locomotion of multilegged animals, the possibility of using real data are strongly limited due to difficulties to capture the motion of real creatures like dogs, horses, insects, spiders, etc. and is sometime impossible like for an imaginary 5-legged character. Thus, Girard et al. in [START_REF] Girard | Computational modeling for the computer animation of legged figures[END_REF][START_REF] Girard | Interactive design of 3d computeranimated legged animal motion[END_REF] propose a fully procedural system to animate multi-legged characters with visually plausible locomotion. Their system is limited to planner terrains. Our work can be considered as an extension into dynamic environments. Johansen in [START_REF] Skovbo | Dynamic Walking With Semi-Procedural Animation[END_REF] adapts existing motion data (biped and quadruped) to the environment using inverse kinematics and footprints prediction. In [START_REF] Mckenna | Dynamic simulation of autonomous legged locomotion[END_REF] they animate a sixlegged character (a cockroach) using reducedlinear articulated body dynamics and gait patterns observed in biology. Wampler et al. in [START_REF] Wampler | Optimal gait and form for animal locomotion[END_REF] present a fully automatic method for generating locomotion and gaits for legged animals based on their shape and by the use of an off-line optimization pass that results in believable animations. In [START_REF] Hecker | Real-time motion retargeting to highly varied usercreated morphologies[END_REF] they use intelligent retargetting of 2-4-6-legged Motion Capture data without any morphological constraint (the user can design their own creature). The objective of our system is to achieve similar results with the procedural techniques in real-time without using any offline components, and with much more control in a dynamic complex environment.

Finally, as our method tries to generate plausible and believable locomotion, we created our controllers in a way that reflects many studies: the effect of limb length and running speed on time of contact and step length in [START_REF] Hoyt | and Department Of Animal. Time of contact and step length: The effect of limb length, running speed, load carrying and incline[END_REF], the relationship between speed and step length in humans in [START_REF] Kuo | A simple model of bipedal walking predicts the preferred speed-step length relationship[END_REF], and animal gaits: galloping, trotting, pacing, etc. in [START_REF] Biren | Body mass distribution & gait mechanics in fat-tailed dwarf lemurs (cheirogaleus medius) & patas monkeys (erythrocebus patas)[END_REF].

Overall Locomotion Controller

Locomotion is the act of moving from one place to another. For most terrestrial animals that means putting one foot in front of the others in a successive way until reaching the designated point of interest (target). During a normal foot movement there are two main phases explained more in [START_REF] Inman | Human locomotion[END_REF]: the stance phase where the foot is blocked on the ground and the swing phase where the foot flies in a parabolic-like curve toward its target without any ground contact. Locomotion cycle is the act of repeating these feet movements, based on a certain rhythm or tempo. Our system follows these principles when generating -procedurally and in real-timethe locomotion of the multi-legged characters. The main advantage of the procedural approach is the reactivity as it adapts to the environment, and the controllability (a list of all controllable parameters is detailed in Section 3.1). Starting from an environment defined by a height map and several obstacles (moving or static), the overall locomotion process is computed by four blocks as shown on Figure 2.

• The character controller is the central main structure that manages the overall locomotion of the multi-legged character (See Section 3.2). It relies on the three other structures to compute the motion of the feet and the overall pelvis displacement.

• The gait manager regulates the feet tempo according to the movement patterns defined by the user (See Section 3.3).

• The 3D path constructor is a utility structure that construct a 3D trajectory of a foot using an efficient discrete representation of the environment that can be easily maintained and updated with dynamic objects (See Section 4).

• The footprints planner evaluates for each foot all the possible targets and trajectories in real-time, and chooses the best couple of footprint target and 3D trajectory (See Section 5).

These four blocks work together in order to generate the movement of the feet and the pelvis in 3D. Now, to obtain the final lower body locomotion movement we apply an inverse kinematics (IK) algorithm. We use a system called the Cyclic Coordinate Descent (CCD) algorithm [START_REF] David | Linear and Nonlinear Programming[END_REF][START_REF] Reklaitis | Linear and Nonlinear Programming[END_REF] which computes the position and orientation of the legs joints that connect the pelvis with its feet. We chose the CCD because of its performance and simplicity: it minimizes the error between the actual end effector position and the desired one by iteratively modifying the angle of the joints. Our system generates the locomotion animations using several input parameters, that the user provides or controls in real-time. All the parameters (except the morphology parameters) can be edited at any time by the user or by any automatic character controller when the focus is to animate a crowd of creatures. An overview of all the parameters follows.

• Multi-legged character morphology: the user provides an initial static skeleton that can be associated with a skinned mesh.

To automatically map the morphology, the user provides the system with the name of each leg hip joint and the name of its end effector (e.g. the joint before the foot).

Out of these inputs our system detects the number of feet, leg sections, the relative hip positions and the initial feet relative positions. The user also needs to provide the system with the desired leg joint limits (for our CCD algorithm), real body center, projected body bounding box, body thickness, and finally the projected foot shape (if it exists) an example is shown on Figure 3.

• Gait/Tempo: using our interface, the user designs for each foot the stance and swing phase cycle. These cycles describe the pattern of the feet movement. The final gait can be symmetrical or asymmetrical.

• Locomotion speed: speed of the movement in meter per second.

•

Step height: the preferred foot step height in meter.

• Feet spacing: the preferred position of each foot, relative to the pelvis.

Character Controller

The character controller is the coordinator of the overall system. It is in charge of two main tasks: managing the movement of the feet and computing the pelvis 3D movement based on the user needs and on the final positions of the feet. Pelvis default movement: the 2D movement of the pelvis on the ZX plan (assuming the Y -axis is up) is calculated based on the speed and orientation. While the computation of the pelvis height is more complex, as shown on Figure 4(b): we first construct a convex hull based on the ground height underneath it and the actual position of the feet. By projecting the multi-legged character center on this convex hull we get the needed pelvis height and by adding the body thickness we get the actual pelvis height. At the same time the character controller adds an oscillating (sinusoidal) movement on this pelvis height, based on the feet phases. The pitch angle of the multi-legged character body is calculated directly from the convex hull. It is the slope of the line segment that contains the projected center of the multi-legged character. Feet movement: Figure 5 explains how the character controller manages the movement of the feet (first task). At each time step, the gait manager informs the character controller about the feet that are going to enter swing phase as detailed in Section 3.3. For each of these feet a preferred footprint target is calculated. As shown on Figure 4(a), the foot step length is calculated based on the current speed, multi-legged character morphology and the foot relative position that the user can impose as illustrated and explained directly on Figure 6. After calculating this foot step length, we use the initial position of the foot to calculate the preferred 3D target position in the environment (See Figure 4(a)).

