Jonas Benhamou
email: jonas.benhamou@safrangroup.com.

Silvère Bonnabel
email: silvere.bonnabel@minesparis.psl.eu.

Camille Chapdelaine
email: camille.chapdelaine@safrangroup.com.

Optimal active sensing control for two-frame systems

Keywords: localization, Lie groups, active estimation, invariant Kalman filtering

This paper provides a complete characterization of the trajectories that maximize the information collected by a moving vehicle, through sensors' measurements, for the recently introduced class of nonlinear "two-frame systems". The information is quantified in terms of the trace of the observability Gramian (OG) along a trajectory. In general, this quantity nontrivially depends on the control inputs and the state trajectory, resulting in a difficult optimal control problem. Herein, we leverage the property of invariant filtering that Jacobians are state-trajectory independent, that is, only depend on the control inputs, which enables us to mathematically derive optimal trajectories in closed form. We illustrate the results numerically on problems from robotics such as 3D robot localization, and 2D simultaneous localization and mapping.

Introduction

The notion of observability of linear systems does not depend on the system's trajectory. By contrast, when turning to nonlinear systems, some trajectories may make the state unobservable, whereas others make it observable. As an extreme example, think of a wheeled robot whose position is measured: if it stays still, its orientation is unobservble, whereas if it is moving in straight line, its orientation is readily recovered. Beyond observability, which is a binary criterion, some trajectories are more informative than others in terms of state estimation of nonlinear systems. In the context of navigation of vehicles, amongst others, it may be useful to know which trajectory brings the most information about the state, to safely move in a constrained environment.

This has prompted previous authors to develop methods for trajectory generation that facilitate state estimation [START_REF] Preiss | Simultaneous selfcalibration and navigation using trajectory optimization[END_REF][START_REF] Salaris | Online optimal active sensing control[END_REF][START_REF] Salaris | Online optimal perception-aware trajectory generation[END_REF], or online parameter estimation [START_REF] Mehra | Optimal input signals for parameter estimation in dynamic systems-survey and new results[END_REF]. This pertains to the theory of optimal experiment design [START_REF] Pukelsheim | Optimal design of experiments[END_REF] as these methods seek to find a control sequence to be applied in order to maximize the information acquired along the trajectory.

As an information measure, it is customary to use the observability Gramian, e.g., [START_REF] Hinson | Path planning to optimize observability in a planar uniform flow field[END_REF]. Some other criteria were used, though, such as Fisher's information, or the covariance matrix associated to the Riccati equation of an extended Kalman filter [START_REF] Rafieisakhaei | On the use of the observability gramian for partially observed robotic path planning problems[END_REF], or measures specifically tailored to the problem, e.g., [START_REF] Murali | Perceptionaware trajectory generation for aggressive quadrotor flight using differential flatness[END_REF] which targets trajectories leading to easier-to-exploit images from the onboard camera. More generally, active simultaneous localization and mapping (SLAM) improves state estimation through control, see e.g., [START_REF] Leung | Active SLAM in structured environments[END_REF].

The most widespread criterion to measure the extent of observability brought by a trajectory is the local OG, introduced by Krener and Ide [START_REF] Krener | Measures of unobservability[END_REF]. In some simple cases, the optimal trajectories might be computed analytically, see [START_REF] Hinson | Path planning to optimize observability in a planar uniform flow field[END_REF], but this is generally not the case, and one needs to resort to numerical methods for trajectory planning [START_REF] Shastry | UAV state and parameter estimation in wind using calibration trajectories optimized for observability[END_REF] or trajectory modification for sensors calibration [START_REF] Preiss | Simultaneous selfcalibration and navigation using trajectory optimization[END_REF]. Despite the drawbacks of Gramian maximization based planning methods [START_REF] Rafieisakhaei | On the use of the observability gramian for partially observed robotic path planning problems[END_REF], the local OG or the closely related Constructibility Gramian [START_REF] Salaris | Online optimal perception-aware trajectory generation[END_REF] remains a relevant tool to establish the most informative trajectories, especially when the system's model is well-known, i.e., the process noise is low.

In this paper, we consider the recently introduced theory of two-frame systems [START_REF] Barrau | The geometry of navigation problems[END_REF], which introduces a large class of navigating vehicles having remarkable properties in terms of state estimation. For those systems, we benefit from the properties of invariant filtering that provide state-trajectory independent Jacobians [START_REF]The invariant extended Kalman filter as a stable observer[END_REF]. This makes the OG independent of the trajectory being followed, and only dependent on the controls, and hence simplifies the associated optimal control problem. Leveraging this property enables us to derive optimal trajectories in closed form that maximize the trace of the OG for two-frames systems.

The main contributions of this paper are:

• The use of the invariant errors to make the observability Gramian independent of the system's state;

• A general analytical computation of the optimal control that maximizes the observability Gramian for a broad class of systems that fall into the theory of two-frames navigating systems;

• Application to three problems: a vehicle that navigates in 3D and estimates its state from position measurements performed by a GNSS, a 2D vehicle equipped with a GNSS mounted on an arm with unknown position in the vehicle (lever arm), and the problem of SLAM for a 2D wheeled robot.

Section II formulates the perception-aware control problem, Section III summarizes the theory of two-frame systems, Section IV calculates the optimal control to apply when facing world frame measurements, Section V addresses the problem when facing vehicle frame measurements, and Section VI illustrates the results numerically.

Optimal perception-aware trajectories

In this section we recall the problem of active sensing. It is a path planning problem where the objective function is an information criterion. Therefore, it will be easier to estimate the system state along the optimized trajectory. Let us consider a general discrete-time dynamical system:

x n+1 = f (x n , u n ), x 0 = x 0 (1)
y n = h(x n ) (2)
where x n ∈ R p represents the state of the system, u n ∈ R m is the control input, y n ∈ R q is the sensor output at time n.

One approach to measure the extent of observability associated to a given trajectory is based on the observability Gramian (OG). The discrete-time OG is defined by:

W K = K n=1 Φ T n H T n H n Φ n (3)
where

H n = ∂h(xn) ∂xn and Φ n = F n Φ n-1 , Φ -1 = I d with F n = ∂f (xn,un)
∂xn and K the estimation horizon. It should be noted that in the case of linear systems, F n neither depends on the state of the system nor the controls. This implies that the OG does not depend on the trajectory: all trajectories are equally informative. However, in the nonlinear case, observability depends on the trajectory, see [START_REF] Hinson | Path planning to optimize observability in a planar uniform flow field[END_REF][START_REF] Krener | Measures of unobservability[END_REF]. The optimal inference problem consists then in finding the trajectories along which the OG is maximum. Thus, the trajectory planning problem writes:

u * = arg max u 0 ,...,u K-1 T r(W K ) subject to ∀n x n+1 = f (x n , u n ) ∀n u n ∈ C n (4)
u * ∈ R pK refers to the sequence of vectors of controls to apply. At each time instant n, the control vector is required to belong to a bounded set C n ⊂ R p . The interest of the above criterion is its simple interpretation in terms of the (linearized) error variables denoted by δx n := x n -xn and δy n := y n -ȳn . Up to the first order, they evolve according to the linearized Jacobians, that is, δx n+1 = F n δx n and δy n = H n x n . As the initial error is unknown, we may assume it be random and isotropically distributed, for instance δx 0 ∼ N (0, I d ). We have then.

Lemma 1. Assume δx 0 ∼ N (0, I d ) denotes the discrepancy between two initial states. We have, neglecting second order terms, the equality

E K n=1 ||δy n (δx 0 )|| 2 2 = T r (W K ) . (5)
This is proved by writing

E K n=1 ||δy n (δx 0 )|| 2 2 = E K n=1 δx T 0 Φ T n H T n H n Φ n δx 0 = K n=1 E T r δx T 0 Φ T n H T n H n Φ n δx 0 = T r K n=1 Φ T n H T n H n Φ n E(δx 0 δx T 0 ) = T r (W K )
where we used the commutation properties of the trace and that E(δx

0 δx T 0 ) = I d .
This provides the following simple interpretation: the optimal sequence of control inputs makes on average the (first-order) discrepancy between the predicted and measured outputs as large as possible, for a random error about which we do not have any information. The trace of the OG is thus a way of measuring the extent of observability gained from the generated trajectory.

