Giulio Romualdi

Nahuel A Villa

Stefano Dafarra

Daniele Pucci

Olivier Stasse

Whole-Body Control and Estimation of Humanoid Robots with Link Flexibility

This article presents a whole-body controller for humanoid robots affected by concentrated link flexibility. We characterize the link flexibility by introducing passive joints at the concentration of deflections, which separate the flexible links into two or more rigid bodies. In this way, we extend our robot model to take link deflections into account as underactuated extra degrees of freedom, allowing us to design a whole-body controller capable to anticipate deformations. Since in a real scenario, the deflection is not directly measurable, we present an observer aiming at estimating the flexible joint state, namely position, velocity, and torque, only considering the measured contact force and the state of actuated joint. We validate the overall approach in simulations with the humanoid robot TALOS, whose hip is mechanically flexible due to a localized mechanical weakness. Furthermore, the paper compares the proposed wholebody control strategy with state-of-the-art approaches. Finally, we analyze the performance of the estimator in the case of different values of hip elasticity.

I. INTRODUCTION

The standard state-of-the-art controllers for bipedal locomotion are designed on several layers [START_REF] Feng | Robust dynamic walking using online foot step optimization[END_REF], [START_REF] Englsberger | Trajectory generation for continuous leg forces during double support and heel-to-toe shift based on divergent component of motion[END_REF], [START_REF] Koolen | Design of a Momentum-Based Control Framework and Application to the Humanoid Robot Atlas[END_REF], with the higher layers, closer to the user, producing instructions for the lower layers which work closer to the robot. In the lowest level, the motor drivers work at high frequency to guarantee either certain torque or acceleration on the robot joints, according to the whole-body controller instructions. In turn, the wholebody controller uses a model of the whole-body dynamics to compute the needed joint actuation for the robot to follow higher level instructions while keeping feasible contact forces on the environment.

The whole-body control layer typically assumes that all robot links are rigid bodies, which models correctly the whole-body motion in most cases [START_REF] Romualdi | A Benchmarking of DCM Based Architectures for Position and Velocity Controlled Walking of Humanoid Robots[END_REF], [START_REF] Hopkins | Compliant locomotion using whole-body control and Divergent Component of Motion tracking[END_REF], [START_REF] Ramuzat | Comparison of Position and Torque Whole-Body Control Schemes on the Humanoid Robot TALOS[END_REF], but becomes problematic with even small link deflections. This problem is common in heavy robot with mechanical weakness in their structure, such as the robot Talos (≈ 100 kg), whose hips bend while walking. At the modeling level, several approaches have been proposed to describe multi-link flexible robots [START_REF] Sayahkarajy | Review of modelling and control of flexible-link manipulators[END_REF]. The lumped-parameter approach separates flexible links into two or more rigid bodies by defining fictitious passive elastic joints [START_REF] Yoshikawa | Modeling of flexible manipulators using virtual rigid links and passive joints[END_REF], [START_REF] Kajita | Balancing a humanoid robot using backdrive concerned torque control and direct angular momentum feedback[END_REF], [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF], [START_REF] Nakaoka | Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms[END_REF]. Using this approach, we extend the robot model to incorporate the flexible degrees 1 Artificial and Mechanical Intelligence, Italian Institute of Technology, Genoa, Italy, (e-mail: name.surname@iit.it) 2 Gepetto Team, LAAS-CNRS, Université de Toulouse, Toulouse, France. (e-mail: name.surname@laas.fr) 3 Machine Learning and Optimisation, University of Manchester, Manchester, UK 4 Artificial and Natural Intelligence Toulouse Institute, Toulouse, France This work was supported by the European Union's Horizon 2020 through the SoftManBot project under Agreement 869855. of freedom as passive joints. Still, it is important to mention that the identification of the correct passive joint placement as well as the elastic parameters is not straightforward and has been the object of several studies [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF], [START_REF] Benallegue | Estimation and stabilization of humanoid flexibility deformation using only inertial measurement units and contact information[END_REF]. Alternatively, the finite element method (FEM) can also be used to model the flexibility of links [START_REF] Ge | A Nonlinear Feedback Controller for a Single-Link Flexible Manipulator Based on a Finite Element Model[END_REF], [START_REF] Subedi | Dynamic Modeling of Planar Multi-Link Flexible Manipulators[END_REF]. However, while providing enhanced modeling capabilities, FEM methods are computationally greedy, which generally forbids their use in time-critical applications.

From the control perspective, link flexibility is often treated with passive elastic joints [START_REF] Kajita | Balancing a humanoid robot using backdrive concerned torque control and direct angular momentum feedback[END_REF], [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF], [START_REF] Nakaoka | Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms[END_REF], [START_REF] Theodore | Robust control of multilink flexible manipulators[END_REF]. To address the elasticity of the robot link, the authors of [START_REF] Kajita | Balancing a humanoid robot using backdrive concerned torque control and direct angular momentum feedback[END_REF], [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF], [START_REF] Nakaoka | Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms[END_REF], [START_REF] Theodore | Robust control of multilink flexible manipulators[END_REF] locally compensate for the effect of deflections by modifying the measured position and velocity of some actuated joints considered in the whole-body controller.

This paper contributes to the design of a control architecture for humanoid robots affected by link elasticity. More precisely, the main contributions follow. i) A whole-body controller allowing for the locomotion of humanoid robots with flexible links. Our controller incorporates a model of flexible links following an approach similar to [START_REF] Kajita | Balancing a humanoid robot using backdrive concerned torque control and direct angular momentum feedback[END_REF], but differently from the state of the art, our design does not perform any local compensation of deflections. ii) As link deflections are not directly measurable, we propose an observer that uses only the measured contact forces and information from actuated joints to estimate the position, velocity and torques of the elastic joints. iii) We validate our approach on the simulated torque-controlled humanoid robot TALOS [START_REF] Stasse | TALOS: A new humanoid research platform targeted for industrial applications[END_REF], and we show how it performs different from the standard rigid-body-based controller.

The paper is organized as follows: Sec. II introduces our notation and recalls some concepts of floating base system, Sec. III details the whole-body controller, Sec. IV discusses the design of our elastic-joint observer, Section V presents the simulation results on the TALOS humanoid robot, and finally, Sec. VI concludes the paper.

II. BACKGROUND

A. Notation

• I m×n and 0 m×n denote the m × n identity and zero matrices. When m = n, we use simply I n , 0 n . • I = (p I , [I]) denotes the inertial frame. p I is the origin and [I] its orientation.

• given A = (p A , [A]) and B = (p B , [B]), then B[A] = (p B , [A]
) has its origin in p B and is oriented as A.

• Given A p C ∈ R 3 and B p C ∈ R 3 , representations of p C in A and B respectively A p C = A R B B p C + A p B = A H B [ B p ⊤ C 1 ] ⊤ . A H B ∈ SE(3)
is the homogeneous transformation and A R B ∈ SO(3) is the rotation matrix.

• the hat operator is • ∧ : R 3 → so(3), where so(3) is the set of skew-symmetric matrices and

x ∧ y = x × y. • the vee operator is • ∨ : so(3) → R 3 .
• given a rotation matrix, log operator returns the associated element of so(3) log : SO(3) → so(3).

• A v C,B ∈ R 3 denotes the relative linear velocity of the frame B with respect to C, expressed in A.

•

A ω C,B ∈ R 3 denotes the relative angular velocity of the frame B with respect to C, expressed in A.

• the velocity of a frame B w.r.t. the frame A is uniquely identified by the twist

A v ⊤ A,B = [ A v ⊤ A,B A ω ⊤ A,B ]. • × * is the dual spatial cross product on R 6 such that A v A,B × * = A ω ∧ A,B 03 A v ∧ A,B A ω ∧ A,B
.

• we drop the suffix I to identify the velocity of a frame with respect to

I, i.e., A v I,B = A v B , A ω I,B = A ω B .

B. Humanoid Robot Model

A humanoid robot is a floating base multi-body system made up of n + 1 links joined by n joints each with one degree of freedom. The robot configuration is fully described by the triplet q = ( I p B , I R B , s) ∈ R 3 × SO(3) × R n where B is the base frame and s the joint positions. The velocity of the floating system is represented by the triplet

ν = ( I v B , I ω B , ṡ) ∈ R 3 × so(3) × R n ∼ = R 3 × R 3 × R n .
The position and orientation of any frame A, attached to some link, is identified by I H A ∈ SE(3), and its twist v A = J A (q)ν is linearly related to the triplet ν by the Jacobian matrix J A (q).

Each link of the robot follows the rigid-body dynamics described by the Euler-Poincaré Equations [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF]

M L L vL + L v L × * M L L v L = M L I R ⊤ L g 0 3×1 + n f k=1 L f k , (1)
where L stands for link frame, M L ∈ R 6×6 is the constant inertia tensor, L v L ∈ R 6 is the link spatial velocity,

n f k=1 L f k ∈ R 6
are the n f wrenches exerted on the link. The proper sensor acceleration is given as [START_REF] Traversaro | Modelling, Estimation and Identification of Humanoid Robots Dynamics[END_REF]Sec. 2.4.4]

α g L := L X L[I] L[I] vL - I R ⊤ L g 0 3×1 . (2)
L X L[I] is the adjoint matrix, α g L is the acceleration obtained by an inertial measurement unit aligned to the link. Combin-ing (1) with (2), we obtain [START_REF] Traversaro | Modelling, Estimation and Identification of Humanoid Robots Dynamics[END_REF]Eq. (2.80)]

ϕ L = n f k=1 L f k , (3a)
ϕ L = M L α g L + 0 3×1 L ω L × * M L 0 3×1 L ω L . (3b)
The complete robot dynamics can be described as [START_REF] Wieber | Modelling and control of legged robots[END_REF] M (q) ν + η(q, ν)

= Bτ + k J ⊤ C k (q)f k . (4)
M (q) ∈ R (n+6)×(n+6) is the mass matrix, η(q, ν) ∈ R n+6 contains the Coriolis, centrifugal and gravitational effects. τ ∈ R n are the joint torques, acting only on joints as indicated by the selection matrix B ∈ R (n+6)×n . We express each contact wrench f k ∈ R 6 in the frame C k [I] located at the contact point and aligned to the inertial frame, J C k (q) is its associated Jacobian. Given a frame Ḡ = (x CoM , [I]) placed at the robot's center of mass (CoM) and aligned to the inertial frame, and applying a given change of coordinates as in [START_REF] Nava | Stability Analysis and Design of Momentum-based Controllers for Humanoid Robots[END_REF], the dynamics (4) can be expressed separating the first 6 rows, referring to the centroidal momentum of the robot Ḡh [START_REF] Orin | Centroidal dynamics of a humanoid robot[END_REF], from the last n rows referring to the actuated joints.