Foot Path Planning

For Each Foot Do Nothing

Stance

Check 3D Trajectory

Already in Swing Phase

Finally, the character controller calls the foot path planner to compute the 3D trajectory that this foot will follow, as described in Section 5. Feet that were already in swing phase may need a new a 3D trajectory (path) for several reasons: a new pelvis orientation (when turning), a new desired relative position of the foot (Figure 6), a new overall speed, or something changed in the environment thus invalidating its current 3D trajectory or its current target. In these cases the character controller processes this foot as if it just started its swing phase, and therefore it is redirected to the previously explained foot path planning phase. Stance feet are blocked on the ground. Pelvis movement refining: After moving the feet, the character controller refines the pelvis movement based on the feet feedback (shown on Figure 7). Each foot can choose a better target that adapts better to the context (see Section 5). In that case, the character controller averages the offset between the preferred footprint targets (Figure 7 in blue) and the effectively chosen one (in green). The controller incorporates this offset when it is quite significant, based on the multi-legged character morphology, otherwise it discards it.

Gait Manager

The role of the gait manager is to organize and visualize the pattern of the feet's cycle and to perform the transition between patterns. Since a locomotion is cyclic, it seemed natural to represent the gait/tempo with circles. As illustrated on Figure 8, each circle represents a foot, with the colored sectors representing the swing phase portion of the foot movement. The feet needle activates sectors and deactivates others based on its current position while turning. Activating a sector means that the corresponding foot should enter in the swing phase. Gaits interpolation/transition During a locomotion cycle any creature changes its gait constantly in order to adapt to the environment. To accommodate for this changes in the gait and to introduce more variety in the locomotion styles, our gait manager allows the transition/interpolation between any needed gaits on the fly. Each swing phase sector has a start and a finish, we compute the transition to another (destination) gait by simply interpolating the start of the source sector toward the start of the destination sector in the direction that does not pass by the finish of the destination sector. On Figure 9(a) we chose the clockwise direction while on Figure 9(b) we chose the counterclockwise one. By doing so, the interpolation does not pass by any area of the circle that the user does not want to. The interpolation of the duration of the swing phase (the size of the sector) is quite straightforward, and the speed of all these processes can be fixed in advance by the user. So, on each step and when the foot is not active (in stance phase), we replace its disk with the new disk calculated in the interpolation process, resulting in seamless and logical interpolation.

3D Path Construction

During the swing phase, each foot has a current position (source) and a desired footprint target. The role of the 3D path construction is to construct the foot 3D trajectory that navigates through the environment, from this current position toward the footprint target without colliding with any obstacle. This 3D path construction is requested several times with different footprint targets by the foot path planner as it will be explained in Section 5. The environment itself can be very complex, and it can include several objects represented by many triangles. That is why we have oriented this path construction process toward a discrete grid-based approach. Indeed, once the environment is converted and represented by our grid, the path computation becomes independent of the object's complexity.

We also have chosen this over a vectorial based path planning because discrete algorithms are easier to implement.

Grid-based representation of the environment. We convert the 3D environment near the animated multi-legged character into two 2D grids: the obstacles' map and the elevations map. The obstacles' map describes the areas of the environment where the feet are allowed to pass, as illustrated on Figure 10(a). The black cells represent the obstacles, and we call them the forbidden cells. The elevations map contains the elevation of the highest obstacle in each cell as shown on Figure 11. These two maps are computed using the terrain heightmap and the objects: the bounding box of each object is voxelized onto the maps. Concave objects are subdivided into convex ones. We pre-compute the map's representation of static objects. While we compute the map's representation of dy- namic objects when they move near our multilegged character. In order to avoid legs crossing, we add for each foot the projection of the other legs into the obstacles' map as forbidden cells. These maps are relatively small as they are only computed around our animated multilegged character, speeding up calculations. Since a foot could not be punctual, we use the bounding box of the foot shape to increase the size of the obstacles in the opposite direction of the foot, as shown on Figure 12. Doing so guarantees the non penetration of the foot with the obstacles.

Start Position Pelvis Preferred Target

Foot Shape and Orientation

Obstacle Contextual new obstacle

Foot Center

Figure 12: We increase each obstacle size using the bounding box of the foot and its orientation in order to better avoid collision.

Path construction. Always with a performance concern, we firstly compute the trajectory from the actual foot position toward the targeted footprint in the ZX plan using the projections of the source and the target on the ZX plan (in black on Figure 11). Our system uses a shortest path computation that processes a Wavefront algorithm combined with the potential field method [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF] for the obstacle avoidance. This method is quite similar to the well-known Dijkstra's shortest path algorithm.

The resulting path off the wavefront algorithm is constructed out of several cells that we call the naive cells of the plan (in yellow on Figure 10). We refine these naive cells, using queries of line of sight (no intersection with a forbidden cell) to eliminate unwanted cells, and we obtain the final cells of the the plan (in red). These cells are used to build the 2D path represented by a parametric Hermite curve as shown on Figure 10)(a).

We sample this 2D curve and elevate it in 3D using the elevations map. Resulting waypoints are used as control points to define a 3D Hermite curve: this curve represents our 3D trajectory through the environment as shown on Figure 10(b).