In general, the optimisation problem is difficult to solve because of the numerous constraints. To overcome this difficulty, several authors [START_REF] Salaris | Online optimal perception-aware trajectory generation[END_REF], [START_REF] Preiss | Simultaneous selfcalibration and navigation using trajectory optimization[END_REF] restrict the study to flat systems [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF]. In this case one may perform parametric (suboptimal) optimization, where the parameters are a number of waypoints for the flat output. In the following, we consider the recently introduced class of two-frames navigating systems, and leverage their properties to show problem (4) is amenable to a more simple optimal control problem, that may be solved analytically.

Reminder: the theory of two-frame systems

The theory of two-frame natural systems [START_REF] Barrau | The geometry of navigation problems[END_REF] is general and encompasses numerous examples from robotics and inertial navigation. The idea is to consider systems consisting of a body that navigates, and whose state is defined by a set of vectors, as well as a variable encoding a change of frame from the world frame to the frame attached to the body. In the present paper, we will focus on the following two-frames systems. The state writes

  R x x   ∈ SO(d) × R dNx × R dNx , x =    x 1 . . . x Nx    , x =    x 1 . . . x Nx    .
R ∈ G = SO(d) with d = 2 or d = 3 is a rotation matrix that encodes the orientation of the frame attached to the navigating body with respect to the world-fixed frame, x is a set of N x vectors of R d , that is, the d-dimensional world in which the body is navigating, which encode physical quantities expressed in the fixed frame, such as the position and velocity of a craft, and x is a set of vectors expressed in the body frame, such as a sensor bias, where the sensor is attached to the body, see Figure 1.

Figure 1: The state of a navigating two-frames system, to be estimated, consists of a rotation matrix R, and vectors, e.g., the position, velocity, or sensor biases, stacked in x (when expressed in the fixed frame) and x (when expressed in the body frame). R transforms vectors of the body frame denoted by uppercase letters to vectors of the world-fixed frame denoted by lowercase letters, and as such encodes the orientation of the craft.

To fix ideas, let us start by giving two examples of interest, before we turn to the unifying theory of two-frame systems.

Examples

The first example is in 2D and the second one in 3D.

First example, from [12]

Consider the classical 2D model of a non-holonomic car, see e.g., [START_REF]The invariant extended Kalman filter as a stable observer[END_REF], or equivalently a wheeled robot or a unicyle. The position of the car in 2D is described by the middle point of the rear wheels axle x n ∈ R 2 , and its orientation (heading) denoted by θ n ∈ R and parameterized by the planar rotation matrix R n of angle θ n that is cos θ -sin θ sin θ cos θ . The car is equipped with a GNSS located at unknown position x n ∈ R 2 in the car frame with respect to point x n (a lever arm to be estimated), which provides the world (fixed) frame position measurements

y n = x n + R n x n ∈ R 2 . ( 6
)
In the schematic diagram below, the triangle is the car and the square is the position measured by the GNSS.

x n θ n x n y n
The discrete-time dynamics of a mobile wheeled robot write:

R n+1 = R n Ω n , x n+1 = x n + R n u n , x n+1 = x n , (7)
The control inputs are the angular velocity Ω n , which can be controlled through steering or via the difference between wheel speeds, and u n that is the average of wheel speeds.

Second example

We consider a body that navigates in 3D, and which is equipped with a GNSS. We consider the acceleration in the body frame as well as the angular velocity to be (possibly high-level) control inputs, and discretize the continuous-time equations, yielding dynamics of the form

     R n+1 = R n Ω n p n+1 = p n + ∆t v n v n+1 = v n + R n u n , (8)
with observation Y n = p n .

Let us now turn to the unifying framework of two-frame systems, that encompasses the two latter examples.

Two-frames general theory

In this paper, we consider the following subclass of discrete-time dynamics amongst the two-frame dynamics of [START_REF] Barrau | The geometry of navigation problems[END_REF]

  R n+1 x n+1 x n+1   =   R n Ω n F F F x n + d n + R n * u n x n   , (9)
where * defines the following operation, called an action

R * x = R *    x 1 . . . x Nx    :=    Rx 1 . . . Rx Nx    , R * x :=    Rx 1 . . . Rx Nx    , (10)
and where the matrix F F F : R dNx → R dNx is of the form

F F F =   
with α ij ≥ 0's real nonnegative numbers, and d n , u n ∈ R dNx . It can be readily noticed that such a matrix commutes with the action, that is

F F F (R * x) = R * (F F F x).
The rotation R maps vectors of the body frame to vectors of the worldfixed frame. It is reasonable to assume that the control inputs are vectors expressed in the body frame, as actuators are attached to the craft. We will therefore consider d n to be a known input, and Ω n , u n to consist of the (possibly high-level) control inputs we will use for trajectory generation. The fact that u n ∈ R dNx , does not preclude the system from being underactuated, since as many components of u n as necessary may be set to 0 (through constraints). The reason for the equation x n+1 = x n in [START_REF] Leung | Active SLAM in structured environments[END_REF] is that the variables of the body frame that we seek to estimate are considered as parameters, as is the case of lever arms (see the first example above) and accelerometer and gyrometer biases in the general theory of two-frames.

Regarding state estimation, the two-frames systems are endowed with sensors providing the vehicle with measurements. In the theory of two-frames systems of [START_REF] Barrau | The geometry of navigation problems[END_REF], they are of two types. Indeed we use either an output map h representing measurements in the fixed frame, or h representing measurements in the frame of the vehicle, defined as fixed-frame:

h(R, x, x) = H H H x x + R * [H H H x x] (12)
body-frame:

h(R, x, x) = R -1 * [-H H H x x] -H H H x x (13)
with H H H x : R dNx → Y and H H H x : R dNx → Y two linear maps that commute with the action of G, that is, are of the form

H H H x =    β 11 I I I d . . . β 1Nx I I I d . . . . . . β Ny1 I d . . . β NyNx I d    (14)
and similarly for H H H x , and where Y = R dNy denotes the output space. In the theory, R also acts on Y = R dNy via term-by-term action similarly to (10).

Link to the first example

We see the first example fits the present framework of two-frames systems indeed, as [START_REF] Rafieisakhaei | On the use of the observability gramian for partially observed robotic path planning problems[END_REF] correspond to [START_REF] Leung | Active SLAM in structured environments[END_REF], with d = 2, N x = 1, N x = 1, hence action * is trivially the matrix-vector product, and F F F = I 2 . The measured output corresponds to fixed-frame measurements [START_REF] Barrau | The geometry of navigation problems[END_REF], with H H H x = H H H x = I 2 .

Link to the second example

We see the second example above fits the present framework of two-frames systems indeed, as [START_REF] Rafieisakhaei | On the use of the observability gramian for partially observed robotic path planning problems[END_REF] correspond to [START_REF] Leung | Active SLAM in structured environments[END_REF],

with d = 3, N x = 2, N x = 0, d n = 0, action * is R * (p, v) = (RP, Rv), and
F F F = I 3 ∆tI 3 0 3 I 3 ,
which is of the form [START_REF] Shastry | UAV state and parameter estimation in wind using calibration trajectories optimized for observability[END_REF]. The measured output corresponds to fixed-frame measurement [START_REF] Barrau | The geometry of navigation problems[END_REF], with H H H x = I 2 .

State-trajectory independent error evolution

The theory of invariant filtering [START_REF] Barrau | Invariant Kalman filtering[END_REF], having its roots in the theory of invariant observers [START_REF] Aghannan | On invariant asymptotic observers[END_REF][START_REF] Bonnabel | Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems[END_REF], and which underlies the recent theory of two-frame systems, uses alternative error variables to recover some properties of linear systems, especially the fact the errors evolve independently from the state trajectory. This notably allows for state-trajectory independent Jacobians, when linearizing the system. When confronted with observations in the fixed frame [START_REF] Barrau | The geometry of navigation problems[END_REF], we use as an alternative state error, the left-invariant error between two states defined as Obs. ( 12)

⇒ e =   R-1 R R-1 * (x n -x) x -(R -1 R) * x  :=   e R e x e x   . ( 15
)
If observations are performed in the body frame instead (13) one should use the right-invariant error defined as Obs. ( 13)

⇒ e =   R R-1 x -(R R-1 ) * x R * (x -x)   :=   e R e x e x   . (16)
If we evolve two state variables through [START_REF] Leung | Active SLAM in structured environments[END_REF], the corresponding left-invariant error evolves as:

e R n+1 = Ω -1 n e R n Ω n e x n+1 = Ω -1 n * [F F F e x n + e R n * u n -u n ] e x n+1 = x -Ω -1 n e R n Ω n * x. (17)
We see the evolution of the error is independent of the state variable (R, x, x) but for the last line. However, if d = 2, rotations commute and the last line becomes e x n+1 = e x n and the error evolution is wholly state-trajectory independent. This is Theorem 5 of [START_REF] Barrau | The geometry of navigation problems[END_REF].