The rate of change of the centroidal momentum of the robot can be expressed by means of the external contact wrenches acting on the system [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF], [START_REF] Traversaro | Modelling, Estimation and Identification of Humanoid Robots Dynamics[END_REF]

Ḡ ḣ = mḡ + k ḠX C k [I] f k , (5)
where mḡ ⊤ = 0 0 -mg 0 0 0 ∈ R 6 is the spatial robot weight and the adjoint matrix ḠX Ci[I] ∈ R 6×6 changes the coordinates of a 6D force from C k [I] to Ḡ.

C. Model of the link flexibility

Let us consider that some robot links are affected by punctual mechanical weakness allowing the apparition of concentrated deformations. We model the extra degrees of freedom introduced by this flexible behavior by splitting each flexible body into two or more rigid links connected by passive virtual joints [START_REF] Nakaoka | Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms[END_REF], [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF]. So, the i-th virtual joint exerts a torque τ i that depends on the deflection s i and its velocity ṡi i.e. γ i : R × R → R as

τ i = γ i (s i , ṡi ). ( 6
)
By numerically differentiating the position of the elastic joint s i over a sampling period d t, we discretize (6) as

τ i [k] = γ i (s i [k], s i [k -1]), (7)
where k ∈ Z is the discretized time. Following [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF], [START_REF] Nakaoka | Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms[END_REF], we assume that γ i models the elastic behavior as

γ i (s i , ṡi ) = -k i s i -d i ṡi , (8)
where k i and d i are, respectively, the stiffness and damping of the i-th elastic joint. Discretizing the joint velocity with

ν, ν q, ν Flexible Joint State Observer Sec. IV G h ref , G ḣref τ * I R ref R , I R ref T I H ref F , F [I] v ref F , F [I] vref F τ ref a , s ref a F [I] f sf , ṡf , τf
τa, sa I HB, vB, sa, ṡa the forward Euler method, we approximate the elastic joint state, namely its position and velocity, by

F [I] vF , F [I] vF
s i [k] = d i s i [k -1] -τ i [k] d t k i d t + d i , (9a)
ṡi [k] = s i [k] -s i [k -1] d t . (9b)

D. Modeling of a floating base system with elastic joints

Let us redefine the robot configuration as:

q = ( I p B , I R B , s a , s f ), ν = I v B , I ω B , ṡa , ṡf (10)
with n a actuated joints s a and n f elastic joints s f , so that the robot posture has n = n a + n f degrees of freedom, and the robot dynamics (4) becomes:

M b ν + η b = k J ⊤ C k b f k , (11a)
M s ν + η s = I na 0 n f ×na τ a + 0 na×n f I n f τ f + k J ⊤ C k s f k . (11b)
where we have separated the base dynamics (subscript b corresponding to the 6 first rows) from the postural dynamics (subscript s corresponding to the n final rows).

III. WHOLE-BODY CONTROLLER

The whole-body controller guarantees the tracking of desired kinematic quantities while ensuring the feasibility of the contact forces. The proposed controller computes the desired actuated joint torques τ a , considering the robot dynamics [START_REF] Nakaoka | Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms[END_REF] and the elastic joint model [START_REF] Yoshikawa | Modeling of flexible manipulators using virtual rigid links and passive joints[END_REF]. The control objective is met by framing the controller as a constrained optimization problem in which low-priority tasks are embedded in the cost function, and high-priority tasks are handled as constraints. In the following equations, the superscript ref denotes the quantities provided by a high-level planner that assumes the robot made of rigid links only.

A. Costs and Constraints

This section introduces the list of low and high priority tasks considered in the optimal control problem.

1) Centroidal momentum task:

To ask for a desired centroidal momentum trajectory, we introduce the task as

Ψ h = Ḡ ḣ * -A c f -mḡ. ( 12
)
where

A c = ḠX C1[I] • • • ḠX Ci[I]
• • • gathers the adjoint matrix for each contact, and the desired value Ḡ ḣ * is chosen from a feedback law to guarantee asymptotic convergence to the desired trajectory

Ḡ ḣ * = mẍ ref CoM Ḡ ḣω ref + mk p CoM 0 3 x ref CoM -x CoM + mk d CoM 0 3 0 3 k hω ẋref CoM -ẋCoM Ḡh ref ω -Ḡh ω (13)
with positive definite gains k p CoM , k d CoM and k hω . 2) Cartesian task: While walking, we often require some link frames to have a specific position and orientation with respect to the inertial frame 3), by specifying the following task

I H ref L = (p ref L , I R ref L ) ∈ R 3 × SO(
Ψ L SE(3) = v * ⊤ L I ω * ⊤ L ⊤ -J L ν -JL ν, (14)
where v *

L = vref L + k d Lp (v ref L -v L ) + k p Lp (p ref L -p L )
and the angular acceleration is set to guarantee almost global stability and convergence of I R L to I R ref L [START_REF] Lynch | Modern Robotics: Mechanics, Planning, and Control[END_REF]:

I ω * L = I ωref L + k d Lω ( I ω ref L -I ω L ) + k p Lω log I R ref L I R ⊤ L ∨ . ( 15
)
k p Lω and k d Lω are positive definite matrices. From ( 14), we introduce the rotational tasks for L, denoted as Ψ L SO(3)

Ψ L SO(3) = 0 3 I 3 Ψ L SE(3) . ( 16
)
3) Floating base dynamics task: The whole-body control considers the base (11a) and joint dynamics (11b) as a set of high-priority tasks. We define the base dynamics task as

Ψ dyn b = η b + M b ν - k J ⊤ C k b (q)f k . (17)
If the robot has under-actuated elastic joints, the joint dynamics should consider the measured (or estimated) joint torques acting on the system as

Ψ dyn s = η s + M s (q) ν - I na 0 n f ×na τ a - 0 na×n f I n f τ f -J ⊤ Cs f. ( 18
) Notice that the last n f rows of (18) represent the underactuated flexible behavior of the robot.

4) Joint position regularization task:

To prevent the controller from computing solutions that generate a huge variation in joint acceleration, we introduce a joint regularization task for both the actuated and elastic joints, as

Ψ sa = s * a -0 na×6 I na 0 na×n f ν (19a) Ψ s f = s * f -0 n f ×6 0 n f ×na I n f ν, (19b)
with s * a equal to

s * a = sref a + k d sa ( ṡref a -ṡa ) + k p sa (s ref a -s a ). (20)
Where s ref a is the desired joint trajectory. k d sa and k p sa are two positive-defined diagonal matrices. On the other hand, we ask for s * f equal to

s * f = -k d s f ṡf -k p s f s f . (21)
k d s f and k p s f are defined positive matrices. Thanks to [START_REF] Orin | Centroidal dynamics of a humanoid robot[END_REF] the controller tries to stabilize the flexible joint position to zero.

5) Joint torque regularization task:

To prevent the controller from providing solutions with large actuated joint torques, we introduce the following task

Ψ τ = τ ref a -τ a . (22)
6) Feasible contact force task: The feasibility of the contact 6D force C k [I] f k is guaranteed by the set of inequalities

Φ f k = Γ C k [I] C k [I] f k -b, Φ f k ⪯ 0, (23)
where Γ C k [I] depends on the robot state q as in [START_REF] Caron | Stability of surface contacts for humanoid robots: Closed-form formulae of the Contact Wrench Cone for rectangular support areas[END_REF], and the inequalities ⪯ hold elementwise.

B. Quadratic programming formulation

We achieve the control objective by designing a constrained quadratic programming (QP) problem of the form

minimize ν, τa, f ∥Ψ T SO(3) ∥ 2 Λ T +∥Ψ R SO(3) ∥ 2 Λ R +∥Ψ s ∥ 2 Λs +∥Ψ τ ∥ 2 Λτ subj. to Ψ • SE(3) = 0 • = {L, R} Ψ h = 0 Ψ dyn • = 0 • = {s, ν} Φ f• ⪯ 0 • = {L, R}. (24)
Here, contact forces f j , the base acceleration B[I] vB , the actuated joint acceleration sa , and the actuated joint torques τ a are considered as optimization variables.

