Relative pose estimation from bearing measurements of three unknown source points

Minh-Duc Hua, Simone De Marco, Tarek Hamel and Randal W. Beard Abstract-This paper unveils a novel discovery that the full relative pose of a monocular camera moving in a three dimensional space can be estimated exploiting bearing measurements of only 3 unknown source points (together with velocity measurements) without any additional knowledge if the camera translational motion is sufficiently exciting. The epipolar constraint commonly used in Computer Vision algebraic algorithms for the determination of the so-called essential matrix (all of them require at least 5 source points) is here exploited in the design of the proposed Riccati observer for pose estimation. One remarkable feature of this work is the determination of an explicit persistence of excitation condition that guarantees uniform observability and, subsequently, (local) exponential stability of the proposed observer. Convincing simulation results are provided to support the proposed approach.

I. INTRODUCTION

Estimating camera motion from a video sequence has many applications in robotics including target tracking, visual odometry, and 3D scene reconstruction. One method to estimate motion from a video sequence is to calculate the essential matrix between consecutive frames. The essential matrix relates the homogeneous image coordinates between frames using the epipolar constraint. After the essential matrix has been determined, it can be decomposed into a rotation and a normalized translation to determine the relative motion of the camera between frames. In order to be robust to noise and feature mismatches, the essential matrix is typically estimated by generating a large number of hypotheses from five-point minimum subsets of matching features, and selecting the best hypothesis using either Random Sample Consensus (RANSAC) [START_REF] Fischler | Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography[END_REF] or Least Median of Squares (LMedS) [START_REF] Rousseeuw | Least median of squares regression[END_REF].

State of the art methods calculate essential matrix hypotheses directly from each five-point minimum subset. One of the best known methods is Nister's algorithm [START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF]. Nister showed that for five matching points, there are potentially ten essential matrices that satisfy the constraints, each corresponding to a real root of a tenth-order polynomial generated from the data. There are many open-source implementations of Nister's five-point algorithm including OpenCV's findEssentialMat function [START_REF]OpenCV 3.1: Open source computer vision library[END_REF]. However, constructing, solving, and extracting the essential matrix from this tenthorder polynomial is complex and can be computationally expensive. Furthermore, since each minimum subset produces up to ten hypotheses, it can be time consuming to score them.

M.-D. Hua, S. De Marco, T. Hamel are with the laboratory I3S, CNRS, Université Côte d'Azur, Sophia Antipolis, France. T. Hamel is also with the Institute Universitaire de France (IUF). Emails: hua(sdemarco, thamel)@i3s.unice.f r. R. W. Beard is with Brigham Young University. Email: beard@byu.edu.

As an alternative to directly calculating essential matrix solutions, some authors [START_REF] Botterill | Fast RANSAC hypothesis generation for essential matrix estimation[END_REF], [START_REF] Botterill | Refining essential matrix estimates from RANSAC[END_REF], [START_REF] Helmke | Essential matrix estimation using gauss-newton iterations on a manifold[END_REF], [START_REF] Ma | Optimization criteria and geometric algorithms for motion and structure estimation[END_REF], [START_REF] Sarkis | A fast and robust solution to the five-point relative pose problem using gauss-newton optimization on a manifold[END_REF] propose solving for the essential matrix using nonlinear optimization algorithms such as Gauss-Newton (GN) and Levenberg-Marquardt (LM). Since the essential matrix has nine entries but only five degrees of freedom, the optimization is performed on the five dimensional essential matrix manifold. There are a number of ways to define the essential matrix manifold. Some authors define the manifold using a rotation and a translation unit vector, which are elements of SO(3) and S 2 respectively [START_REF] Botterill | Refining essential matrix estimates from RANSAC[END_REF], [START_REF] Ma | Optimization criteria and geometric algorithms for motion and structure estimation[END_REF]. Others define the manifold using two elements of SO(3) [START_REF] Helmke | Essential matrix estimation using gauss-newton iterations on a manifold[END_REF], [START_REF] Sarkis | A fast and robust solution to the five-point relative pose problem using gauss-newton optimization on a manifold[END_REF]. The computational requirements of the resulting scheme are significantly less than Nister's five point algorithm. However, one weakness of optimization-based solvers is that they only find one of the ten possible essential matrices at a time. Finding all solutions requires additional optimization runs with different initialization points. The optimization method is also sensitive to initial conditions, which can cause the optimizer to fail to produce a valid solution.

After the essential matrix between images is found, it must then be decomposed into a rotation and a normalized translation. Given an essential matrix, there are four possible rotation-translation pairs [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF]. The correct rotation-translation pair is typically determined using the Cheirality check that ensures that matching features are in front of both cameras. However, the Cheirality check is sensitive to noise in the image and frequently returns the wrong decomposition.

In this paper we propose a novel observer to estimate the 3D relative pose using the recently introduced Riccati observer design framework [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF]. We show that the proposed observer is locally exponentially stable if the motion of the camera is sufficiently exciting. The key contribution consists in using the relative pose of the camera as system state and the epipolar constraints (involving the essential matrix) of three unknown source points. This approach is different from existing works that estimate first the essential matrix and then decompose it to obtain the relative orientation and only the relative normalized translation.

The remainder of the paper is organized as follows. The problem is formally stated in Section II. The Riccati observer and its theoretical properties are reviewed in Section III. The application of the observer to relative pose estimation and the associated stability analysis are provided in Section IV. Simulation results are presented in Section V and the conclusions are in Section VI.

II. PROBLEM STATEMENT

A. Mathematical notation

• {e 1 , e 2 , e 3 } denotes the canonical basis of R 3 and the identity matrix and the null matrix of R n×n are denoted as I n and 0 n , respectively. The closed ball in R n of radius r is denoted as B n r . Let (•) × denote the skew-symmetric matrix associated with the cross product, i.e.

u × v = u × v, ∀u, v ∈ R 3 . Let π u I 3 -uu ⊤ , ∀u ∈ S 2
, denote the projection operator onto the plane orthogonal to u.

• We write f (x, y, t) = O(|x| k1 |y| k2 ) with k 1 ≥ 0 and k 2 ≥ 0 if ∀t : |f (x, y, t)|/(|x| k1 |y| k2 ) ≤ γ < ∞ in the neighbourhood of (x = 0, y = 0). If f depends only on x and t then we write f (x, t) = O(|x| k ) if ∀t : |f (x, t)|/|x| k ≤ γ < ∞ in the neighbourhood of x = 0.

B. Problem formulation

Consider a robotic vehicle equipped with a monocular camera observing n (n ≥ 3) source points with unknown 3D coordinates. Without loss of generality and to make the presentation clear we focus only on the most difficult case of n = 3.

Let us now introduce some notation. Let {A} (resp. {B}) denote the reference (resp. current or body) frame attached to the camera at the reference (resp. current) view. The orientation of frame {B} w.r.t. frame {A} is represented by a rotation matrix R ∈ SO(3). Let ξ ∈ R 3 (resp. ξ ∈ R 3 ) denote the position of frame {B} w.r.t. frame {A} expressed in the body frame {B} (resp. frame {A}). One verifies that ξ = Rξ. The dynamics of the camera pose (R, ξ) are given by

Ṙ = RΩ × ξ = -Ω × ξ + V (1)
where V ∈ R 3 and Ω ∈ R 3 denote the camera's linear and angular velocities expressed in the body frame {B}.

Assume that the vehicle is equipped with a linear velocity sensor (e.g., a Doppler sensor) that measures the linear velocity V together with an Inertial Measurement Unit (IMU) that measures of the angular velocity Ω. We also consider the situation where a collection of 3 unknown source points is always observed by the camera so that their bearings can be directly obtained from the camera images. Let pim i (i = 1, 2, 3) (resp. p im i ) denote the calibrated projective coordinates of the 3 source points onto the camera plane expressed w.r.t. the frame {A} (resp. frame {B}) (see Fig. 1). Denoting Pi ∈ R 3 (resp. P i ∈ R 3 ) the 3D coordinates of the i-th source point w.r.t. the frame {A} (resp. frame {B}) expressed in {A} (resp. frame {B}), and by zi (resp. z i ) its third component, one verifies that pim i = Pi /z i (resp. p im i = P i /z i ). Instead of using the perspective outputs typically used in computer vision algorithms, we use bearing outputs

pi := pim i |p im i | = Pi | Pi | ∈ S 2 , p i := p im i |p im i | = P i |P i | ∈ S 2 (2)
that correspond to the projection onto a virtual unit spherical image plane and differ from the perspective outputs only by the scaling. Using the relations

P i = R ⊤ Pi -ξ = R ⊤ ( Pi -ξ),
the following epipolar constraint can be deduced

p⊤ i Rξ × p i = 0, (i = 1, 2, 3) (3)
which, using the essential matrix definition E := Rt × with t := ξ/|ξ|, can be expressed as p⊤ i Ep i = 0. Instead of estimating the essential matrix E from the epipolar constraints and then decomposing it into a rotation R and a normalized translation t like in traditional algebraic approaches, we will estimate directly the pose (i.e. R and ξ) by also exploiting the dynamic equations [START_REF]OpenCV 3.1: Open source computer vision library[END_REF]. Note that in contrast with the illdefinition of the essential matrix when the translation vector ξ vanished (i.e. ξ = 0), the pose is always well defined.

