A Sensitivity-Aware Motion Planner (SAMP) to Generate Intrinsically-Robust Trajectories

Simon Wasiela 1 , Paolo Robuffo Giordano 2 , Juan Cortés 1 and Thierry Siméon 1

Abstract-Closed-loop state sensitivity [START_REF] Giordano | Trajectory generation for minimum closed-loop state sensitivity[END_REF], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF] is a recently introduced notion that can be used to quantify deviations of the closed-loop trajectory of a robot/controller pair against variations of uncertain parameters in the robot model. While local optimization techniques are used in [START_REF] Giordano | Trajectory generation for minimum closed-loop state sensitivity[END_REF], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF] to generate reference trajectories minimizing a sensitivity-based cost, no global planning algorithm considering this metric to compute collision-free motions robust to parametric uncertainties has yet been proposed. The contribution of this paper is to propose a global control-aware motion planner for optimizing a state sensitivity metric and producing collision-free reference motions that are robust against parametric uncertainties for a large class of complex dynamical systems. Given the prohibitively high computational cost of directly minimizing the state sensitivity using asymptotically optimal sampling-based tree planners, the proposed RRT*-based SAMP planner uses an appropriate steering method to first compute a (near) time-optimal and kinodynamically feasible trajectory that is then locally deformed to improve robustness and decrease its sensitivity to uncertainties. The evaluation performed on planar/full-3D quadrotor UAV models shows that the SAMP method produces low sensitivity robust solutions with a much higher performance than a planner directly optimizing the sensitivity.

I. INTRODUCTION

Modern motion planners can generate feasible and globally-optimal paths for high-dimensional systems and complex constraints/environments. However, most planners do not consider (and, even less, exploit) the unavoidable presence of the feedback controller that will track the planned trajectories of the robot, thus leading to poor intrinsic robustness and potential loss of optimality due to the controller actions at runtime.

The idea of merging planning with control for generating "robust planners" or more "global controllers" for dealing with the robustness problem in a more comprehensive way is however not completely novel in the robotics community. For example, the "feedback motion planning" approach introduced in [START_REF] Tedrake | Lqrtrees: Feedback motion planning via sums-of-squares verification[END_REF] exploits the possibility of constructing explicit Lyapunov functions, but assuming a LQR controller and linearizing the robot dynamics around a nominal trajectory. This idea was extended in [START_REF] Majumdar | Funnel libraries for real-time robust feedback motion planning[END_REF] by considering a polynomial form for the robot dynamics/control strategy. This method is however computationally heavy and it requires the off-line computation of a finite set of motion primitives with their associated "funnels" where the system state is guaranteed to evolve. Also, in the control-aware motion Fig. 1. Solutions with their uncertainty tube (in green) computed by our SAMP method between the same start/goal states of a quadrotor dynamical model for small/high parameters uncertainties (top/bottom, respectively) planner [START_REF] Tognon | Control-aware motion planning for task-constrained aerial manipulation[END_REF], an Inverse Kinematics controller is used as local planner to connect randomly sampled states within a RRT algorithm. Although this method relies on some coupling between planning and control, uncertainties/disturbances are not explicitly considered. Other robust planning techniques integrate a reasoning on uncertainty models. In particular Belief -planning is a popular alternative for providing a principled and general framework for dealing with uncertainties, especially using POMDPs [START_REF] Koenig | A robot navigation architecture based on partially observable markov decision process models[END_REF] [START_REF] Candido | Minimum uncertainty robot path planning using a pomdp approach[END_REF]. However, despite recent progress in efficient point-based solvers, planning in beliefspace still suffers from computational bottlenecks that limit applicability to low dimensional problems.

Work has also been carried out on the control community side to propose "less local" controllers, resulting in the popular Model Predictive Control (MPC) technique [START_REF] Camacho | Model predictive control[END_REF] that iteratively replans an optimal (and feasible) trajectory with a feedback from the current robot state (that may deviate from the desired one because of disturbances/uncertainties). Robust versions of MPC-based schemes have also been investigated [START_REF] Houska | Robust optimization for mpc[END_REF] for providing a robustness layer to the classical MPC formulation.

Approaches based on contraction theory [START_REF] Manchester | Robust direct trajectory optimization using approximate invariant funnels[END_REF]- [START_REF] Singh | Safe motion planning with tubes and contraction metrics[END_REF] allow to synthesize a specific robust controller that guarantees the trajectory remains within an uncertainty tube. These meth-ods, however, remain specific to the synthesized controller.

Recently introduced, the notion of closed-loop sensitivity, [1], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF], captures the sensitivity of any robot/controller pair against variations in the model parameters. This concept has been used for generating locally optimal trajectories under actuation and state constraints. However, it has never been used in a global planning framework, taking into account the presence of obstacles.

In this respect, this paper proposes a control-aware sampling-based planner call SAMP, that considers metrics introduced in [1], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF] in order to plan robust collisionfree motions with a low cost w.r.t. the sensitivity-based metrics. This approach differs from [START_REF] Majumdar | Funnel libraries for real-time robust feedback motion planning[END_REF] and [START_REF] Manchester | Robust direct trajectory optimization using approximate invariant funnels[END_REF]- [START_REF] Singh | Safe motion planning with tubes and contraction metrics[END_REF] by considering general forms of controllers and robot dynamics, and moreover emphasizes the generation of low sensitivity trajectories. In addition, the proposed approach allows to also take into account uncertainties in the robot input, which may be relevant when considering actuation constraints.

The rest of this paper is organized as follows. Section II recalls the main notions of the considered closed-lood sensitivity metrics. Section III then first discusses the difficulties associated with using the sensitivity in classical global planners, and then presents our SAMP method. Simulations in different environments taking into account the full dynamics of a 3D quadrotor are provided in Sect. IV. Finally we draw some conclusions and future directions in Sect. V.

II. SENSITIVITY-BASED METRIC

In this section, we briefly recall the notion of closed-loop state sensitivity introduced in [1], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF] and later extended in [START_REF] Brault | Tube-based trajectory optimization for robots with parametric uncertainty[END_REF], and discuss how this quantity can be used as a suitable metric for our planner.

We consider a generic robot dynamical model q = f (q, u, p), q(t 0 ) = q 0 ,

where q ∈ R nq is the state vector, u ∈ R nu the input vector, and p ∈ R np the vector containing (possibly uncertain) parameters of the robot model. We also consider a reference trajectory r d (a, t) parameterized by a vector a (e.g., the control points of a B-spline) and time. The reference trajectory r d (a, t) is tracked by the robot by means of a controller in the generic form

ξ = g(ξ, q, a, p c , k c , t), ξ(t 0 ) = ξ 0 u = h(ξ, q, a, p c , k c , t), (2)
where ξ ∈ R n ξ are the internal states of the controller (e.g., an integral action), k c ∈ R n k the controller gains and p c ∈ R np the vector of nominal robot parameters used in the control loop. Vector p c may differ from the real p due to potential uncertainties in the knowledge of some robot parameters.

As discussed in [START_REF] Giordano | Trajectory generation for minimum closed-loop state sensitivity[END_REF], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF], it is possible to quantify how variations in p (w.r.t. the 'nominal' p c ) will affect the states q(t) and inputs u(t) behavior for the closed-loop system (1-2) by means of the state sensitivity matrix

Π(t) = ∂q(t) ∂p p=p c ∈ R nq×np , (3)
and input sensitivity matrix

Θ(t) = ∂u(t) ∂p p=p c ∈ R nu×np , (4)
These two matrices can be easily computed (numerically by forward integration) for the robot/controller pair (1-2) and they can be used for optimization purposes. For instance, one may try to find a reference trajectory r d (a, t) that minimizes a norm of Π(t) (and/or of Θ(t)) for producing a 'motion plan' that would be intrinsically robust against uncertainties in the robot parameters p.