In [START_REF] Stellato | OSQP: An Operator Splitting Solver for Quadratic Programs[END_REF], the tracking of the left Ψ L SE(3) and right Ψ R SE(3) feet pose is treated as high-priority SE(3) tasks [START_REF] Subedi | Dynamic Modeling of Planar Multi-Link Flexible Manipulators[END_REF]. The Centroidal momentum tracking is considered as a highpriority task [START_REF] Benallegue | Estimation and stabilization of humanoid flexibility deformation using only inertial measurement units and contact information[END_REF]. We also consider the base [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF] and joints dynamics [START_REF] Traversaro | Modelling, Estimation and Identification of Humanoid Robots Dynamics[END_REF] as high-priority tasks. To prevent the controller from asking for a high motion of the upper body, we introduce two SO(3) tasks, one associated with the chest and the other with the waist orientations, respectively, denoted Ψ T SO(3) and Ψ R SO [START_REF] Koolen | Design of a Momentum-Based Control Framework and Application to the Humanoid Robot Atlas[END_REF] . In both cases, we ask the z coordinates of the link frames to remain parallel to the gravity vector. The postural conditions of the actuated and elastic joints [START_REF] Wieber | Modelling and control of legged robots[END_REF] are considered as low priority tasks. We regularize the desired actuated joint torques as a low priority task Ψ τ [START_REF] Lynch | Modern Robotics: Mechanics, Planning, and Control[END_REF]. Finally, to guarantee feasible contact forces for the feet, we add the task [START_REF] Caron | Stability of surface contacts for humanoid robots: Closed-form formulae of the Contact Wrench Cone for rectangular support areas[END_REF], denoted respectively as Φ f L and Φ f R .

IV. ELASTIC JOINT STATE OBSERVER

Solving the optimal control problem (24) requires a priorknowledge of the position, velocity and torque of all elastic joints. These values are fully known in simulation. However, in the real scenario, they must be estimated. We present an algorithm that computes the state of elastic joints considering the contact wrenches acting on the robot feet and the actuated joint state. Let us assume:

• the link flexibility is located on the robot leg;

• the robot leg is described by a kinematic chain where the first node represents the robot base; • all the flexible links are modeled with m f elastic joints described by ( 8) and ( 9); • each elastic joint is connected to its child joint with a fake link having zero mass and inertia; • there exists m a actuated links and joints between the latest elastic joint and the robot foot.

Let us consider a single leg; we denote the waist of the robot by L 0 , all virtual links and elastic joints by L f i and s f i with 1 ≤ i ≤ m f ; and all actual robot links and actuated joints by L a i and s a i with m f +1 ≤ i ≤ m a +m f , respectively. We also name the frame associated with each link as the link itself. We assume that each link is subject to only two wrenches, exerted by the parent and child links. With an abuse of notation, we consider the environment as the child of the foot L a ma+m f . For the dynamics of the links (3), we denote by ϕ i the wrenches exerted on the actual robot links and on the fictitious flexible links.

A. Forward kinematics

Given the angular velocity and the proper sensor acceleration of the foot L a ma+m f , we can compute ϕ i with m f + 1 ≤ i < m a + m f , recursively. Given the link L a i , its body angular velocity writes as

i ω i = i R i+1 i+1 ω i+1 + i ω i+1,i , m f + 1 ≤ i < m a + m f (25
) where i+1 ω i+1 is the angular velocity of the child link. i ω i+1,i is the velocity of L a i with respect to L a i+1 written in L a i . Similarly, we recall that α g L a i can be computed recursively by considering the child joint position s a i+1 velocity ṡa i+1 and acceleration sa i+1 , and α g L a i+1

as in [START_REF] Traversaro | Modelling, Estimation and Identification of Humanoid Robots Dynamics[END_REF]. We want to underline that if an IMU is mounted on the robot feet, ϕ ma+m f can be derived from the sensor readouts. Otherwise, we propose two possible solutions: i) In the case of low swing foot velocity and acceleration, we suggest considering ma+m f ω ma+m f and α g L a ma equal to zero. ii) Assuming that an IMU is mounted on the robot waist. We suggest setting α g L a ma and ma+m f ω ma+m f at time t equal to the one predicted by the forward kinematics at the previous step t -d t. To guarantee the convergence of the algorithm, the choice of d t becomes crucial. Here, we suggest setting d t small enough to capture the evolution of the system dynamics.

B. Inverse dynamics propagation

Assuming that ϕ L a i has been computed as in Sec. IV-A and considering [START_REF] Ge | A Nonlinear Feedback Controller for a Single-Link Flexible Manipulator Based on a Finite Element Model[END_REF] with n f = 2, the dynamics of L a i holds

ϕ i = i f λ(i),i + i f i+1,i , m f + 1 ≤ i < m a + m f , (26)
where λ(i) gives the parent link of L a i , i f λ(i),i is the spatial force exerted by the parent link λ(i) to L a i whose coordinates are expressed in L a i . i f i+1,i is the spatial force exerted by L a i+1 to L a i expressed in L a i . We now reorganize [START_REF] Huynh | Metrics for 3D rotations: Comparison and analysis[END_REF] to reveal its recursive structure:

i f λ(i),i = ϕ i -i f i+1,i (27a) = ϕ i + i X i+1 i+1 f i,i+1 (27b)
We notice that by projecting ( 27) into the joint motion axis i s ∈ R 6 [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF], [START_REF] Traversaro | Modelling, Estimation and Identification of Humanoid Robots Dynamics[END_REF], the torque acting on the joint s a i writes as

τ a i = i s ⊤ i f λ(i),i ∈ R.
If the system is equipped with joint torque sensors, τ a i can be directly measured. Using this information, we attempt to improve the estimation of the 6D force i f λ(i),i as follows:

i f c λ(i),i = (1 -β)τ am i +β i s ⊤ i f λ(i),i i s + I 6 -i s i s ⊤ i f λ(i),i . (28)
Here, i f c λ(i),i denotes the corrected 6D force and τ am i is the measured joint torque. We choose the parameter β ∈ [0, 1] according to our trust among measurement and calculations. At its extremes, β = 0 considers only the measured torques, and β = 1 just takes the computed torques.

C. Elastic joint state estimation

Propagating the inverse dynamics in Sec. IV-B, we recursively compute the wrench acting on the joint s a 1 . Assuming a neglected mass and inertia for the flexible link L f m f , i.e., M L f m f ≈ 0 6×6 , we write the flexible link dynamics as

m f f m f -1,m f = -m f f m f +1,m f = m f X m f +1 m f +1 f m f ,m f +1 , (29)
where m f X m f +1 depends on the position of the joint s a m f +1 obtained in Sec. IV-B. Given [START_REF] Dafarra | Dynamic complementarity conditions and whole-body trajectory optimization for humanoid robot locomotion[END_REF], we can compute the elastic joint torque

τ f m f by projecting m f f m f -1,m f into the joint motion axis m f s τ f m f = m f s ⊤ m f f m f -1,m f . ( 30
)
Combining [START_REF] Dantec | Whole Body Model Predictive Control with a Memory of Motion: Experiments on a Torque-Controlled Talos[END_REF] with the discretized elastic joint position (9a) we estimate the elastic joint position s f m f . Applying the same approach, we can estimate the elastic joint torque τ f i and position s f i , with 1 ≤ i ≤ m f -1 At each time step, applying the procedure of Sec. IV-A, IV-B, Eq. ( 29) and [START_REF] Dantec | Whole Body Model Predictive Control with a Memory of Motion: Experiments on a Torque-Controlled Talos[END_REF], we estimate the elastic joint state. The result is finally considered by the whole-body controller to compute the desired actuated joint torques.

V. RESULTS

In this section, we test the control strategy presented in Sec III on a simulated version of the TALOS humanoid robot [START_REF] Stasse | TALOS: A new humanoid research platform targeted for industrial applications[END_REF] -see Fig. 1. TALOS's hip flexibility has a significant impact on its leg control and, as a result, its balance and locomotion [START_REF] Ramuzat | Comparison of Position and Torque Whole-Body Control Schemes on the Humanoid Robot TALOS[END_REF]. Since the hip flexibility is due to the vertical linkage following previous work [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF], we model the flexibility along the pitch and roll axis for each leg. As a result, we have n f = 4 while n a = 32. The architecture takes (on average) less than 1 ms to evaluate its output. The OSQP [START_REF] Stellato | OSQP: An Operator Splitting Solver for Quadratic Programs[END_REF] library is used to solve the optimization problem. The simulations are obtained by integrating the robot dynamics and the joint elasticity as in Fig. 2.

To validate the performance of the proposed architecture, we present two main experiments. First, the proposed strategy is compared with a state-of-the-art task-based inverse dynamics algorithm that considers rigid links [START_REF] Romualdi | A Benchmarking of DCM-Based Architectures for Position, Velocity and Torque-Controlled Humanoid Robots[END_REF] in the case of different stiffness parameters k. From now on, the proposed control approach is called TSID-Flex and the state-of-the-art controller TSID-Rigid. Second, we analyze the performance of the TSID-Flex in the case of different stiffness. In both scenarios, the desired robot CoM, footsteps, and actuated joint trajectories are computed offline1 . The robot walks 1 meter forward with a step length of 20 cm, starting with the right foot. The first and last steps are 10 cm long. The double support lasts 0.2 s, while the single support lasts 1.2 s.

A. Comparison between TSID-Flex and TSID-Rigid

In Table I, we summarize the results of the control strategies for different stiffness parameters k. Labels success and failure mean that the associated controller is either able or not to ensure the robot's balance while walking. In all the experiments presented in this section, the damping parameter is arbitrarily set to b = 2 √ k. To compare the two controllers, we decided to perform two main experiments. In the former, we choose a set of stiffness parameters such that both whole-body controllers guarantee the balance while walking. In the latter, we decrease the value of the stiffness parameter. Namely:

-Experiment 1 k = 1 × 10 4 N m rad -1 ; -Experiment 2 k = 3 × 10 3 N m rad -1 .
1) Experiment 1: Fig. 3 shows the CoM tracking performance obtained with TSID-Flex and TSID-Rigid in terms of tracking error. The TSID-Flex controller seems to show good tracking performance, and the CoM error is kept below 2 mm. On the other hand, the TSID-Rigid induces a higher error on the CoM tracking. Similar considerations hold also for the tracking of the centroidal angular momentum.