III. THEORETICAL BACKGROUND: BRIEF RECALL OF A RICCATI OBSERVER DESIGN FRAMEWORK

The observer proposed in this paper is based on the Riccati design framework recently developed in [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF]. The following nonlinear system (a particular case of systems studied in [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF]) is investigated:

Ẋ = A(t)X + U + O(|X| 2 ) + O(|X||U |) Y = C(X, t)X + O(|X| 2 ) (4) with state X = [X ⊤ 1 , X ⊤ 2 ] ⊤ , X 1 ∈ B n r , X 2 ∈ R n , output Y ∈ R m , A(t) ∈ R 2n×2n a continuous matrix-valued function uniformly bounded w.r.t. t in the form A(t) = A 1,1 (t) 0 n A 2,1 (t) A 2,2 (t) (5)
and C(X, t) ∈ R m×2n a continuous matrix-valued function uniformly continuous w.r.t. X and uniformly bounded w.r.t. t. Then, apply the input

U = -P C ⊤ D(t)Y (6)
with P ∈ R 2n×2n a symmetric positive definite matrix solution to the following continuous Riccati equation (CRE):

Ṗ = AP + P A ⊤ -P C ⊤ D(t)CP + S(t) (7)
with P (0) ∈ R 2n×2n a symmetric positive definite matrix, D(t) ∈ R m×m bounded continuous symmetric positive semi-definite, and S(t) ∈ R 2n×2n bounded continuous symmetric positive definite. Then, from Theorem 3.1 and Corollary 3.2 in [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF], X = 0 is locally exponentially stable when both matrices D(t) and S(t) are larger than some constant positive matrix and the pair (A ⋆ (t), C ⋆ (t)), with A ⋆ (t) A(t), C ⋆ (t) C(0, t), is uniformly observable. Definition 1 (uniform observability) The pair (A(t), C(t)) is called uniformly observable if there exist δ, µ > 0 such that ∀t ≥ 0 The matrix W (t, t + δ) is the so-called observability Gramian.

W (t, t + δ) 1 δ t+δ t Φ ⊤ (s, t)C ⊤ (s)C(s)Φ(s, t)ds ≥ µI 2n (

IV. POSE OBSERVER DESIGN FROM BEARING MEASUREMENTS OF THREE UNKNOWN SOURCE POINTS

A. Observer derivation

The proposed observer has the following form

Ṙ = RΩ × -Rσ R× ξ = -Ω × ξ + V -σ ξ (9)
with initial conditions R(0) ∈ SO(3), ξ(0) ∈ R 3 , and with σ R , σ ξ ∈ R 3 the innovation terms to be designed thereafter.

The following error variables are defined

R := R⊤ R, ξ := ξ -ξ (10)
Then, the objective of observer design consists in stabilizing ( R, ξ) about (I 3 , 0). From (1), ( 9) and ( 10) one verifies that the error system is given by

Ṙ = -Ω × R + RΩ × + σ R× R ξ = -Ω × ξ + σ ξ (11)
For analysis purposes let us assume that ξ, Ω and V remain bounded for all time, which is a completely reasonable assumption for the considered applications.

The following step involves developing first order approximations of the error system [START_REF] Sarkis | A fast and robust solution to the five-point relative pose problem using gauss-newton optimization on a manifold[END_REF] and of the measurement equations ( 2) and (3) in order to obtain the system in the form (4). From the Rodrigues' formula, the first order approximation of R is given by

R = I + λ × + O(|λ| 2 ) ( 12
)
with λ ∈ B 3 2 equal to twice the vector part of the quaternion associated with the attitude error matrix R. One then deduces from the first equation of ( 11), (12) and the identity

a × b × -b × a × = (a × b) × , ∀a, b ∈ R 3 that in first order approximations λ = -Ω × λ + σ R + O(|λ| 2 ) + O(|λ||σ R |) (13)
Now let us develop first order approximations of the measurement equations ( 2)-(3). From the epipolar constraint (3) one deduces

0 = p⊤ i R R( ξ + ξ) × p i = p⊤ i R(I + λ × )( ξ + ξ) × p i + O(|λ| 2 ) = -p ⊤ i R( ξ × p i ) × λ -p⊤ i Rp i× ξ + p⊤ i R ξ× p i + O(|λ| 2 ) + O(|λ|| ξ|) = p⊤ i R ξ× p i -p⊤ i R( ξ × p i ) × p⊤ i Rp i× λ ξ + O(|λ| 2 ) + O(|λ|| ξ|)
or equivalently

p ⊤ i R ξ× p i = p⊤ i R( ξ × p i ) × p⊤ i Rp i× λ ξ + O(|λ| 2 ) + O(|λ|| ξ|) (14)
From ( 13), the second equation of [START_REF] Sarkis | A fast and robust solution to the five-point relative pose problem using gauss-newton optimization on a manifold[END_REF], and ( 14), one obtains in first order approximations the system in compact form [START_REF] Fischler | Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography[END_REF] with

                           X := λ ξ , U := σ R σ ξ , Y :=   p⊤ 1 R ξ× p 1 p⊤ 2 R ξ× p 2 p⊤ 3 R ξ× p 3   A(t) := -Ω × 0 3 0 3 -Ω × C(X, t) =   p⊤ 1 R( ξ × p 1 ) × p⊤ 1 Rp 1× p⊤ 2 R( ξ × p 2 ) × p⊤ 2 Rp 2× p⊤ 3 R( ξ × p 3 ) × p⊤ 3 Rp 3×   (15)
The fact that X = [X ⊤ 1 , X ⊤ 2 ] ⊤ with X 1 := λ ∈ B 3 2 , X 2 := ξ ∈ R 3 , together with the particular form of the matrix A as [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF], allows one to obtain the expression of the innovation terms from the input U calculated according to (6) and [START_REF] Helmke | Essential matrix estimation using gauss-newton iterations on a manifold[END_REF] where the matrices D and S, involved in the CRE [START_REF] Helmke | Essential matrix estimation using gauss-newton iterations on a manifold[END_REF], are chosen larger than some constant positive matrix.

B. Observability and stability analysis

According to [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF] the equilibrium X = 0 is locally exponentially stable, provided that the pair (A ⋆ (t), C ⋆ (t)), with A ⋆ (t) = A(t) and C ⋆ (t) := C(0, t), is uniformly observable. By setting X = 0 in the expression of C(X, t) in (15), and using ξ = Rξ and RP i = Piξ with i = 1, 2, 3, one obtains

C ⋆ =   p⊤ 1 ( ξ × Rp 1 ) × R p⊤ 1 (Rp 1 ) × R p⊤ 2 ( ξ × Rp 2 ) × R p⊤ 2 (Rp 2 ) × R p⊤ 3 ( ξ × Rp 3 ) × R p⊤ 3 (Rp 3 ) × R   =     1 | P1||P1| P ⊤ 1 ( ξ × P1 ) × R -1 | P1||P1| P ⊤ 1 ξ× R 1 | P2||P2| P ⊤ 2 ( ξ × P2 ) × R -1 | P2||P2| P ⊤ 2 ξ× R 1 | P3||P3| P ⊤ 3 ( ξ × P3 ) × R -1 | P3||P3| P ⊤ 3 ξ× R     =      -| P1 | ξ⊤ |P1| π p1 R ξ⊤ |P1| p1× R -| P2 | ξ⊤ |P2| π p2 R ξ⊤ |P2| p2× R -| P3 | ξ⊤ |P3| π p3 R ξ⊤ |P3| p3× R      (16)
The following theorem settles a persistence of excitation condition for the uniform observability of the matrix pair (A ⋆ , C ⋆ ) and, subsequently, the local exponential stability of the error dynamics.

Theorem 1 Assume that the bearings p1 , p2 , p3 of the 3 observed source points are linearly independent. Assume that the camera translational motion is sufficiently exciting in the sense that for all time there exist δ, β > 0 such that ∀i = 1, 2, 3

Π i (t, t + δ) := 1 δ t+δ t ξ(s) ξ ⊤ (s) | Pi -ξ(s)| 2 ds ≥ βI 3 ( 17
)
Assume also that ξ, Ω and V remain uniformly bounded for all time. Then, the pair (A ⋆ , C ⋆ ) is uniformly observable. By choosing the matrices D and S involved in the CRE [START_REF] Helmke | Essential matrix estimation using gauss-newton iterations on a manifold[END_REF] larger than some constant positive matrix, one ensures that the equilibrium ( R, ξ) = (I 3 , 0) of the error system is locally exponentially stable.

Proof: To prove that the pair (A ⋆ (t), C ⋆ (t)) is uniformly observable, let us compute the observability Gramian W (t, t+ δ) as defined by [START_REF] Ma | Optimization criteria and geometric algorithms for motion and structure estimation[END_REF]. First, in view of the expression of A in (15) one verifies that the transition matrix associated with the state matrix A ⋆ (i.e. A) is of the form

Φ(s, t) = R⊤ (s) 0 3 0 3 R⊤ (s) (18) with R(s) ∈ SO(3) the solution to d ds R(s ≥ t) = R(s)Ω(s) × , R(s = t) = I 3
One deduces from (16) that

C ⋆⊤ C ⋆ = R ⊤ 0 3 0 3 R ⊤ M R 0 3 0 3 R (19)
with

M :=      i=1,2,3 | Pi | 2 π pi ξ ξ⊤ |P i | 2 π pi i=1,2,3 | Pi |π pi ξ ξ⊤ |P i | 2 pi× - i=1,2,3 | Pi |p i× ξ ξ⊤ |P i | 2 π pi - i=1,2,3 pi× ξ ξ⊤ |P i | 2 pi×     
Using (18), ( 19) and the fact that R⊤ (s) = R ⊤ (s)R(t) one then deduces the observability Gramian

W (t, t + δ) = 1 δ t+δ t Φ ⊤ (s, t)C ⋆⊤ (s)C ⋆ (s)Φ(s, t)ds = R ⊤ (t) 0 3 0 3 R ⊤ (t) 1 δ t+δ t M (s)ds R(t) 0 3 0 3 R(t) (20)
with

1 δ t+δ t M (s)ds =     i=1,2,3 | Pi | 2 π pi Π i (t, t+δ)π pi i=1,2,3 | Pi |π pi Π i (t, t+δ)p i× - i=1,2,3 | Pi |p i× Π i (t, t+δ)π pi - i=1,2,3 pi× Π i (t, t+δ)p i×     (21)
and Π i (t, t + δ) defined in (17). From now on, the shortened notation Π i is used in the place of Π i (t, t + δ) for convenience. Since Π i are symmetric and positive definite by assumption (c.f. Eq. ( 17)), they can be decomposed as

Π i = Q ⊤ i ∆ i Q i , with ∆ i = diag([β i,1 , β i,2 , β i,3 ]
) and β i,j ≥ β (∀j = 1, 2, 3) using (17), and some Q i ∈ SO(3). Thus,