A recent extension [START_REF] Brault | Tube-based trajectory optimization for robots with parametric uncertainty[END_REF] has also shown how to leverage the quantities Π and Θ for evaluating the tubes of perturbed state/input trajectories given a range for the parametric deviations. In particular, assume that each uncertain parameter p i lies in the range p i ∈ [p ci -δp i , p ci + δp i ] and consider the diagonal weight matrix W = diag(δp 2 i ). Letting ∆p = pp c , an ellipsoid in the parameter space centered at p c and with semi-axes δp i has equation

∆p T W -1 ∆p = 1 (5)
from which, as explained in [START_REF] Brault | Tube-based trajectory optimization for robots with parametric uncertainty[END_REF], it is possible to obtain the resulting ellipsoids in the state space

∆q T (ΠW Π T ) -1 ∆q = 1 (6)
and in the input space

∆u T (ΘW Θ T ) -1 ∆u = 1. (7)
Here ∆q stands for ∆q = qq nom where q nom is the state behavior in the nominal case (p = p c ), and analogously for ∆u. Equation ( 6) can be used to define a sensitivity metric that can capture the deviations in the states given the (known) parametric ranges δp i . To this end, as discussed in [START_REF] Brault | Tube-based trajectory optimization for robots with parametric uncertainty[END_REF], in this work we consider the largest eigenvalue of the kernel matrix Π(t)W Π(t) T as sensitivity norm, since this will represent the largest (worst-case) deviation in the state space. In particular, letting λ i (t) be the eigenvalues of Π(t)W Π(t) T , we approximate the max(•) operation with the p-norm

λ max (t) ≈ λ i (t) p 1/p (8)
for a large enough p. The total cost function c[π] for a trajectory π is then defined as

c[π] = t f t0 λ max (τ )dτ. (9)
Minimization of this cost will minimize the largest deviation of the state tube along the whole motion.

A second possible use of (6-7) is to evaluate the radius r i (t) ≥ 0 of the perturbed tubes along directions of interest in the state and/or input spaces, see [START_REF] Brault | Tube-based trajectory optimization for robots with parametric uncertainty[END_REF] for a detailed derivation. The quantities r i (t) can then be used to bound components of the states q(t) or inputs u(t) over time. For example, letting q i,nom (t) the behavior of the i-th state component in the non-perturbed case (p = p c ), one has that

q i,nom (t) -r i (t) ≤ q i (t) ≤ q i,nom (t) + r i (t) (10)
for a perturbed state q i (t), and an analogous derivation is also possible for the inputs u i (t). Availability of bounds 10 can be exploited for, e.g., planning a trajectory that will ensure robust collision avoidance for the envelope of perturbed states q for any value of the parameters p in the considered range.no

III. SENSITIVITY-AWARE MOTION PLANNING

We will now present our SAMP approach that consists of two stages: it first plans a (near) time-optimal, collisionfree and kinodynamically feasible trajectory that is then locally optimized for minimizing the cost function ( 9) via a 'shortcut' technique classically used to smooth solutions of randomized planners [START_REF] Geraerts | Creating high-quality paths for motion planning[END_REF]. In the first part of this section, we will justify the choice of this approach and in the second part we will detail our implementation.

A. General approach 1) Sensitivity Computation: As explained in Sec.II, the state/input sensitivity matrices Π(t) and Θ(t) are computed by forward integration. This computation is performed along the reference trajectory using a given time step. Thus, depending of the execution time of the trajectory, the evaluation of the sensitivity (and computation of the associated uncertainty tube) can be time consuming, in the order of hundred milliseconds for a hundred points over the trajectory, and up to several seconds for long trajectories requiring thousands of evaluations.

Also note that the evaluation of the sensitivity is done along a reference trajectory, taking into account both the dynamics of the system and the controller. Therefore, in cases where this trajectory is not feasible with respect to the system dynamics, there is a risk that despite the presence of a robust controller, the tracking error will diverge, making the sensitivity computation impossible. Thus, this justifies the requirement for the SAMP method of computing kinodynamically feasible trajectories in order to avoid such diverging behavior.

2) Sensitivity-Aware Motion Planner (SAMP): One may consider to use standard asymptotically-optimal randomtree planners like RRT * [START_REF] Karaman | Optimal kinodynamic motion planning using incremental sampling-based methods[END_REF] or SST * [START_REF] Li | Asymptotically optimal sampling-based kinodynamic planning[END_REF] for computing robust motions directly minimizing the sensitivity metrics. However such solution would be too inefficient for the reasons explained below.

Regarding RRT * , each iteration is based on an extension phase towards a random sample followed by a rewiring phase, both considering a neighborhood of the sampled state. Since this neighborhood consists of a growing set of possibly many vertices, and as the two phases mentioned above require to compute the sensitivity for each feasible trajectory among this set, this leads to a prohibitive computation time per iteration.

Moreover, it is important to emphasise that the sensitivity is not invariant for non-modified edges since Π depends on the path taken from the root node to the source node of the edge. Therefore, unlike a distance or time cost which are invariant on non-modified edges, the sensitivity must be recomputed for an edge each time one of its parents is changed. Thus, when the rewiring is performed and the cost of the rewired nodes is updated, the sensitivity must also be recalculated on the edges. This re-computation can lead to sub-optimal connections and therefore a re-connection is necessary.

Consequently, the computation time of a RRT * iteration becomes rapidly too high (up to tens of seconds), leading to a very large running time with a slow convergence of the algorithm as shown in Fig. 2. In order to improve the speed of random tree planners one needs to limit the number of sensitivity path-costs computed per iteration. This is actually the case for the SST * algorithm that indeed results in a much higher performance than RRT * as shown in Fig. 2. However, the convergence time of SST * still remains very long because in the absence of a steering method many iterations must be performed in order to reach a goal region accurately. We therefore propose a new approach called SAMP (Sensitivity-Aware Motion Planner) described in Algorithm 1. SAMP is aimed at producing an intrinsically-robust and controlaware motion with a good sensitivity-cost in a much faster manner than the conventional planners mentioned above. It consists in planning a (near) time-optimal, kinodynamically feasible and uncertainty-robust trajectory using a modified version of RRT * (SARRT * ) to guarantee this robustness and then performing a local optimization of the sensitivity of the computed solution. This second step is currently performed using a random 'shortcut' algorithm [START_REF] Geraerts | Creating high-quality paths for motion planning[END_REF]. 9) corresponds to the integration of the sensitivity over the execution time of the trajectory, it is then noticeable that starting from an initial (near) time-optimal trajectory helps to produce initial solutions of rather good quality wrt. the sensitivity cost. In order to produce a near time-optimal and kinodynamically feasible trajectory, the SARRT * planner currently uses as steering method the Kinosplines defined in [START_REF] Boeuf | Motion planning[END_REF]. To ensure kinodynamic feasibility, this method considers the kinodynamic states of the system and two given states are connected using a bang-bang snap control. Regarding time optimality, the local optimization used to steer between the two states aims to reach full speed as quickly as possible and to maintain it as long as possible. This steering method, as well as the sensitivity-based metric, is applicable to a large class of dynamic systems, which makes the proposed framework very general. Note that, instead of using the costto-go as distance metric, an efficient state-space quasi-metric is employed as defined in [START_REF] Boeuf | Motion planning[END_REF].