In fact, TSID-Flex ensures a smaller angular momentum error with respect to the TSID-Rigid. One reason for this behavior is that the TSID-Rigid assumes the robot composed only by rigid link, hence all the joints are considered to be actuated. This assumption is generally valid in the case of stiff k, but it does not hold if k decreases. Fig. 5 presents the left foot trajectory error when the whole-body controller is TSID-Flex or TSID-Rigid. The angular error is given by the angle of the axis-angle representation between the foot orientation and the desired orientation [START_REF] Huynh | Metrics for 3D rotations: Comparison and analysis[END_REF]. Since TSID-Rigid does not consider link deformations, the controller assumes a wrong foot orientation when the robot is on single support -for 2.5 s ≤ t ≤ 5.5 s in Fig. 5.

2) Experiment 2: The centroidal quantity tracking problem discussed in Experiment 1 worsens at lower values of the stiffness parameter k. Fig. 4 shows the CoM tracking performance of the two controllers. In Fig. 4 we also present the tracking of the angular momentum. The TSID-Flex is still capable of ensuring good performance. On the other hand, the TSID-Rigid does not consider the elastic joint state. As a consequence, this leads to a non-negligible error on the robot CoM. In order to keep the balance, the TSID-Rigid controller requires high variations of the robot's foot orientation and at t ≈ 10 s the robot falls -Fig. 6.

B. Performance of the TSID-Flex in the case of different k

We now benchmark the performance of the TSID-Flex controller for different stiffness parameters k, namely k = 10 kN m rad -1 , 5 kN m rad -1 , 3 kN m rad -1 , 2 kN m rad -1 , 1 kN m rad -1 , and 0.5 kN m rad -1 . Fig. 7 shows the CoM tracking error for different values of k. The controller ensure acceptable performance for all values of k, i.e., the error is always below 3 mm. However, the lower the stiffness, the higher the tracking error. This increase in tracking error is caused by the elastic joint state estimation error. Fig. 8 presents the estimation error of an elastic joint. Similar considerations hold also for the other three joints. When the robot switches from single support to double support, the estimated torque associated with the elastic joint has a spike -at t ≈ 8.5 s, 11 s and 13.5 s in Fig. 8a. As a consequence, the error propagates in the estimation of the elastic joint position and velocity -see Fig. 8b and8c. This behavior is caused by the discontinuity of the contact force. To mitigate this effect, we may perform a smother transition between single and double support.

VI. CONCLUSIONS AND FUTURE WORK

This paper contributes to the design of a whole-body QP controller for a humanoid robot affected by link flexibility. We model the flexibility by introducing equivalent passive joints that simulate the motion caused by the viscoelastic link deformation. We then considered the passive joints position and velocity as state of the floating base system dynamics. Thanks to this choice, unlike other state-of-the-art approaches, we develop a whole-body controller that explicitly considers link flexibility in the stabilization problem. The paper also details the design of an estimator that aims at computing the elastic joint state in real-time. The proposed approach is validated in a simulated version of the TALOS humanoid robot, where its hip flexibility has a significant impact while performing locomotion tasks. Moreover, the architecture is then compared with a state-of-the-art whole-body controller that considers all links rigid. In future work, we plan to mitigate the contact forces discontinuity by performing a smoother transition between contiguous support phases. We also plan to make a comparison with other state-of-the-art controllers that consider the flexibility of the robot link [START_REF] Villa | Torque Controlled Locomotion of a Biped Robot with Link Flexibility[END_REF].

Another interesting research direction is to consider the link flexibility in the generation of reference centroidal trajectories [START_REF] Romualdi | Online Non-linear Centroidal MPC for Humanoid Robot Locomotion with Step Adjustment[END_REF], [START_REF] Murooka | Centroidal Trajectory Generation and Stabilization Based on Preview Control for Humanoid Multi-Contact Motion[END_REF] and whole-body-motion [START_REF] Dafarra | Dynamic complementarity conditions and whole-body trajectory optimization for humanoid robot locomotion[END_REF], [START_REF] Dantec | Whole Body Model Predictive Control with a Memory of Motion: Experiments on a Torque-Controlled Talos[END_REF].

Fig. 1 :

 1 Fig. 1: TALOS walking with the proposed controller.

Fig. 2 :

 2 Fig. 2: Controller architecture.

  Fig 2 presents the input and output quantities of the Whole-body controller.

  Angular Momentum -z.

Fig. 3 :

 3 Fig. 3: Centroidal momentum tracking: comparison between TSID-Rigid and TSID-Flex -k = 1 × 10 4 N m rad -1

Fig. 4 :

 4 Fig. 4: Centroidal momentum tracking: comparison between TSID-Rigid and TSID-Flex -k = 3 × 10 3 N m rad -1 .

  Foot tracking angular error.

Fig. 5 :

 5 Fig. 5: Foot tracking: TSID-Rigid and TSID-Flex comparisonk = 1 × 10 4 N m rad -1 .

Fig. 6 :

 6 Fig. 6: Foot tracking: TSID-Rigid and TSID-Flex comparisonk = 3 × 10 3 N m rad -1 .

7 :

 7 CoM Tracking performance in the case of different k.

  Elastic joint torque error.

  Elastic joint position error.

  Elastic joint velocity error.

Fig. 8 :

 8 Fig. 8: Elastic joint state estimation error.

TABLE I :

 I Controller outcome in the case of different stiffness parameter k.

The whole-body reference trajectories are provided by: github.com/loco-3d/multicontact-api/tree/v2.1.0


Nadiia Yakovenko

Baptiste Amouroux

Magali Albignac

Fabrice Collin

Clément Roux

Anne-Françoise Mingotaud

Pierre Roblin

Christophe Coudret
email: coudret@chimie.ups-tlse.fr

Alexandra Ter Halle

Alexandra Ter-Halle
email: ter-halle@chimie.ups-tlse.fr

Top-down synthesis of luminescent microplastics and nanoplastics by incorporation of upconverting nanoparticles for environmental assessment

Keywords: nanoplastic, UCNPs, lanthanide, powdering, milling, top-down synthesis, environmentally relevant, polymer

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The scientific community has recently invested crucial effort in evaluating the impact of plastic on ecosystems, and this has allowed us to realize, in addition to the fact that all compartments of the earth were concerned, that the pollution reached the nanoscale. Our experience in engineered nanomaterials enables us to envisage that nanoplastics have unique physical and chemical properties (e.g., size, shape, composition, and reactivity). The very nature of nanoplastics (NPs, sizes are below 1 µm) [START_REF] Alimi | Microplastics and Nanoplastics in Aquatic Environments: Aggregation, Deposition, and Enhanced Contaminant Transport[END_REF][START_REF] Gigault | Current opinion: What is a nanoplastic?[END_REF][START_REF] Ter Halle | Nanoplastic in the North Atlantic Subtropical Gyre[END_REF] being polydisperse, polymorphic, and mainly made of C, H, and O, similar to natural organic matter, further complicates the analysis of NP behavior by traditional colloid science.

In recent decades, much attention has been given to the study of the occurrence and potential negative effects of microplastics (MPs, ranging from 1 µm to 5 mm). [START_REF] Horton | Microplastics in freshwater and terrestrial environments: Evaluating the current understanding to identify the knowledge gaps and future research priorities[END_REF] MPs were analyzed in organisms ranging from zooplankton [START_REF] Cole | Microplastic Ingestion by Zooplankton[END_REF] to marine mammals. [START_REF] Zantis | Marine mammals and microplastics: A systematic review and call for standardisation[END_REF] Laboratory studies to investigate MP bioaccumulation and immunotoxicity in living organisms reported their biological impact on organisms (see references [START_REF] Anbumani | Ecotoxicological effects of microplastics on biota: a review[END_REF][START_REF] Alimba | Microplastics in the marine environment: Current trends in environmental pollution and mechanisms of toxicological profile[END_REF][START_REF] Galloway | Interactions of microplastic debris throughout the marine ecosystem[END_REF] for review). However, the fate and role of NPs remain poorly understood because of the lack of suitable analytical methods for monitoring their uptake, transport, and translocation and for understanding their potential adverse effects on living organisms. [START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF] To date, most of the evaluation studies have been run on model NPs using commercially available polystyrene (PS) nanospheres. [START_REF] Larue | A Critical Review on the Impacts of Nanoplastics and Microplastics on Aquatic and Terrestrial Photosynthetic Organisms[END_REF][START_REF] De Sa | Studies of the effects of microplastics on aquatic organisms: What do we know and where should we focus our efforts in the future?[END_REF] The distinction between model NPs and NPs-the nanoscale fraction of plastic debris generated in the environment-is often omitted even if their physicochemical and biological behavior are drastically different. [START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF] Model NPs are rather easy to handle and characterize but, lacking surface chemistry, polydispersity and an irregular shape similar to the one formed in the environment, they are of limited interest for ecotoxicological study. [START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF] A few recent studies proposed a top-down approach (i.e., producing smaller particles from larger objects) [START_REF] Reverberi | New trends in the synthesis of nanoparticles by green methods[END_REF] to prepare more environmentally relevant model NPs. Magrì et al. described the use of laser ablation in water to obtain polyethylene terephthalate nanoparticles with an average size of 100 nm and a surface chemical composition similar to plastic exposed to UV light in the environment. [START_REF] Magri | Laser Ablation as a Versatile Tool To Mimic Polyethylene Terephthalate Nanoplastic Pollutants: Characterization and Toxicology Assessment[END_REF] Astner et al. produced model MPs and model NPs (in the range 360-390 nm) made of polybetaine adipate-coterephthalate and low-density polyethylene by applying a sequence mimicking environmental weathering. [START_REF] Astner | Mechanical formation of micro-and nano-plastic materials for environmental studies in agricultural ecosystems[END_REF] El Hadri et al. produced NPs ranging from 20 nm to 1000 nm by applying blade grinding followed by planetary ball milling. [START_REF] Hadri | Fate of nanoplastics in the environment: Implication of the cigarette butts[END_REF] Model NP production from MPs collected in the environment was also proposed to prepare particles with a surface chemistry even closer to that of native nanoplastics. Their toxicological evaluation demonstrated that their impact on marine organisms was superior to that of PS nanospheres. [START_REF] Baudrimont | Ecotoxicity of polyethylene nanoplastics from the North Atlantic oceanic gyre on freshwater and marine organisms (microalgae and filter-feeding bivalves)[END_REF] Even if the plastic particles obtained by the top-down methods reproduce the main physicochemical characteristics of NPs formed in the environment, their tracking remains challenging during biological evaluation tests, making it difficult to monitor their ingestion, transport, and translocation in living organisms.