Π i -βI 3 = Q ⊤ i (∆ i -βI 3 )Q i = Γ ⊤ i Γ i with Γ i = (∆ i -βI 3 ) 1 2 Q i , ∀i = 1, 2, 3. Thus, one obtains | Pi | 2 π pi (Π i -βI 3 )π pi | Pi |π pi (Π i -βI 3 )p i× -| Pi |p i× (Π i -βI 3 )π pi -p i× (Π i -βI 3 )p i× = | Pi | 2 π pi Γ ⊤ i Γ i π pi | Pi |π pi Γ ⊤ i Γ i pi× -| Pi |p i× Γ ⊤ i Γ i π pi -p i× Γ ⊤ i Γ i pi× = | Pi |π pi Γ ⊤ i -p i× Γ ⊤ i | Pi |Γ i π pi Γ i pi× ≥ 0 (22)
Using ( 21) and ( 22) one deduces

1 δ t+δ t M (s)ds ≥ β N (23)
with

N :=     i=1,2,3 | Pi | 2 π pi i=1,2,3 | Pi |p i× - i=1 ,2,3 | Pi |p i× i=1,2,3 π pi 
   = i=1,2,3 Ni Ni := | Pi | 2 π pi | Pi |p i× -| Pi |p i× π pi = Z⊤ i Zi Zi := | Pi |π pi pi× (24)
In view of (20) and ( 23), the uniform observability condition (8) involving the observability Gramian matrix W (t, t + δ) is satisfied if the constant symmetric matrix N

given by ( 24) is positive definite. Thus, one only needs to prove that the equation (p 1 , p2 , p3 )), one deduces that ν 1 = 0, which in turn implies that pi × ν 2 = 0, ∀i = 1, 2, 3. Using again the fact that the triplet (p 1 , p2 , p3 ) is linearly independent, one deduces that ν 2 = 0. The remainder of the proof then directly follows by application of Theorem 3.1 and Corollary 3.2 in [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF].

ν ⊤ N ν = 0, with ν = [ν 1 , ν 2 ] ⊤ ∈ R 6 , implies that ν = 0 (i.e. ν 1 = ν 2 = 0) is the unique solution. One has 0 = ν ⊤ N ν = 3 i=1 ν ⊤ Ni ν = 3 i=1 | Zi ν| 2 ⇒ Zi ν = 0, ∀i = 1, 2, 3 ⇒ | Pi |π pi ν 1 + pi × ν 2 = 0, ∀i = 1, 2, 3 ⇒ pi × -| Pi |p i × ν 1 + ν 2 = 0, ∀i = 1, 2, 3 ⇒ ∃β i ∈ R s.t. ν 2 = β i Pi + Pi × ν 1 , ∀i = 1, 2, 3 ⇒ β i Pi -β j Pj + ( Pi -Pj ) × ν 1 = 0, ∀i = j, i, j ∈ {1, 2, 3} ⇒ ν ⊤ 1 ( Pi -Pj ) × ( Pi × Pj ) = 0 ⇒ ν ⊤
V. SIMULATION RESULTS The performance of the proposed observer is demonstrated through simulation results performed with Matlab Simulink. In the simulated scenario, the camera's position ξ expressed in frame {A} follows a periodical trajectory varying in all 3 components: ξ(t) = [15sin(πt/6), 15sin(πt/3), -5 + 2sin(πt/2)] ⊤

The camera's attitude is generated so that it is also strongly varying as shown in The matrices S and D -1 involved in the CRE (7) are interpreted as covariance matrices of the additive noise on the system state and output respectively, and the observer is tuned in a similar way like Kalman-Bucy filters. The following parameters are chosen: P (0) = 0.1I 6 , D = 100I 3 , S = diag(0.1I 3 , I 3 ).

Two simulations will be reported next, where the first one corresponds to the case of ideal measurements (i.e. no noise) and the second one evaluates the effects of measurement noise. For both simulations, the following initial estimation errors are considered: ξ(0) = [4, 5, -5] ⊤ , q R(0) = [q 0 , q⊤ v ] ⊤ (0) = [0.9119, -0.3079, -0.1673, -0.2135] ⊤ (the unit quaternion associated with R(0), corresponding to errors in roll, pitch and yaw Euler angles of 40(deg), 10(deg), 30(deg), respectively).

A. Simulation 1 -ideal measurements

Simulation results for this noise-free case are illustrated by Figures 2345. Figures 2 and4 show the fast convergence of the estimated position and attitude (represented by Euler angles) to the real values. In a different perspective, Figures 3 and5

B. Simulation 2 -noisy measurements

The same scenario as in Simulation 1 is simulated but now with noise added in the measurements of Ω, V and of the source point bearings. More precisely, white Gaussian noise with variance of 2(deg/s) ≈ 0.035(rad/s) and 0.2(m/s) are introduced on the measurements of Ω and V , respectively. We also try to simulate the image noise by first adding uncorrelated white Gaussian noise with variance of 0.01 to all components of the source point bearings and then re-normalizing them onto S 2 to obtain noisy bearing measurements. Simulation results reported in Figures 6789show that the ultimate time evolution of the estimated variables follows very near to the corresponding real variables and the ultimate estimation errors remain relatively small.

VI. CONCLUSIONS

The classical problem of pose estimation from bearing measurements is re-visited. However, unlike the classical problem of essential matrix estimation that requires at least 5 source points, we have shown that 3 source points would be sufficient to estimate the camera full pose if the camera translational motion is sufficiently exciting. To our knowledge, this discovery is a new result. By adopting the recent Riccati observer design framework developed in [START_REF] Hamel | Riccati observers for the nonstationary PnP problem[END_REF], we have proposed a nonlinear Riccati observer with the support of rigourous observability and stability analysis together with convincing simulation results.

Fig. 1 .

 1 Fig. 1. Intuitive representation of 3D coordinates Pi and P i (i = 1, 2, 3), of planar projective coordinates pim i and p im i , and of spherical projective coordinates pi and p i of the 3 source points.

1 |Fig. 2 .

 12 Fig. 2. (Simulation 1) Real and estimated positions (expressed in frame {B}) ξ, ξ (m) versus time (s).

Fig. 3 .

 3 Fig. 3. (Simulation 1) Norm of the position error | ξ| (m) versus time (s).

Figure 4 .

 4 The (unknown) 3D coordinates expressed in frame {A} of the 3 observed source points are given by P1 = [2, 4, 2.5] ⊤ , P2 = [-4.5, 1, 1.5] ⊤ , P3 = [-1, -1.5, 0.6] ⊤ .

Fig. 5 .

 5 Fig. 5. (Simulation 1) Imaginary part qv of the unit quaternion associated with the attitude error R versus time (s).

  illustrate the observer performance by showing the convergence to zero of the estimation errors in terms of the norm of the position error | ξ| and the imaginary part qv of the unit quaternion associated with the attitude error R.

Fig. 6 .

 6 Fig. 6. (Simulation 2) Real and estimated positions (expressed in frame {B}) ξ, ξ (m) versus time (s).

Fig. 7 .

 7 Fig. 7. (Simulation 2) Norm of position error | ξ| (m) versus time (s).

Fig. 8 .

 8 Fig. 8. (Simulation 2) Real and estimated attitudes represented by roll, pitch and yaw Euler angles ( • ) versus time (s).

Fig. 9 .

 9 Fig. 9. (Simulation 2) Imaginary part qv of the unit quaternion associated with the attitude error R versus time (s).

  8) with Φ(s, t) the transition matrix associated with A, i.e. such that d ds Φ(s, t) = A(s)Φ(s, t) with Φ(t, t) = I 2n .

Acknowledgment: This work was supported by the French ANR Astrid CONGRE (ANR-18-ASTR-0006) and FUI GreenExplorer projects.


Sujing Wang
email: sjwang4@ustc.edu.cn

Maria Cabrero-Antonino

Sergio Navalón

Chen-Chen Cao

Antoine Tissot

Iurii Dovgaliuk

Jérôme Marrot

Charlotte Martineau-Corcos

Liang Yu

Hao Wang

William Shepard

Hermenegildo García
email: hgarcia@qim.upv.es

Christian Serre
email: christian.serre@ens.fr

A Robust Titanium Isophthalate Metal-Organic Framework for Visible Light Photocatalytic CO 2 Methanation

SUMMARY

Isophthalic acid (IPA), a feedstock linker, has been considered so far to build series of topical metal-organic frameworks (MOFs) of diverse structures with various di-and trivalent metal ions, such as CAU-10(Al), owing to its facile availability, unique connection angle/mode and a wide scope of functional groups attached. Constructing MOFs from IPA and tetravalent metals, typically Group 4 metals, would be of a great interest due to expected higher chemical stability. In particular, titanium-IPA frameworks possessing photoresponse is alluring, in relation to the known challenge of synthesizing new Ti-MOFs. Here, we have synthesized the first Ti-IPA MOF, denoted as MIP-208, via a solvothermal process that efficiently combines the use of preformed Ti 8 oxoclusters and in situ acetylation of 5-NH 2 -IPA linker. MIP-208 has helical chains of cis-connected corner-sharing TiO 6 polyhedra as the inorganic building units, which are interconnected to each other leading to a 3D ultramicroporous framework. Solid-solution mixed linkers strategy was then successfully applied resulting in a series of multivariate MIP-208 structures with tunable chemical environment and sizable porosity. Finally, the excellent thermal and hydrolytic stabilities of MIP-208 allowed its use for the photocatalytic carbon dioxide (CO 2 ) methanation, showing the best result among the pure MOF catalysts. Ruthenium oxide nanoparticles were further photodeposited on MIP-208 forming a highly active and selective composite catalyst, MIP-208@RuO x , to largely improve the photocatalytic performance, which features a notable visible light response, an excellent stability and recycling ability.