2) Sensitivity-Aware RRT * (SARRT * ): Algorithm 2 provides the pseudo-code of the SARRT * planner. It is a variant of RRT * , modified in order to guarantee the robustness of the computed solution wrt. the state/input uncertainties. Indeed, in addition to providing a kinodynamic (near) time-optimal trajectory the algorithm verifies that the returned trajectory is collision-free wrt. the state uncertainties and also that inputs remain in their allowed bounds. However, as mentioned in Sec.III-A.2, the number of sensitivity computation within the RRT * must remain as limited as possible to avoid a too long computing time. For this reason, trajectory robustness is checked in a lazy way, only when a better solution is found, and reconnecting the nodes optimally if necessary.

The first stage applies the unmodified RRT * (line 2-5) where the Extend procedure performs the collision checking, optimal connection, and rewiring phase for the given sample x rand as in [START_REF] Karaman | Optimal kinodynamic motion planning using incremental sampling-based methods[END_REF]. Then, the algorithm checks if a new solution with a better cost has been found (line 6). If so, it computes the uncertainties along the the trajectory (line 7-8).

Next, function TubesCC (line 9) performs a robust Collision Checking by first considering the state uncertainties (computed using eq.( 6)) and then 'growing' the geometry of the robot consequently. It also considers the uncertainties on the different actuators by checking that the tube associated with each input remains in its feasibility domain. An example of infeasible input is presented in Fig. 3, where the tube (red) around the reference trajectory (blue) exceeds the maximum allowed input (red). This TubeCC function checks if the motion is valid or not considering the uncertainties. In case of collision, it also determines the first colliding tree node (state) along the path. This node is used to perform an optimal re-connection of the tree starting from it.

This re-connection is performed by the RobustExtend function (line 13) that follows the same principle as the Extend fucntion in the standard RRT * , excepted that it only considers for the optimal connection and rewiring stages the neighborhoods of nodes reached by a robust trajectory wrt. the state/input uncertainties. For this, it maintains a set denoted X collide of all states already found by TubeCC to be invalid wrt. the uncertainty tubes during the previous iterations. This set is unique for each node of the tree. It is first incremented for x collide by its current parent if x collide is eventually found at the current iteration. Then, all the states of the tree from x collide are disconnected, and for each of them an Extend procedure is executed considering the associated X collide set. As in RRT * , an optimal re-connection Fig. 3. Profile of the nominal input 1 (in blue) for a quadrotor controlled in rotation per minute (rpm), and its uncertainty tube (in green), along the trajectory considering the maximum and minimum allowed inputs (in red) is first tried. If no re-connection is found for a node, then the node is deleted from the tree. Otherwise, the rewiring phase is carried out considering the X collide set.

The output of SARRT * is a (near) time-optimal trajectory, feasible both in terms of kinodynamic constraints and also with respect to uncertainties. It is however not optimised in terms of the sensitivity. Therefore, its robustness can be further improved.

Algorithm 2 SARRT * [x init , x goal ] 1: V ← {x init , x goal } ; E ← ∅; sol ← ∅; i ← 0; 2: while i < N do 3: T ← (V, E); 4:
x rand ← Sample(i); i ← i + 1;

5:

T ← Extend(T, x rand ); end if 16: end while 17: return sol 3) Sensitivity-Aware shortcut (SAshortcut): In order to improve the sensitivity-based cost function, a local optimization is performed using a simple variant of the 'shortcut' smoothing algorithm [START_REF] Bohm | Cop: Control & observability-aware planning[END_REF], called SAshortcut and described in Algorithm 3. The reason for this variant, as already mentioned in Sec.III-A.2, is that the sensitivity of a trajectory portion is not invariant but instead depends on the previous portions since the start of the motion.

The algorithm first randomly samples two states along the trajectory (line 3). Then, it computes the Kinospline between the two samples (line 6) and verifies that it is collision-free (line 7). Unlike the classical shortcut, where only the cost of the old and the new portions are compared, the whole trajectory must be reconsidered in order to incrementally compute the sensitivity and the related uncertainty "U " using the GetSensi procedure (line 8-9). Finally, if the new solution has a lower sensitivity cost (line 10), then a robust collision checking using the uncertainty tubes is performed before updating the trajectory portion in case of success (line [START_REF] Chou | Model error propagation via learned contraction metrics for safe feedback motion planning of unknown systems[END_REF][START_REF] Singh | Robust online motion planning via contraction theory and convex optimization[END_REF].

Algorithm 3 SAshortcut [sol SARRT * ] 1: {sol Π , cost Π } ← {sol SARRT * , GetSensi(sol SARRT * )} ; 2: while i < N do 3: {x 1 , x 2 } ← SampleOnT raj(sol Π ); 4:
π start ← sol Π (x init , x 1 ); 5:

π end ← sol Π (x 2 , x goal ); 6: π ← Steer(x 1 , x 2 ); 7:
if CollisionF ree(π) then 8:

sol new ← π start + π + π end ; 9:
{cost new , U } ← GetSensi(sol new );

IV. SIMULATION RESULTS

To evaluate the effectiveness of our approach, we considered the model of a 3D quadrotor as described in [START_REF] Bohm | Cop: Control & observability-aware planning[END_REF]. The control inputs are the squared rotor speeds of the quadrotor u = [ω 2 1 ω 2 2 ω 2 3 ω 2 4 ]. We consider p = [k f k τ g x g y ] as the uncertain parameters of our model where k f and k τ are coefficients associated with the dynamics of the propellers, and g x and g y are the location of the centers of mass in the x-axis and y-axis of the quadrotor body frame, which may be uncertain due to the presence of on-board sensors for example. The values of this vector considered in the simulations are either δp low = [10%, 10%, 5cm, 5cm] or Note that the tubes computed from δp low/high remain an approximation, but that are accurate enough for well chosen δp values. Finally, the tracking controller considered in this work is the so-called Lee (or geometric) controller [START_REF] Lee | Geometric tracking control of a quadrotor uav on se(3)[END_REF].

Let X C = [x y z θ ϕ ψ] ∈ R 6 be the configuration of the quadrotor. We therefore define X K = [X C ẊC ẌC ] ∈ R 18 as the kinodynamic state considered for motion planning.

The performances of the SAMP algorithm were evaluated in three different environments, also considering different parametric uncertainties. For the 2D environments (U-shape and 2-Way) we forced the drone to evolve on a plane by producing Kinosplines only between samples in it and by not allowing displacements along Z. The sensitivityoptimal trajectories used as references to evaluate SAMP were computed using an SST * as implemented in OMPL [START_REF] Sucan | The open motion planning library[END_REF], by specifying a goal state X goal , and allowing a tight goal region around it for all the components of X K .

Table I gathers the average computing times of SST * and of the two core procedures of SAMP (SARRT * , SAshortcut), as well as the average final cost found by SST * and SAMP. The mean values associated with SST * were obtained over 3 runs (due to the very high computing time) while those associated with SAMP are averaged over 10 runs.

1) 2D U-Shape Environment: Fig. 4 shows how SAMP is able to produce solutions that mimic the optimal one found by the SST * in the so called U-shape environment considering the δp low parametric uncertainty vector. Furthermore, according to the results shown in Table .I and in Fig. 4, SAMP produces solutions with sensitivity costs close to the optimal reference found by SST * . Also note the huge time saving of the SAMP method compared to the SST * .

Finally, we note that the average computing time of SARRT * in this case is low compared to the other environments. This is mainly due to the fact that in this case the time-optimal trajectory is far from the obstacles and therefore few re-connections via the SARRT * 's RobustExtend method are performed when extending the tree.