Recent synthesis strategies, such as [START_REF] Reverberi | New trends in the synthesis of nanoparticles by green methods[END_REF] C enrichment, [START_REF] Sander | Assessing the environmental transformation of nanoplastic through (13)C-labelled polymers[END_REF] rare metals, [START_REF] Mitrano | Synthesis of metal-doped nanoplastics and their utility to investigate fate and behaviour in complex environmental systems[END_REF] or simple staining methods with fluorescent dyes, [START_REF] Maes | A rapid-screening approach to detect and quantify microplastics based on fluorescent tagging with Nile Red[END_REF] were used to obtain label model MPs and NPs. Fluorescence labeling is a widely developed strategy, but most fluorescent tags operate with UV-blue to green excitation light, showing low tissue penetration and toxicity to living organisms and potentially leading to background autofluorescence, thus preventing their use in deep-tissue imaging. Dyes can also suffer from photobleaching with prolonged exposures [START_REF] Paganin-Gioanni | Fluorescence imaging agents in cancerology[END_REF] or can have solvatochromic properties, making the emission spectra dependent on the polarity of the surrounding environment. [START_REF] Greenspan | Spectrofluorometric studies of the lipid probe, nile red[END_REF] Another limitation is the potential costaining of the organic matter present in the sample under focus, resulting in false detection. As an alternative to fluorescence labeling, Mitrano et al. proposed the use of heavy metals as tracers and inductively coupled plasma-mass spectrometry (ICP-MS) for indirect plastic monitoring.

Polyacrylonitrile particles doped with Pd were accurately assessed in a complex matrix (activated sludge from wastewater treatment plant), and no Pd leaching from particles was observed. [START_REF] Mitrano | Synthesis of metal-doped nanoplastics and their utility to investigate fate and behaviour in complex environmental systems[END_REF] This strategy was found to be very efficient but is not well adapted for monitoring MPs and NPs in living organisms because of the use of a destructive analytical technique.

The objective of the present study is to synthesize environmentally relevant models of MPs and NPs in terms of polydispersity, shapes and surface properties that could be easily visualized and tracked in the tissues of living organisms. Polyethylene (PE) was selected as a model polymer because it is the most commonly used polymer and the most frequently found polymer in the environment. However, especially at the nanoscale, this material is very stealthy and poorly detectable using nondestructive methods. For easy imaging, lanthanidebased upconverting particles (UCNPs) were incorporated into the polymer matrix. Because of the spectroscopic features of the lanthanide family, these nanomaterials can accumulate near-infrared (NIR) energy and release it as visible emission as a line spectrum typical of the emitting element (thus erbium gives rise to a bright green visible emission). [START_REF] Himmelstoss | A critical comparison of lanthanide based upconversion nanoparticles to fluorescent proteins, semiconductor quantum dots, and carbon dots for use in optical sensing and imaging[END_REF] They are commonly used as background-free luminophores in imaging or anti-counterfeiting applications [START_REF] Kermorgant | Evaluation of upconverting nanoparticles towards heart theranostics[END_REF] or as local sources of light ("nanolamps") [START_REF] Amouroux | A photochemical determination of luminescence efficiency of upconverting nanoparticles[END_REF] in material or biological applications. [START_REF] Dacosta | Lanthanide upconversion nanoparticles and applications in bioassays and bioimaging: a review[END_REF][START_REF] Park | Upconverting nanoparticles: a versatile platform for wide-field two-photon microscopy and multi-modal in vivo imaging[END_REF][START_REF] Yang | Principles and application of up-converting phosphor technology[END_REF][START_REF] Kang | Lanthanide-doped upconversion nanomaterials: recent advances and applications[END_REF] Moreover, highly photostable (no blinking, no bleaching) [START_REF] Wu | Non-blinking and photostable upconverted luminescence from single lanthanidedoped nanocrystals[END_REF] and requiring lower power density than 2-photon fluorophores, these background-free persistent fluorophores have become an attractive probe for imaging purposes, including thick samples. Furthermore, UCNPs derived from rare element salts (NaREF4) can be prepared in a very monodispersed manner depending on the choice of rare earth elements and the preparation protocol. We chose to develop a top-down approach for the synthesis of labeled PE by incorporating highly monodispersed, hydrophobic and less than 30 nm UCNPs into bulk PE material and by using powdering to produce labeled nanometric particles with irregular shapes and a variety of sizes. The particles were characterized with many analytical techniques to obtain full information about their physicochemical properties.

Materials and methods

Chemicals: PE pellets (High-Density PE, CAS 9002-88-4) were purchased from Sigma Aldrich (Saint Louis, MO, USA). The polymer used was characterized by a melt flow index (MFI) of 12 g/10 min, a melting point between 125 and 140 °C and a density of 0.952 g.mL -1 at 25 °C. Rare earth (RE) chloride hydrates, ammonium fluoride, oleic acid (OA) and 1-octadecene (ODE) were purchased from Alfa Aesar and used as received. All other organic solvents were of spectroscopic grade and used as received. Deionized water (ρ= 18 MΩ cm -1 ) obtained from an Aquadem apparatus was used in this study.

Upconverting nanoparticle synthesis:

In a three-necked flask, a solution of YCl3.6H2O (2.4 g, 7.9 mmol), YbCl3.6H2O (1.35 g, 3.5 mmol) and ErCl3.6H2O (0.088 g, 0.23 mmol) in water (10 mL) was added to an OA:ODE mixture (35:190 mL:mL), and water was distilled off under an Ar flush. The resulting cloudy suspension was brought to 160 °C until clear (ca 1 h) and then cooled under an Ar flush. NH4F (2.37 g, 64 mmol, 80 mL, 0.8 M) and NaOH (1.86 g, 46.6 mmol, 80 mL, 0.58 M) were dissolved separately in MeOH. These two solutions were added simultaneously using syringe pumps at 0.4 mL.min -1 to the vigorously stirred RE(OA)3:OA:ODE solution. Once the addition was completed, the methanol was removed under an Ar flush at 100 °C. The reaction mixture, after careful degassing, was then brought to 310 °C as fast as possible (ca 20 °C.min -1 ) and kept at this temperature for 90 min. After cooling, an equivalent amount of ethanol was added, and particles were collected by centrifugation at 9000 g for 10 min. They were purified by repeating twice the following sequence: pellet redissolution in cyclohexane/addition of the same volume of ethanol/centrifugation. UCNP incorporation into PE : PE pellets were milled using a RETSCH ZM 200 ultracentrifugal mill. The obtained polymer powder with an average size of 200 µm was dissolved in boiling oxylene (≥ 99.0% (GC grade); Sigma Aldrich, Saint Louis, MO, USA), where UCNPs (10 wt.% PE) were subsequently introduced. The mixture was homogenized by magnetic stirring at 146 °C for 20 min. Precipitation of the synthesized polymer material was achieved by rapid cooling in an ice bath. UCNPs that were not incorporated into PE were removed by washing the polymer with cyclohexane (≥99.7% (HPLC grade); Sigma Aldrich, Saint Louis, MO, USA). Before future use, the polymer was dried overnight at 40 °C. As a negative control, a second batch of polymers containing only PE (Blank-PE) was prepared following a similar protocol.

Synthesis of the labelled micro-and nanoparticles: Bulk materials Blank-PE and Upcon-PE

were exposed to a cryogenic grinder (SPEX™ SamplePrep 6775 Freezer/Mill™, Delta Labo, Avignon, France). First, samples were precooled for 5 min, and then grinding was carried out in 10 cycles of 1 min with cooling between cycles for 2 min in liquid nitrogen. The grinding rate was 10 coups per second (CPS). The obtained polymer powder was dispersed in ethanol (≥99.8% (HPLC grade); Sigma Aldrich, Saint Louis, MO, USA) and sonicated for 15 min at room temperature to avoid particle aggregation. Then, particle size separation was performed in 6 sequential filtration steps through a metal mesh with a cutoff of 500 µm (Retsch, Haan, Germany) and NITEX nylon mesh (Dominique Dutscher SAS, Brumath, France), with pore sizes of 200 μm, 100 μm, 50 μm, 11 μm, and 1 μm. Wet milling was performed in a 20 mL glass vial filled with approximately 50 g of ZY-E (yttrium stabilized zirconium oxide) beads, 1.0-1.2 mm (Sigmund Lindner GmbH (SiLi), Warmensteinach, Germany), 0.1 to 0.5 g of MPs, and approximately 10 mL of absolute ethanol. The milling process lasted 48, 60, and 120 days under 80 rpm stirring on a horizontal ROLLER 6 digital apparatus (IKA, Staufen, Germany).

Eventually, MPs and NPs were separated by the 6 steps of subsequent filtration described above and rinsed with cyclohexane.