INTRODUCTION

Metal-organic frameworks (MOFs) or porous coordination polymers (PCPs) have been recognized as one of the most intensively investigated heterogeneous catalysts for liquid phase reactions, due to their porous tunable structure, high surface area, and acidic/basic sites or metal nodes with versatile and spatially controlled distribution. [START_REF] Dhakshinamoorthy | Catalysis and photocatalysis by metal organic frameworks[END_REF][START_REF] Xu | Metal-Organic Framework Composites for Catalysis[END_REF][START_REF] Garc Í A | Metal -Organic Frameworks: Applications in Separations and Catalysis[END_REF] Organic ligand plays an essential role in the formation of the MOF structure and introduction of corresponding property. 4; 5 As organic synthesis evolved significantly during past decades, almost unlimited organic compounds have been applied in the MOF fabrication via the linker-engineering strategy, resulting in a vast combination of structures and functions. [START_REF] Desai | N-donor linker based metal-organic frameworks (MOFs): Advancement and prospects as functional materials[END_REF][START_REF] Zhang | Macrocycle-based metal-organic frameworks[END_REF][START_REF] He | Multifunctional metal-organic frameworks constructed from meta-benzenedicarboxylate units[END_REF] However, time and effort requiring organic synthesis for linker preparation undoubtedly limits the reproduction and application of costly MOF candidates. 9; 10 In this regard, commercially available organic linkers from industry are of great interests due to their easy availability at large scale and low cost. 11; 12 Benzene-dicarboxylic acid isomers are representative compounds fulfill the aforementioned demand currently produced in billion-kilogram scale annually. The three isomers, namely ortho-phthalic acid (phthalic acid), meta-phthalic acid (isophthalic acid, IPA) and para-phthalic acid (terephthalic acid), are all important feedstock for preparation of a large number of industrial products. 13; 14 In their use of constructing MOFs, phthalic acid alone tends to form materials of low dimensionality, mainly due to the deficient separation and steric hindrance between the adjacent carboxylate groups. [START_REF] Ohtani | Water-triggered macroscopic structural transformation of a metal-organic framework[END_REF][START_REF] Padmanabhan | An unprecedented two-dimensional polymeric [Zn(OOC-C6H4-COO)2]n2 -[+H3N -(CH2)3 -NH3+]n system bearing one-dimensional chain of zinc(II) bis(phthalate) dianions held by propane-1,3-diammonium cations: Crystal structure, thermal and fluorescent properties[END_REF][17] On the contrary, terephthalic acid is one of the most popular carboxylate linkers from the very beginning of the MOF journey. [START_REF] Li | Coordinatively Unsaturated Metal Centers in the Extended Porous Framework of Zn3(BDC)3•6CH3OH (BDC = 1,4-Benzenedicarboxylate)[END_REF] The ideal configuration with suitable molecular length, perfect separation of coordination sites and varied connection modes make it adaptable to MOF structures with any dimension built with almost every sort of metal element. [START_REF] Banerjee | Recent Advances in s-Block Metal Carboxylate Networks[END_REF][START_REF] Pagis | Lanthanide-Based Metal Organic Frameworks: Synthetic Strategies and Catalytic Applications[END_REF][START_REF] Aguirre-Díaz | Group 13th metal-organic frameworks and their role in heterogeneous catalysis[END_REF] In the case of isophthalic acid, the unique angle (120°) and the medium distance between the two carboxylate groups enable a large degree of manipulation in synthesizing MOFs compared with the other two isomers. Numerous metal-IPA frameworks have been identified with nodes ranging from alkaline earth [START_REF] Kang | Solvothermal synthesis and characterization of new calcium carboxylates based on cluster-and rod-like building blocks[END_REF] , to divalent and trivalent transition metals [START_REF] Bourne | Self-Assembly of Nanometer-Scale Secondary Building Units into an Undulating Two-Dimensional Network with Two Types of Hydrophobic Cavity[END_REF][START_REF] Vodak | Metal-organic frameworks constructed from pentagonal antiprismatic and cuboctahedral secondary building units[END_REF][START_REF] Barthelet | VIII(H2O)]3O(O2CC6H4CO2)3•(Cl, 9H2O) (MIL-59): a rare example of vanadocarboxylate with a magnetically frustrated three-dimensional hybrid framework[END_REF][START_REF] Qazvini | A Robust Ethane-Trapping Metal-Organic Framework with a High Capacity for Ethylene Purification[END_REF] , as well as rare earth [27][START_REF] Wang | [END_REF][START_REF] Mihalcea | Occurence of an Octanuclear Motif of Uranyl Isophthalate with Cation-Cation Interactions through Edge-Sharing Connection Mode[END_REF] or main group elements [START_REF] Vougo-Zanda | Influence of Ligand Geometry on the Formation of In-O Chains in Metal-Oxide Organic Frameworks (MOOFs)[END_REF][START_REF] Bu | A 4-connected anionic metal-organic nanotube constructed from indium isophthalate[END_REF][START_REF] Panda | Structural isomerism leading to variable proton conductivity in indium(iii) isophthalic acid based frameworks[END_REF] , and even mixed metal systems [START_REF] Che | Heteropolynuclear Metamagnet Showing Spin Canting and Single-Crystal to Single-Crystal Phase Transformation[END_REF][START_REF] Zhang | Topology Analysis and Nonlinear-Optical-Active Properties of Luminescent Metal-Organic Framework Materials Based on Zinc/Lead Isophthalates[END_REF][START_REF] Zhang | New Heterometallic Carboxylate Frameworks: Synthesis, Structure, Robustness, Flexibility, and Porosity[END_REF] , in which the meta-side of the coordinated carboxylate groups on the benzene ring frequently faces the structure void. IPA offers thus an obvious and facile opportunity of tuning the chemical environment of the MOF pore via introduction of functional groups on the meta-side.

Correspondingly, distinct organic functions on the 5-and 4-position of the benzene ring have been presented in the reported examples, showing great interests and potential in fine-tuning the structure-property correlation [START_REF] Mccormick | Coordination Polymers of 5-Alkoxy Isophthalic Acids[END_REF][START_REF] Chen | Substituent effect of R-isophthalates (R = -H, -CH3, -OCH3, -tBu, -OH, and -NO2) on the construction of CdII coordination polymers incorporating a dipyridyl tecton 2,5-bis(3-pyridyl)-1,3,4-oxadiazole[END_REF][START_REF] Du | R-Isophthalate (R = -H, -NO2, and -COOH) as modular building blocks for mixed-ligand coordination polymers incorporated with a versatile connector 4-amino-3,5-bis(3-pyridyl)-1,2,4-triazole[END_REF][START_REF] Chen | Ultrafast water sensing and thermal imaging by a metal-organic framework with switchable luminescence[END_REF] .

It is noteworthy that isophthalic acid is compatible for the MOF assembly with every transition metal subgroup but not the Group 4 elements in the periodic table, which indicates, to the best of our knowledge, no report of IPA based-MOF example built with titanium (Ti), zirconium (Zr), hafnium (Hf) or rutherfordium (Rf). On the other hand, Ti and Zr, the most common Group 4 metals, have attracted major attention of the MOF research community during the past years, owing to their high natural abundances, low toxicities, strong interactions with oxygen sites and thus significantly enhanced stabilities once in the MOF structures. 40; 41 More recently, the potential of MOFs has been expanded to other areas of catalysis, particularly in photocatalysis. [START_REF] Dhakshinamoorthy | Metal-Organic Framework (MOF) Compounds: Photocatalysts for Redox Reactions and Solar Fuel Production[END_REF] The strong interaction between the positively charged metal nodes and negatively charged antenna linkers in rigid configurations makes MOFs suitable in promoting photoinduced charge separation. [START_REF] Dhakshinamoorthy | Metal-Organic Framework (MOF) Compounds: Photocatalysts for Redox Reactions and Solar Fuel Production[END_REF] Hence, an electron transfer from the organic linker in its excited state to the metal node upon light absorption appears a general photochemical step, especially in the case of a favorable overlapping between the lowest occupied crystal orbit (LOCO) with the atomic orbitals of the metal cations. [START_REF] Alvaro | Semiconductor Behavior of a Metal-Organic Framework (MOF)[END_REF] For this reason, MOFs are among the most promising porous photocatalysts, particularly for the production of solar fuels.

Taking into account the prevalent role that MOFs and derivative materials in photocatalysis, Ti-MOFs have attracted considerable attention due to the known activity of TiO 2 and Ti-oxo clusters to undergo photoinduced charge separation. [START_REF] Nasalevich | Enhancing optical absorption of metal-organic frameworks for improved visible light photocatalysis[END_REF] In general, Ti-MOFs are not only promising candidates as photo-responsive materials, [START_REF] Zhu | Titanium-based metal-organic frameworks for photocatalytic applications[END_REF] but also show great potential in gas separation, 46; 47 proton conduction [START_REF] Wahiduzzaman | A High Proton Conductive Hydrogen-Sulfate Decorated Titanium Carboxylate Metal-Organic Framework[END_REF] and clinical applications. [START_REF] Pinto | Tuning Cellular Biological Functions Through the Controlled Release of NO from a Porous Ti-MOF[END_REF] Nevertheless, direct synthesis of Ti-MOFs still remains an important challenge, [START_REF] Assi | Titanium coordination compounds: from discrete metal complexes to metal-organic frameworks[END_REF] despite the gradually increased number of reported Ti-MOF structures over the past decade, in most of the cases attained through serendipity. It is admitted that an infinite number of Ti-O units is expected to facilitate photoinduced charge separation and charge mobility, in comparison with MOFs based on discrete inorganic units with a low number of Ti-O bonds. One and two dimensional (1D and 2D) arrays of Ti-O units should thus in principle improve the semiconducting properties of the Ti-MOFs. [START_REF] Tachikawa | Photoinduced Charge Separation in Titania Nanotubes[END_REF] In this context, the photoactive Ti-MOFs that have been prepared so far are those with 1D chain or nanowire of Ti-O building units, whose photo-responsive behaviors have been proven to be closer than those compounds constructed of discrete Ti-O clusters to that of bulk TiO 2 . [START_REF] Wang | A phase transformable ultrastable titanium-carboxylate framework for photoconduction[END_REF] As shown in Figure 1, there are only three kinds of 1D Ti-O inorganic building units reported up to date, namely the TiO chains in MIL-91, [START_REF] Serre | Synthesis, Structure and Properties of Related MicroporousN,N'-Piperazinebismethylenephosphonates of Aluminum and Titanium[END_REF] the Ti 6 O 9 nanowires in MIP-177-HT, [START_REF] Wang | A phase transformable ultrastable titanium-carboxylate framework for photoconduction[END_REF] ZSTU-1, 2 and 3, [START_REF] Li | Tunable titanium metal-organic frameworks with infinite 1D Ti-O rods for efficient visible-light-driven photocatalytic H2 evolution[END_REF] and the TiO chain in DGIST-1. [START_REF] Keum | Titanium-Carboxylate Metal-Organic Framework Based on an Unprecedented Ti-Oxo Chain Cluster[END_REF] Therefore, as these MOFs are all constructed from sophisticated ligands, it would be of great interest to discover IPA type linker-based Ti-MOFs with infinite Ti-O building units. It would not only expand the scope of Ti-MOFs structural and chemical diversity, but also might lead to improved photocatalytic performance.