2) 2D 2-way Environments: This environment referred to Fig. 1 where the two sets of uncertainties δp low and δp high are considered.

As shown in Fig. 1, in the presence of small uncertainties SAMP produces a trajectory allowing to use the narrow passage. However, while considering large uncertainties, the time optimal trajectory which goes through the narrow passage cannot be taken as a collision is found by considering the uncertainty tube as shown in red in the Fig. 1. Nevertheless, SAMP is able to find the fastest trajectory apart from those passing through this passage.

In both cases the average cost of the solutions produced by SAMP is close to that found by SST * as shown in Table I, where 2-Way low refers to the presence of small uncertainties and 2-Way high to the presence of large uncertainties. The average computing time of the SST * is equivalent in both cases as it is performed on exactly the same environment. However, note that the average computing time of the SARRT * is longer than in the U-shape case because the time-optimal trajectories must pass through a location where many collisions may occur. Moreover, in the 2-Way high case the SARRT * 's computing time is longer than in the 2-Way low . This is because more iterations are needed before considering a neighborhood deprived of nodes passing through the narrow corridor. Finally, we see that the SAshortcut computing time remains of the same order of magnitude for both cases as that of the U-shape, and again SAMP produces a solution much faster than SST * .

3) 3D Environment: The uncertainty set considered in this full 3D environment is δp low . As shown in Fig. 5, once again SAMP produces trajectories that mimic the optimal obtained by the SST * . It is interesting to note the impact of the SAshortcut procedure on the Z component in addition to the impact on the X Y components already seen in the previous 2D environments. The average computing time of the shortcut is similar to other environments according to Table I. Again, the SAMP method produces a near-optimal trajectory much faster than conventional optimal planners. Fig. 5. Trajectories produced by SARRT * (green), SAMP (red), and SST * (dashed black) in a 3D environment (top). A profile view along the YZ plane is given (middle) as well as the evolution of the sensitivity (red) and the optimal one found by the SST * (dashed blue) (bottom) during the execution of SAMP.

V. CONCLUSION

We have presented a method called SAMP for planning control-aware trajectories robust to parametric uncertainties. The proposed framework has been specifically designed to handle global planning using sensitivity-based metrics from [START_REF] Giordano | Trajectory generation for minimum closed-loop state sensitivity[END_REF], [START_REF] Brault | Robust trajectory planning with parametric uncertainties[END_REF], [START_REF] Brault | Tube-based trajectory optimization for robots with parametric uncertainty[END_REF] in a more efficient way than the classical asymptotically optimal sampling-based tree planners. Simulations have shown that SAMP generates near-optimal trajectories with significant time savings, and which can be robustly executed by the considered robot for any value of the uncertain parameters in a given range. However, this computing time is still far from being applicable to real-time motion planning. Future work will therefore focus on improving computing time by considering learning techniques in order to estimate the sensitivity cost and uncertainties associated with a trajectory.

Fig. 2 .

 2 Fig. 2. Evolution of the sensitivity as a function of the running time of an SST * (in green), and of an RRT * (in red)

6 :

 6 sol new ← CheckF orSolution(T );

7 :

 7 if sol new f ound then 8:U ← GetU ncertainties(sol new );9:{valid, x collide } ← T ubesCC(sol new , U );

Fig. 4 .

 4 Fig. 4. Three trajectories (red, green, blue) produced by SARRT * (top), and the three trajectories resulting from SAMP (middle), with the evolution of their respective sensitivity (bottom). The dashed black trajectory (top, middle) and the dashed purple cost (bottom) correspond to the sensitivityoptimal reference found by SST * .

10 :

 10 if CostBetterT han(cost new , cost Π ) then 11: if T ubesCC(sol new , U ) then 12: {sol Π , cost Π } ← {sol new , cost new } ;

	13:	end if
	14:	end if
	15:	end if
	16:

i ← i + 1; 17: end while 18: return sol Π

* This work was supported by the project ANR-20-CE33-0003 "CAMP"


William Garat

Nicolas Le Moigne
email: nicolas.le-moigne@mines-ales.fr

Stephane Corn
email: stephane.corn@mines-ales.fr

⁎⁎ Johnny Beaugrand

Anne Bergeret

Swelling of natural fibre bundles under hygro-and hydrothermal conditions: Determination of hydric expansion coefficients by automated laser scanning

Keywords: A, natural fibres B, Fibre deformation D, Moisture

The effect of humidity conditions on the moisture content and dimensional variations of natural fibre bundles from several botanical origins with contrasting biochemical and structural characteristics is investigated. Results highlight wide variations in water uptake and swelling behaviour of fibre bundles according to plant species. Two main swelling mechanisms are identified: (i) a microscopic swelling due to the sorption of bound water in the cell walls and the middle lamella, and (ii) a macroscopic swelling related to the formation of free water in pores and lumens, which induces anisotropic deformation of bast fibre bundles cross section. Cross-sectional hygro-and hydroexpansion coefficients are determined and studied in relation with the structural features of plant fibre bundles. These results constitute key data for the predictive modelling of "in-service" mechanical behaviour of biocomposites.

Introduction

Composite materials are subjected to various environmental stresses such as humidity, temperature or UV radiation. This exposure to environmental conditions is one of the primary causes of their accelerated ageing. These ageing processes require special attention in order to evaluate the evolutionary behaviour of composite materials in servicelife, and particularly those of biocomposites reinforced with natural fibres [START_REF] Péron | Study of hygroscopic stresses in asymmetric biocomposite laminates[END_REF]. Indeed, the dimensional stability and the mechanical behaviour of natural fibres are strongly influenced by hygro-and hydrothermal conditions [START_REF] Duigou | In-situ evaluation of flax fibre degradation during water ageing[END_REF], related to relative humidity (RH) and immersion conditions, respectively.

Depending on moisture content, natural fibres may contain water in two states: (i) the water bound to the different biopolymers constituting the cell walls and the middle lamella, primarily via the formation of hydrogen bonds with hydroxyl groups eOH and (ii) the free water which is in the micro-and macropores of cell walls, filling the voids (lumens) and retained by capillary forces [START_REF] Engelund | A critical discussion of the physics of wood-water interactions[END_REF][START_REF] Placet | Influence de traitements hygrothermiques sur les propriétés mécaniques de composites à fibres végétales[END_REF][START_REF] Placet | Influence of environmental relative humidity on the tensile and rotational behaviour of hemp fibres[END_REF][START_REF] Fredriksson | On wood-water interactions in the over-hygroscopic moisture range-mechanisms, methods, and influence of wood modification[END_REF]. Fig. 1 schematically shows the swelling processes occurring in wood cells according to moisture content. As the relative humidity increases, water vapour penetrates and is absorbed into the cell walls and the middle lamella via the formation of hydrogen bonds, thereby inducing significant cell swelling. At a certain moisture content in the cells, which depends on the plant species considered, the bound water saturates the cell walls and the middle lamella: the water saturation point of the plant cells is then reached, ∼20-40% for wood cells [START_REF] Zelinka | A solution thermodynamics definition of the fiber saturation point and the derivation of a wood-water phase (state) diagram[END_REF]. Finally, cellular cavities (lumens and porosities) are filled by the free water until total saturation.