If we evolve two state variables through [START_REF] Leung | Active SLAM in structured environments[END_REF], the corresponding rightinvariant error evolves as:

e R n+1 = e R n e x n+1 = F F F n e x n + d n -e R n d n e x n+1 = RΩ n * (x -x). ( 18
)
We see the evolution of the error is independent of the state variable (R, x, x) but for the last line. However, if d = 2, rotations commute and the last line becomes e x n+1 = Ω n * e x n and the error evolution is wholly state-trajectory independent. This is Theorem 5 of [START_REF] Barrau | The geometry of navigation problems[END_REF].

State-trajectory independent prediction error

Instead of using the usual prediction error y -ȳ = h(R, x, x) -h( R, x, x) of state estimation, called innovation in the literature devoted to Kalman filtering, the theory of invariant filtering advocates the use of alternative innovation variables, that also reflect the prediction error. The innovation variable z associated to fixed-frame output y, i.e., Obs. [START_REF] Barrau | The geometry of navigation problems[END_REF], writes:

z = R-1 * (y -H H H x x) -H H H x x ( 19
)
while the innovation variable z associated to body-frame output Y, i.e., Obs. ( 13), writes:

z = R * (Y + H H H x x) + H H H x x (20)
The interest of those alternative innovation variables, is the strong link they bear with the alternative errors we have defined. Indeed,

Error (15) ⇒ z n = H H H x e x n + H H H x e R n * e x n , (21)
Error ( 16)

⇒ z n = -H H H x e x n -H H H x e R n -1 * e x n . (22)
This result of [START_REF] Barrau | The geometry of navigation problems[END_REF] is easily proved. It will allow for state-independent Jacobian of the output map h, that we are about to leverage to simplify our perception-aware control problem.

Optimal solutions to active sensing control with world frame measurements

This section addresses the problem of optimal perception-aware trajectory generation for the class of two-frames systems we have considered. Our main result is to prove that they can be computed explicitly for this general enough class of systems modeling vehicles that navigate.

Our goal is to maximize the global information acquired over a given future horizon, i.e., K-steps trajectory. This is measured as the trace of the observability Gramian matrix, see problem (4). To get this matrix using our alternative errors, we need the Jacobians resulting from linearizing the error system. In Subsection 4.3, we will relate this to the original control objective based on standard linear errors in the original variables, as classically used by the extended Kalman filter.

In this section we consider the case of measurements performed in the world frame. In the next, we will consider body-frame measurements.

Linearized error system and observability Gramian

We would like, for (R, x, x) close to ( R, x, x), a first-order (i.e., linearized) approximation to the left-invariant error [START_REF] Barrau | Invariant Kalman filtering[END_REF]. Recall that we may relate a rotation matrix with a vector through the matrix exponential R-1 R = exp((ξ R ) × ) where

d = 3 ⇒ ξ R ∈ R 3 , (ξ R ) × =   0 -ξ R 3 ξ R 2 ξ R 3 0 -ξ R 1 -ξ R 2 ξ R 1 0   and d = 2 ⇒ ξ R ∈ R, (ξ R ) × = ξ R J with J = 0 -1 1 0
. Using a first order expansion of the matrix exponential we have e R = R-1 R ≈ I d + (ξ R ) × and the linearized error is then defined as (ξ R , ξ x , ξ x ), where we let ξ x = x -x and ξ x = x -x. Substituting those approximations in [START_REF] Bonnabel | Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems[END_REF] we find in the 2D case

d = 2 ξ R n+1 = ξ R n ξ x n+1 = Ω -1 n * [F F F ξ x n + Ju n ξ R n ] ξ x n+1 = ξ x n , (23)
and in the 3D case d = 3 by not including x in the state

ξ R n+1 = Ω -1 n ξ R n ξ x n+1 = Ω -1 n * [F F F ξ x n -u n × ξ R n ] (24)
where we also used that Ω

-1 n (ξ R n ) × Ω n = (Ω -1 n ξ R n ) × .
This yields the linearized system (hence the Jacobian) associated to the dynamics, when written in the invariant variables. Regarding the Jacobian of map h, it is related to the linearization of the innovation, which yields from (21) neglecting second order terms

δz n = H H H x ξ x n + H H H x ξ x n . ( 25
)
Our goal is to find the control inputs Ω n , u n 's that maximize the trace of the observability Gramian associated to the linearized systems (23), (25) in 2D and ( 24)-(25) in 3D. Recalling the derivation [START_REF] Pukelsheim | Optimal design of experiments[END_REF], this is equivalent to maximizing K k=1 E (||δz k || 2 ), for a random isotropically distributed initial error.

Optimal trajectory generation

Recalling that u ∈ R dNx is a multi-vector, we take as constraints that at all time i, each component U k i of the input must remain in a predefined bounded set C ik ⊂ R Nx .

Our first main theoretical result states that, in the case of world-frame measurements, optimal solutions are obtained by maintaining a fixed orientation R n and taking the control inputs collinear and of maximum authorized norm.

Theorem 1. Consider a two-frames system with dynamics (9) and observations in the fixed frame given by [START_REF] Barrau | The geometry of navigation problems[END_REF], such that the β ij 's in [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF] related to H H H x (only) are either all nonnegative or all nonpositive. Moreover, suppose that:

1. Either d = 2, that is, the vehicle evolves in 2D space, 2. or d = 3 and there is no variable x in the state, that is, the state consists of variables (R, x).

Then, assume we may take Ω * k = I d and u * k such that the scalar product across components and times satisfies

u k i .u l j = max C ik ||u k i || max C jl ||u l j || for 0 ≤ i, j ≤ K -1, 1 ≤ k, l ≤ N x .
This defines an optimal trajectory. Proof. We begin noticing from (24) the variable ξ x is fixed when d = 2, and when d = 3 we supposed it was not part of the state. As the components of the error are assumed initially independent, this yields

E (||δz k || 2 ) = E (||H H H x ξ x k || 2 )+E (||H H H x ξ x k || 2 ).
As a result all we need is to maximize

K k=1 E (||H H H x ξ x k || 2 ).
Without loss of generality let us focus on the 3D case (24). By recursion we may prove

ξ R k = Ω -1 k-1 . . . Ω -1 0 ξ R 0 = (Π k-1 0 Ω j ) -1 ξ R 0 ξ x k = (Π k-1 0 Ω l ) -1 * F F F k ξ x 0 - k j=1 (Π k-1 l=j-1 Ω l ) -1 * F F F k-j u j-1 × ξ R j-1 (26)
where we have used that Ω * and F F F commute. We see the goal boils down to finding the control sequence that maximizes

E||H H H x k j=1 (Π k-1 l=j-1 Ω l ) -1 * F F F k-j u j-1 × (Π k-1 0 Ω j ) -1 ξ R 0 || 2 . Lemma 2. Let u 1 , u 2 ∈ R d with d = 2, 3, let Ω ∈ SO(d) and let ξ ∼ N (0, I d ). We have E(u 1 × ξ . u 2 × Ωξ) ≤ (d -1)||u 1 || ||u 2 ||. Equality is obtained for Ω = I d and u 2 = γu 1 , γ > 0. For d = 3 it is unique. For d = 2 the maximum is attained for all combinations Ω ∈ SO(2), u 2 = γΩu 1 , γ > 0.
Proof. Using the properties of mixed product followed by triple product expansion we find u

1 × ξ . u 2 × Ωξ = (u 1 .u 2 )(ξ.Ωξ) -(u 1 .ξ)(u 2 .Ωξ) = (u 1 .u 2 )T r(ξ T Ωξ)-T r(u T 1 ξξ T Ω T u 2 ), whose expectation equals (u 1 .u 2 )T r(ΩEξξ T ))- T r(u T 1 E(ξξ T )Ω T u 2 ) = T r(Ω)(u 1 .u 2 ) -u T 1 Ω T u 2 := g(u 1 , u 2 , Ω). We write that max ||u 1 ||=||u 2 ||=1,Ω∈SO(d) g = max ||u 1 ||=1 max Ω∈SO(d) max ||u 2 ||=1
g As g(u 1 , u 2 , Ω) = (T r(Ω)u 1 -Ωu 1 ).u 2 the method of Lagrange multipliers yields that max ||u 2 ||=1 g is attained for u

* 2 = [T r(Ω)u 1 -Ωu 1 ]/||T r(Ω)u 1 - Ωu 1 ||, and g(u 1 , u * 2 , Ω) = ||T r(Ω)u 1 -Ωu 1 || = 1 + T r(Ω) 2 -2T r(Ω)u T 1 Ωu 1 .
To conclude we use two facts. First, letting θ be the rotation angle associated to Ω, we have T r(Ω) = 1 + 2 cos θ for d = 3. Then, we have cos θ ≤ Ωu 1 .u 1 ≤ 1, for ||u 1 || = 1. This stems from using the Rodrigues formula and the triple product expansion that proves Ωu.u = 1 + (1 -cos(θ))((v.u) 2 -1) ≥ 1 + (1 -cos(θ))(-1) = cos θ, with v the unit vector encoding the rotation axis. Thus g 2 ≤ 1 + T r(Ω) 2 -2T r(Ω) cos θ = 2 + 2 cos θ. This proves g ≤ 2 with equality attained if and only if cos θ = 0, that is, Ω = I 3 .