Footprints and Feet Path Planning

In Section 3.2, we explained that starting from the initial foot position the character controller calculates a preferred footprint target according to the locomotion parameters and the surrounding environment (Figure 11). But these calculations do not take into consideration the actual state of the foot and its preferences (e.g. the preferred target calculated by the character controller can be too close to an obstacle). That is why in this step we calculate and assign to each foot the best trajectory toward the best target in the current environment.

Potential Footprints

Our algorithm evaluates several footprint targets by exploring the potential cells around the preferred footprint (the one computed by the character controller in Figure 11). Several operations are done to do that. In both maps (the obstacles' map and the elevations map), we firstly increase the size of the obstacles using the foot shape (See Figure 12 and Figure 11 in red). Secondly, to eliminate possible intersection with other feet we fill the obstacles' map with the position of these other feet. Thirdly, we identify all the -potential-cells that may accept a footprint: all the non-forbidden cells that are not a size increase ones (Figure 13). For each potential target, we calculate a score based on several criteria's: distance to the preferred footprint, difference of elevation between the footprint and the surrounding cells, leading or not to a feet crossing, etc. The combination of all these criteria's provides the final cell's score (footprint score) that we normalize between 0 and 1. Figure 13 shows these scores. Avoidance-wise. During the 3D path construction (See Section 4), the forbidden cells in the obstacles' map designate the parts of the environment that the 2D trajectory is going to avoid and go around. This 2D trajectory passes through cells that have an elevation. The 3D path constructor uses this elevation to construct the final 3D trajectory. The system uses this obstacles' map to process an obstacle: either going around it or going over it. If the system wants to go around an obstacle, it just needs to add it to the obstacles' map as forbidden cells. In that way, the resulting 2D trajectory will definitely go around it. While if the system wants to go over an obstacle, it does not need to do anything special as the obstacle is already present in the elevations map. And the obstacle will be avoided accordingly based on its elevation. In order to achieve this avoidance distinction in an optimal way in a complex environment (height map plus obstacles), our system discretize the elevations map on the Y -axis into slices (for instance slices of 10cm). For each slice, the system fills the obstacles' map with the cells that have higher elevation than this slice. In that way, all calculated trajectories will go over all obstacles, objects and pieces of the environment that have an elevation lower or equal to the slice's elevation. At the same time, all calculated trajectories will go around all obstacles, objects and pieces of the environment that have an elevation higher than the chosen slice's elevation.

Best Pair of Footprint and Path

So each trajectory can go around or over each obstacle, which generates multiple trajectories toward a target. Each one of these trajectories has a score. Our search space is all the possible trajectories that go from the starting point toward all the possible targets. Out of this multitude of possibilities, our main algorithm picks up the best couple (target-trajectory) in an intelligent way.

Our algorithm (result shown on Figure 13) loops on all the possible targets based on their score (in a descending way). For each one of these targets, the algorithm first generates the 2D trajectory and scores it, if the score of the couple (target-trajectory in 2D) is better than the current best couple found till now, it continues. Then it generates the 3D trajectory based on the 2D one and scores it, if the score of the new couple (target-trajectory in 3D) is better than the current best couple, then it tags this new couple as the best one and continues. The algorithm continues evaluating the couples until it reaches a couple (target-trajectory in 3D) with a score worse than the current best one, in this case it stops and the current best couple is the best one. So, at the end of the algorithm we obtain the best footprint target and 3D trajectory that this foot should follow. We calculate the length of this 3D trajectory, and using the needed time given by the character controller, we move the foot on the curve at a constant speed.

Trajectory scoring. We generate for each created 2D/3D trajectory a score based on the application specifications, like the total length of the path, the acceleration and the curve tangents profile, etc. For instance, the system shown in the accompanying video prefers trajectories that are more straight (direct) with less curvature, a preference observed in biomechanics as it minimizes the energy cost [START_REF] Alexander | Optima for animals[END_REF][START_REF] Alexander | Principles of animal locomotion[END_REF].

Level of Details techniques (LOD)

Our main algorithm (Section 5.2) execution time can be controlled easily when simulating many multi-legged characters in order to accelerate the simulation itself. This comes from the ability of controlling the number of evaluated couples (target-trajectory) and from the ability of controlling the size of the slices used when avoiding obstacles. But in order to always have a plausible locomotion, we start applying this LOD after we find the first valid couple (target-trajectory), in this way we ensure that the algorithm will assign to each foot a valid trajectory toward a valid target (although it is not the best couple).

In our simulation and for off-screen characters, the algorithm stops directly when finding this first couple. While for the on-screen characters, we limit the number of couples evaluated after finding this first couple in a linear way based on the distance of that multi-legged character from the camera. In this way for far on-screen characters the algorithm stops also when it finds the first couple. In the same way, for off-screen characters the algorithm tries to go over any obstacle using no slices (not going around any obstacle). While for the on-screen characters, we increase the size of the slices (for example 10cm, 15cm...no slices) in a linear way based on the distance of that character from the camera. This LOD can be regarded as: the more the character is far away from the camera (or off-screen) the less our algorithm worries if the chosen couple is comfortable. In next section

Results

In our implementation, the environment is represented by two classes of objects: the terrain and the obstacles. A height map defines the terrain and is loaded without any pre-processing analysis (See Figure 11 ). The obstacles are mainly represented by crates with various sizes and orientations randomly generated on the terrain (See Figure 1 The results show that our system is well adapted for real-time applications. In Figure 14 and in the accompanying video we show many morphologically different characters animated in real-time using our test machine (Core i7 2.7 GHZ, 8 GB RAM, 6870 ATI Radeon HD with 1GB vRam). Table 1 shows average computation time for 100 8-legged characters.

In LOD we make sure that all of the characters are in the field of view of the camera, while in LOD Zoomed we zoom on one character and make sure that at least 50% of the characters are shown on the screen. As we can observe, the maps preparation is fixed, as it is pre-computed once, for all characters at the same time. In LOD Zoomed the system loses some computation time per character as there are more characters doing full search for the best couple, while in the same time gains computation time in the IK systems as we do not calculate the IK for the off-screen characters.