Several measurement techniques are used to quantify moisture content in natural fibres as a function of relative humidity. In particular, (i) DVS (Dynamic Vapour Sorption) which is a gravimetric technique that measures the amount of water absorbed by a sample as a function of relative humidity at equilibrium (or water activity a w = P/ P 0 with P the water vapour pressure at the surface of the considered substrate and P 0 the saturation vapour pressure of water). Several authors have used this technique to determine water sorption isotherms from a dry sample, or desorption isotherms from a water-saturated sample [START_REF] Bourmaud | Exploring the potential of waste leaf sheath date palm fibres for composite reinforcement through a structural and mechanical analysis[END_REF][START_REF] Guicheret-Retel | Creep behaviour of single hemp fibres. Part II: Influence of loading level, moisture content and moisture variation[END_REF][START_REF] Hill | The water vapor sorption behavior of natural fibers[END_REF]; (ii) WRV (Water Retention Value) which is a centrifugation and weighing method developed for wood pulps to determine their water retention value after immersion. WRV is determined according to ISO 23714 standard [START_REF]Pulps -Determination of water retention value (WRV)[END_REF].

The processes of water sorption and swelling of natural fibres are complex because of their biochemical, structural and morphological features. In this regard, these processes are influenced by several factors, such as the lumen size, the microfibrillar angle (MFA) [START_REF] Burgert | Tensile and compressive stresses in tracheids are induced by swelling based on geometrical constraints of the wood cell[END_REF], cellulose crystallinity, accessibility to hydroxyl groups, amorphous polysaccharides/biopolymers amounts and their relative hydrophilic/ hydrophobic character. Moreover, swelling, water organization and its interactions with cell walls biopolymers are highly heterogeneous when considering the different scales of natural fibres structures. In the case of (ligno-)cellulosic substrates, a high hysteresis between the sorption and desorption isotherms is usually observed. The origins of this phenomenon are still debated in literature and although there is still no consensus, several authors attribute it to softening and relaxation processes related to water interactions with amorphous biopolymers present in the cell walls [START_REF] Salmén | On the origin of sorption hysteresis in cellulosic materials[END_REF][START_REF] Lu | Demonstration of the "conditioning effect" in soil organic matter in support of a pore deformation mechanism for sorption hysteresis[END_REF] and their glass transition temperature. Based on molecular modelling of water molecules interactions with crystalline cellulose -hemicelluloses -lignin models, Charlier and Mazeau [START_REF] Charlier | Molecular modeling of the structural and dynamical properties of secondary plant cell walls: influence of lignin chemistry[END_REF] and Kulasinski et al. [START_REF] Kulasinski | Water adsorption in wood microfibril-hemicellulose system: role of the crystalline-amorphous interface[END_REF] showed that water mobility and its organization are heterogeneous at the microscale within cell wall structures, e.g. higher density of sorbed water molecules is found at cellulose/hemicellulose interfaces [START_REF] Kulasinski | Water adsorption in wood microfibril-hemicellulose system: role of the crystalline-amorphous interface[END_REF]. At the macroscale, Karmaker et al. [START_REF] Karmaker | Effect of water uptake on some physical properties of jute fibres[END_REF] studied the swelling of jute fibre bundles cross-sections in water. Dry jute fibre bundles were compared to the same fibre bundles immersed for two weeks in water. Optical microscopy observations showed that cross-sectional swelling was heterogeneous along the fibre bundles and that maximum swelling occurred in the regions where the transverse dimensions of the fibre bundles were the smallest. According to the authors, in these thinner areas, the elementary fibres would be more packed within the fibre bundle so that during water sorption, the swelling of each elementary fibre causes a larger increase in diameter in these areas.

All these studies highlight the interest of accurately measuring the swelling of natural fibres in relation with their moisture content and structural features. These data are essential to better describe and understand the swelling processes of natural fibres that directly influence the processing and the in-service life behaviour of biocomposites. In particular, it should be pointed out that the swelling of natural fibres can induce modification of the stress state at the fibre/matrix interface in biocomposites, and contribute to an improvement in interfacial adhesion [START_REF] Duigou | Hygroscopic expansion: a key point to describe natural fibre/polymer matrix interface bond strength[END_REF][START_REF] Réquilé | Interfacial properties of hemp fiber/ epoxy system measured by microdroplet test: effect of relative humidity[END_REF], but also generate stress levels that can locally damage the matrix, and thus cause an irreversible degradation of the composite [START_REF] Regazzi | Reversible and irreversible changes in physical and mechanical properties of biocomposites during hydrothermal aging[END_REF].

The aim of this work is to characterize, under variable and controlled hygro-and hydrothermal conditions (i.e. 20%, 50%, 73% RH and immersion, respectively), the moisture content and the dimensional variations of fibre bundles from several botanical origins with contrasting biochemical, structural and morphological characteristics. An automated laser scanning technique [START_REF] Garat | Analysis of the morphometric variations in natural fibres by automated laser scanning: towards an efficient and reliable assessment of the cross-sectional area[END_REF], conducted either in a climate chamber or in an immersion cell, is used to evaluate the variations in the cross-sectional dimensions of the fibre bundles in relation with their moisture content determined by DVS and WRV measurements. Swelling processes are discussed, and hygro-and hydroexpansion coefficients are determined and studied in relation with the specific structural features of the different plant fibre bundles.

Experimental

Selection of fibre bundles

Fibre bundles were extracted from various phylogenetic plant species with distinct functions in planta. These fibre bundles have contrasting morphometric characteristics as described in Garat et al. [START_REF] Garat | Analysis of the morphometric variations in natural fibres by automated laser scanning: towards an efficient and reliable assessment of the cross-sectional area[END_REF], microstructures and biochemical compositions [START_REF] Bourmaud | Towards the design of highperformance plant fibre composites[END_REF]. Three of them belong to eudicots and are commonly called 'bast fibres'. These fibre bundles are weakly lignified (< 5%), have a low MFA (< 10°) and a support function of the stem in planta: flax Linum usitatissimum (Alizée, Grandvilliers, Picardie, 2012), hemp Cannabis sativa (Fedora 17, Champagne Ardenne, 2012) as annual fibre crop (after retting) and nettle Urtica dioica (2014, Lorraine) as a perennial herbaceous plant. The other two species are from perennial monocots. These fibres are highly lignified (> 9%) and have an higher MFA (> 20°): sisal Agave sisalana (FRD, Fibre Recherche Développement®, Troyes) coming from the leaf of agave and palm tree Phoenix dactylifera (Al-Ahsa, Saudi Arabia) present in the form of mats of foliar sheath surrounding palm tree stems. The biochemical compositions and average MFA of these different batches of fibre bundles are given in Table 2.

Characterization of fibre bundles under controlled hygro-and hydrothermal conditions 2.2.1. Determination of moisture content

The determination of the water sorption isotherms and moisture contents at given RH for the different plant fibre bundles were carried out by Dynamic Vapor Sorption (DVS) analysis. These tests were conducted over a relative humidity range of 0 to 90% RH, at RH intervals of 10% with a sample quantity of about 5 mg.

A centrifugation method adapted from ISO 23714 [START_REF]Pulps -Determination of water retention value (WRV)[END_REF] was used to evaluate the Water Retention Value (WRV). Batches of 0.3 g of fibre bundles were immersed in deionized water for 5 min at 23 °C and then placed in tubes to be centrifuged at 3000 g for a period of 30 min, and finally weighed to determine the mass M f (in g) of fibres. WRV values were determined according to Eq. (1) after drying for 10 min at 105 °C in a Sartorius-MA160 infrared moisture analyser (Grosseron SAS, Coueron, France) and weighing of the anhydrous mass of fibres M s (in g).