If d = 2, only the trace is modified and we find g = 1, that is, the maximum is non uniquely attained for arbitrary Ω, letting u 2 = u * 2 . In particular it is attained for Ω = I 2 and u * 2 = u 2 .

Let us go back to our control objective of maximizing E||H H H

x k j=1 (Π k-1 l=j-1 Ω l ) -1 * F F F k-j u j-1 × (Π k-1 0 Ω j ) -1 ξ R 0 || 2 .
We need to recall we are dealing with multivectors, and that H H H x is of the form [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF], where the β ij 's have the same sign. We may expand the squared norm, and we recover a linear combination with positive coefficients of scalar products of the form E(ũ 1 × ξ . ũ2 × Ωξ), using our assumption on F F F and using that for rotation matrices A, B we have

(Aũ 1 ) × (Aξ) . (B ũ2 ) × (Bξ) = ũ1 × ξ . (A T B ũ2 ) × (A T Bξ).
We may bound those terms using the lemma, and to get an equality we take all Ω k to be I d . If the condition of the theorem

u k i .u l j = max C ik ||u k i || max C jl ||u l j
|| is met, it implies from Cauchy-Schwarz inequality the collinearity of the components and we see from the lemma the sum reaches its maximum.

For instance, if we consider Example 2 with dynamics (8) and constraints ||u n || ≤ γ, we see the optimal trajectory is obtained by fixing the orientation, and taking full acceleration. Note that, there is no requirement regarding R n apart from it being fixed: the maximizing acceleration could be lateral if constraints permit. Besides, we note that such a system is not flat, as the vehicle's orientation is not related to the trajectory, contrary to wheeled robots.

Remark 1. The optimal trajectory of the theorem is not necessarily unique mathematically. However, when d = 3, the proof shows the condition Ω * k = I d is inevitable. Moreover, when d = 2, the other maximizing trajectories are in practice not feasible (they violate the constraints) and in any case more complicated to achieve while not improving the objective.

Connection to the original variables

We now relate this to the original Gramian maximization problem. Indeed, the fact that the trajectory be optimal for given modified error and innovation, does not a priori mean it is optimal in the problem's original error variables, or more prosaically speaking using the conventionally defined Jacobians in (4). It turns out, though, that in the present case both are equivalent.

Proposition 1. The optimal trajectories of Theorem 1 are optimal too for the problem (4) using the original variables of the problem.

Proof. The proof is based on the use of the equivalent formulation [START_REF] Pukelsheim | Optimal design of experiments[END_REF]. Indeed, we see that if we take a two-frame system above, we have z = R-1 * (y -ȳ), where ȳ = H H H x x -R * [H H H x x] denotes the predicted output. As a result ||z|| = ||y -ȳ|| so that both objectives coincide.

Optimal solutions to active sensing control with body frame measurements

In this section, we first provide the counterpart of Theorem 1 for body-frame measurements. Then, we step back and discuss the trace criterion.

Optimal trajectory generation

When facing body-frame measurements, one should use right-invariant errors, leading to the error system [START_REF] Durrant-Whyte | Simultaneous localization and mapping: part i[END_REF]. Along the lines of the theory of two-frames, we see the two following cases lead to a state-independent error:

1. Either d = 2, that is, the vehicle evolves in 2D space, 2. or d = 3 and there is no variable x in the state, that is, the state consists of variables (R, x), which are the hypotheses we used in Theorem 1. In the first case, the rotations commute, leading to autonomy of the error evolution. In the case of body-frame measurements, it turns out that all trajectories bring equal information, exactly as is the case for linear systems. This is due to the linearized error system being virtually independent of the controls.

Theorem 2. Consider a two-frame system with dynamics (9) and observations in the body frame given by [START_REF]The invariant extended Kalman filter as a stable observer[END_REF]. Then, under any of the two assumptions above, all trajectories yield an equal trace of the observability Gramian, in other words all trajectories are optimal for problem (4).

Proof. The proof is in two steps. First we show the result using the Jacobians related to the right-invariant error, as a natural consequence of the fact the error evolution is (almost) independent of the controls. Then, we relate the result to the Jacobians expressed using the original variables.

As before, we may linearize the innovation (22) which yields

δz k = -H H H x ξ x k -H H H x ξ x k .
Both ξ x and ξ x evolve independently, so that we have

E (||δz k || 2 ) = E (||H H H x ξ x k || 2 ) + E (||H H H x ξ x k || 2 )
. The latter term is independent of the controls, see [START_REF] Durrant-Whyte | Simultaneous localization and mapping: part i[END_REF]. Regarding ξ x , we see from [START_REF] Durrant-Whyte | Simultaneous localization and mapping: part i[END_REF] that in the case where it is present in the state, that is, when d = 2, it evolves as ξ x n+1 = Ω n * ξ x n . Recalling we are dealing with multi-vectors and the form of the matrix [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF], and that rotations are isometries, we see

E (||H H H x ξ x k || 2
) is also independent of the controls. This proves all trajectories yield the same trace of the Gramian, when using the Jacobians of invariant filtering, that is, those related to the right-invariant error.

If we turn to the original variables of the problem, we may exploit the link between problem (4) and relation [START_REF] Pukelsheim | Optimal design of experiments[END_REF]. Besides, we see that

z = R * (Y -Ȳ), where Ȳ = R-1 * b -R-1 * H H H x x -H H H x x
, denotes the predicted output. As a result ||z|| = ||Y -Ȳ|| so that both objectives coincide.

Discussion and further results

Whether in the case of fixed-frame or body-frame observations, we were able to derive closed-form expressions for the trajectories that maximize the information for a large class of problems, in the sense of the trace of the OG criterion, for which we provided a probabilistic interpretation in terms of averaged observed linearized error. In this subsection, we would like to step back and reflect on the limitations of this widespread criterion.

Albeit useful, and allowing for closed-form expressions, the trace criterion might prove insufficient for a number of applications revolving around navigation of 2-frames systems. Indeed, as shown in [START_REF] Pukelsheim | Optimal design of experiments[END_REF], the trace is related to an average over random (unknown) initial error. A trajectory that optimizes such a criterion does not preclude the risk of falling into an unobservable configuration for a particular initial error, though. As a result, the following objective shall also be interesting for a variety of applications max u 0 ,...u K-1

min ||ξ 0 ||=1 K k=1 ||δz k (ξ 0 )|| 2 2 , (27)
which coincides with max u 0 ,...u K-1 min i λ i (W K ), i.e., maximizing the smallest eigenvalue of the Gramian as seen from ( 5), and advocated in [START_REF] Salaris | Online optimal perception-aware trajectory generation[END_REF]. We see that in 3D a uniformly accelerated motion fails to allow for estimation of the vehicle's angle around the acceleration axis, and as such it may be optimal for the Gramian's trace criterion but not for the latter. Deriving the optimal trajectories in closed form -as we just did -for this max-min eigenvalue may prove involved, and is left for future research.

There is some overlap between the objectives, though.

Corollary 1. a) Assume that d = 2 and there is no body-frame-expressed variables x to be estimated in the state. Then the optimal trajectory of Theorem 1 solves the minimium eigenvalue optimization problem (27) in the case of fixed-frame measurements. b) In the case of body-frame measurements, under the assumption of Theorem 2 all trajectories are optimal in the sense of (27).

