Ahmed Khalifa

Olivier Kermorgant

Salvador Dominguez

Philippe Martinet

An Observer-based Longitudinal Control of Car-like Vehicles Platoon Navigating in an Urban Environment

In this paper, we study longitudinal motion control of car-like vehicles platoon navigating in an urban environment with minimum communication links. To achieve a higher traffic flow, a constant-spacing policy between successive vehicles is commonly used but this is at a cost of an increased number of communication links as any vehicle information must broadcast to all its followers. Therefore, we propose a distributed observerbased control law that depends both on communicated and measured information. Our formulation allows designing the control law directly in the curvilinear coordinates. Internal and string stability analysis are conducted. We provide simulation results, through dynamic vehicular mobility simulator, to illustrate the feasibility of the proposed approach and corroborate our theoretical findings.

I. INTRODUCTION

Car sharing systems have attracted attention in recent years to overcome the urban transportation problems. However, cars redistribution among all stations is one of the main issues. One of the proposed solutions is to use vehicle platoons guided by professional drivers. A leader vehicle, driven by a human, comes to pick up the vehicles, then joins them to a platoon system, and finally drops off the vehicles over the stations so that the vehicles are well distributed.

In urban environments, the platoon motion has to be represented with respect to a path that the vehicle intends to follow. The curvilinear coordinates are usually used instead of absolute ones to represent the vehicle motion [START_REF] Barfoot | Motion planning for formations of mobile robots[END_REF].

The global architecture of platoon control systems consists of two modules: longitudinal and lateral controllers. The former has two sub-modules including Upper-Level Controller (ULC), which provides the desired vehicle linear acceleration, and Lower-Level Controller (LLC), which generates the required engine throttle/brake to achieve this desired acceleration. Similarly, the lateral controller generates the required steering angle to achieve the desired lateral states. In this research, we are interested in designing the longitudinal ULC.

The framework of the longitudinal platoon control consists of four components, including Vehicle Dynamics, Information Flow Topology (IFT), Distributed Controller, and Spacing Policy [START_REF] Li | An overview of vehicular platoon control under the four-component framework[END_REF].

For the vehicle dynamics, different models are used in the literature, including linear and nonlinear models [START_REF] Yue | Nonlinear platoon control of arduino cars with range-limited sensors[END_REF]. Three linear models are frequently used including single integrator model [START_REF] Lin | Optimal control of vehicular formations with nearest neighbor interactions[END_REF], double integrator model [START_REF] Lemos | Local and global path generation for autonomous vehicles using splines[END_REF], third order model [START_REF] Zegers | Consensus control for vehicular platooning with velocity constraints[END_REF]. Despite the huge amount of relevant literature to date, few studies in the literature have handled the problem of platooning in the urban environment/path coordinates, e.g. [START_REF] Avanzini | A control strategy taking advantage of inter-vehicle communication for platooning navigation in urban environment[END_REF]. In these studies, the authors assume a first-order model for the vehicle dynamics which has many defects. In our previous work [START_REF] Khalifa | Vehicles platooning in urban environment: Consensus-based longitudinal control with limited communications capabilities[END_REF], [START_REF]Vehicles platooning in urban environments: Integrated consensus-based longitudinal control with gap closure maneuvering and collision avoidance capabilities[END_REF], a second order longitudinal dynamic model is used and we have shown that one can design a controller to achieve control objectives in the path (operational) coordinates explicitly.