= - × WRV M M M (%) 100 f s s (1)
Fig. 1. Swelling and water state according to moisture content in wood cells (adapted from [START_REF]Workshop Companion[END_REF]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

48.8 ± 2.0%, followed by sisal, flax and hemp fibre bundles with moisture contents of 60.4 ± 1.0%, 60.9 ± 0.7% and 62.8 ± 0.7%, respectively. It is noted that nettle fibre bundles are highly hygroscopic, with moisture content in immersion of 112.2 ± 1.2%, almost twice as much as flax and hemp fibre bundles.

Variations in moisture contents of the different fibre bundles in immersion conditions should primarily come from morphological differences, especially the amount and size of open pores and lumens. The contrasting biochemical composition between the plant species considered may also induce substantial differences in water sorption of the cell walls under controlled hygrothermal conditions. Pejic et al. [START_REF] Pejic | The effects of hemicelluloses and lignin removal on water uptake behavior of hemp fibers[END_REF] studied the influence of non-cellulosic biopolymers such as lignin on the water sorption of hemp fibre bundles, and showed that the decrease in lignin content tends to increase the water retention capacity of hemp fibres. Fig. 2 shows that the palm fibre bundles having the highest lignin contents (Table 2) also have low moisture content in overall. In contrast, the very high moisture content of nettle fibre bundles above 80% RH compared to other bast fibres could also originate from a poor retting inducing high pectin content and stronger hygroscopic character (Table 2).

Moisture data obtained from DVS and WRV measurements will now be used to analyse the morphological and dimensional variations of the fibre bundles under the different hygro-and hydrothermal conditions according to their moisture content.

Morphological and dimensional variations as a function of moisture content

Morphological variation: Evolution of the cross-sectional shape factor

Fig. 3 shows the evolution of the median cross-sectional shape factor (α), i.e. the median value calculated from the ratios maximum diameter/ minimum diameter of each cross-section of the fibre bundle considered, for the different plant fibre bundles and according to the humidity conditions. The cross-sectional shape factor is a morphometric indicator of the non-circularity of fibre bundles cross-section [START_REF] Garat | Analysis of the morphometric variations in natural fibres by automated laser scanning: towards an efficient and reliable assessment of the cross-sectional area[END_REF], and its evolution is studied here according to the moisture content in each type of fibre bundles. In the case of palm and sisal fibre bundles, moisture content has little influence on shape factors that remain close to 1, i.e. these fibre bundles retain substantially circular shaped cross-sections upon swelling. Indeed, their median shape factors remain similar, about 1.20 ± 0.06 and 1.29 ± 0.14 respectively for palm and sisal, over the entire range of moisture content and related humidity conditions (from 20% RH to immersion conditions). It can therefore be concluded that their cross-sectional swelling is isotropic.

On the other hand, bast fibre bundles have higher and more dispersed median cross-sectional shape factors, which are influenced by the moisture content, in particular in immersion conditions. The crosssectional swelling of these fibre bundles is anisotropic because it is characterized by an increase in the minimum diameter much greater than that of the maximum diameter, leading to a significant decrease in their cross-sectional shape factor under immersion conditions (corresponding to moisture content > 60% and hence much greater than the saturation point of the cell walls, in the domain where free water is formed). Indeed, the increase in the minimum diameter for flax and hemp fibre bundles is about twice that of the maximum diameter, and up to three times higher for nettle fibre bundles. The hemp, flax and nettle fibre bundles thus have median cross-sectional shape factors in immersion of 1.82 ± 0.26, 1.92 ± 0.42 and 1.98 ± 0.69 respectively, which implies an increase in their circularity under these humidity conditions. This result is contrasting with macroscopic wood specimens for which bound water plays a predominant role in their dimensional changes. Within a stem, the elementary fibres that grow on the periphery of the central lacuna, between the cortical parenchyma and the phloem/xylem, are grouped into packed fibre bundles. Therefore, we assume that this mode of intrusive growth of elementary fibres [27], as

Dimensional measurements

A fibre dimensional analysis system (FDAS, Diastron Ltd, Hampshire, UK) was used to measure the cross-sectional dimensions of fibre bundles with a nominal length of 30 mm and typical diameters ranging from 50 to 250 µm at 50% RH (see also in Ref. [START_REF] Garat | Analysis of the morphometric variations in natural fibres by automated laser scanning: towards an efficient and reliable assessment of the cross-sectional area[END_REF]), and thus estimate their cross-sectional area (CSA) that is generally variable along plant fibre bundles. This experimental method based on automated laser scanning of the object, as well as the sample preparation protocol, have been described in detail in a previous study [START_REF] Garat | Analysis of the morphometric variations in natural fibres by automated laser scanning: towards an efficient and reliable assessment of the cross-sectional area[END_REF]. During the measurement, the fibre bundle sample is put under slight tension to maintain it straight, and then translated and rotated, so as to collect up to 600 values of apparent diameter per revolution for each cross-section. Each analysed cross-section were separated by a longitudinal pitch of 560 μm. In the end, about 25 000 values of apparent diameters are collected for each fibre bundles allowing to take into account the irregularity of the fibre bundles along their length and the non-circularity of their cross-section. As detailed in Garat et al. [START_REF] Garat | Analysis of the morphometric variations in natural fibres by automated laser scanning: towards an efficient and reliable assessment of the cross-sectional area[END_REF], an elliptical model of the fibre bundle cross-sections using statistical data processing of apparent diameters was applied in order to estimate their median CSA.

In order to analyse the influence of relative humidity on the crosssectional dimensions of the fibre bundles, the dimensional analysis apparatus (FDAS) was placed in a climatic chamber supplied by ETS (Electro-Tech Systems Inc, Pennsylvania, USA) to control the temperature at 23 °C and relative humidity at 20, 50 and 73%. An immersion cell (DSM, Dynamic Swelling Module, Diastron Ltd., Hampshire, United Kingdom) was implemented on the FDAS device to measure the cross-sectional dimensions of the fibre bundles immersed in deionized water at 23 °C. It should be noted that under immersion conditions, the device allows the dimensional analysis of the sample for only one cross-section at a time. For each plant species and each hygro/ hydrothermal condition, up to 20 fibre bundles were characterized after being acclimatized 13 h at 23 °C and at the studied RH (i.e. 20, 50, 73%) or at 50% RH before immersion.

Results and discussion

Influence of hygro/hydrothermal conditions on moisture content

The equilibrium moisture content of the different plant fibre bundles as a function of the RH level was quantified from the sorption curves obtained by DVS analysis. Whatever the plant species considered, the shape of the curves is identical and of sigmoid type (type II); it results from the combination of different sorption mechanisms. Indeed, the sorption isotherms can be divided into three zones [START_REF] Hill | The water vapor sorption behavior of natural fibers[END_REF][START_REF] Fengel | Woodchemistry, ultrastructure, reactions[END_REF][START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF]. In the first zone related to Langmuir's mode (RH < 10%), the water is mainly 'chemically' absorbed and bound via hydrogen bonds to the hydroxyls of amorphous cellulose and hemicelluloses and the carboxyls of pectins, present in the cell walls and the middle lamella. In the second zone that can be described by Henry's law (10% < RH < 65%), the water is rather 'physically' sorbed within the porous microstructure of the fibres where the water can freely diffuse. Finally, beyond 65% RH, water concentration is high which involves the formation of free water loosely bonds in the macropores and lumens until water saturation of fibre bundles.