Proof. In the 2D case, the last expression of (26) becomes

k j=1 (Π k-1 l=j-1 Ω l ) -1 * F F F k-j Ju j-1 ξ R 0 .
Maximizing the expectation of its squared norm w.r.t. ξ R 0 ∼ N (0, 1) or its squared norm with (ξ R 0 ) 2 = 1 yields identical problems. For body-frame measurements we have independence w.r.t. the controls.

Numerical illustrations

We now illustrate the theoretical results for the two motivating examples of Section 3.1, plus a 2D SLAM example.

Navigation with GNSS measurement in 3D

In the case of 3D navigation as presented in section 3.1.2, the position of the system is directly observed. However, the attitude of the vehicle must be estimated. The quality of the estimation depends on the chosen trajectory. To illustrate the result of Theorem 1 of Section 4.2, we compare the evolution of the OG's trace over time for the optimal trajectory and an arbitrary trajectory with same (maximum) energy u. The results in Figure 2 confirm the theorem.

Navigation with online GNSS lever arm calibration in 2D

One of the advantages of the two-frame theory is the possibility to apply invariant filtering to state variables expressed in the body frame. The modelling of a wheeled robot with unknown lever arm between the GNSS and the middle point of the rear axle can be found in Section 3.1.1. In Theorem 1 of Figure 2: On the left is shown the evolution of the OG trace for the optimal trajectory in blue and an arbitrary trajectory in orange. On the right, the two trajectories in 3D.

Section 4.2, optimal trajectories were computed. To illustrate the result, we compute the trace of the OG of the optimal trajectory and compare it with another trajectory with identical energy. The results in Figure 3 confirm the theorem.

2D Simultaneous localization and mapping

Consider the 2D SLAM problem with two unknown landmarks and observations of the relative distance between the vehicle and these points [START_REF] Durrant-Whyte | Simultaneous localization and mapping: part i[END_REF]. The state vector is: Ξ n = (θ n , x n , p 1 n , p 2 n ) where θ n is the vehicle's heading, x n ∈ R 2 is the position of the vehicle, and p i n ∈ R 2 , i ∈ {1, 2} denotes the position of the unknown landmarks. The system's equations are [START_REF] Barrau | The geometry of navigation problems[END_REF]:

         θ n = θ n-1 + ω n x n = x n-1 + u n R(θ n-1 )e 1 p i n = p i n-1 y i n = R(θ n ) T (p i n -x n ) , (28)
the system can be linearized as follows:

ξ n = F n ξ n-1 and z n = ȳn -y n = H n ξ n (30)

with

F n =     1 0 2,1 0 2,1 0 2,1 u n JR n e 1 I 2,2 0 2,2 0 2,2 0 2,1 0 2,2 I 2,2 0 2,2 0 2,1 0 2,2 0 2,2 I 2,2     , H n = JR T n (p 1 n -x n ) -R T n R T n 0 2,2 JR T n (p 2 n -x n ) -R T n 0 2,2 R T n , (31)
Using these error variables, it is not obvious at the first sight that the Gramian does not depend on the controls and that all trajectories are equally informative as ω n implicitly acts on R n and x n . However, the system (28) obviously fits into the class considered here, and applying Theorem 2, we find that all trajectories lead to identical objective. This is similar to the linear case, where the Gramian depends neither on the controls nor on the state.

Conclusion

In this paper we have considered the problem of optimal active sensing, and we have derived the optimal trajectories in closed form for a large subclass of two-frame systems, that model vehicles that navigate in 2D and in 3D, leveraging the properties of invariant filtering and the isometry properties of rotation matrices. Although the results are not surprising (straight lines with full acceleration allow for distinguishing a heading error best), they have the merit of being quite general. They are nontrivial to derive, in that it is rare to solve explicitly optimal control problems outside of the LQR framework, and they have indeed necessitated the combination of various ingredients: a) the discrete time, which is not widespread in Gramian-based active sensing (in continuous time we generally have no closed form for the Gramian itself), b) a recent general theory, which highlights systems whose properties are akin to those of linear systems, c) a proof which is not straightforward and heavily relies on the properties of scalar and cross products, d) the detour via [START_REF] Pukelsheim | Optimal design of experiments[END_REF] while directly maximizing the trace-as is done by numerical methods-would have been more difficult analytically.

In the future, we would like to derive the counterpart for the max-min criterion (27) for the same class of systems. Besides, we would like to consider more general problems that do not perfectly fit into the invariant filtering theory, such as navigation with online calibration of gyrometers' and accelerometers' biases. In this case, the trajectory remains partially statetrajectory independent, which may lead to speedups of numerical methods, in the spirit of [START_REF] Maidens | Symmetry reduction for dynamic programming[END_REF].

α 11 I

 11 I I d . . . α 1Nx I I I d . . . . . . α Nx1 I I I d . . . α NxNx I I I d   

Figure 3 :

 3 Figure 3: On the left is shown the evolution of the OG trace for the optimal trajectory in blue and an arbitrary trajectory in orange. On the right, the two trajectories in the 2D plane are plotted.

Figure 4

 4 confirms the result.

Figure 4 :

 4 Figure 4: On the left, the traces of the OG for the two trajectories are identical over time. On the right, the two trajectories are plotted with the two unknown landmarks.


Etienne Mondesert

Anne-Sophie Bargnoux

Florence Portet

Dalila Laoudj-Chenivesse

Sandrine Arbogast

Stéphanie Badiou

Jean-Frédéric Brun

Nils Kuster

Eric Raynaud De Mauverger

Jean-Paul Cristol
email: jp-cristol@chu-montpellier.fr

Cystatin C for kidney function assessment in patients with facioscapulohumeral muscular dystrophy

Keywords:

On first examination samples, mean eGFRcys was significantly lower than mean eGFRcreat of 25.5 and 17.9 ml/min/1.73 m2 using CKD-Epi and EKFC equations, respectively. 53.5% (CKD-Epi) and 59.1% (EKFC) of agreement were obtained when using eGFRcys instead of eGFRcreat with reclassifications occurring mainly towards more severe stages. Age was correlated with cystatin C but not with creatinine, MM was correlated with creatinine but not with cystatin C. eGFR decreases > 1 ml/min/1.73 m2 were more important when using eGFRcys instead of eGFRcreat (CKD-Epi: 37.5 vs 15.4%, p < 0.001; EKFC: 34.6 vs 20.2%, p < 0.01).

Conclusion:

Cystatin C which is independent of MM appears as a promising candidate biomarker for CKD diagnosis and follow-up in FSHD patient.

---

A B S T R A C T

Background and aims:

Muscle mass (MM) impairment observed in facioscapulohumeral muscular dystrophy (FSHD) may bias estimated glomerular filtration rate (eGFR) based on creatinine (eGFRcreat). eGFR based on cystatin C (eGFRcys), produced by all nucleated cells, should be an interesting alternative. Main objectives were to compare eGFRcreat and eGRFcys for chronic kidney disease (CKD) staging and for annual eGFR evolution. Secondary objective was to analyse creatinine, cystatin C with measured MM.

Material and methods:

During 4 years, 159 FSHD patients having one or more creatinine and cystatin C measurements (total samples: n = 379), with MM determination by bio-impedancemetry during their follow-up were included. eGFR were determined with CKD-Epi and EKFC equations.