Three main platoon control objectives have to be achieved. First, the platoon can navigate in an urban environment (curvilinear path). Second, achieving position and velocity tracking for follower vehicles. Third, guarantee of the string stability, i.e., the error signals will attenuate when propagating downstream the vehicle string. The string instability may result in a rear-end collision between any predecessor/follower vehicles. In the literature, there are six main approaches in designing a platoon controller, including Linear [START_REF] Chehardoli | Third-order leaderfollowing consensus protocol of traffic flow formed by cooperative vehicular platoons by considering time delay: constant spacing strategy[END_REF], Optimal [START_REF] Fardad | Sparsity-promoting optimal control for a class of distributed systems[END_REF], H ∞ -based [START_REF] Gao | Robust control of heterogeneous vehicular platoon with uncertain dynamics and communication delay[END_REF], Sliding Mode [START_REF] Wu | Distributed sliding mode control for multi-vehicle systems with positive definite topologies[END_REF], Model predictive [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF], and Consensus-based controller [START_REF] Zegers | Consensus control for vehicular platooning with velocity constraints[END_REF]. In [START_REF] Khalifa | Vehicles platooning in urban environment: Consensus-based longitudinal control with limited communications capabilities[END_REF], [START_REF]Vehicles platooning in urban environments: Integrated consensus-based longitudinal control with gap closure maneuvering and collision avoidance capabilities[END_REF], we present a consensus-based control algorithm that considers the motion of variable velocity platoon.

Several spacing policies are provided in the literature [START_REF] Swaroop | A comparision of spacing and headway control laws for automatically controlled vehicles1[END_REF]. The constant distance spacing policy is used commonly due to its capability to achieve very high traffic capacity. However, to achieve string stability, more communication links are required to transmit leading vehicle information to all the following vehicles in the platoon.

The Information Flow Topology (IFT) represents the inter-vehicle communication topology that the vehicles use to acquire the information from its surrounding vehicles. The Predecessor-Leader Following (PLF) topology [START_REF] Swaroop | A comparision of spacing and headway control laws for automatically controlled vehicles1[END_REF] is commonly used in the literature in which the leader communicates with all the vehicles in broadcast, and every vehicle also considers information from its predecessor to compute the control action. However, for either a low-cost onboard communication module that has limited bandwidth capabilities or for robust communication, high-speed twochannel communication is not always feasible or preferred. In this study, the topology assumption is called Hybrid PLF, in which only the leader broadcasts its information to all the vehicles. On the opposite, position or velocity information of a predecessor is not communicated but rather measured by its direct follower through a range sensor, e.g. laser. Therefore, we propose an observer-based control technique. A distributed longitudinal controller designed in the path coordinates is the basis of the proposed control strategy and its role is to achieve position and velocity consensus among the platoon members. An observer is then introduced in the loop to estimate the predecessor velocity. The merit of using the predecessor velocity in the control is twofold. It provides a faster convergence of the platoon with safety, and also results in a smoother acceleration of the motion of the vehicle.

In the literature, several observer-based consensus control algorithms are introduced for multi-agent systems in [START_REF] Wen | Observer-based consensus of second-order multi-agent systems without velocity measurements[END_REF]- [START_REF] Ding | Observer-based eventtriggering consensus control for multiagent systems with lossy sensors and cyber-attacks[END_REF] and for platoon systems in [START_REF] Yan | Consensus based platoon algorithm for velocity-measurement-absent vehicles with actuator saturation[END_REF]. However, these works use the observer to estimate the velocity of either the ego agent or the other vehicles by using the communication links among the agents. Moreover, they do not consider the string stability constraint during the controller design and analysis. In our study, the observer does not require any communication link with other vehicles/agents in the platoon which adds complexity in the design and analysis in order to guarantee both the platoon internal and string stability.

The main contributions of this paper are as follows:

• A distributed observer-based control algorithm is designed without the need to communicate with the preceding vehicle. • The proposed algorithm can achieve higher traffic flow (by using constant spacing policy) with a minimum number of required communication links (by estimating the predecessor velocity with an observer). • Conditions for both internal and string stability are provided.

• The proposed framework is validated via realistic simulation studies which are conducted in dynamic vehicular mobility simulator that considers real vehicle dynamics and platooning staff in an urban environment.

The rest of this paper is organized as follows. Section II formulates the platooning problem with kinematic and dynamic models expressed in both body and path coordinates. In Section III, the proposed control law is presented, and both internal and string stability are analyzed. Simulation studies are given in Section IV. Conclusions are made in Section V.