Conclusions

We presented a system capable of procedurally generating believable locomotion animation of several multi-legged characters (like arachnids, insects, or any imaginary n-legged robots/creatures) in real-time with no a priori motion data nor any information about the environment. Our system is quite generic and can be applied on a variety of morphologies. In runtime the user can control many parameters like the the gait, the speed, the direction, etc.

Nevertheless, the current system assumes that the feet of the character can be represented by a point. Our model does not include the feet's metatarsus, which has a great impact on the naturalness of the locomotion when considering more complex characters. As a future work, it will be interesting to study this point, specially for bipeds/humans. And to study the addition of a flexible spine-like structure, which is essential when simulating quadrupeds [START_REF] Coros | Locomotion skills for simulated quadrupeds[END_REF]. We are also planning on improving the pelvis behavior to incorporate dynamic-like balance reactions. Our system can serve as input to higher level characters' controllers that would like to provide more animations than only the locomotion, like touching objects for discovering the environment, studying insect's behavior, etc.

Figure 1 :

 1 Figure 1: (a) Spider model avoids crates flagged as forbidden and steps on others flagged as safe (b) Ant model on a height map (c) Imaginary 5-legged Robot.

Figure 2 :

 2 Figure 2: System Overview

Figure 3 :

 3 Figure 3: Spider Model: (a) internal representation of the spider model with its IK system, in pink is the projected bounding box, in blue the feet orientation. (b) the actual spider 3D model.

Figure 4 :

 4 Figure 4: The vertical slice in the environment can contain objects and obstacles, (a) computation of the preferred footprint target for each foot. The naive target position is calculated by querying the environment about the elevation of the 2D point resulted from the initial position + the step length (b) Computation of the pelvis' height using several convex hulls that covers the environment slice that interests us

Figure 5 :

 5 Figure 5: Main job of the character controller per foot: based on the current foot state it can take different decisions

Figure 6 : 8 -

 68 Figure 6: 8-Legged Character Feet Spacing Interface

Figure 7 :

 7 Figure 7: Feet feedback: During the computation of the pelvis displacement, we use the feet feedback.

Figure 8 :

 8 Figure 8: An example of a four feet's gait as shown by the gait manager. With this interface, the user can edit the pattern.

Figure 9 :

 9 Figure 9: Interpolations Options: (a) deciding to do a clockwise position interpolation based on the actual disposition of the gait disks (b) deciding to do a counter-clockwise position interpolation

Figure 10 :

 10 Figure 10: (a) We firstly construct the path in a 2D voxel-based representation of the environment (yellow dots) and then generate the final curve (in black) with a selection of waypoints (red dots). (b) Using the elevations map and the previous 2D trajectory we construct the 3D trajectory.

Figure 11 :

 11 Figure 11: An example of our environment discretization: pink spheres represents the ground, yellow spheres contain the corner of an obstacle, green spheres contain an obstacle edge, blue spheres are an obstacle interior and red spheres are an obstacle size increase.

Figure 13 :

 13 Figure 13: Potential targets: the color varies from green (best target) (score = 1) to blue (worst target) (score = 0). Crossed cells are the cells processed by our algorithm, in pink possible trajectories, in black the chosen one.

  and 14). These crates can be static or can move on the terrain (See the accompanying video). Without changing anything in the system, we have tested it on different kinds of terrain (flat surface, smooth terrain, regular stairs, etc.) with different kinds of obstacles (static crates and moving ones). The size of the discretization maps used in our tests is 70 × 70 with 7cm cells, which consumes little memory and is precise enough since the maps describe only the environment close to each animated character. The animated multi-legged characters gaits are inspired from biology studies[START_REF] Wilson | Stepping patterns in tarantula spiders[END_REF][START_REF] Blake | Efficiency And Economy in Animal Physiology[END_REF]

Figure 14 :

 14 Figure 14: Simulation snapshot.

Table 1 :

 1 Average computation time for 100 8-legged characters

		Total for Characters Maps Preparation CCD IK Systems Average FPS
	LOD	0.026s	0.004s	0.013s	~30fps
	LOD Zoomed	0.028s	0.004s	0.006s	~30fps
	No LOD	0.045s	0.004s	0.013s	~18fps
	we speak about the actual performance of our
	real-time system.


S Soum

V Carlotti
email: carlotti@enscbp.fr.

S Rejsek-Riba

S Lewandowski
email: simon.lewandowski@onera.fr.

S Remaury

Solé
email: sole@map-coatings.com.

M Planes

C Le Coz

A Soum

S Carlotti

V Rejsek-Riba

S Solé

Polydimethylsiloxane/Additive Systems for Thermal and Ultraviolet Stability in Geostationary Environment

  L'archive ouverte pluridisciplinaire

I. Introduction

T HE space environment around the Earth is exceptionally complex and has a significant impact on the lifetime and performance of any orbiting satellite. Space irradiation is an important research area in aerospace industry. High-energy particles induce damage to materials and electronic components in spacecrafts [START_REF] Grossman | Space Environment Effects on Polymers in Low Earth Orbit[END_REF][START_REF] Özçam | Effect of Ultraviolet/ozone Treatment on the Surface and Bulk Properties of Poly(Dimethyl Siloxane) and Poly(Vinylmethyl Siloxane) Networks[END_REF][START_REF] Xiao | Optical Degradation of Silicone in ZnO/Silicone White Paint Irradiated by <200-keV Protons[END_REF]. Silicone resins are often used and play a key role as coatings for space materials due to their high chain mobility and flexibility. For example, polydimethylsiloxane (PDMS) is a silicone polymer known to possess a high thermal stability and good resistance to oxidation and hydrolysis. The low glass transition temperature of polysiloxane resins makes them attractive for a wide variety of common applications such as adhesives and fabric finishing agents, but they are also increasingly considered in high-value aerospace applications, in particular as varnishing or encapsulating [START_REF] Colas | Silicones: Preparation, Properties and Performances[END_REF][START_REF] Andriot | Silicones in Industrial Applications[END_REF]. For these specific applications, PDMS resins have to be qualified in optical, thermal, and outgassing properties [START_REF] Guillaumon | Development of a New Silicone Adhesive for Space Use: Mapsil® QS 1123[END_REF]. To consider their use as thermal control coating, it is necessary to improve their thermal and photostability in harsh environment.