Herein, we report an ultramicroporous Ti-MOF, denoted as MIP-208 (MIP stands for the Materials of the Institute of porous materials from Paris), the first MOF composed of Group 4 metal and IPA type linker. In the crystal structure of MIP-208, helical chains of cis-connected corner-sharing TiO 6 polyhedra are held in place by the in situ

RESULTS AND DISCUSSION

Twenty-four different Ti-MOFs prepared from direct-synthesis have been reported to date. The Ti sources used to produce these MOFs can be classified into two kinds:

simple Ti complexes and polynuclear TiO clusters. Compared with simple Ti complexes, such as TiCl 3 , TiCl 4 , and Ti(iPrO) 4 , Ti-oxoclusters are known to slow down the reaction rate and avoid the undesirable hydrolytic formation of TiO 2 , owing to the gradual Ti-O connection rearrangements of the oxocluster precursor for most cases. Thus, it usually leads to highly crystalline Ti-MOF materials constituted by large single crystals. [START_REF] Keum | Titanium-Carboxylate Metal-Organic Framework Based on an Unprecedented Ti-Oxo Chain Cluster[END_REF][START_REF] Yuan | A single crystalline porphyrinic titanium metal-organic framework[END_REF][START_REF] Padial | Hydroxamate Titanium-Organic Frameworks and the Effect of Siderophore-Type Linkers over Their Photocatalytic Activity[END_REF] In this regard, a Ti 8 O 8 cluster with formate and acetate terminals (Ti 8 AF) was selected here, due to its facile preparation, easy handle, suitable stability and reactivity. [START_REF] Wang | Toward a Rational Design of Titanium Metal-Organic Frameworks[END_REF] Regarding the reaction solvent, it is well-recognized that the conventional ones, such as dimethylformamide (DMF) and diethylformamide (DEF), should be replaced by greener solvents to decrease the toxicity and harm to human and environment. Based on our previous success of preparing Group 4 metal MOFs, formic acid, acetic acid and acetic anhydride, which have been shown to efficiently slow down the reaction leading to highly crystalline products, were selected as solvents. 5-NH 2 -IPA was selected as the linker due to the documented benefit of the amino group presence in increasing the CO 2 adsorption and enhancing visible light photoresponse. 59; 60 Initial reaction attempts indicated that the aforementioned three acidic solvents can interfere with decomposition of the Ti 8 AF precursor, while offer a good solubility to the 5-NH 2 -IPA linker. However, all attempts did not generate any crystalline solid.

Various additives known to either facilitate the linker exchange or promote the Ti-O connection rearrangement were tested, such as water, methanol, ethanol, acetone, and DMF. A highly crystalline product of a dark yellow or light brown color was formed only when methanol was used in the mixture of acetic acid and acetic anhydride.

Acetic acid was found to be not necessary for the product formation, but to accelerate the reaction rate. An in situ amidation took place between the amino group of 5-NH 2 -IPA and acetic anhydride, generating the 5-acetamide-IPA (5-Aa-IPA) as the final framework building linker (Figure S1). Solid state NMR data collected on the as-synthesized MIP-208 sample demonstrated the presence of 5-Aa-IPA within the structure (Figures S2 andS3).

It is likely that the in situ acetylation step is critical to yield the crystalline material, which possibly slows down the reaction rate significantly. A control experiment using the preformed 5-Aa-IPA as linker under the same reaction conditions did not lead to MIP-208, but to an amorphous solid. Therefore, this solvothermal synthesis is likely to benefit synergistically from the rearrangement of the Ti 8 AF cluster and in situ linker formation, which is a rare case for MOF preparation reported so far. However, it could provide us a valuable and even possibly rational strategy for further synthesizing new Ti-MOFs through a control of the reaction rate from both inorganic and organic precursor conversions.

MIP-208 was obtained as microsize single crystals, ruling out its structure determination from laboratory diffraction. Thus Synchrotron diffraction single-crystal data of MIP-208 was collected with a microfocused X-ray equipment on the Proxima 2A beamline (Synchrotron SOLEIL, France) [START_REF] Duran | -31 -PROXIMA 2A -A New Fully Tunable Micro-focus Beamline for Macromolecular Crystallography[END_REF] running along the c-axis. Unlike the linear 1D Ti-O SBUs in MIL-91, [START_REF] Serre | Synthesis, Structure and Properties of Related MicroporousN,N'-Piperazinebismethylenephosphonates of Aluminum and Titanium[END_REF] DGIST-1 [START_REF] Keum | Titanium-Carboxylate Metal-Organic Framework Based on an Unprecedented Ti-Oxo Chain Cluster[END_REF] and MIP-177-HT, [START_REF] Wang | A phase transformable ultrastable titanium-carboxylate framework for photoconduction[END_REF] the helical TiO chain in MIP-208 is the first example of this type of SBU observed in Ti-MOFs, to our knowledge (Figure 1). Al(OH) chain with similar configuration was discovered previously in the CAU-10 MOF series (CAU stands for Christian-Albrechts-University), [START_REF] Reinsch | Structures, Sorption Characteristics, and Nonlinear Optical Properties of a New Series of Highly Stable Aluminum MOFs[END_REF] with μ 2 -OH group as the sharing-corner.

Notably, the replacement of bridging OH in trivalent metal-based MOFs by oxo group in similar MOFs built with tetravalent metal could lead to a dramatic difference in terms of structural flexibility and related properties, as documented in MIL-47 and MIL-53 compounds (MIL stands for Materials from Institute Lavoisier). 63; 64 In particular, the presence of OH or oxo groups in Ti-O materials has been demonstrated

to be critical to their performance in photocatalysis, [START_REF] Leshuk | Photocatalytic Activity of Hydrogenated TiO2[END_REF][START_REF] Chen | Black titanium dioxide (TiO2) nanomaterials[END_REF][START_REF] Liu | Titanium Dioxide Nanomaterials: Self-Structural Modifications[END_REF] which highlights the

The neighboring TiO chains are interconnected via pairs of 5-Aa-IPA linkers in an up and down mode to avoid as much as possible the steric hindrance (Figure 2b), resulting in a 3D framework with double-walled 1D channels running along the c-axis (Figure 2c). One can expect its micropores to be accessible for guest molecules as reported before for CAU-10 compounds constructed from IPA bearing the bulkiest functional groups, 62; 68 due to large size and thus considerable steric hindrance of the acetamide groups facing the pores. As reported before for CAU-10-OMe and CAU-10-Br, MIP-208 did not show any accessibility for nitrogen at 77 K, but a noticeable porosity for CO 2 (Figure S4).

A large number of functionalized IPAs are commercially available chemicals offered at low prices. The good compatibility of IPA derivatives regarding their behavior in coordination chemistry has been documented, [START_REF] Chen | Substituent effect of R-isophthalates (R = -H, -CH3, -OCH3, -tBu, -OH, and -NO2) on the construction of CdII coordination polymers incorporating a dipyridyl tecton 2,5-bis(3-pyridyl)-1,3,4-oxadiazole[END_REF] thus offering a great promise to tune the chemical environment of the pore in MIP-208 structure via a solid-solution mixed linkers strategy. Solid-solution strategy is a powerful alternative way to introduce multiple functional groups together in the MOF structures, especially when the direct synthesis of isostructural MOF built with a single linker bearing specific functional group is challenging. [START_REF] Deng | Multiple Functional Groups of Varying Ratios in Metal-Organic Frameworks[END_REF][START_REF] Foo | Functional Hybrid Porous Coordination Polymers[END_REF][START_REF] Helal | Multivariate metal-organic frameworks[END_REF] This strategy suits perfectly the case of As shown in Figure 3b, the inclusion of the second linker did not generate noticeable changes on the overall structural long-range order, as evidenced by their highly similar PXRD patterns. On the contrary, the porosity of the resulting materials was altered, as deduced from the CO 2 adsorption isotherms collected at 273 K (Figure 3c).

For instance, the sample with 50 mol% IPA substitution ratio led to a significant enhancement of CO 2 uptake (50% at both 0.15 and 1 bar) compared with the pure MIP-208, likely due to the much less steric hindrance of lighter H over the acetamide group. A similar observation was noticed for the 5-Me-IPA mixed sample (33 mol%).

In the case of 5-Br-IPA derivative (38 mol%), a slight decrease of uptake in comparison with the parent compound has been observed, probably attributable to a much larger atomic weight of Br atom, in spite of its lower steric hindrance. When the slightly bulkier tBu group but with similar molecular weight was introduced to replace the acetamide one, the sample shows limited drop in pore volume mainly due to the small linker substitution ratio (17 mol%). It is worth noting that even PDA, which has a five-membered heterocyclic aromatic moiety, could be included in the MIP-208 framework, though the substitution ratio is low (11 mol%), probably due to the similar angle of the two carboxylate groups and the presence of basic N-site on the heterocyclic ring. Therefore, solid-solution strategy is here an efficient way to expand and modify the chemical variety of the MIP-208 structural cavity, leading to a series of MTV-MIP-208 materials.

Stability of MOF is a practical concern in applications. For instance, photocatalytic methanation requires a combination of good thermal, hydrolytic and UV irradiation stability of the MOF catalyst, since the catalysis is normally carried out at a temperature close to 200 °C, under UV irradiation and in the presence of generating stoichiometric water as the side product. Therefore, the stability of MIP-208 was tested before applying it to the photocatalytic methanation. The results from both temperature-dependent PXRD measurement and thermogravimetric analysis (TGA)

support the high stability of MIP-208 (Figure S6). Its ordered crystal structure is stable up to 325 °C in air, which is comparable to MIP-177-HT, and clearly surpass MIL-91 and DGIST-1 (unstable above 200 °C). Furthermore, the activated MIP-208 sample hardly adsorb water when exposed in air for a long time (less than 5%), considerably much less than the Al-CAU-10 series do. This enhanced hydrophobicity of MIP-208 is thus associated to the bridging oxo group effect, in comparison with the OH in CAU-10, as well as the limited hydrophilic character of acetamide.