Introduction

Muscle mass (MM) is one of the key determinants of plasma creatinine levels, since this later is the product of muscle metabolism. Therefore, important MM variations could impact creatinine levels and induce variation of estimated glomerular filtration rate based on creatinine (eGFRcreat) without any variation of kidney function. This pitfall of creatinine is well recognized and limits its use in specific populations presenting physiological MM variations such as paediatrics, geriatrics or people with high muscular mass [START_REF] Delanaye | Serum Creatinine: Not So Simple![END_REF]. MM is also altered in pathological conditions such as Duchenne's muscular dystrophy (DMD), amyotrophic lateral sclerosis (ALS) or Facioscapulohumeral muscular dystrophy (FSHD). FSHD is a myopathy characterized by a slow but steady loss of MM, involving certain welldefined muscle groups [START_REF] Statland | Facioscapulohumeral Muscular Dystrophy[END_REF]. It is a genetic inherited disease caused by a deletion on the 4q35 locus containing the D4Z4 locus repetition in 95% of cases (type I FSHD) or by mutations on the SMCHD1 gene in<5% of cases (type II FSHD) [START_REF] Daxinger | Genetic and epigenetic contributors to FSHD[END_REF]. As a result, DUX4 gene translation is derepressed and DUX4 protein plays a critical role in the pathology by creating oxidative stress and mitochondrial dysfunction [START_REF] Turki | Functional muscle impairment in facioscapulohumeral muscular dystrophy is correlated with oxidative stress and mitochondrial dysfunction[END_REF]. Clinical symptoms onset generally occurs in the first two decades of life resulting in early abnormally low MMs. Thus, kidney function and chronic kidney disease (CKD) management with eGFRcreat can be challenging in this population, at all ages. Cystatin C, which is far less sensitive to MM variation, appears as a potent kidney function biomarker in this specific population. Estimated glomerular filtration rate (eGFR) using cystatin C (eGFRcys) had been already suggested for CKD diagnosis and follow-up of other myopathies and neuromuscular diseases, such as DMD or ALS [START_REF] Screever | Kidney Function in Patients with Neuromuscular Disease: Creatinine Versus Cystatin C[END_REF][START_REF] Braat | Renal function in children and adolescents with Duchenne muscular dystrophy[END_REF][START_REF] Tetsuka | Utility of cystatin C for renal function in amyotrophic lateral sclerosis[END_REF]. Compared to these conditions, life expectancy of patients with FSHD is higher and so they should be more concerned by CKD. Moreover, patients with FSHD has cardiac involvement [START_REF] Ducharme-Smith | Cardiac Involvement in Facioscapulohumeral Muscular Dystrophy (FSHD)[END_REF][START_REF] Blaszczyk | Subclinical myocardial injury in patients with Facioscapulohumeral muscular dystrophy 1 and preserved ejection fraction -assessment by cardiovascular magnetic resonance[END_REF] and sarcopenic obesity [START_REF] Vera | Sarcopenic Obesity in Facioscapulohumeral Muscular Dystrophy[END_REF], justifying the need for renal monitoring in this population. To the best of our knowledge, no recommendations nor literature had been yet published on the prevalence and management of CKD in the context of FSHD.

Main objective was to compare eGFRcreat and eGRFcys for chronic kidney disease (CKD) staging. Second objectives were to follow the kidney functions through years using assess eGFRcreat or eGFRcys and to analyse creatinine, cystatin C, with age and muscle mass (MM) measured by bio-impedancemetry (BI).

Materiel and Methods:

Patients

During 4 years (June 2018 to June 2022), all adult patients with FSHD-1 followed at the Montpellier University Hospital Clinical Physiology Outpatient Department were included. All patients had a muscle mass determination by BI. Clinical data, Vignos scale score [START_REF] Vignos | Management of progressive muscular dystrophy in childhood[END_REF], medication history and genetic results were collected from medical records. We included only patients suffering from type I FSHD and excluded patients suffering from type II FSHD. This observational study was approved by the local ethic committee (Comit´e de Protection des Personnes de Montpellier) and was declared under the number 2014-A00153-44. The study was conducted in accordance with the Declaration of Helsinki and its later amendments and the European guidelines for good clinical practice and was registered at clinicaltrials.gov (NCT02622438).

Laboratory measurements and kidney function classification

Lithium heparin and dry tubes were collected from routine biological annual follow-up, then centrifuged 10 min at 2000g. Plasma creatinine was measured by an isotope dilution mass spectrometry (IDMS) traceable enzymatic method on COBAS 8000® (Roche diagnostics, Meylan, France) [START_REF] Pi´eroni | A multicentric evaluation of IDMS-traceable creatinine enzymatic assays[END_REF][START_REF] Kuster | Enzymatic creatinine assays allow estimation of glomerular filtration rate in stages 1 and 2 chronic kidney disease using CKD-EPI equation[END_REF][START_REF] Delanghe | Focusing on the clinical impact of standardization of creatinine measurements: a report by the EFCC Working Group on Creatinine Standardization[END_REF]. Serum cystatin C was measured by Particle-Enhanced Turbidimetric Immunoassay (PETIA) using Diasys® (DiaSys Diagnostic Systems GmbH, Holzheim, Germany) reagents on a BS 480® apparatus (Mindray, Shenzen, China). The assigned values of the calibrators have been made traceable to the ERM®-DA471/IFCC reference material [START_REF] Delanaye | Calibration and precision of serum creatinine and plasma cystatin C measurement: impact on the estimation of glomerular filtration rate[END_REF][START_REF] Bargnoux | Multicenter Evaluation of Cystatin C Measurement after Assay Standardization[END_REF]. Total intra-laboratory coefficients of variation are below 1.8% for creatinine and 2.8% for cystatin C. eGFR was determined by chronic kidney diseaseepidemiology collaboration (CKD-Epi) equations, with 2021 version for creatinine (CKD-Epicreat) [START_REF] Inker | Chronic Kidney Disease Epidemiology Collaboration New Creatinine-and Cystatin C-Based Equations to Estimate GFR without Race[END_REF] and 2012 cystatin C equation (CKD-Epicys) [START_REF] Inker | Estimating glomerular filtration rate from serum creatinine and cystatin C[END_REF], and by european kidney function consortium (EKFC) equations, with 2021 equation for creatinine (EKFCcreat) [START_REF] Pottel | Development and Validation of a Modified Full Age Spectrum Creatinine-Based Equation to Estimate Glomerular Filtration Rate: A Cross-sectional Analysis of Pooled Data[END_REF] and 2023 cystatin C equation (EKFCcys) [START_REF] Pottel | Cystatin C-Based Equation to Estimate GFR without the Inclusion of Race and Sex[END_REF]. Classification of CKD was based on the KDIGO definition [START_REF] Levey | The definition, classification, and prognosis of chronic kidney disease: a KDIGO Controversies Conference report[END_REF]. Decline in kidney function (eGFR of the current year-eGFR of year n + 1) was defined following the 2012 KDIGO guidelines [START_REF] Stevens | Kidney Disease: Improving Global Outcomes Chronic Kidney Disease Guideline Development Work Group Members Evaluation and management of chronic kidney disease: synopsis of the kidney disease: improving global outcomes 2012 clinical practice guideline[END_REF]. Normal decline was considered for values < 1 ml/min/1.73 m2, moderate for values comprised between 1 and 5 ml/min/1.73 m2, accelerated for values > 5 ml/min/1.73 m2.

Bio-impedance analysis

Bio-impedance analysis (BIA) was done using a biacorpus RX spectral® device (MEDICAL HealthCare GmbH, Karlsruhe, Germany), measuring corporal resistance at an alternative current of 50 kHZ. Body fat mass (FM) (kg) and fat-free mass (FFM) (kg) were calculated according to the manufacturer's equations. Anthropomorphic parameters (weight, height) were obtained by using standardized procedures. MM (kg) was calculated from the equation developed by Janssen et al. [START_REF] Janssen | Low relative skeletal muscle mass (sarcopenia) in older persons is associated with functional impairment and physical disability[END_REF], MM index (MMI) (kg/m2) was given by the formula MM/height2. MM below 15 kg (women) or 20 kg (men) and MMI below 5.5 kg/m2 (women) or 7 kg/m2 (men) were considered as low [START_REF] Cruz-Jentoft | Sarcopenia: revised European consensus on definition and diagnosis[END_REF].

Statistical analysis

Passing-Bablok regression analysis was used to compare the eGFRcreat to the eGFRcys. The scatter of differences was visualised by means of Bland-Altman plot [START_REF] Altman | Measurement in Medicine: The Analysis of Method Comparison Studies[END_REF]. Kolgomorov-Smirnov test was used to assess normality. The difference between quantitative data was analysed by Student's t-test for sex-adjusted anthropomorphic parameters comparisons or paired Student's t-test for eGFR comparisons. Chi-2 and MacNemar's test were performed to compare qualitative data. For all comparisons, alpha significance level was set at 5% and a pvalue under 0.05 was considered statistically significant. The concordance between CKD classification of patients using creatinine and cystatin C was assessed using weighted Cohen's κ-test coefficient (wκ) with value < 0.