The degradation mechanisms of PDMS under space conditions were described by several groups [START_REF] Delman | Photodecomposition of Polymethylsiloxane[END_REF][START_REF] Smeenk | Space Environmental Testing of Flexible Coverglass Alternatives Based on Siloxanes[END_REF][START_REF] Zhang | Damage Effects and Mechanisms of Proton Irradiation on Methyl Silicone Rubber[END_REF][START_REF] Hill | Molecular Weight Changes and Scission and Crosslinking in Poly (Dimethyl Siloxane) on Gamma Radiolysis[END_REF][START_REF] Jochem | Effects of 400 keV Electrons Flux on Two Space Grade Silicone Rubbers[END_REF]. Their degradation under UV irradiation depends on the UV wavelength range and the dioxygen presence. Characteristics bonds of PDMS (Si-H, Si-C, Si-O, and C-H) are subject to homolytic cleavage and formation of free radicals. These highly reactive radicals can undergo various reactions such as recombination, disproportionation, and hydrogen abstraction. It is proposed that photon absorption by unsaturated bonds formed in the material leads to further photochemical reactions and eventually to the degradation of the PDMS resin by causing several inter-and/or intramolecular rearrangements [START_REF] Vasilets | Photooxidation of Siloxane Polymers Under Vacuum Ultraviolet Irradiation[END_REF][START_REF] Fischer | Degradation Mechanism of Silicone Glues Under UV Irradiation and Options for Designing Materials with Increased Stability[END_REF]. Consequently, improved UV stability might be accomplished by limiting the formation of uncontrolled species, which bring important degradations. There are mainly two strategies to limit the material degradation against solar irradiation: the use of additives or protective coatings. In the last few years, several methods were investigated to restrict the damage caused by UV irradiation, namely the use of additives such as silsesquioxanes [START_REF] Lin | U.S Patent Application for Silsesquioxane Polymers, Method of Synthesis, Photoresist Composition, and Multilayer Lithographic Method[END_REF] and inorganic [START_REF] Chiavari | Organosilane Coatings Applied on Bronze: Influence of UV Radiation and Thermal Cycles on the Protectiveness[END_REF] or organic copolymers [START_REF] Kotlík | Acrylic Copolymer Coatings for Protection Against UV Rays[END_REF]. To protect the PDMS films, some researchers have also envisaged the deposition of ZnO or ZnO-doped quantum dots in the form of multilayers to obtain a UV-blocking material [START_REF] De Lima | Transparent UV-Absorbers Thin Films of Zinc Oxide: Ceria System Synthesized via Sol-Gel Process[END_REF][START_REF] Girigoswami | Studies on Polymer-Coated Zinc Oxide Nanoparticles: UV-Blocking Efficacy and in Vivo Toxicity[END_REF]. Another drawback of PDMS results from the insufficient thermal stability of the polymer when used as thermal control coating [START_REF] Hanu | Thermal Stability and Flammability of Silicone Polymer Composites[END_REF]. This thermal stability could be improved by incorporating silica nanoparticles within the matrix or cross-linking the matrix with polymethylmethoxysiloxane [START_REF] Han | Improvement of Thermal Resistance of Polydimethylsiloxanes with Polymethylmethoxysiloxane as Crosslinker[END_REF][START_REF] Chen | Thermal Stability, Mechanical and Optical Properties of Novel Addition Cured PDMS Composites with Nano-Silica Sol and MQ Silicone Resin[END_REF]. Loading levels between 10 and 50% are necessary to obtain significant effects on the PDMS properties, but it is accompanied with a loss of transparency.

Other additives were reported in the literature as UV stabilizers, i.e., hindered amine light stabilizers (HALSs) [START_REF] Bojinov | Synthesis of Novel Bifunctional Hindered Amine-UV Absorber Polymer Stabilizers[END_REF], ultraviolet absorbers (UVAs) [START_REF] Dobashi | Dependence of Ultraviolet Absorbers' Performance on Ultraviolet Wavelength[END_REF], and carbazoles [START_REF] Sawicki | Synthesis and Ultraviolet Absorption Spectra Studies of 2,3-and 3,4-Disubstituted Carbazoles[END_REF]. HALSs, derived from 2,2,6,6-tetramethyl piperidine, are efficient light stabilizers that prevent the degradation of various polymers such as polyethylene. The mechanism of UV inhibition by HALS and UVA differs due to their chemical structures. In contrast, carbazoles and their polymeric derivatives act as UV absorbents due to their aromatic structure [START_REF] Tsuyoshi | Synthesis and Properties of 1,8-Carbazole-Based Conjugated Copolymers[END_REF]. In this work, different UV stabilizers were mixed before the crosslinking of the PDMS by hydrosilylation [START_REF] Lewis | Platinum Catalysts Used in the Silicone Industry[END_REF][START_REF] De | Hydrosilylation of Siloxanes in the Presence of Pt(II) and Pt(IV) Complexes[END_REF]. The thermal and optical properties of PDMS were studied under UV irradiation at room temperature and under vacuum.

II. Experimental Part

A. Materials PDMS was provided by MAP as a bicomponent material: base and curing agent. The base contains predominantly vinylterminated PDMS and Pt catalyst (Karstedt). The curing agent is a mixture of vinyl-terminated PDMS and hydride-terminated PDMS. Reference PDMS samples were prepared by mixing the base with the curing agent in a 10:1 ratio by weight. UV stabilizers are commercially available and purchased from Sigma Aldrich: 3,6-dibromo-9Hcarbazole (carbazole 1), bis(2,2,6,6-tetramethylpiperidin-4-yl)decanedioate (HALS 1), bis(2,2,6,6-tetramethyl1(octyloxy)piperidin-4yl) decanedioate (HALS 2), 2-(tert-butyl)-6-(5-chloro-2H-benzo[d] [1,2,3] triazol-2-yl)-4-methylphenol (UVA 1), and (2-hydroxy-4methoxyphenyl) (2-hydroxyphenyl)methanone (UVA 2). The additive solutions were prepared at 10 mg∕ml in n-heptane, dispersed into the base, and sonicated. Afterward, the mixture was homogenized by using a mechanical dispersion. Then, the solvent was removed under vacuum for 2 h, and the mixture was added to the curing agent for 15 h at 70°C to obtain the final cross-linked materials. The prepared formulations contained 0.5 to 2.0 wt% of additives. The thickness of the prepared samples was measured with a sliding caliper and was approximately 100 μm.