Hydrolytic stability of MIP-208 was tested by soaking it in liquid water at room temperature for three days and heating it in boiling water for 8 h separately. The PXRD patterns of the tested samples are identical to that of the as-synthesized one (Figure 4a), supporting the robustness of the crystalline long-range order. CO 2 adsorption data (Figure 4b) confirm the good hydrolytic stability of MIP-208, with minor differences between the sample before and after water treatments. It is worth noting that MIP-208 is among the most hydrothermally stable Ti-MOFs, comparable to that of the Ti-bisphosphonate MIL-91, outperforming all Ti-carboxylate MOFs reported so far. As MIP-208 possesses 1D TiO chain building block, a good photoresponsive activity is expected. Photocatalytic conversion of CO 2 to methane was thus selected as the model reaction, since this reaction could generate value-added product from CO 2 emissions. Promising progress of using MOF-based heterogeneous catalysts for CO 2 conversion to methane has been achieved lately. [START_REF] Ding | Carbon capture and conversion using metal-organic frameworks and MOF-based materials[END_REF] For instance, MOFs supported nickel and platinum nanoparticles (NPs) could promote the thermal methanation under harsh conditions, such as high pressures and temperatures. Photocatalytic methanation by MOF-based catalysts has however been hampered for a long time by the poor selectivity until the recent success reported when a HKUST-1@TiO 2 core-shell composite was applied under UV irradiation, but with rather low efficiency. [START_REF] Li | Integration of an Inorganic Semiconductor with a Metal-Organic Framework: A Platform for Enhanced Gaseous Photocatalytic Reactions[END_REF] Lately, the first example of pure photoactive MOF towards catalytic methanation has been reported, which involves ZnO inorganic bricks in a nitrogen-rich framework exhibiting a CH 4 generation productivity of 30 mol•g catalyst -1

after 24 h at 215 ºC under UV light irradiation. A notable enhancement of activity was attained via supporting Cu 2 O NPs on the Zn-MOF to form a composite catalyst, which led to a CH 4 production of 45 mol•g catalyst -1 under the same condition. [START_REF] Cabrero-Antonino | Design of cost-efficient and photocatalytically active Zn-based MOFs decorated with Cu2O nanoparticles for CO2 methanation[END_REF] Those breakthroughs are undoubtedly encouraging. However, the photocatalytic activity of MOF materials is however very low. Thus developing efficient and selective MOF-based photocatalyst for methanation is of great interests, yet remains a challenge.

Pure MIP-208 sample was first tested for the photocatalytic methanation reaction. A CH 4 production of about 40 µmol/g catalyst -1 could be achieved using MIP-208 as the catalyst in the mixture of CO 2 (0.25 bar) and hydrogen (1.05 bar) under UV irradiation (Xenon lamp, 1350 W×m -2 ) at 200 ºC after 22 hours (Figure S7). This result highlights the positive influence of 1D TiO chain SBU in MIP-208 regarding its photo-response, since the benchmark NH 2 -MIL-125 built with discrete Ti 8 clusters exhibits negligible photocatalytic activity under similar reaction conditions. [START_REF] Cabrero-Antonino | Design of cost-efficient and photocatalytically active Zn-based MOFs decorated with Cu2O nanoparticles for CO2 methanation[END_REF] Furthermore, in comparison with the state-of-the-art MOF photocatalyst for methanation, [START_REF] Cabrero-Antonino | Design of cost-efficient and photocatalytically active Zn-based MOFs decorated with Cu2O nanoparticles for CO2 methanation[END_REF] an enhancement of 33 % in efficiency was noticed for MIP-208, which suggests that MIP-208 could be the most efficient pure MOF photocatalyst for methanation up to now, to our knowledge.

pattern compared with that of the pure MIP-208 could be attributed to the low Ru loading and the evenly distributed small particle size, as determined by TEM images.

The MIP-208@RuO x solid was also characterized by X-ray photoelectron spectroscopy (XPS, Figure S12). The C1s spectrum shows the presence of aromatic carbons (284.4 eV). In addition, a band centred at 285.9 eV attributable to the C-N bond of the amide groups is observed, together with another centred at 289 eV that can be assigned to the carbonyl of both the amide and the carboxylate groups. The amide group is clearly observed in the N1s band centred at about 399 eV as well. The broad O1s band is mainly due to the presence of oxygen atoms in the carboxylate and amide groups, together with the Ti-O and Ru-O bonds. Ti2p spectrum can be assigned to the presence of Ti 4+ ion. Ruthenium is recognized by the bands appearing at about 280.7 and 282.4 eV that can be assigned to the Ru 3d 5/2 of RuO 2 and hydrated RuO 2 , respectively. 79; 80 The expected Ru 3d 3/2 band appearing at higher binding energies (~4.1 eV) (Supplementary Figure 12a) is, however, difficult to observe, since it overlaps with the C-N contribution of the acetamido groups. [START_REF] Wenderich | Methods, Mechanism, and Applications of Photodeposition in Photocatalysis: A Review[END_REF]; 80 XPS analysis also can provide an estimation of the valence band maximum of the MIP-208@RuO x solid by measuring the lowest energy electron band of the material referred to the work function of the instrument. As shown in Supplementary Figure 13, a valence band potential of -1.51 eV was estimated. This value together with the 2.92 eV bandgap measured from the Tauc plot of the diffuse reflectance UV-Vis spectrum, gives a value for the LUCO of -2.874 eV.

The photocatalytic performance of the MIP-208@RuOx composite in CO 2 methanation was then evaluated under the same reaction condition as that used for testing the pristine MOF. As expected, a selective CO 2 conversion into CH 4 (800 µmol/g catalyst -1 ) as the only product was observed. Specifically, CO and C 2 H 6 amounts were under the detection limit of our micro-GC. This CH 4 production value is 20 times higher compared with that obtained using the pure MIP-208 sample. Control experiments using MIP-208@RuO x in the dark under the same reaction condition showed negligible CH 4 production.

Similar time-conversion plot, with a slightly lower CH 4 formation rate, was also measured using simulated sunlight irradiation (1 Sun) (Figure 5a). The difference between the CH 4 production using the xenon lamp or the simulated sunlight can be attributed to the lower power of the simulated sunlight (1000 W×m -2 ) compared to the Xenon lamp (1350 W×m -2 ). This similarity in the temporal CO 2 conversion indicates that a major percentage of the MIP-208@RuO x photoresponse must derive from the visible region. Otherwise, a more noticeable decrease in photocatalytic activity, much larger than that expected from the power ratio, should have been observed since the UV content of the simulated sunlight is very minor compared with the xenon lamp.

Cycling stability of the MIP-208@RuO x composite as photocatalyst was determined by performing a series of consecutive reuses following the temporal evolution of CO 2 conversion and CH 4 formation. These stability tests are important considering the conditions of the photocatalytic methanation, particularly the presence of H 2 O and the reaction temperature of 200 °C. The results presented in Figure 5b show coincident plots for the temporal CH 4 evolution upon seven consecutive runs, supporting the stability of the material under the conditions of the photocatalytic reaction. It is noteworthy that this highly selective and constant methanation activity promoted by RuO x has hardly been achieved before, as CO has been proven as the side product in this reaction, which led to a considerable deactivation of the catalyst. [START_REF] Albero | Titanium-Perovskite-Supported RuO2 Nanoparticles for Photocatalytic CO2 Methanation[END_REF] Additionally, irradiation of the composite in argon atmosphere using the Xe lamp or solar simulator at 200 ºC revealed the evolution of a tiny amount of CO 2 , corresponding to the decomposition of a small fraction of the IPA linkers present in the material (<0.05 mol CO 2 /mol ligand). This small CO 2 evolution from MIP-208@RuO x compares favourably with the stability of the most robust MOFs reported as photocatalysts under similar irradiation conditions. [START_REF] Santiago-Portillo | Long-Term Photostability in Terephthalate Metal-Organic Frameworks[END_REF] The crystallinity of the MIP-208@RuO x sample after seven times cycling in catalysis was well-maintained, as no notable change in the PXRD pattern was observed (Figure S14). Furthermore, the solid-state [START_REF] Hillesheim | Phthalic Acid and Derivatives[END_REF] C-NMR spectra of the fresh and seven-time reused MIP-208@RuO x were coincident (Figure S15). DF-TEM images of the catalyst after cycling were also similar to those of the fresh sample, without any evidence of RuO x particle growth or agglomeration (Figure S16). Thus, both photocatalytic activity and characterization of the seven-times used MIP-208@RuO x sample indicate the stability of the material under irradiation conditions. To determine the origin of CH 4 , an experiment using 13 C-labelled CO 2 was carried out, monitoring the isotopic composition of CH 4 evolved by mass spectrometry. The results presented in Figure S17 show the peaks corresponding to [START_REF] Hillesheim | Phthalic Acid and Derivatives[END_REF] CH 4 appearing at 17 Dalton. It is important to note that injection of a sample before irradiation does not show in the chromatogram any peak at the retention time of 13 CH 4 .

To shed light about the main operating mechanism of the photocatalytic methanation, H 2 was replaced by dimethylaniline as sacrificial electron donor. Two possible operating mechanisms have been reported for the photocatalytic CO 2 methanation, either the so-called photothermal pathway or the photoinduced charge separation. [START_REF] Wenderich | Methods, Mechanism, and Applications of Photodeposition in Photocatalysis: A Review[END_REF] In the photothermal mechanism, light energy is converted into local heat on the metal NPs and thus temperature increases on the NP surface promotes the thermal conversion of CO 2 and H 2 into the product. [START_REF] Wenderich | Methods, Mechanism, and Applications of Photodeposition in Photocatalysis: A Review[END_REF] This local temperature increase is undetectable by measurements of the macroscopic system due to the low RuO x proportion. In the photocatalytic mechanism, light absorption leads to charge separation with generation of electrons in the conduction band and holes in the valence band that causes CO 2 reduction and H 2 oxidation, respectively. In the photoinduced charge separation state, the use of an electron donor better than H 2 should equally result in CO 2 conversion to methane, with an even higher reaction rate than using H 2 as reagent. On the contrary, this electron donor should not be adequate for the photothermal reaction. In the present study, dimethylaniline (0.76 V vs Ag/AgCl) was selected as electron donor. [START_REF] Albero | Graphene supported NiO/Ni nanoparticles as efficient photocatalyst for gas phase CO2 reduction with hydrogen[END_REF] The results showed that CH 4 is formed in the presence of dimethylaniline even at higher rate than in the presence of H 2 as reagent, reaching a CH 4 production rate in 5 h of 250 µmol/g catalyst -1 compared to 180 µmol/g catalyst -1 when using H 2 as reagent. This behavior is consistent with the prevalent operation of a photocatalytic mechanism.