Granulometry analysis:

The MP size distribution was determined using a Mastersizer MS3000 (Malvern Panalytical, UK). The polymer powder was dispersed in ethanol using a HYDRO MV device with stirring at 2500 rpm. The refractive indices used for ethanol and particles were 1.36 and 1.52, respectively, with an absorption index for particles of 0.1. The results are presented as the mean number of particles obtained from 5 measurements. Differential scanning calorimetry: DSC was performed with a DSC1 Mettler Toledo apparatus equipped with an HSS8 sensor. Samples were weighed (between 5 and 7 mg) and sealed in 40 mL aluminum pans. They were heated from 20 °C to 160 °C at 5 °C/min with an empty aluminum pan as the reference. Dry nitrogen with a flow rate set at 20 mL/min was used as the purge gas. Each sample was analyzed in triplicate. The first heating cycle was used to erase all processing, thermal, mechanical, crystallization, and shear history of the samples. The melting temperature and crystallinity were obtained from the second heating cycle. The crystallinity was calculated from the expression:

Crystallinity (%) = (ΔHm/ ΔHmref)*100 ( 1
)
where ΔHm is the melting enthalpy of the analyzed samples, and ΔHmref is the melting enthalpy of 100% crystalline PE (293 J/g).

Small-angle X-ray scattering (SAXS) analysis: SAXS measurements were performed on a XEUSS 2.0 laboratory source equipped with a PILATUS 1 M pixel detector (DECTRIS) and an Xray source provided by a GeniX3D with a fixed wavelength based on Cu Kα radiation (λ = 1.54 Å). The distance between the sample and the detector was fixed at 1216.5 mm, giving a q range starting from 0.005 Å -1 to 0.5 Å -1 , assuming that q is the scattering vector equal to 4π/(λsin θ) with a 2θ scattering angle. The distance was calibrated in the small-angle region using silver behenate (d001 = 58.34 Å). For solid samples, the dry MP powder was placed between two Kapton windows with 1 mm of the optical pathway and placed on the motorized sample holder. For liquid samples, particle dispersion was injected into a capillary tube with a diameter of 1.5 mm and then placed under the beam in the conditions. To remove scattering and absorption from air, a primary vacuum was applied to the entire instrument. To maximize the signal at high angles, 6 acquisitions of 600 seconds were averaged. All scattering curves were corrected for the empty cell contribution or for the solvent, divided by the transmission factor, acquisition time and optical path to obtain SAXS curves in absolute units (cm -1 ).

Transmission electron microscopy energy-dispersive X-ray spectroscopy: NP dispersions in ethanol were prepared with polyvinylpyrrolidone (PVP) 0.4% (w/v) to ensure particle adhesion on the top of the formvar/carbon-coated copper grid. One drop of 20 µL of sample solution was placed on the top of the grid. After drying, TEM/EDX analysis was performed using a JEOL cold-FEG JEM-ARM200F operated at 200 kV equipped with a probe Cs corrector reaching a spatial resolution of 0.078 nm. EDX spectra were recorded on a JEOL CENTURIO SDD detector.

Nanoparticle tracking analysis

Results

Large-scale monodisperse UCNP synthesis

Using classical thermal coprecipitation to prepare hydrophobic oleate-coated NaREF4 UCNPs, we implemented a new protocol to achieve the synthesis of monodisperse 20 nm UCNPs over a multigram scale. This approach allows the use of a single batch of phosphors to be further dispatched in multiple assays, therefore maintaining a homogeneous luminescence response over all samples. [START_REF] Wilhelm | Water dispersible upconverting nanoparticles: effects of surface modification on their luminescence and colloidal stability[END_REF][START_REF] Jiao | Controllable Synthesis of Upconversion Nanophosphors toward Scale-Up Productions[END_REF] As we noticed that a slow and separate introduction of fluoride and sodium sources via syringe pumps to the rare earth oleate mixture led to a dramatic size reduction, [START_REF] Amouroux | Importance of the Mixing and High-Temperature Heating Steps in the Controlled Thermal Coprecipitation Synthesis of Sub-5-nm Na(Gd-Yb)F4:Tm[END_REF] we decided to adapt this strategy to the efficient protocol described by Zhang et al., known to yield 40 nm NaYF4-based UCNPs. [START_REF] Li | An efficient and user-friendly method for the synthesis of hexagonalphase NaYF(4):Yb, Er/Tm nanocrystals with controllable shape and upconversion fluorescence[END_REF] W collected regular hexagonal prism-shaped erbium-doped NaYF4:Yb30% and Er2% UCNPs (NaREF4, where RE= 2% Er; 30% Yb; 68% Y) with an average size of 21.8±1.3 nm even on an 11.6 mmol scale: up to 3 g of bright green emissive UCNPs could be obtained in this way. XRD analysis was consistent with the pure hexagonal phase (Figure 1). We chose to incorporate UCNPs into the polymer using a dissolution/precipitation procedure, which allows one to work on smaller quantities than melt mixing (Figure 2a). The UCNP:PE ratio was chosen by comparison with a standard nanocomposite formulation involving HDPE. [START_REF] Gao | Preparation and Properties of High-Density Polyethylene/Silica Composites[END_REF] Briefly, PE was solubilized in boiling o-xylene in the presence of UCNPs (10% in weight). Cooling the colorless solution at 0 °C led to the quantitative precipitation of the polymer. Excitation of the collected whitish solid with a 976 nm continuous-wave (CW) laser induced the bright green luminescence typical of Er 3+ , indicating that UCNPs were successfully incorporated in bulk PE (Upcon-PE). A similar test was found to be negative for UCNP-free synthesis, Blank-PE.

The powdering process consisted of a first step to cryogenic grinding. We selected this method because it is a process where thermally sensitive and elastic substances are generally successfully processed. The effect of several parameters was considered, including precooling time, grinding time, and oscillation frequency. Experiments reveal a nonmonotonous effect of these parameters on abrasion efficiency. At this stage of the powdering process, particles at the nanoscale were not detected by nanoparticle tracking analysis (NTA). Both powders, with or without UCNPs, were named µ-Upcon-PE and µ-Blank-PE, respectively. The size distribution obtained by granulometric analysis showed that approximately 90% of the particles were smaller than 15 µm (Figure 2b). The incorporation of the UCNPs did not significantly impact the size distribution of the powder (Figure S1).

To reach the nanoscale, we proceeded to further size reduction using a low-energy wet-driven milling technique. The method is simple, economical, sustainable and, similar to the first step, provides the advantage of the preservation of the polymer structure. [START_REF] Trofa | Nanoparticles Synthesis in Wet-Operating Stirred Media: Investigation on the Grinding Efficiency[END_REF] It consists of preparing a dispersion of the powder in a solvent, adding zirconium beads and then slowly rolling the solution. Ethanol was chosen as the solvent because synthesis in water, which would be more appropriate for future use in toxicological tests, was not possible. PE is very hydrophobic and does not wet in water. Furthermore, PE floating in water and contact with zirconium beads used for milling were not favored. After the two-stage powdering process, the particles were fractionated by sequential filtration in ethanol (using a 6 mesh cutoff: 500 µm, 200 μm, 100 μm, 50 μm, 11 μm, and 1 μm). This easy-to-implement protocol is aimed at preventing rapid clogging of the filter mesh during filtration without resorting to a more complex process, such as tangential flow filtration (TFF). [START_REF] Dehghani | Tangential flow microfluidics for the capture and release of nanoparticles and extracellular vesicles on conventional and ultrathin membranes[END_REF] Nanoscale particles (nano-Upcon-PE and nanoblank-PE)

were produced with a yield of approximately 6%. The distribution of the particles by nanoparticle tracking analysis (NTA) showed that approximately 90% of the particles were smaller than 201 nm for nanoblank-PE and 117 nm for nano-Upcon-PE (Figure 2c or Figure SI2 for details). The incorporation of UCNPs led to smaller particles; we have no explanation for this result.

Chemical integrity of PE

Attenuated total reflection Fourier transform infrared (ATR-FTIR) analysis of the samples (on bulk, micrometric and nanometric particles) was performed to determine if any PE degradation occurred during synthesis or powdering. An important side reaction could be PE oxidation leading, in particular, to the formation of carbonyl moieties, which is easily detected by an intense absorption band in the region of 1650-1850 cm -1 . [START_REF] Chiellini | Oxo-biodegradable carbon backbone polymers -Oxidative degradation of polyethylene under accelerated test conditions[END_REF] FTIR spectra were obtained after each processing step and compared to the spectra of raw PE pellets. No absorption band due to the presence of oxidation was detected (Figure S3). This result indicates that synthesis and grinding processes do not cause any chemical modification of PE.

Effectiveness of UCNPs incorporation in PE

During the synthesis of the bulk material, the dissolution/precipitation procedure does not allow control of the UCNP rate of incorporation. This amount was measured by two distinct techniques: differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA). DSC was used to accurately determine the PE crystallinity at each stage of the preparation. The melting enthalpy (ΔHm), onset temperature (Tonset), and exothermic peak temperature (Tpeak) of crystallization were obtained from the second heating cycle (Table 1). If a polymer is mixed with a material that does not melt in the same temperature range, DSC is a way to record in what proportion it is added. [START_REF] Gao | Preparation and Properties of High-Density Polyethylene/Silica Composites[END_REF] We selected the second cycle because this measurement does not record the history of the material, such as the way it was cooled (however, the use of the enthalpy recorded at the first cycle gave the same proportion of incorporated UCNPs, data not shown).