B. Initial Characterizations

Preliminary characterizations were carried out to validate the industrial cross-linking process and the resulting properties of the new silicone material.

Thermogravimetric analysis (TGA) of PDMS samples (10-15 mg) was performed on a TA Instruments Q50 under nitrogen atmosphere using platinum pans. Temperature profiles ranged from 25 to 600°C with a heating rate of 10°C • min -1 .

Differential scanning calorimetry (DSC) measurements were carried out on a TA Instruments DSC Q100 LN2 (t∕T modulation) by heating from -150 to 150°C with a ramp of 5°C • min -1 and with samples weighted around 5 mg. The analyses were made with helium atmosphere using aluminum pans; the instrument was calibrated with indium sample.

ATA Instrument RSA3 (dynamic mechanical analysis, DMA) was used to study the thermomechanical properties of the PDMS samples under nitrogen atmosphere from 0 to 200°C at a heating rate of 5°C • min -1 . The measurements were performed in compression mode at a frequency of 1 Hz, an initial static force of 0.5 N, and strain sweep of 0.1%.

C. Aging and Postcharacterizations

UV irradiation was performed in a vacuum chamber coupled to a short arc xenon lamp (Oriel solar simulator, 1000 W). This equipment allows reproducing the solar irradiation distribution (250-2500 nm) close to the UV solar conditions. During the UV experiment, the pressure was maintained under 10 -5 mbar, and a water-cooling system was used to control the temperature of the sample holder (25-30°C). A black paint was recovering the vacuum chamber to inhibit the reflection of UV radiations. The experiment time was in a range 450-500 of equivalent solar hours (ESH) corresponding to six months of geostationary flight. The change of the transmittance after UV irradiation allows estimating the UV resistance. The initial materials should have a high initial transmittance (greater than 90%) in the wavelength range from 250 to 1000 nm. The change in transmittance between 250 and 400 nm was used to estimate the UV stability of the composites, due to the fact that this wavelength range corresponds to 90% of the UV doses emitted by the sun.

Ultraviolet-visible-near-infrared (UV-vis-NIR) spectroscopy analyses were performed before and after photoaging (250-1000 nm) on a Perkin Elmer Lambda 900 double beam combined with an integrating sphere of 150 mm diameter in a spectral range from 250 to 1000 nm. The spectra were obtained with a scanning speed of 400 nm∕ min and a resolution of 1 nm.

To estimate the degradation of the materials all over the solar spectrum and for different wavelength ranges, the following formulas of solar transmittance were used [START_REF]Space Product Assurance Availability Analysis[END_REF][START_REF] Ohnishi | Measurements of Solar Absorptance and Thermal Emittance of Lunar Simulants[END_REF]:

Ts R 1000 250 Tsλ • Is • dλ R 1000 250 Is • dλ (1a) Ti R 1000 250 Tiλ • Is • dλ R 1000 250 Is • dλ (1b) Tf R 1000 250 Tfλ • Is • dλ R 1000 250 Is • dλ (1c)
where Ts the mean spectral transmittance of the material, Is the spectral irradiance of the sun (watt∕micrometer∕meter 2 , W∕μm∕ m 2 ), Ti is the initial mean transmittance before irradiation, and Tf is the final mean transmittance after irradiation [dλ

1 nm].
ΔTs is the level of degradation on each wavelength range in percent:

ΔTs% Ti -Tf ( 2
)
and Δg is the gain in UV stability in percent:

Δg% ΔTs Neat PDMS -ΔTs PDMS∕additive (3)

D. Evolution of the Optical Properties During the Photoaging

Optical performance is the main parameter in the space domain that defines the quality of the UV protective coating. To investigate the optical performance, an unexposed composite coating was compared to a sample that has received 450-500 ESH. To meet the requirements of the optical properties, the maximum loading level of additives employed in the PDMS matrix was adjusted to 2.0 wt%. The photostability of the PDMS/additive was investigated by performing photoaging experiments based on a specifically designed procedure for the characterization of aerospace coatings. For these experiments, thin PDMS films (100 μm thickness) containing 0.5 to 2.0 wt% of UV additives were prepared, and the transmittance was measured before and after UV irradiation. The UV photoaging tests were conducted through two irradiation sessions. To calculate more precisely the improvement in UV stability of the films, Eq. (1a) was employed for two wavelength ranges (250-1000 nm and 250-400 nm). These calculations permit to obtain the mean transmittance of the materials on the wavelength range, which was used to compare the behavior of each composite under UV irradiation. The initial mean transmittance Ti is calculated for each film before irradiation Eq. (1b). Afterward, the final mean transmittance Tf is also calculated after irradiation [Eq. (1c)]. Equation ( 2) was used to obtain the level of degradation on each wavelength range. Finally, the gain in UV stability (Δg) was estimated with Eq. (3).