Results

Population

159 patients were included, 55 were seen only once while 104 had several annual follow-ups. Characteristics at the first and last medical examination are presented in Table 1. 42 patients had 4 samples, 30 patients had 3 samples, 30 patients had 2 samples, and 55 patients had 1 sample (total: 379 samples). Mean age at inclusion was 49 ± 15.3 years at last medical examination, and 52.8% of patients were men. Most of them received an anti-oxidant supplementation including vitamins and oligo-elements [START_REF] Passerieux | Effects of vitamin C, vitamin E, zinc gluconate, and selenomethionine supplementation on muscle function and oxidative stress biomarkers in patients with facioscapulohumeral dystrophy: a double-blind randomized controlled clinical trial[END_REF]. Among cardiovascular comorbidities, hypertension was the most prevalent. Considering the 379 samples, most of creatinine plasma levels (54.1%) were below physiological norms (woman: 45-84 μmol/L, man: 59-104 μmol/L), while most of cystatin C serum levels (76.8%) were between physiological norms (<50 years: 0.53-0.92 mg/ L, >50 years: 0.58-1.02 mg/L). Anthropomorphic and BI parameters are listed in Table 2. On all samples, mean MMI was 8.2 kg/m 2 for men and 6.3 kg/m 2 for women. Proportion of low MMI was above 20% and was not statistically different between women and men, whereas proportion of high percentage of FM was above 50% and was significantly higher in women compared to men.

On first and last examination, 9.4% and 9,6% respectively of patients had a BMI < 18.5kg/m 2 , while 35.9 and 30.8% of patients had a BMI > 24.9 kg/m 2 .

Comparison between eGFRs, Correlations

All analyses of this section were performed on the 159 first examination samples. Regression between CKD-Epicreat (x) and CKD-Epicys (y) gave the following equation: y = 0.44x + 41 (Pearson's coefficient, r = 0.462, p < 0.0001) (Fig. 1A). Regression between EKFCcreat (x) and EKFCcys (y) gave close results (y = 0.34 + 55.3, r = 0.447, p < 0.0001) (Fig. 1B). Bland-Altman's plots (Fig. 2A and2B), showed a mean positive difference of 25.5 ml/min/1.73 m2 for CKD-Epicreat compared to CKD-Epicys and a mean positive difference of 17.9 ml/min/1.73 m2 for EKFCcreat compared to EKFCcys. Comparison between mean values of eGFRcreat and eGFRcys with CKD-Epi and EKFC is reported in Fig. 3. eGFRcreat appeared significantly higher than eGFRcys for CKD-Epi and EKFC (p < 0.0001). Furthermore, CKD-Epicreat appeared significantly higher than EKFCcreat (119 ± 19.4 vs 110.8 ± 21.7 ml/min/1.73 m2, p < 0.0001), whilst CKD-Epicys and EKFCcys were comparable (93.5 ± 18.6 vs 92.7 ± 16.4 ml/min/1.73 m 2 , p = 0.143). Correlations between serum/plasma levels of the two markers and age (Fig. 4A and4B) showed that creatinine plasma levels were not correlated with age (r = -0.041, p = 0.983) by contrast to cystatin C serum levels that were positively correlated with age (r = 0.452, p < 0.0001). Regarding MM and MMI (Fig. 4C,4D,4E,4F), a positive correlation with creatinine was found (r = 0.678 and r = 0.712, p < 0.0001), whereas cystatin C wasn't correlated with the two parameters (r = 0.041 and r = 0.062, p = 0.608 and p = 0.438). FM positively correlated with cystatin C (r = 0.259, p < 0.001) but not with creatinine (r = -0.130, p = 0.102) (Fig. 4G,4H).

CKD classification

The percentages of agreement between CKD classification based on eGFRcys and eGFRcreat were 53.5% for CKD-Epi (95%CI = 45.7;61.2) and 59.1% for EKFC (95%CI = 51.5;66.8). Concordance based on eGFRcreat and eGFRcys determined by CKD-Epi equations (Table 3A) and EKFC equations (Table 3B) was also evaluated by wκ and ρc cations for both equations (CKD-Epi: wκ = 0.066, 95%CI = -0.012;0.134, ρc = 0.244, 95%CI = 0.162;0.322, EKFC: wκ = 0.147, 95%CI = 0.013;0.250, ρc = 0.300, 95%CI = 0.199;0.394). Use of eGFRcreat resulted in 94.3 (CKD-Epicreat) and 84.3% (EKFCcreat) of patients classified in G1 CKD stage and 5.7 (CKD-Epicreat) and 15.7% (EKFCcreat) of patients in G2 stage, while eGFRcys classified 52.2 (CKD-Epicys) and 54.7% (EKFCcys) of patients in G1 stage, 44.7 (CKD-Epicys) and 43.4% (EKFCcys) in G2 stage and 3.1 (CKD-Epicys) and 1.9% (EKFCcys) in G3a stage. A clear tendency of reclassification towards more severe stages of CKD with eGFRcys was observed, for 44 (CKD-Epicreat) and 35.8% (EKFCcreat) of patients. Most of them corresponded to transitions from G1 to G2 stage, then from G1 to G3a, then from G2 to G3a. Considering worsened classifications with CKD-Epicys, 50% concerned patients aged over 60 vs 19.1% for other patients (p < 0.0001). By contrast, reclassifications towards less severe stages of CKD were observed for few patients, which all transitioned from G2 to G1 stages and with an eGFRcreat around 90 ml/min/1.73 m 2 . Supplementary Table 1 presents concordance between the two equations (CKD-Epi and EKFC) using the same marker. The percentages of agreement between were 88.7% (95%CI: 83.8;93.6) for creatinine and 95.6% (95%CI: 89.9;97.5) for cystatin C between CKD classification based on EKFC and CKD-Epi equations. For creatinine there was an impact on classification with a fair agreement (wκ = 0.396) observed. For cystatin C, an almost perfect agreement (wκ = 0.919) was observed.

Renal function evolution

Renal function evolution (Table 4) could be assessed for 104 patients with 2 to 4 annual samples. Longitudinal follow-up was impacted by the marker employed for eGFR calculation: median eGFRcreat yearly evolutions were equal to+ 0,25 (CKD-Epi) and + 0.13 (EKFC) ml/min/1.73 m 2 /year (interquartile range: -0.68;1.57 and -2.42;2.81) while median eGFRcys yearly evolutions was equal to -0.41 (CKD-Epi) and -0.45 (EKFC) ml/min/1.73 m 2 /year (interquartile range -2.42;2.81 and -1.81;2.17). There were significantly more patients classified in moderate or rapid decline with CKD-Epicys compared to CKD-Epicreat (37.5 vs 15.4%, p < 0.01, McNemar's test). The same observation was made for EKFCcys vs EKFCcreat (34.7 vs 20.2%, p < 0.01).

Discussion

This study evidenced discrepancies in estimating renal function between creatinine and cystatin C in patient suffering from FSHD. Creatinine plasma levels are abnormally low (with a median value below the lower limit of normal ranges), potentially in relation with FSHD muscle impairment, and overestimate eGFRcreat. By contrast, cystatin C serum levels were mainly situated within normal ranges and seemed to be less affected by the pathology. eGFRcys was significantly lower than eGFRcreat resulting in reclassification towards more severe stages of CKD. This scheme was observed for 44% (CKD-Epi) and 35.8% (EKFC) of patients, with a mean eGFRcys-eGFRcreat difference of 25.5 (CKD-Epi) and 17.9 (EKFC) ml/min/1.73 m2. The main discrepancy lies in mixing up G1 with G2, where cystatin C equations significantly categorizes more patients in G2 than creatinine. Cystatin C appears as an interesting alternative endogenous biomarker, with a mean CKD-Epicys at 93.5 ml/ min/1.73 m2 and a mean EKFCcys at 92.7 ml/min/1.73 m2 close to mean measured GFR in healthy adults [START_REF] Delanaye | Normal reference values for glomerular filtration rate: what do we really know?[END_REF] while only a small proportion of patients have an eGFRcreat with both equations below 90 ml/min/1.73 m2. CKD-Epicreat overestimates EKFCcreat by 8.2 ml/min/1.73 m2 (Fig. 3) resulting in a fair to poor concordance in CKD classification (Supplementary Table 1). This observation is in agreement with the results published by Pottel et al. [START_REF] Pottel | Cystatin C-Based Equation to Estimate GFR without the Inclusion of Race and Sex[END_REF] since the greatest difference between CKD-Epi and EKFC was observed in young people. The 33 (20.8%) patients aged 15-35 years in our cohort had a significantly higher CKD-Epicreat-EKFCcreat difference compared to the 126 (79.2%) patients aged over 35 years (13.3 ml/min/1.73 m 2 vs 6.9 ml/min/1.73 m 2 , p < 0.0001).