The occurrence of photoinduced charge separation and the reaction of this photoinduced charge separated state with electron donors was additionally confirmed by photocurrent measurements. A thin film of MIP-208@RuO x was further deposited on a transparent FTO electrode. Starting from a polarization potential of 1.6 V, the MIP-208@RuO x electrode was submitted to a decreasing bias potential in a single electrochemical cell using tetrapropylammonium tetrafluoroborate in acetonitrile as electrolyte. The current density clearly increased at each voltage upon illumination of the photoanode with the output of a Xe lamp (Figure S18). Furthermore, the presence of methanol as electron donor increases significantly the photocurrent, indicating that this reagent is acting as sacrificial electron donor, becoming oxidized and donating electrons to the MIP-208@RuO x in its excited state.

In summary, benefiting from a synergetic adjustment including both the preformed Ti 8 AF cluster rearrangement and the in situ linker formation that slows down the reaction, MIP-208, the first Ti-IPA MOF, was synthesized with high crystallinity and tunable scale. In its crystal structure, isostructural to CAU-10, constituted by cis-connected corner-sharing TiO 6 polyhedra extends along the c-axis resulting in a 1D helical chain inorganic building unit. Multivariate MIP-208 structures with tunable chemical environment and sizable porosity could be achieved by adopting the solid-solution mixed linker strategy. This material was proven to be the best photocatalyst for CO 2 methanation regarding activity and selectivity among all the pure MOFs, owing to its excellent stability and photoresponse. Further improvement of catalytic activity was achieved by photodeposition of RuO x . The resulting MIP-208@RuO x composite exhibits under the same conditions an efficiency enhancement about 20 times in comparison with that of the pure MOF. The composite catalyst displays a good stability and reusability. These results illustrate the continuous interest in the synthesis of novel Ti-MOFs that could lead to improved photocatalyst generations, suitable for the production of solar fuels and photoinduced CO 2 methanation.

generated 5 -

 5 photoresponse, reaching at 200 °C a productivity of 0.8 mmol g catalyst -1 in 22 h.

Figure 1 .

 1 Figure 1. Comparison of 1D Ti-O inorganic building units reported. a) Linear TiO chain in MIL-91. b) Linear Ti 6 O 9 nanowire in MIP-177-HT, ZSTU-1, 2 and 3. c) Linear TiO chain in DGIST-1. d) Helical TiO chain in MIP-208 reported here.

  to determine its crystal structure. The coordination environment of Ti 4+ ion and the overall framework connection could be solved precisely, despite that the accurate position of the acetamide group could not be located due to its disorder. Assisted by the solid-state NMR characterization, a formula of Ti(μ 2 -O)(5-Aa-IPA) could be concluded for MIP-208. It crystallizes in tetragonal I4 1 /amd space group (No. 141), associated with unit-cell parameters of a = b = 21.8190 Å, and c = 10.9380 Å. As shown in Figure 2a, each Ti 4+ ion is in an octahedral coordination geometry with four carboxylate oxygen atoms and two μ 2 -O oxo groups. The adjacent TiO 6 polyhedra adopt a cis-connected corner-sharing mode to connect each other, giving rise to a 1D infinite chain secondary building unit (SBU)

  Figure 3a). Several representative IPA type dicarboxylic acids, including IPA, 5-NO 2 -IPA, 5-Me-IPA, 5-Br-IPA, 5-tBu-IPA, 3,5-pyrazole-dicarboxylic acid (PDA) and camphoric acid, were selected as the substitution linkers, aiming to replace the 5-Aa-IPA spacer in the MIP-208 framework. All the tested linkers except for 5-NO 2 -IPA and camphoric acid could reach certain substitution ratios, ranging from 11 mol% to 50 mol% (FigureS5), as deduced from the NMR data. However, the reason for the unsuccessful inclusion of 5-NO 2 -IPA and camphoric acid is not clear yet and still under investigation.

Figure 3 .

 3 Figure 3. Illustration and characterizations of MTV-MIP-208 materials. (a) Illustration of MTV-MIP-208 structure with multiple functional groups (the blue ball stands for the acetamide group, the green ball stands for the other functional groups, and all hydrogen atoms are omitted for clarity) (b) PXRD patterns of the parent and MTV-MIP-208s (λ Cu ≈1.5406 Å). (c) CO 2 adsorption isotherms collected at 273 K for parent and MTV-MIP-208 solids.

Figure 4 .

 4 Figure 4. Results of water stability tests. (a) PXRD patterns and (b) CO 2 adsorption isotherms collected at 298 K for the MIP-208 samples before and after water treatments.

Figure 5 .

 5 Figure 5. (a) Temporal evolution of CH 4 during the photocatalytic CO 2 reduction with MIP-208@RuO x as the catalyst using a Xenon lamp or a solar simulator. b) Reusability of MIP-208@RuO x in the photocatalytic CO 2 methanation (seven times cycling) using a xenon lamp as irradiation source. Reaction conditions: catalyst (15 mg), H 2 (1.05 bar), CO 2 (0.25 bar), temperature (200 ºC), irradiation source a xenon lamp (1350 W×m -2 ) or solar simulator (1000 W×m -2 ).

ACKNOWLEDGEMENTS

CMC is grateful for financial support from the Institut Universitaire de France (IUF) and the Paris Ile-de-France Region -DIM "Respore". S.N. thanks financial support by the Fundación Ramón Areces (XVIII Concurso Nacional para la Adjudicación de Ayudas a la Investigación en Ciencias de la Vida y de la Materia, 2016), Ministerio de Ciencia, Innovación y Universidades RTI2018-099482-A-I00 project and Generalitat Valenciana grupos de investigación consolidables 2019 (AICO/2019/214) project. H.G. thanks financial support to the Spanish Ministry of Science and Innovation (Severo Ochoa and RTI2018-098237-CO2-1) and Generalitat Valenciana (Prometeo2017/083).

importance of TiO chains in MIP-208 structure over its analogues with the lower valence metal ions. In order to further improve the CH 4 formation rate, incorporation of metal NPs onto MIP-208 to form a composite catalyst was considered, since it is well-documented that photocatalytic methanation generally requires the presence of a metal or metal oxide with hydrogenation activity such as nickel, copper, rhodium, palladium, or ruthenium. [START_REF] Ulmer | Fundamentals and applications of photocatalytic CO2 methanation[END_REF] We initially selected RuO x NPs as the composite component, owing to the fact that Ru species generally show the highest activity in CO 2 methanation under milder conditions despite their limited selectivity compared with other active elements. 76; 77 In addition the oxide form instead of elemental metal NPs features elevated stability to facilitate easy handling during catalysis process. The MIP-208@RuO x composite was obtained following the post-synthetic photo-deposition of RuO x NPs on MIP-208 by photocatalytic reduction of water soluble KRuO 4 using methanol as sacrificial electron donor (Figure S8). The success of RuO x photodeposition by photocatalytic perruthenate reduction evidenced further the photoactivity of MIP-208 material, as it has been proven that an evidence supporting the occurrence of photoinduced charge separation upon irradiation of a solid is the observation of photodeposition onto the material of metal NPs from aqueous solutions. [START_REF] Wenderich | Methods, Mechanism, and Applications of Photodeposition in Photocatalysis: A Review[END_REF] After deposition, determination of the Ru content and average particle size in the composite was carried out. Energy-dispersive X-ray spectroscopy (EDX) analysis of MIP-208@RuO x showed a Ru content of 0.76 wt%. Field emission scanning electron microscopy (FESEM) images of MIP-208@RuO x displayed a similar crystallite morphology as that of the pristine MOF, without obvious free RuO x particles (Figure S9). Dark-field scanning transmission electron microscopy (DF-STEM) measurements revealed the presence of small RuO x NPs (1.4±0.14 nm). The homogeneous distribution of RuO x NPs all over MIP-208 was further confirmed by DF-STEM coupled to an EDX detector (Figure S10). Preservation of the crystalline structure of MIP-208 after RuO x photodeposition was confirmed by PXRD data (Figure S11). The lack of additional peaks in the composite

EXPERIMENTAL PROCEDURES

Synthesis of MIP-208 (small scale). To a 23 mL Teflon reactor, Ti 8 AF cluster solid (220 mg) was added followed by adding acetic anhydride (5 mL) and acetic acid (5 mL). The mixture was stirred at room temperature for 20 minutes before 5-NH 2 -IPA (362 mg, 2 mmol) was added once. After stirring for 10 minutes, MeOH (0.5 mL) was added and the overall mixture was stirred at room temperature for another 10 minutes. Afterwards, the reaction was heated in an oven at 180 °C for 48 hours. When the reaction was cooled to room temperature, the dark yellow/light brown product was collected by filtration, washed with acetone for air dry.

Yield: 330 mg (average of five parallel reactions).

Synthesis of MIP-208 (scale-up).

To a 125 mL Teflon reactor, Ti 8 AF cluster solid (3.5 g) was added followed by adding acetic anhydride (25 mL) and acetic acid (25 mL). The mixture was stirred at room temperature for 20 minutes before 5-NH 2 -IPA (3.62 g, 20 mmol) was added once. After stirring for 10 minutes, MeOH (3.5 mL) was added and the overall mixture was stirred at room temperature for another 10 minutes. Afterwards, the reaction was heated in an oven at 180 °C for 48 hours. When the reaction was cooled to room temperature, the dark yellow/light brown product was collected by filtration, washed with acetone for air dry. Yield:

g (average of five parallel reactions).