II. PROBLEM STATEMENT

A. Vehicle Kinematic Model

The kinematic model of a car-like vehicle can be simplified to that of a bicycle model [START_REF] Khalifa | Vehicles platooning in urban environment: Consensus-based longitudinal control with limited communications capabilities[END_REF], see Fig. 1.

1) Cartesian Coordinates: Let {V }, O v -x v y v , represents the vehicle body-fixed reference frame with its x-axis in the vehicle's forward direction and its origin at the center of the rear axle, O v . The configuration of the vehicle, with respect to the world-fixed inertial reference frame, {G}, O -xy, is represented by the generalized coordinates [x, y, θ] T ∈ R 3 , where its position is given by [x, y] T and its orientation in the global frame is represented by θ. δ is the steering angle in the body frame. Since the wheels cannot slip sideways, the vehicle velocity is v in the vehicle x-direction, and zero in the y-direction. Consider the distance L from the rear wheel along the orientation of the vehicle to the front wheel. Then the kinematic model is given by

ẋ = v cos(θ), (1a) ẏ = v sin(θ), (1b) θ = v L tan(δ) (1c)
2) Path Coordinates Model: Let [s, r, ψ] T represents the curvilinear coordinates. The tracking path, C, defined in the Global Frame, can be represented as a function of several items including path length s (curvilinear abscissa) at the closest point M to O v , the angular deviation, ψ = θ -θ c , where θ c is the angle between the path tangent at M and the x-axis, and the lateral deviation, r, which is the signed orthogonal distance from the center of the rear axle, O v , to the closest point on the path, M . Then, the kinematic model in the path coordinates is represented by

ṡ = v cos(ψ) 1 -rκ(s) , (2a)
ṙ = v sin(ψ), (2b)
ψ = v tan(δ) L - κ(s) cos(ψ) 1 -rκ(s) (2c)
where κ(s) is the path curvature at M .

B. Car-like Vehicle Longitudinal Dynamic Model

The vehicle longitudinal dynamic behavior is determined by several factors including the engine, drive line, brake system, aerodynamics drag, tire friction, gravitational force, etc [START_REF] Wu | Distributed sliding mode control for multi-vehicle systems with positive definite topologies[END_REF]. For the vehicle model, we assume: Assumption 1. The rigidity and symmetry of vehicle body.

Assumption 2. The integration of driving and braking inputs into one control input. Assumption 3. The pitch and yaw motions effects are neglected.

The general longitudinal dynamic equation is given by

M v a + C v v + G v = F v (3)
where The longitudinal LLC being an inverse model compensation technique, it follows

a = v, M v , C v , G v
F v = M v µ + C v v + G v ( 4
)
where µ is theinput signal after system linearization which has to be designed. By applying this LLC, the second order dynamic model can be written in the body coordinates as

a = µ (5)
When platooning in an urban environment, the relation between the dynamics in the path and Cartesian coordinates will be derived such that the controller can be designed in the path coordinates. The block diagram of longitudinal controller with the dynamic model is given in Fig. 2.

To find a relation between the control signals in the path and body coordinates, let's recast (2a) as

ṡ = Jv ( 6
)
where J is given by

J = cos(ψ) (1 -rκ) . (7)
The acceleration in path coordinates, η, is thus given as

η = Ja + Jv (8)
where η = q = s and q = ṡ are the vehicle acceleration and velocity in the path coordinates respectively. Consequently, if the mapping from the control signal (i.e., desired acceleration) in the path coordinates, u, to that in the body coordinates, µ, is given by

µ = 1 J u -Jv , (9)
then, the vehicle longitudinal dynamic model in the path coordinates is given by

ṡ = q, (10a) q = η = u. (10b)

C. Platoon Longitudinal Model

Consider a platoon of N + 1 vehicles, as illustrated in Fig. 3, including a leader (with index 0) and N followers (with index i; i = 1, 2, . . . , N ). The vehicles can share information by either sensor-based or communication-based link. It is known that the vehicles involved in the car sharing system are often with the same type of small passenger vehicles, or have very close dynamics, hence we can assume Assumption 4. All the vehicles have equal dynamics. The in-platoon vehicle dynamics in the path coordinates can be given by