Muscle mass, which is a well-recognized factor of variation of creatinine plasma levels, is lower in patients with FSHD compared to healthy controls [START_REF] Statland | Facioscapulohumeral Muscular Dystrophy[END_REF]. BIA revealed that sexadjusted mean MM and MMI in our cohort are lower than those measured in another study conducted in middle-age healthy subjects (6.2 kg/m 2 vs 7.7 kg/m2 for women, 8.2 kg/m 2 vs 10.5 kg/m 2 for men) [START_REF] Tichet | Prevalence of sarcopenia in the French senior population[END_REF]. Moreover, in agreement with previous reports [START_REF] Vera | Sarcopenic Obesity in Facioscapulohumeral Muscular Dystrophy[END_REF], we observed sarcopenic obesity in our cohort as 14.5% and 24% of patients exhibited low MMI combined with high %FM at first and last examination. This low MM could explain the present discrepancies of our cohort because creatinine is correlated to MM and MMI by contrast to cystatin C (Fig. 4). In accordance with recent publications [START_REF] Laucyte-Cibulskiene | Increased fat mass index is associated with decreased glomerular filtration rate estimated from cystatin C Data from Malmö Diet and Cancer cohort[END_REF][START_REF] Tanabe | Association between visceral fat accumulation and decline in the estimated glomerular filtration rate based on cystatin C in the Japanese urban population: the KOBE study[END_REF], a weak but significant correlation was found between cystatin C and FM which may contribute to the observed differences between eGFR. In our study cystatin C serum levels are correlated with age (r = 0.437), as demonstrated by previous studies [START_REF] Pottel | Estimating glomerular filtration rate for the full age spectrum from serum creatinine and cystatin C[END_REF], while creatinine plasma levels are not (r = 0.014). A possible explanation could be that cystatin C have a much faster increase than creatinine for patients aged over 50 years [START_REF] Pottel | Cystatin C-Based Equation to Estimate GFR without the Inclusion of Race and Sex[END_REF] who represent 49.7% of our population. As pointed in introduction, no prior study has evaluated the interest of cystatin C measurement in patients suffering from FSHD. A work concerning 39 patients suffering from DMD and other myopathies [START_REF] Screever | Kidney Function in Patients with Neuromuscular Disease: Creatinine Versus Cystatin C[END_REF] has shown similarities notably that mean eGFRcys was lower than mean eGFRcreat. Interestingly, another study concerning children with DMD showed that eGFRcys was a lost closest to measured GFR than eGFRcreat [START_REF] Braat | Renal function in children and adolescents with Duchenne muscular dystrophy[END_REF]. However, DMD and FSHD have very different physiopathology and mean ages (31 and 15.5 years) were inferior compared to the mean observed in our cohort (50.7 years).

Our results demonstrate that biological kidney function assessment based only on eGFRcreat is not satisfying in FSHD settings and that cystatin C seems required for optimal renal follow-up of this population. Based on creatinine, median eGFR yearly evolution was positive (CKD-Epicreat: +0.25 ml/min/1.73 m2/year, EKFCcreat: +0.13 ml/min/1.73 m2/year), while based on cystatin C highlighted a negative median eGFR (CKD-Epicys: 0.41 ml/min/1.73 m2/y, EKFCcys: 0.45 ml/min/1.73 m2/y). Cystatin C therefore seems to reflect the physiological decline of glomerular filtration rate starting at age 40, although it is probably underestimating the phenomenon as measured GFR decline had been evaluated at 0.89 ml/min/1.73 m2/year in healthy living potential kidneys donors [START_REF] Pottel | Glomerular Filtration Rate in Healthy Living Potential Kidney Donors: A Meta-Analysis Supporting the Construction of the Full Age Spectrum Equation[END_REF]. Longitudinal follow-up using cystatin C identified 14 patients with a rapid eGFRcys decline (>5ml/year/1.73 m2) which is hidden with eGFRcreat use. These patients could be eligible for nephrology consultation and risk factors should be checked such as albuminuria to creatininuria ratio (ACR) assessment, blood pressure measurement, diabetes screening. These patients should receive a nephroprotective strategy.

Strengths of the study are the relatively high number of patients included for a rare disease and the longitudinal eGFR follow-up. This study has some limitations: First, FSHD is a slowly progressing disease, thus we included a wide range of clinical phenotypes with heterogeneous muscular impairment. Second, we didn't have ACR and gold standard GFR measurement available. Third, all patients are relatively healthy from a nephrological point of view and no CKD patients with eGFRcreat < 60 ml/min/1.73 m 2 were included. Finally, MM and FM were assessed by BIA which could be dependent of both BI apparatus and statistical model. That is not the case for dual-energy X-ray absorptiometry (DEXA) or computed tomography which could appears as better methods. However, this same BIA device was previously validated against DEXA [START_REF] Guiraudou | Body composition measured by bioimpedance segmental (BIAS) analysis and sprint performance in rugby players[END_REF], and was used by trained and certified technician personnel during the study. Furthermore, BIA is considered as a valid method to evaluate patients and to monitor changes in fat and lean mass over time in patients with dystrophy [START_REF] Grilo | Validity of bioelectrical impedance to estimate fat-free mass in boys with Duchenne muscular dystrophy[END_REF][START_REF] Mok | Assessing Change in Body Composition in Children with Duchenne Muscular Dystrophy: Anthropometry and Bioelectrical Impedance Analysis versus Dual-Energy X-Ray Absorptiometry[END_REF].

In summary, eGFRcys, unlike eGFRcreat, showed that FSHD subject seems concerned by kidney function degradation throughout aging given the more important observed decline. If an eGFRcys below 60 ml/ min/1.73 m 2 is confirmed on two samples with three-month interval, the patient could be addressed to a nephrologist and have GFR measurement with gold-standard methods (iohexol or radionuclides). Nevertheless, lack of standardization of measured GFR still needs to be assessed and there is also a need for an endogenous filtration marker or a panel of filtration markers [START_REF] Speeckaert | Measured Glomerular Filtration Rate: The Query for a Workable Golden Standard Technique[END_REF]. In this context, the interest of cystatin C, measurable from only one blood sample and with an available standard, still remains strong. Although cystatin C appears independent of MM, its values can be potentially influenced by several factors like FM, inflammation, thyroid function which are not all taken into account in this study [START_REF] Benoit | Cystatin C as a biomarker of chronic kidney disease: latest developments[END_REF]. ACR is also an important marker of CKD and should be added to eGFRcys for kidney function monitoring [START_REF] Miller | Optimal Use of Biomarkers for Chronic Kidney Disease[END_REF].

In conclusion, clinicians should be advised to avoid creatinine measurement in FSHD patients which seems to overestimates GFR and underestimates decline in kidney function. The use of cystatin C for CKD diagnosis and follow-up appears suitable in this population. To a greater extend, this finding could apply to other diseases with significant MM loss, but further studies are needed to validate cystatin C use in these settings.

CRediT authorship contribution statement Etienne Mondesert:

Data curation, Formal analysis, Writingoriginal draft.

  2 indicating poor agreement, value > 0.8 indicating very good agreement. Lin's concordance correlation coefficient (ρc) was also determined, indicating poor agreement for values < 0.8, and substantial agreement for values > 0.95. Statistical analyses were performed using XLSTAT® software, version 2016.06.35661 (New York, USA) and R® software, version 4.1.2 (R Foundation for Statistical Computing, Vienna, Austria).

Acknowledgments

None.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

Data will be made available on request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


During the manufacture of dairy powders, pasteurization is the only thermal step that destroys vegetative and potentially pathogenic microorganisms. However, bacterial spores present in raw milk persist. These can germinate, grow, form biofilms on stainless steel surfaces (specifically in heat exchangers and evaporators). In addition, the conditions encountered (temperature, a w ) are optimal for germination and growth of thermophilic spores. These spore-forming bacteria may alter the final product during rehydration or storage. It is therefore necessary to evaluate their presence in the dairy powder industries in France. For more information, please contact : louis.delaunay@univ-brest.fr

Figure 1 :Figure 3 :

 13 Figure 1: Count of thermophilic spores after heat treatment at 80°C for 10 min (TSC) and 106°C for 30 min (HRTSC). 1: Whole milk powder (N=8); 2: Skim milk powder (N=7); 3: Casein powder (N=2); 4: Caseinate powder (N=4); 5: Whey powder (N=8); 6: Permeate powder (N=4); 7: Protein powder (N=10); 8: Infant Formula (N=19). Central point: median; box: quartile: bar; percentile; Δ: extreme or atypical points