III. Results and Discussion

Cross-linked PDMS were synthesized and characterized in the presence of several UV stabilizers, which are illustrated in Fig. 1. Two different chemical structures of HALS only differing by the moiety on their piperidine unit were studied. HALS 1 possesses a hydrogen atom on the piperidine nitrogen, which can form hydrogen bonds with the oxygen atoms in the PDMS chains. In contrast, HALS 2 bears an alkoxy chain, which can act as a plasticizer of the PDMS. The efficiency of HALS is explained by a regenerative process, which was described by McCusker and Gijsman [START_REF] Mccusker | A UVA/HALS Primer: Everything You've Ever Wanted to Know About Light Stabilizers-Part II[END_REF][START_REF] Gijsman | The Mechanism of Action of Hindered Amine Light Stabilizers[END_REF]. HALSs are made of secondary or tertiary cyclic amine units and of ester groups. After oxidation of the amine moiety, the formed nitroxide can act as radical scavenger. More important, the nitroxy radicals can react with polymeric alkyl radicals to form alkyloxy hindered amine derivatives, which are postulated to further react with alkylperoxy radicals, forming dialkylperoxides and regenerating the active nitroxy radicals [START_REF] Balabanovich | Thermal Stability and Thermal Decomposition Study of Hindered Amine Light Stabilizers[END_REF]. In addition, HALS can be used to enhance flame retardancy of polymeric materials. Two commercial UV stabilizers, i.e., 2; 2 0 -dihydroxy-4-methoxybenzophenone (UVA 1) and 2-tert-Butyl-6(5-chloro-2H-benzotriazol-2yl 4-methylphenol (UVA 2), were also used. UVAs absorb a photon to reach an excited state. Phenolic types can undergo a rapid hydrogen transfer to a heteroatom in their first excited singlet state to form a vibrationally excited ground state species, which can lose thermal energy and transfer back the hydrogen to reform the fundamental state. This cycle is completed within nanoseconds for most UVAs [START_REF] Pickett | Review and Kinetic Analysis of the Photodegradation of UV Absorbers[END_REF][START_REF] Razaz | Synthesis of Anthraquinone Containing Acrylates and Their Applications as UV-Absorber and UV-Curable Monomers[END_REF]. The photon absorption by the PDMS chain is therefore expected to be decreased. Finally, carbazole-based compounds present also an interest as UV stabilizers for PDMS resins because they are wellknown fluorescent labels in biology and possess an excellent UV absorption [START_REF] Van Dijken | Carbazole Compounds as Host Materials for Triplet Emitters in OLEDs: Polymer Host for High Efficiency LEDs[END_REF].

A. Hindered Amine Light Stabilizers as Polydimethylsiloxane Additives

All systems (PDMS/additive), with a variation of 0.5-2.0 wt% UV stabilizers added, were characterized by TGA, DSC, and DMA to determine their thermal properties, elastic modulus, and cross-linking process before UV irradiation. The temperature of cross-linking was recorded by DSC analysis on the neat PDMS. The thermogram revealed an exothermic event at 80°C corresponding to the formation of a three-dimensional network. To confirm the efficiency of the curing process (15 h at 70°C), DSC analyses were performed on neat crosslinked PDMS, and no exothermic peak during the first temperature ramp was observed at 80°C. In the same way, the features of the crosslinked PDMS/additives were investigated. After the curing process, no exothermic signal was recorded in the presence of the HALS. The glass transition temperature of the PDMS was unchanged at -120°C. Elastic modulus measured by DMAwas also shown to be identical for the neat and composite PDMS at about 10 6 Pa and in the expected range of elastomeric materials [START_REF] Meththananda | The Relationship Between Shore Hardness of Elastomeric Dental Materials and Young's Modulus[END_REF].

TGA analyses revealed that the amount of added HALS has an impact on the thermal stability of the material (Fig. 2). The main degradation temperature of the neat PDMS recorded at Td 5% 360°C corresponds to the depolymerization of PDMS backbone through random chain scission reactions and formation of low-molar-mass cyclic siloxane species [START_REF] Camino | Polydimethylsiloxane Thermal Degradation Part 1. Kinetic Aspects[END_REF][START_REF] Camino | Thermal Polydimethylsiloxane Degradation. Part 2. The Degradation Mechanisms[END_REF]. The degradation temperature at 5% weight loss for the silicone resin with 0.5 wt% HALS 2 was close to the initial PDMS (Fig. 2). No influence on the thermal stability was observed with higher amounts of HALS 2. In contrast, HALS 1 showed a significant effect on the thermal stability with a Td 5% up to 395°C.

UV-vis-NIR spectroscopy was then performed to observe any difference in UV stability between the neat silicone resin and the resins containing 0.5 to 2.0 wt% of HALS 1 and 2 (Fig. 3). The change in UV transmittance of the PDMS films before irradiation is presented Fig. 3a. Concerning the neat PDMS, the transmittance before irradiation is relatively constant throughout the wavelength region from 250 to 1000 nm with a transmittance of around 92%. Regardless of the additives used, the optical properties before irradiation were similar, and the initial transmittance measured was above 90%. After the UV irradiation (450-500 ESH), a loss of transmittance equal to 32% at 250 nm is observed for the neat PDMS (Fig. 3b). Its degradation is induced by the homolytic bond cleavage on the main chain leading to inter-and/or intramolecular rearrangements. These photochemical reactions explain the yellowing of the neat resin. This coloring could also be caused by the Karstedt's catalyst because, under specific UV irradiation, colloidal Pt species are formed, leading to a loss of optical transparency [START_REF] Lewis | Platinum-Catalyzed Hydrosilylation-Colloid Essential Step[END_REF]. Regardless of the nature and the amount of HALS investigated, a higher UV degradation was observed in the wavelength ranges of 250-1000 and 250-400 nm for all materials except for the PDMS containing 0.5 wt% of HALS 1 (Fig. 3b). An average increase of 2.6% in UV stability was determined for the latter in the wavelength range from 250 to 400 nm. The radicals formed from this additive are expected to trap radicals coming from the polymer degradation. Gijsman et al. reported that the efficiency of HALSs as UV stabilizers depends on the presence of dioxygen and that high O 2 levels are beneficial [START_REF] Gijsman | The Mechanism of Action of Hindered Amine Light Stabilizers[END_REF]. The UV irradiation was performed here under high vacuum (10 -5 mbar), meaning that only traces of dioxygen are present and that the efficiency of HALS is expected to be lower. Besides, the two studied structures only differ by the moiety on their piperidine unit. HALS 1 possesses a hydrogen atom at the piperidine nitrogen, whereas HALS 2 bears an alkoxy chain. The bond enthalpy is higher for N-H bond than N-O bond (i.e., 93 versus 55 kcal • mol -1 ), meaning that the homolytic cleavage of the N-O bond is much easier [START_REF] Balabanovich | Thermal Stability and Thermal Decomposition Study of Hindered Amine Light Stabilizers[END_REF]. The degradation of PDMS with HALS 2 is more pronounced and can be explained by an easier formation of radicals that contribute to the degradation instead to trap the radicals formed by the degradation of the PDMS. This hypothesis was confirmed by an increased loading of additives, whatever the structure, which leads to a pronounced degradation and as also observed for polypropylene [START_REF] Kaci | Kinetic Study of Hindered Amine Light Stabilizer Photografting in Poly(Propylene) Films Under Natural Weathering and Accelerated UV Conditions: Effect of Additive Concentration[END_REF].