Fig. 2 shows the evolution of the moisture content of fibre bundles as a function of relative humidity (DVS measurements) and in immersion (WRV measurements). Between 0 and 50% RH, the moisture content of the various plant species is low and similar, i.e. 5.3 ± 0.4% on average. Between 50 and 73% RH, there is an increase in moisture content up to about 13.7 ± 1.7% regardless of the plant species considered. On the other hand, in the vicinity of the saturation point of the cell walls (moisture content of about 20-40% as determined for wood cells) and under immersion conditions (formation of free water in the pores and lumens), the moisture content increases considerably and differences are observed depending on the plant fibre species. The palm fibre bundles have the lowest moisture contents in immersion, well as the progressive drying of the cells at the end of growth, after harvesting and cutting of the stems or leaves, cause or at least contribute to the anisotropic swelling phenomenon observed during rehydration of bast fibre bundles in immersion conditions. Thus, the lower the thickness of the cell walls, the greater the flattening effect during desiccation and the greater will be the decrease in the shape factor of bast fibre bundles during rehydration. Consequently, the geometry of their cross-section tends towards a more circular shape.

Dimensional variation: Evolution of the median cross-sectional area (CSA) and swelling mechanisms

The influence of the moisture content on the dimensional variations of the fibre bundles, and in particular on their cross-sectional swelling, was analysed. The cross-sectional swelling G s of a fibre bundle is defined as follows (Eq. ( 2)):

= - × G S S S (%) 100 s i 0 0 (2)
where S i is the median CSA of the fibre bundle considered for a given humidity condition (in μm 2 ) and S 0 its median CSA at 50% RH (in μm 2 ). Fig. 4 shows the evolution of the median cross-sectional swelling of the different plant fibre bundles as a function of their moisture content. For moisture contents less than 5% (i.e. at 20% RH, see Fig. 2), a slight shrinkage is observed, ranging from -1.2 ± 0.1% to -3.3 ± 1.0% for the palm and sisal fibre bundles, respectively. The flax and hemp fibre bundles have shrinkage of -2.8 ± 0.6% and -3.0 ± 1.8%, respectively. This phenomenon is more pronounced in the case of nettle fibre bundles, reaching -6.9 ± 2.9%. For moisture contents between 10.8% and 14.7% depending on the type of fibre bundles (i.e. at 73% RH, see Fig. 2), the swelling is relatively low for all plant species (< 10%) with low dispersion. In this range of moisture contents ranging from 5% to 15% (corresponding to relative humidity ranging from 20% RH to 73% RH), the swelling of fibre bundles is mainly associated with processes occurring at the level of cell walls and middle lamella, and related to the sorption of bound water by the biopolymers.

Under immersion conditions, the median cross-sectional swelling increases considerably and large differences are observed depending on the plant species. There is also greater dispersion on the swelling values. The palm fibre bundles had the lowest swelling, 19.3 ± 7.1%, followed by the sisal fibre bundles with 43.2 ± 5.9%. The hemp, flax and nettle fibre bundles showed greater swelling, 49.0 ± 10.4%, 57.2 ± 19.9% and 75.1 ± 22.6%, respectively. The anisotropic swelling observed in immersion for bast fibre bundles (flax, hemp and nettle) contributes to the increase of their cross-sectional area, and hence to the measured median cross-sectional swelling. A secondary process is implied here in the swelling of fibre bundles. Indeed, the formation of free water in large quantities in the lumens and porosities appears to contribute significantly to the cross-sectional swelling of fibre bundles, due to the morphological variations described above, and possibly induced by the hydric pressure exerted on the cell walls.

The swelling of the fibre bundles from the different plant species is therefore governed by two main processes (Fig. 4): (i) a microscopic swelling, at the scale of cell walls and middle lamella, due to the sorption of bound water, (ii) a macroscopic swelling related to the formation of free water in the pores and lumen (by capillary condensation at high RH or during immersion) which induces anisotropic deformation of bast fibre bundles and an additional increase in their crosssectional swelling. Contrary to wood specimens, the water saturation point of the cell walls in natural fibre bundles is thus difficult to determine because, at high relative humidity between 80 and 99% RH, the swelling of the cell walls due to sorption of bound water and the morphological variations related to the formation of free water by capillary condensation processes are concomitant. Therefore, we rather assume a continuum of swelling processes occurring at the microscopic and macroscopic scales without a water saturation point of the cell walls being clearly defined. Experiments in an intermediate relative humidity range (i.e. between 80 and 99% RH) would nevertheless be interesting in order to complete these experimental data. It should be noted that the measuring devices used for this study cannot be used in this range of relative humidity because of the risks of water condensation which would damage the apparatus. A rise in temperature above 23 °C could allow limiting condensation and investigating higher relative humidity (potentially up to 90% RH). However, thermally induced phenomena, as polymer relaxation processes, would also have to be considered.

Determination of hydric expansion coefficients and relation with structural features

The study of the morphological and dimensional variations of the fibre bundles as a function of their moisture content makes it possible to determine their hydric expansion coefficients, defined in this study as the cross-sectional swelling G s (%) induced by a variation of 1% in moisture content Δm (%). Therefore, it is the surface expansion coefficients β s of the fibre bundles cross-section that are determined.

Besides, we have previously shown that the swelling of fibre bundles results from two phenomena that strongly depend on hygro-and hydrothermal conditions. The swelling of the cell walls and the middle lamella due to sorption of bound water is the main process at stake under moderate hygrothermal conditions. In immersion conditions or when the relative humidity is sufficiently high to generate the formation of free water (by capillary condensation), significant morphological variations also contribute to the swelling of bast fibre bundles. It is therefore necessary to differentiate the cross-sectional hygroexpansion coefficient β s,hygro (Eq. ( 3)) and the cross-sectional hydroexpansion coefficient β s,hydro (Eq. ( 4)), defined as follows in this study:

= β G m From 20 to 73% RH (%) Δ (%) s hygro s , (3)
= β G m In immersion conditions (%) Δ (%) s hydro s , (4)
Fig. 5 shows the cross-sectional hygroexpansion coefficients β s,hygro for the different plant fibre bundles, calculated between 20 and 73% RH at 23 °C. This coefficient varies from 0.42 to 1.70 per change in moisture content depending on the plant species. For comparison, Le Duigou et al. [START_REF] Duigou | Hygroscopic expansion: a key point to describe natural fibre/polymer matrix interface bond strength[END_REF] determined a radial hygroexpansion coefficient for elementary flax fibres (from Flax Tape, Lineo®) of 1.14 per change in moisture content based on diameter measurements conducted in an Environmental Scanning Electron Microscope (ESEM, at 23 °C, steam pressure 5-10 mbar, from 20% to 98% RH). Assuming the isotropic radial expansion of an equivalent cylinder undergoing the same surface hygroexpansion coefficient β s , a straightforward geometric calculation leads to Equation ( 5) which allows estimating an equivalent radial hygroexpansion coefficient β r for flax fibre bundles in this study:

= + - β β 1 1 r s (5)
Fig. 4. Median cross-sectional swelling of the fibre bundles (normalized to the median CSA at 50% RH) as a function of moisture content at 23 °C for the different plant species. Proposal of two concomitant swelling mechanisms: (i) microscopic swelling of cell walls and middle lamella due to the sorption of bound water, (ii) macroscopic swelling related to anisotropic deformation of bast fibre bundles cross section, possibly induced by hydric pressure of free water in the pores and lumen. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The resulting β r value is 0.63 for flax fibre bundles. This value is lower than that obtained by Le Duigou et al. [START_REF] Duigou | Hygroscopic expansion: a key point to describe natural fibre/polymer matrix interface bond strength[END_REF] for an elementary flax fibre. The origin of the flax fibre may explain that difference, at least in part. Indeed one can hypothesize that the flax bundles and elementary flax fibres compared here are distinct in term of biochemistry and thermal history. In particular, the flax tape used in Le Duigou et al. [START_REF] Duigou | Hygroscopic expansion: a key point to describe natural fibre/polymer matrix interface bond strength[END_REF] has been industrially modified, and could be more sensitive to moisture induced swelling due to structural degradation or variation in biochemical composition. Besides, the extent of swelling of a fibre bundle should be different from an elementary fibre. Indeed, the swelling of elementary fibres included within a bundle is more restricted due to intercellular cohesion with surrounding elementary fibres as compared to an isolated elementary fibre that can swell more freely. More investigations in that way would shed light on those differences.