The melting points were similar regardless of the stage of the process; this result confirms the absence of PE chemical modification after the incorporation of UCNPs or powdering. Before engagement in the dissolution/precipitation step, the PE raw pellet and raw powder had similar melting enthalpies. After cryogenic grinding, the enthalpy of µ-Blank-PE was slightly superior, that is, the polymer was slightly more crystalline at this stage (85% crystallinity compared to 83% initially). Knowing that UCNPs are thermodynamically stable in the hexagonal phase up to 700 °C, [START_REF] Thoma | The sodium fluoride-lanthanide trifluoride systems[END_REF] their presence did not contribute to the enthalpy recorded between 20 °C and 160 °C. This result led us to a UCNP-content in µUpcon-PE of 6.4 ± 1.2 wt.%. This rate was confirmed by thermogravimetric analysis, which gave a value of 5.7% (Table S1).

Table 1. DSC analysis of raw PE pellets, µ-Blank-PE and µ-Upcon-PE. The melting enthalpy, onset temperature and exothermic peak temperature were obtained from the second heating cycle (average values ± SD). The corresponding degrees of crystallinity (Xc) were calculated as the ratio between ΔHm of the sample and the reference ΔHmref value of a 100% crystalline form of PE (293 J g -1 ). [START_REF] Dehghani | Tangential flow microfluidics for the capture and release of nanoparticles and extracellular vesicles on conventional and ultrathin membranes[END_REF][START_REF] Chiellini | Oxo-biodegradable carbon backbone polymers -Oxidative degradation of polyethylene under accelerated test conditions[END_REF]

Sample

ΔHm, (J g -

Study of UCNPs dispersion within PE

The way UCNPs were incorporated in PE was investigated in detail with particles at the microscale by small-angle X-ray scattering (SAXS). Details of the applied models for SAXS are given in the Supporting Information. The SAXS curve of µ-Blank-PE (Figure 3a and Figure S4)

shows a constant decay with a small inflection of approximately 0.05 Å -1 , reflecting the correlation distance between crystalline parts present in the PE matrix. The SAXS curve of µ-Upcon-PE follows the same decay as the SAXS curve of µ-Blank-PE plus the contribution of UCNPs (Figure 2a). The first result confirms that the UCNPs do not alter the organization of PE. The contribution of UCNPs was investigated to depict whether the UCNPs were either adsorbed on the surface or only partially incorporated into PE. To discriminate the last case, a simulated SAXS curve was obtained by a linear combination of the two contributions and compared to the experimental curve (Figure 3b). Data were plotted in Log I(q)q 4 as a function of Log q to magnify the differences. No linear combination of the SAXS curves of PE and UCNPs was found to fit the experimental SAXS data of Upcon-PE. The difference observed between the curves suggests that another term contributes to the SAXS curves. In the case where UCNPs are distributed in the PE matrix, it is necessary to consider the scattering cross terms, which reflect the interactions between atoms of UCNPs and PE. A simplified model was applied to describe µ-Upcon-PE, and a two-component model was applied to describe the organization at large distances (see "Applied models for SAXS analysis" and Figure S5 in the Supporting Information for more details). The states of the surfaces of µ-Blank-PE and µ-Upcon-PE were found to be similar, indicating that the incorporation of UCNPs does not induce any modification of the surface particle properties. This inference suggests that UCNPs are not adsorbed onto the surface of PE but rather fully incorporated inside the PE matrix.

This result is interesting since PE is not prone to easily accept the incorporation of other materials, and fabrication of PE composites generally requires a more complex process than a simple dissolution/precipitation procedure such as the one proposed here. [START_REF] Xu | Ultrafast Fabrication of Graphene-Reinforced Nanocomposites via Synergy of Steam Explosion and Alternating Convergent-Divergent Flow[END_REF][START_REF] Hu | The fabrication and characterization of high density polyethylene composites reinforced by carbon nanotube coated carbon fibers[END_REF] The effective incorporation of UCNPs into PE prevents UCNPs from leaching and enables their detectability regardless of the treatments or experiments performed (washing, purification, use in vivo, etc.).

a) b)

Figure 3. Small-angle X-ray scattering (SAXS) analysis of a) µ-Blank-PE (blue curve), µ-Upcon-PE (green curve) and UCNPs alone (yellow curve), where Log I(q) is plotted as a function of Log q; b) µ-Upcon-PE spectrum and a simulated spectrum of a linear combination of the scattering curves of µ-Blank-PE and UCNPs, considered two independent contributions; Log I(q)q 4 is plotted as a function of Log q.

Colloidal properties of labelled nanoplastics

Electron microscopy

As discussed earlier, NTA allowed us to conclude that the particles prepared at the nanoscale were very polydisperse and mostly smaller than 200 nm. The shape of the particles was assessed by electron microscopy (TEM) and showed that the particles were indeed polydisperse and polymorphic (Figure 4a). The incorporation of UCNPs did not modify their shapes (data not shown). The heterogeneous shapes obtained by this top-down process are in accordance with a recent study. [START_REF] Hadri | Nanoplastic from mechanically degraded primary and secondary microplastics for environmental assessments[END_REF] The darker dots in the particles were supposed to be UCNPs because they were expected to be highly dense, and their size was approximately 20 nm (Figure 4a). The images showed that the UCNPs were rather well dispersed within PE

Zeta potential

The surface charge of the produced micro-and nanometric particles was obtained by zetapotential (-potential) measurements. At the micro-and nanoscales, the particles were found to be negatively charged (Table S2). The presence of incorporated UCNPs did not significantly modify the particle charges. The zeta potentials for the micro-and nanometric particles were approximately -70 mV and -60 mV, respectively. These values were in accordance with a recent study using the top-down process; all the model NPs were negatively charged, and those in PE had a zeta potential of -38 ±2.4 mV at a similar pH. [START_REF] Hadri | Nanoplastic from mechanically degraded primary and secondary microplastics for environmental assessments[END_REF] It is expected that most NP surfaces would bear functional groups such as carboxylic acids or hydroxyl groups resulting from the oxidation of the polymer; consequently, at neutral pH, they are expected to be negatively charged. The model NPs prepared are also negatively charged.

This parameter is important, as the surface charge of nanoparticles dramatically affects their behavior in the environment or in vivo. [START_REF] Hotze | Nanoparticle Aggregation: Challenges to Understanding Transport and Reactivity in the Environment[END_REF]

Emission properties of labeled nanoplastics

The emission properties of the UCNPs once incorporated in PE were investigated.

Fluorescence measurements (exc. = 980 nm) of the dispersed particles in ethanol showed no emission for Blank-PE, while Upcon-PE showed three emission peaks at approximately 525, 550 and 655 nm, attributed to the Er 3+ transitions 2 H11/2  4 I15/2, 4 S3/2  4 I15/2 and 4 F9/2  4 I15/2. [START_REF] Amouroux | Importance of the Mixing and High-Temperature Heating Steps in the Controlled Thermal Coprecipitation Synthesis of Sub-5-nm Na(Gd-Yb)F4:Tm[END_REF] . These specific emission peaks were observed with µ-Upcon-PE and nano-Upcon-PE (Figure 5).

To check whether the UCNPs maintained their functionality once encapsulated in PE, the emission properties of all the prepared particles were investigated under upconversion conditions. Under continuous-wave laser irradiation at 980 nm, the pristine Er-UCNPs displayed typical green visible luminescence. The recorded spectrum of the dispersed UCNPs in cyclohexane showed three emission peaks at approximately 525, 550 (green) and 655 nm (red), attributed to the Er 3+ transitions 2 H11/2  4 I15/2, 4 S3/2  4 I15/2 and 4 F9/2  4 I15/2, respectively. [START_REF] Amouroux | Importance of the Mixing and High-Temperature Heating Steps in the Controlled Thermal Coprecipitation Synthesis of Sub-5-nm Na(Gd-Yb)F4:Tm[END_REF] (Figure 5). As expected, blank PE particles dispersed in ethanol did not show any signal in this spectral range upon excitation under similar conditions. However, Upcon-PE (micro-and nanoscale) displayed an emission spectrum similar to that of UCNPs: the upconversion property was not lost in the PE incorporation/grinding and milling process. upconverted light that could be collected between 490 and 600 nm and 630 and 710 nm, respectively. The absence of any emission from the blank PE particle indicates that no 2photon processes occur in pure PE and justifies the use of UCNPs as a background-free luminescent tag for plastic particles. The band ratio G/R was found differ from that recorded at lower fluence. It is known that the power dependence of the emission lines is rather nonlinear; therefore, the band ratio typically observed at very low fluence, such as with a CW 980 nm laser, can be different when using a more intense pulse laser, which is commonly used to trigger 2-photon fluorescence emission. This spectral alteration is also a signature of the presence of UCNPs and can be used to identify tagged particles in more complex environments.

Conclusion

This top-down synthesis allows the production of model plastic particles within the micro-and nanometric range, which present heterogeneity in size and shape that is absent for most of the particles formed by the bottom-up approaches. This method does not allow us to control the amount of UCNPs incorporated in the polymer, as the parameters of the precipitation process are difficult to control, but the UCNP content reaches 6% by weight. Using a series of physicochemical characterizations, we demonstrated that the UCNPs were incorporated within the polymer, not localized at its surface (SAXS), and well dispersed (TEM-EDX). The powdering process does not alter the chemical structure or the morphology of the polymer (ATR-FTIR and DSC). The mixture of micro-and nanometric objects formed can easily be separated by filtration; thus, a range of sizes can be selected for a given application. The nanometric particles were formed in a small proportion (6% yield), but the remaining micrometric particles can be engaged in another long process of wet milling for further production of smaller particles.

The incorporation of the UCNPs did not alter their luminescent properties, and the labeled PE particles could be easily monitored with a standard two-photon microscope. This feature offers the possibility to easily detect these model NPs in complex organic matrices, while the detection of plastic itself is always difficult.