Typical synthesis of MTV-MIP-208. To a 23 mL Teflon reactor, Ti 8 AF cluster solid (220 mg) was added followed by adding acetic anhydride (5 mL) and acetic acid (5 mL). The mixture was stirred at room temperature for 20 minutes before a linker mixture of 5-NH 2 -IPA (1 mmol) and the secondary IPA ligand (1 mmol) was added. After stirring for 10 minutes, MeOH (0.5 mL) was added and the overall mixture was stirred at room temperature for another 10 minutes. Afterwards, the reaction was heated in an oven at 180 °C for 48 hours. When the reaction was cooled to room temperature, the dark yellow/light brown product was collected by filtration, washed with acetone for air dry.

Photocatalytic methanation tests.

The photocatalytic CO 2 methanation experiments were carried out using a quartz photoreactor (51 mL) equipped with a heating mantle to control the desired temperature. In a typical experiment the powder photocatalyst (15 mg) is placed as a bed in the reactor, and then the system purged first with H 2 and later with CO 2 until a ratio of 4 to 1 is obtained. Subsequently, the photoreactor is heated at 200 ºC, and then the photocatalyst was irradiated using a Xe lamp (300 W) or a solar simulator. At the desired reaction time an aliquot was sampled from the photoreactor and analyzed in an Agilent 490 MicroGC equipped with two channels and thermal conductivity detectors. One channel equipped with a MolSieve 5Å column allows analysis of H 2 , O 2 , N 2 and CO, while the other channel equipped with a Pore Plot Q column allows determining CO 2 , CH 4 and short chain hydrocarbons. Quantification was performed using calibration plots employing commercially available gas mixtures.

X-ray crystallographic

DECLARATION OF INTERESTS

The authors declare no competing interests.


Pascal Ratet
email: pascal.ratet@cnrs.fr

Symbiosis signaling: Solanaceae symbiotic LCO receptors are functional for rhizobium perception in legumes

A new study shows that plant receptor genes necessary for the ancient and widespread symbiosis with arbuscular mycorrhizal fungi were co-opted in legume plants without modifications to establish the evolutionary more recent and more specific symbiosis with their bacterial rhizobium partners.

Text (1117 words):

Some plants are able to grow in nitrogen poor soils because they can establish symbiosis with rhizobia or frankia (actinobacteria) bacteria able to reduce the non-limiting atmospheric gaseous nitrogen (N 2 ) to NH 3 to the benefit of their host. The capacity to establish this symbiosis also called root nodule symbiosis (RNS) is confined to the Rosid I clade including actinorhizal plants, Parasponia and legumes [START_REF] Soltis | Chloroplast gene sequence data suggest a single origin of the predisposition for symbiotic nitrogen fixation in angiosperms[END_REF]. The RNS was extensively studied using the legume-rhizobia symbiosis and generally requires a specific signaling between both organisms with excretion of flavonoids in the rhizosphere by the plant and production of specific lipo-chitooligosaccharides (LCO or Nod factors for the RNS) by the rhizobia that are recognized by plant specific receptors belonging to the LysM kinase receptor family (LysM-RLK; 2). Genetic studies have highlighted the central role of the Nod factor dependent signaling in most of the legume rhizobia interactions even if variations around this theme exist [START_REF] Fabre | Nod Factor-independent nodulation in Aeschynomene evenia required the common plant-microbe symbiotic toolkit[END_REF]. RNS requires organogenesis and lead to the formation of the nodule organ (generally on the root), hosting the symbiotic bacteria and in which nitrogen reduction takes place. Actinorhizal nodules present a single central vascular bundle that makes them resembling modified lateral roots [START_REF] Franche | Actinorhizal symbioses: Recent advances in plant molecular and genetic transformation studies[END_REF][START_REF] Froussart | Recent advances in actinorhizal symbiosis signaling[END_REF]. Legume nodules are divided in two main types depending on the presence or absence of an apical meristem (indeterminate versus determinate nodules) and have peripheral vascular tissues surrounding the central infected tissues.

Genetic studies have suggested that Legume-rhizobia symbiotic signaling evolved from AM symbiosis (6) because some legume mutants are affected in both symbiosis and mycorrhizal fungi are able to produce Nod factor-like (LCO) molecules. It is the mechanism that allowed AM symbiotic genes, and more precisely LCO receptors, to be used for the RNS in legumes that is address in this issue by Girardin et al., [START_REF] Girardin | LCO receptors involved in arbuscular mycorrhiza are functional for rhizobia perception in legumes[END_REF].

Genes that are common to both symbioses define the common symbiosis signaling pathway (CSSP) [START_REF] Parniske | Arbuscular mycorrhiza: the mother of plant root endosymbioses[END_REF]. However it is now evident that AM symbiosis requires a more complex signaling than the RNS one, because AM fungi produce mixtures of chitooligosaccharides (CO) and LCO molecules [START_REF] Maillet | Fungal lipochitooligosaccharide symbiotic signals in arbuscular mycorrhiza[END_REF][START_REF] Genre | Short-chain chitin oligomers from arbuscular mycorrhizal fungi trigger nuclear Ca2+ spiking in Medicago truncatula roots and their production is enhanced by strigolactone[END_REF] perceived by probably different LysM-RLK. This suggests that nodule forming plants have recruited only part of the AM signaling pathway to develop RNS. In agreement with these genetic approaches, phylogenetic studies suggest that RNS has evolved only once, around 100 million years ago (MYA) using components of the AM symbiotic signaling which itself appeared 400 MYA (10 -12). However, how these AM symbiotic genes and more precisely how the LysM-RLKs were recruited in legume plants to establish the RNS remained unknown. In Legume plants the Nod factor receptors (NFP, Lyk3 in Medicago truncatula and NFR1, NFR5 in Lotus japonicus) are expressed in root epidermis and in the nodule and are thought to recognize specifically the cognate Nod factor. NFP and NFR5 belong to the LysM-RLK LYRIA clade and mutants in these genes are unable to establish RNS but are not affected for AM symbiosis. Paralogs of these genes are expressed in arbuscule containing cells of the root suggesting a role in the fungal symbiosis [START_REF] Gomez | Medicago truncatula and Glomus intraradices gene expression in cortical cells harboring arbuscules in the arbuscular mycorrhizal symbiosis[END_REF][START_REF] Rasmussen | Intraradical colonization by arbuscular mycorrhizal fungi triggers induction of a lipochitooligosaccharide receptor[END_REF]. The presence of two LCO receptors in legume plants suggests a gene duplication in the legume ancestors (or nodule forming plant ancestor) followed by a neo-functionalization of one of them for the RNS. With this hypothesis the receptor protein should have acquired the specificity for the cognate Nod factor perception and the promoter region should have evolved to specifically express in epidermis, in root hairs and during nodule organogenesis.

In contrast to this hypothesis, Girardin et al., [START_REF] Girardin | LCO receptors involved in arbuscular mycorrhiza are functional for rhizobia perception in legumes[END_REF] elegantly demonstrate that the promoter and the proteins encoded by the LYRIA receptor genes necessary for arbuscular mycorrhiza recognition in Solonaceae plants (AM symbiosis) are functional for the rhizobium recognition in legume and thus were recruited without modification during evolution for the acquisition of symbiosis with nitrogen fixing bacteria (RNS). The key experiments of this study are described below.

The tomato and petunia SlLYK10 and PhLYK10 genes are unique LYRIA genes (homolog to NFP) and the corresponding sllyk10 and phlyk10 mutants are required for AM symbiosis establishment. Furthermore specific LCOs binding by membrane fractions containing these receptors and the high affinity (Kd in the range of 20 nM) to LCOs compared to COs (Kd higher than 1 M) demonstrate that the Solanaceae receptors are genuine LCO receptors with a LCO affinity comparable to the legume ones [START_REF] Broghammer | Legume receptors perceive the rhizobial lipochitin oligosaccharide signal molecules by direct binding[END_REF].

In order to know if the promoters of these Solanaceae genes are still functional in legume plants, the authors first show that promoter::GUS (proSlLYK10 and proPhLYK10) constructs are specifically up-regulated at early stages of arbuscule development in root cortical cells. Secondly they expressed these constructs in legume plants together with the proNFP::GUS construct and showed that the three constructs are similarly expressed. This demonstrates that the AM symbiotic Solanaceae promoters contain all the information (cisregulatory elements) required for expression in legume nodules. Furthermore the authors expressed the NFP coding sequence from either the NFP or SlLYK10 promoters and could complement the nfp mutant with this construct. This confirms that the cis-regulatory elements from the Solanaceae AM symbiotic promoters are sufficient to express the receptor in the nodules.

The authors also searched for and identified a common cis-regulatory element in the LYRIA promoter region from dicotyledonous plant species including nodule-forming and non-nodule forming plants. A minimal SiLYK10 promoter retaining this sequence is expressed in young nodules suggesting that the recruitment of LYRIA genes for RNS did not require modification of the cis-regulatory sequences in the symbiotic promoters. Lastly the authors expressed the SlSILK10 and PhSILK10 coding sequences under strong promoters in the legume nfp or nfr5 mutants and were able to complement the mutations showing that the Solanaceae proteins have the capacity to support RNS.

This work provides clear genetic evidence via the characterization of petunia and tomato mutants that members of the LysM-RLK (LYRIA phylogenetic group) promote efficient AMF root penetration and are required for proper arbuscule development. In addition it allows proposing an evolutionary scenario in which an ancestral LYRIA gene involved in AM establishment was recruited directly for a role in nodule organogenesis and rhizobial colonization without a need for cis-regulatory elements and protein sequence modification.

In addition, these results raise two hypotheses for the biological role of these proteins.

First, because they participate to intracellular accommodation in the two symbioses, the role of these LYRIA proteins in plants might be to host microbial partners intracellularly.

Secondly the complementation of the nfp and nfr5 mutants by the tomato and petunia proteins raises the question of the specificity of Nod factor recognition by the legume proteins. We can thus suppose that the Solanacea proteins are recognizing specifically the general LCO structure but not the CO molecules as shown in this work. However they probably do not participate to the specific recognition of the LCO decoration, that are responsible for the rhizobium strain specificity. Other LysM-RLK like Lyk3 (M. truncatula) or NFR1 (L. japonicus) or NF hydrolases ( 16) might be responsible for this specificity.