B. Ultraviolet Absorbers and Dibromocarbazole as Polydimethylsiloxane Additives 2; 2 0 -Dihydroxy-4-methoxybenzophenone (UVA 1), 2-tert-Butyl-6(5-chloro-2H-benzotriazol-2yl 4-methylphenol (UVA 2), and 3,6dibromo-9H-carbazole were studied as UV stabilizers for PDMS resins. In contrast to the HALS additives, UVAs were specifically developed for space applications and other polymers because their UV inhibition mechanisms do not require oxygen. Unfortunately, no miscibility between those UVAs and the PDMS could be obtained, resulting in heterogeneous and not usable materials.

The cross-linking process of a silicone resin containing 1.0 wt% of 3,6-dibromo-9H-carbazole was followed by DSC. A curing process of 15 h at 70°C was again appropriate. The elastic modulus measured by DMA at 10 6 Pa was not modified by addition of the carbazole derivative. Halogens (Cl and Br) are well known to be good flame retardants and were used for decades in electrical engineering applications, textile treatments, product construction, and polyurethane foams to prevent ignition or to delay and slow the spread of fire by interrupting or disturbing the combustion process [START_REF] Avondo | The Role of Phosphorus and Bromine in Flame Retardancy[END_REF][START_REF] Hutzinger | Environmental Chemistry of Flame Retardants Part 1. Introduction and Principles[END_REF][START_REF] Hutzinger | PBDD and PBDF from Brominated Flame Retardants: Combustion Equipment, Analytical Methodology and Synthesis of Standards[END_REF]. Bromine atoms in dibromocarbazole are also expected here to induce higher thermal stability than neat PDMS. The PDMS/dibromocarbazole thermogravimetric analysis showed a strong increase (about 100°C) of the degradation temperature as compared to the neat PDMS, with a Td 5% close to 460°C (Fig. 4).

UV-vis-NIR spectroscopy was then performed before and after UV irradiation on a PDMS prepared with 1.0 wt% of dibromocarbazole. The bond enthalpy is relatively low for C-Br (68 kcal • mol -1 ), and the homolytic cleavage can occur quite easily. The change in UV transmittance of the PDMS films before irradiation is presented in Fig. 5. The transmittance of the neat PDMS before irradiation is rather constant throughout the wavelength region of 250-1000 nm with a transmittance around 90%. In contrast, the transmittance before UV irradiation for a silicone resin containing 3,6-dibromo-9H-carbazole is 30% lower for short wavelength as compared to the reference material and is explained by the light absorbance of the carbazole derivative at ∼280-nm [START_REF] Li | Novel Carbazole Derivatives with Quinoline Ring: Synthesis, Electronic Transition, and Two-Photon Absorption Three-Dimensional Optical Data Storage[END_REF].

After the UV irradiation, a loss of transmittance is observed between 250 and 400 nm, going down to 7% at 250 nm for the neat PDMS (Fig. 5) and 3% for the PDMS/1.0 wt% dibromocarbazole. The difference of transmittance between neat PDMS shown Figs. 3, 5 is due to an irradiation time difference: 450 and 500 ESH, respectively. The degradation of the silicone resin with 1.0 wt% of dibromocarbazole was not satisfying, with a calculated average decrease of 14.5%.

IV. Conclusions

Different additives (hindered amine light stabilizer (HALS), ultraviolet absorber (UVA), and dibromocarbazole) for silicone resins were investigated with the aim to enhance the thermal and UV stabilities of cross-linked polydimethylsiloxane (PDMS). The chemical compatibility between the UV stabilizers and the polymer was shown to be a key feature. HALS and dibromocarbazole yielded homogeneous materials when UVAs did not. Addition of 0.5 wt% of HALS 1 can slightly improve both the thermal stability (from 360 to 395°C) and the UV stability (up to 2.6%). Increasing the amount of this additive led to a decrease of the UV stability and a yellowing of the resin. The use of a dibromocarbazole revealed the efficiency of an aromatic/halogen structure as a thermal stabilizer of a PDMS resin.

Fig. 1

 1 Fig. 1 Structures of studied UV stabilizers for cross-linked PDMS.

Fig. 2

 2 Fig. 2 Thermal degradation profile of PDMS with HALS 1 and 2 under nitrogen.

Fig. 3

 3 Fig. 3 UV spectra (250-1000 nm range) of PDMS films prepared with HALS 1 and 2: a) before, and b) after UV irradiation (450 ESH).

Fig. 4

 4 Fig.4Thermal degradation of PDMS with 1.0 wt% of dibromocarbazole under nitrogen atmosphere.

Article in Advance / PLANES ET AL. Downloaded by UNIVERSITA DEGLI STUDI DI MILANO on December 22, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.A33484

Article in Advance / PLANES ET AL.

Downloaded by UNIVERSITA DEGLI STUDI DI MILANO on December 22, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.A33484

Acknowledgments

This work was supported by a cofunding between Centre National d'Études Spatiales, the French Space Agency, and ONERA-The French Aerospace Lab. The authors also thank MAP Coatings, for their participation in this project, and the Laboratoire de Chimie des Polymères Organiques, where most of the experiments have been performed.