Viguié et al. [START_REF] Viguié | Analysis of the hygroexpansion of a lignocellulosic fibrous material by digital correlation of images obtained by X-ray synchrotron microtomography: application to a folding box board[END_REF] assessed hygroexpansion coefficients of dense lignocellulosic fibrous networks (with folding board as case study). They found hygroexpansion coefficients of 0.13 and 0.098 in the cross direction (from 20% to 50% RH), at the macroscopic and mesoscopic scales (based on digital correlation of images obtained by X-ray microtomography) respectively. Joffre et al. [START_REF] Joffre | A method to measure moisture induced swelling properties of a single wood cell[END_REF] determined a local hygroexpansion coefficient (transverse to the microfibrils within the cell walls) for wood fibres (spruce, Picea abies) of 0.45 per change in moisture content (23 °C, from 47% to 80% RH) based on X-ray microtomography measurements (Synchrotron, Grenoble) and a numerical identification method.

Although all these hygroexpansion coefficients are of the same order of magnitude, these different results illustrate the importance of the measuring method and relative humidity range but also the dimensions and the nature of the object under consideration. Indeed, according to the scale (macro/meso/micro) and the type of plant fibres under study, the induced variations in microstructure and biochemical composition will have an influence on the value of the hygroexpansion coefficient. Note also the strong anisotropy of the microstructures encountered in plant fibres that generates very significant differences between the transverse and longitudinal hygroexpansion coefficients, the latter being very low especially for plant fibres with a low MFA (e.g. 0.037 for wood fibres in Joffre et al. [START_REF] Joffre | A method to measure moisture induced swelling properties of a single wood cell[END_REF]).

Similarly, the cross-sectional hydroexpansion coefficients β s,hydro were determined for the different plant fibre bundles and are reported in Table 1. In general, we found that these coefficients are lower than the cross-sectional hygroexpansion coefficients β s,hygro , in particular for fibre bundles which have high moisture content and/or wide morphological variations in immersion conditions, case of flax and nettle (Figs. 3 and4). This indicates that sorption of bound water within the cell walls and the middle lamella plays a predominant role in the swelling of fibre bundles. It is also noticeable that, for phylogenetically distant species such as palm and hemp, the measured values of β s,hygro and β s,hydro expansion coefficients are similar.

Table 2 proposes a correlation (Pearson correlation coefficient, r) between the cross-sectional hygroexpansion coefficient β s,hygro and the available structural features of the cell walls of the plant fibre bundles, i.e. their biochemical composition and their MFA. This analysis shows a strong (negative) linear relationship of the β s,hygro coefficient with the lignin content and the MFA (i.e. Pearson coefficients r of -0.76 and -0.78, respectively). Thus, high lignin contents and high MFA as those found in palm fibre bundles tend to limit the water retention capacity and reduce the cross-sectional swelling (Fig. 4), hence decreasing the cross-sectional hygroexpansion coefficient (0.42 for palm fibre bundles). Conversely, the low lignin content and low MFA found in flax fibre bundles are correlated with a higher cross-sectional hygroexpansion coefficient (1.67). Statistical approaches such as multivariate analyses on larger sets of data in terms of cell walls biochemical composition, MFA and integrating other descriptors such as the composition/structure of the middle lamellae, the size of the lumens… could help to better understand the contribution of these different microstructural parameters to the hygroexpansion of plant fibre bundles.

Conclusion

The effect of hygro-and hydrothermal conditions on moisture content and dimensional variations of different plant fibre bundles (palm, sisal, hemp, flax, nettle), with contrasting biochemical and structural characteristics, was studied by DVS and WRV measurements and an automated laser scanning of cross-sectional dimensions conducted in a climate chamber (20%, 50%, 73% RH) and an immersion cell. The results evidenced different processes of water uptake and swelling according to the plant species and the hygro/hydrothermal conditions: (i) a microscopic swelling, at the scale of the cell walls and the middle lamella, due to the sorption of bound water and (ii) a macroscopic swelling related to the formation of free water in pores and lumens (by capillary condensation at high RH or during immersion) which induces anisotropic deformation of bast fibre bundles and contributes to their cross-sectional swelling. These results made it possible to determine the cross-sectional hygro-and hydroexpansion coefficients of the different plant fibre bundles studied, ranging from 0.42 to 1.70. A correlation between the cross-sectional hygroexpansion coefficient and some structural characteristics of the fibre bundles cell walls (biochemical composition and MFA) was found, thus demonstrating a dependence on lignin content and MFA. These results open interesting perspectives for the predictive modelling of "in-service" mechanical behaviour of biocomposites that would better take into account the swelling of natural fibres in relation with their structural features.

CRediT authorship contribution statement

Table 2

Correlation (Pearson coefficient, r) between cross-sectional hygroexpansion coefficients β s,hygro (20-73% HR; 23 °C) and composition in cellulosic and non-cellulosic polysaccharides (by HPLC analysis), lignin (Klason method) and MFA (average values taken from literature [START_REF] Bourmaud | Towards the design of highperformance plant fibre composites[END_REF][START_REF] Baley | Fibres naturelles de renfort pour matériaux composites[END_REF][START_REF] Bodros | Study of the tensile properties of stinging nettle fibres (Urtica dioica)[END_REF][START_REF] Müssig | Testing methods for measuring physical and mechanical fibre properties (plant and animal fibres)[END_REF]) of the fibre bundles from the different plant species.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Moisture content of fibre in relative humidity (DVS) and in immersion (WRV) conditions at 23 °C for the different plant species. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5 .

 5 Fig. 5. Determination of cross-sectional hygroexpansion coefficients β s,hygro of fibre bundles measured between 20% and 73% RH at 23 °C for the different plant species. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

William Garat :

 Garat Methodology, Validation, Investigation, Writingreview & editing. Nicolas Le Moigne: Conceptualization, Methodology, Writing -original draft, Writing -review & editing, Visualization. Stephane Corn: Conceptualization, Methodology, Writing -review & editing. Johnny Beaugrand :: Conceptualization, Resources, Writingreview & editing. Anne Bergeret: Conceptualization, Supervision, Funding acquisition.

Table 1

 1 Cross-sectional hygroexpansion β s,hygro and hydroexpansion β s,hydro coefficients of fibre bundles at 23 °C for the different plant species.

	Plant species	β s, hygro	β s, hydro
	Palm	0.42	0.47
	Sisal	1.49	0.84
	Hemp	0.92	0.90
	Flax	1.67	1.07
	Nettle	1.70	0.72

Acknowledgements

William Garat thanks Montpellier SupAgro for funding his PhD work. The authors thank Arnaud Day (Fibres Recherche Développement, Troyes, France) for the supply of nettle fibre bundles, and Yann Leray and Steve Bucknell (Diastron Ltd., Hampshire, Great Britain) for their technical support on FDAS and DSM devices. The authors thank the French research cluster CNRS-INRA 'Symbiose' SYnthon et Matériaux BIOSourcés for mission funding.