ṡi = q i , (11a) qi = η i = u i , (11b)
where the transformation between the two coordinates can be implemented using [START_REF]Vehicles platooning in urban environments: Integrated consensus-based longitudinal control with gap closure maneuvering and collision avoidance capabilities[END_REF]. For the vehicles string, the inter-vehicle distance, d i = s i-1 -s i , is the curvilinear distance between vehicle i and its predecessor i-1, d i0 = s 0 -s i , is the curvilinear distance between vehicle i and leader, d r,i is the desired inter-vehicle distance between vehicles i and i-1, and d r,i0 is the desired inter-vehicle distance between vehicles i and 0. For high traffic capacity, the constant spacing policy method is used, i.e. d r,i is constant.

D. Control Objectives

Under the hybrid IFT given in Fig. 4, the platoon is required to move in an urban environment (curvilinear path) and to track a variable velocity leader. Each follower i can obtain leader information (i.e., position, velocity, and acceleration s 0 , q 0 , η 0 ) and predecessor position information (i.e., s i-1 ). Under these circumstances, ∀ i = 1, . . . , N , the following goals have to be achieved:

• estimation of the preceding vehicle velocity to be used in the distributed control law; • convergence of the velocity estimation, position and velocity tracking errors; • string stability.

III. CONTROL DESIGN

This section starts by presenting the proposed observerbased control algorithm. After that, the platoon internal stability analysis is presented. Finally, the string stability is analyzed.

A. Observer-based Control Law

The information flow structure among the platoon can be modeled by a graph in which the node dynamics are described by [START_REF] Chehardoli | Third-order leaderfollowing consensus protocol of traffic flow formed by cooperative vehicular platoons by considering time delay: constant spacing strategy[END_REF]. The graph topology is described by an adjacency matrix M = [m ij ] N ×N , in which the element m ij = 1 if vehicle i can obtain information from vehicle j; otherwise, m ij = 0. To include the leader (with index 0) in the network, an augmented graph is used. The Laplacian matrix L = [l ij ] ∈ R N ×N associated with the graph is given by

l ij = -m ij if i = j, and l ij = N k=1 m ik if i = j. Remark 1.
For the Predecessor -Leader Following (PLF) information flow topology, the elements of the adjacency matrix is given as

m ij = 1 if j = i -1, and m ij = 0 otherwise.
Let us consider a full PLF-based control law given as [START_REF] Fardad | Sparsity-promoting optimal control for a class of distributed systems[END_REF] u i (t) = η 0 + g c,2 e q,i0 (t) + g c,1 e s,i0 (t) + g o,1 e s,i (t) + g o,2 e q,i (t),

where e s,i0 (t) = s 0 (t)-s i (t)-d r,i0 , e q,i0 (t) = q 0 (t)-q i (t), e s,i (t) = s j (t)-s i (t)-d r,i , and e q,i (t) = q j (t)-q i (t).

G c = [g c,1 , g c,2 ] T , G o = [g o,1 , g o,2
] T are controller parameters to be designed. Remark 2. In the proposed hybrid PLF, the information required to apply this control law ( 12) is available except the velocity of the predecessor or the velocity error between the ego vehicle i and the predecessor velocity, e q,i , which has to be estimated using an observer.

For observer design, let us define two states z 1,i = e s,i which is known (i.e., measured by the sensor mounted on vehicle i), and z 2,i = e q,i which is unknown and we need to estimate. The state equations that relates these two states are given by ż1,i = z 2,i , ż2,i = ėq,i .

The state z 2,i can be estimated using the following observer

ż1,i = ẑ2,i + h 1 (z 1,i -ẑ1,i ), ż2,i = h 2 (z 1,i -ẑ1,i ). ( 14
)
Including the observer output in the control law [START_REF] Fardad | Sparsity-promoting optimal control for a class of distributed systems[END_REF], then the i th observer-based control law is proposed as [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF] u i (t) = η 0 + g c,2 e q,i0 (t) + g c,1 e s,i0 (t) + g o,1 ẑ1,i (t) + g o,2 ẑ2,i (t).