The proposed method is compatible with a large variety of polymers, while the UCNP rate of incorporation in the polymer matrix is not easily controllable and certainly depends on the polarity and affinity of the UCNPs for the polymer. However, this method also offers the possibility to use formulated polymers, such as commercialized plastic, to produce plastic particles even closer to those present in the environment. This process is also compatible with weathered polymers.

The prepared particles are very good environmentally relevant models because of their polydispersity and heterogeneity. These two properties are very scarcely considered in actual toxicological evaluation studies because most of the particles used are produced with a bottom-up method. [START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF] Because of the extreme photostable and NIR-activable luminescent properties of the prepared particles, strategies based on their use can pave the way for a deeper understanding of the fate of plastic particles in the trophic chain.

:

  NTA analyses were performed at 25 °C using a NanoSight LM10 (Malvern Instruments, Ltd, UK) instrument equipped with an sCMOS camera. Samples were prepared by NP dispersion in ultrapure Milli-Q water (50 mg/L). Measurement of each sample was performed in triplicate, consisting of three records of 60 s. The results are presented as the mean size of the particles ± SD. NTA is known to provide accurate measurements even for polydisperse nanoparticles and was thus used without any complementary measurement by dynamic light scattering measurements.Fluorescence measurement: Fluorescence spectra were acquired with a Fluorolog-3-2iHR320 modular spectrofluorometer (HORIBA Jobin Yvon SAS, Palaiseau, France). The standard light source of the fluorimeter was substituted by a 980 nm continuous-wave laser operating at a power of 1.5 W. Emission was collected between 400 nm and 750 nm. Two-photon microscopy: Samples were imaged on a Zeiss LSM-710 two-photon microscope equipped with a Ti: Sapphire Chameleon Vision II laser (Coherent, Santa Clara, California) operating at 980 nm (4% of max power for UCNPs and 50% for µ-Blank-PE and µ-Upcon-PE samples, 140 fs pulses, 80 MHz repetition rate) and a Zeiss Plan-Apochromat 63x/1.40 oil immersion lens. Upconverted light was collected between 490 and 600 nm for the green band and 630 and 710 nm for the red band. Images were processed using ImageJ (Fiji) software.Zeta-potential measurements: Zeta-potential measurements for MPs and NPs were carried out at 25 °C on a Zetasizer Nano-ZS (Malvern Instruments, Ltd, UK) equipped with a He-Ne laser (λ = 633 nm) at an angle of 173°. Samples were prepared by dispersion of particles in 10 mM NaCl solution to provide a minimum level of conductivity in the samples, following ISO and ASTM standard guides. Before analysis, the pH of every sample was measured. The zeta potential and standard deviation (SD) were obtained from 5 measurements of 11 runs of 10 seconds using the Smoluchowski model.

Figure 1 .

 1 Figure 1. Transmission electron microscopy (TEM) and X-ray diffraction characterization of the prepared NaYF4:Yb30%,Er2%, A: TEM image (scale bar is 100 nm). B: Size distribution obtained from analysis of TEM images (Gaussian fit in red). C: XRD diffractogram of the particles (red lines show the characteristic peaks of hexagonal NaYF4 (JCPDS 00-016-0334)). Incorporation of UCNP in PE and powdering

Figure 2 .

 2 Figure 2. a) General scheme of upconverting nanoparticles (UCNPs) incorporated into polyethylene (PE) followed by a two-step powdering process to prepare micro-and nanometric particles. b) Granulometric analysis of the micrometric fraction obtained after cryogenic grinding, expressed as a percentage and cumulative percentage as a function of size. c) Nanoparticle tracking analysis (NTA) of the nanometric fraction obtained after wet milling and cascade filtration. Characterization of labelled micro-and nanoplastics

  particles. The identification of UCNPs was confirmed by high-resolution TEM coupled to energy-dispersive X-ray elemental analysis (TEM-EDX), as the dark dots showed the homogeneous distribution of F, Y and Yb (see Figure4b and FigureS6for more details). This result showed that the chemical integrity of UCNPs was maintained during the powdering process, as well as their sizes and shapes.

Figure 4 .

 4 Figure 4. a) Transmission electron microscopy (TEM) bright-field images of nano-Upcon-PE. The three images illustrate that the particles were polydisperse and polymorphic. b) Highangle annular dark-field (HAADF) TEM of a single nano-Upcon-PE, along with energydispersive X-ray characterization of yttrium (Y), fluorine (F) and ytterbium (Yb).

Figure 5 .

 5 Figure 5. Upconversion emission spectra of UNCPs, Blank-PE, and Upcon-PE micro-and nanoparticles; normalized fluorescence intensity as a function of the wavelength (exc. = 980 nm). Acronyms: lanthanide-based upconverting nanophosphors (UCNPs), polyethylene (PE), raw PE (Blank-PE), UCNP-doped PE particles prepared at the microscale (µ-Upcon-PE), UCNPdoped PE particles prepared at the nanoscale (nano-Upcon-PE).

Figure 6 .

 6 Figure 6. I) Bright-field image of particle aggregates: a) UCNPs; b) µ-Blank-PE; c) µUCNPs-PE. II) Green and red emission under an NIR irradiation observer for a) UCNPs and c) µUCNPs-PE; b) µ-Blank-PE remains nonemissive.

Figure S6 .

 S6 Figure S6. TEM image and energy-dispersive X-ray spectra of Upcon-PE nanoparticles. The box 700 indicates where the measurement was taken on UCNPs (top) and on the PE matrix (bottom). 701 Fluorine, yttrium, and ytterbium were detected in UCNPs but were not present in the PE 702 matrix. 703 704

Acknowledgments

The authors acknowledge the Université Paul Sabatier and CNRS for financial support. C. C., B. A. and A.-F. M. gratefully acknowledge the Agence Nationale de la Recherche (Grant ANR-15-CE09-0020 BLINK) for financial support. We would like to thank Teresa Hungria (The Raimond Castaing Microcharacterization Centre) for the TEM/EDX images, Corinne Routaboul (The Infrared Spectroscopy Service, Institute of Chemistry of Toulouse (ICT)) for help with ATR-FTIR analysis, Rodica Chiriac (Thermal Analysis Platform (PLAT)) at the Laboratory of Multimaterials and Interfaces (LMI) for the TGA analysis, and Christine Rey-Rouch (Service Analyze et Procédés -SAP, Chemical Engineering Laboratory (LGC)) for granulometry analysis.

Supporting Information

Figure S3

. Attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR) of raw PE and at different stages of particle synthesis were recorded using a Thermo Nicolet 6700 spectrometer equipped with a diamond crystal attenuated total reflectance accessory and a deuterated triglycine sulfate (DTGS) detector. Background and sample spectra were recorded as the average of 16 scans in the spectral range of 650-4000 cm -1 at a resolution of 4 cm -1 . Recorded data were corrected to obtain transmission-like spectra using the ATR Thermo correction (the refractive index considered as 1.5). The carbonyl signal for all analyzed samples was integrated in the 1650-1850 cm -1 region. a) FTIR spectra of raw PE material and Blank-PE and Upcon-PE samples recorded after every processing step of MP synthesis (synthesis of bulk material, cryogenic grinding, ZB milling); b) zoom around the carbonyl region, showing the absence of a carbonyl band, which signifies that no oxidation occurs for PE during the synthesis of MPs.

Applied models for SAXS analysis

The SAXS curves can be described by the following equation written with the SASView program (SasView, http://www.sasview.org/): 𝐼(𝑞) 𝑔𝑙𝑜𝑏𝑎𝑙 = 𝐼(𝑞) 𝑝𝑜𝑤𝑒𝑟𝑙𝑎𝑤 + 𝐼(𝑞) 𝑠𝑝ℎ𝑒𝑟𝑒 * 𝑆(𝑞) ℎ𝑎𝑟𝑑𝑠𝑝ℎ𝑒𝑟𝑒 (1)

2 * 𝑆(𝑞) where A and B are scaling factors, and the sphere is described by the volume V, radius r and contrast . The nanospheres interact with each other, and the conditions where S(q) = 1 at small angles were not verified. The form factor S(q) is described here with a "hard sphere interaction model" described by Percus-Yevick (J.K. Percus, J. Yevick, J. Phys. Rev., 1958, 110, 1). When UCNPs are distributed in the PE matrix, the scattering cross terms that reflect the interactions between the atoms of UCNPs and PE must be considered. However, we can apply the simplified model proposed in Equation ( 1) to describe Upcon-PE and a two-component model to describe the organization of PE MPs at large distances as follows:

where the second term represents a Lorentzian function to describe the interaction between the crystalline clusters inside the plastic matrix. q0 is the position of the peak corresponding to the averaged distance separating the clusters (q0 = 0.05 Å -1  d = 125 Å). The most interesting term is the exponential parameter P in Eq. ( 1) and (3), which describes the state of the particle surface. For both curves, this value is close to 3.7, corresponding to a slightly rough surface. The absence of a difference in the value of P suggests that the presence of UCNPs does not modify the state of the PE MP surface. Thus, these results suggest that UCNPs are absorbed inside the PE matrix. Figure S5. SAXS spectra of Blank-PE and Upcon-PE, along with superimposed fitting curves (blue and green, respectively). The experimental SAXS curves are plotted in Log I(q) as a function of Log q.

Table S1. The thermal stability of the samples was analyzed using thermogravimetric analysis (TGA). The apparatus was a TGA/SDTA 851 Mettler Toledo device. Approximately 15 mg of each sample was placed in 150 µL aluminum pans with a pierced cover. The analysis was performed from 30 to 900 °C at a heating rate of 10 °C/min under a nitrogen atmosphere with a rated flow of 40 mL/min. For each sample, a blank (analysis with empty crucible) was carried out under the same conditions as for the sample. All samples were analyzed in duplicate.

Sample