Let us define the error state vector of vehicle i as

f i = [f 1,i , f 2,i ] T with f 1,i = e s,i0
and f 2,i = e q,i0 . Then, the i th closed-loop state space model is given by

ḟi (t) = Af i (t) + B i (t), (16)
where

A = 0 1 0 0 , B = 0 1 , and (17)
i = -g c,2 e q,i0 -g c,1 e s,i0 -g o,1 ẑ1,i -g o,2 ẑ2,i .

Exploiting that z 1,i = e s,i = e s,i0 -e s,j0 and let ẑi = [ẑ 1,i , ẑ2,i ] T , then ( 14) can be rewritten concisely as

żi = A -HCẑ i + HC f 1,i -f 1,j (18)
where

H = [h 1 , h 2 ] T and C = [1, 0, 0]. Defining G c = 1 2 KQ 1 , G o = 1 2 KQ 2 , K = [k 1 , k 2 ],
and [START_REF] Wen | Observer-based consensus of second-order multi-agent systems without velocity measurements[END_REF] can be recast as

Q 1 and Q 2 ∈ R 2×2 , then
(19) i (t) = - 1 2 KQ 1 f i (t) - 1 2 KQ 2 ẑi (t), Let f = [f T 1 , . . . , f T N ] T , ẑ = [ẑ T 1 , . . . , ẑT N ]
T , then the system collective dynamics are given after some manipulations by:

(20) ḟ = I N ⊗ A -I N ⊗ 1 2 BKQ 1 f -I N ⊗ 1 2 BKQ 2 z ż = I N ⊗ (A -HC) ẑ + D ⊗ HC f (21)
where I N denotes (N × N ) identity matrix, D = L + diag{1, 0, 0}, and the operator ⊗ denotes Kronecker product.

B. Internal Stability

The closed loop system dynamics represented in ( 20) and ( 21) can be recast in the form

Ẋ = A c X (22)
where X = [f T , z T ] T , and

A c = I N ⊗ A -1 2 BKQ 1 -I N ⊗ 1 2 BKQ 2 D ⊗ HC I N ⊗ (A -HC) (23)
For the internal stability of the proposed platoon observerbased control system let us present the following theorem.

Theorem III.1. Consider the distributed control of a platoon of connected car-like vehicles with dynamics defined in (2), ( 3) -(9), and (11), with assumptions 1 -4, and the observerbased control algorithm proposed in ( 14) and ( 15) is applied to it. Then, the system is asymptotically stable as long as the following conditions are met.

1) Find Γ ∈ R 2×2 to satisfy (A -HC)Γ -Γ(A -BK) = -λ i HC (24)
where λ i is the i th eigenvalue of the matrix D.

2) The gain matrices Q 1 and Q 2 must satisfy

1 2 Q 1 , 1 2 Q 2 = I 2 Γ -1 (25)
3) Constraints on the gain vector K are

k 1 > 0 and k 2 > 0 (26) 4) λ max 1 2 ΓBKQ 2 < λ min -(A -HC) (27)
Proof. For the PLF information flow topology, D is an upper-triangular matrix and its eigenvalues, λ i , are real.

Therefore, for vehicle i, the matrix A c can be represented as

Āc = A -1 2 BKQ 1 -1 2 BKQ 2 λ i HC A -HC (
Q Āc Q-1 = A -BK -1 2 BKQ 2 0 A -HC + 1 2 ΓBKQ 2 (29) Therefore, eig Āc = eig {A -BK} ∪ eig A -HC + 1 2 ΓBKQ 2 ,
where eig{M} denotes the eigenvalues of M. To ensure (A -HC + 1 2 ΓBKQ 2 ) < 0, condition (27) must be satisfied. Moreover, the characteristic equation of the matrix (A -BK) can be expressed as

det SI 2 -(A -BK) = S 2 + k 2 S + k 1 ( 30
)
According to Routh-Hurwitz stability criterion [START_REF] Dorf | Modern control systems[END_REF], this polynomial (30) is Hurwitz stable under condition (26). This implies that all eigenvalues of Āc and consequently A c have negative real parts.

Remark 3. Adding observer to the control loop increases the complexity of the string stability analysis. Therefore, the next corollary is proposed in order to reduce the number of design parameters which in turn simplifies both the observer-based control design and string stability analysis.

Corollary III.2. The proposed platoon observer-based control system is asymptotically stable provided that the design parameters are selected as

k 1 = p 2 c , (31a) k 2 = 2p c , (31b) h 1 = 2p o , (31c) h 2 = p 2 o , (31d) p o = γp c , ∀γ, p c , and p o > 0, (31e) γ > 1 2 and γ = 1, (31f)
where the desired eigenvalues of controller part (A -BK) are assigned at -p c , and that of the observer part, (A -HC), are placed at -p o . γ is a constant that determines the distance between the poles of the controller and the observer.

Proof. By choosing K and H as given in conditions (31a) -(31d), all the poles of (A -BK) and (A -HC) are placed at -p c and -p o respectively. Hence, one can choose the eigenvalues to be different by satisfying (31f). Thus, solution of (24), Γ, exists, and consequently, condition (25) is met. Checking stability conditions of (30), and setting the value of k 1 and k 2 , as given in (31a) -(31b), we can satisfy condition (26).

The design of the algorithm parameters as in (31a) -(31e) has another merit that is the number of the design parameters are reduced to γ and p c . Substitution (31a) -(31e) into (27), after some algebraic manipulations, the roots of A -HC + 1 2 ΓBKQ 2 are

1 2 -r ± 1 2 √
4r -5 which are negative definite under a condition for γ given in (31f).

C. String Stability

In addition to internal stability, the platoon controller has to guarantee string stability; otherwise, the error signals will amplify when propagating downstream the vehicle string, which may result in rear-end collision.

Definition III.3. L 2 String Stability means that the energy (represented by the L 2 norm) of the position error of vehicle i, e s,i , is smaller than the energy of the position error of proceeding vehicle j = i -1, e s,j [START_REF] Eyre | A simplified framework for string stability analysis of automated vehicles[END_REF]. Since

Ḡ(S) = Ei(S) Ej (S) ∞ = sup es,j ∈L2 es,i 2 es,j 2
, the string stability

is guaranteed if Ḡ(S) ∞ < 1. (32)
For simplicity of notation and analysis, let us define the following errors e i (t) = s j (t) -s i (t) -d r,i , ėi (t) = q j (t)q i (t), e i0 (t) = s 0 (t)-s i (t)-d r,i0 , and ėi0 (t) = q 0 (t)-q i (t).

Theorem III.4. Consider the platoon system presented in Theorem III.1 and the observer-based control algorithm proposed in [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF] and its parameterization given in Corollary III.2. Then, the platoon is string stable as long as the conditions (31a) -(31e) are satisfied and the choice of factor γ is constrained by

γ ≥ 9 12 . (33)
Proof. Let us start with

ëi = η j -η i . (34)
Substituting ( 11) and ( 15) into (34), considering the relations e i = e i0 -e j0 and ėi = ėi0 -ėj0 , and after some algebraic manipulations, we get

ëi = -g c,2 ėi -g c,1 e i -g o,1 ẑ1,i -g o,2 ẑ2,i +g o,1 ẑ1,j +g o,2 ẑ2,j . (35)
The observer model given [START_REF] Gao | Distributed reduced-order observer-based approach to consensus problems for linear multi-agent systems[END_REF] can be represented in the form of transfer functions as

Ḡz1 (S) = Ẑ1,i (S) E j (S) = h 1 S + h 2 S 2 + h 1 S + h 2 , Ḡz2 (S) = Ẑ2,i (S) E j (S) = h 2 S S 2 + h 1 S + h 2 . ( 36
)
Writing (35) in the S-domain along with (36) yields:

Ḡ(S) = g o,1 Ḡz1 + g o,2 Ḡz2 S 2 + g c,2 S + g c,1 + g o,1 Ḡz1 + g o,2 Ḡz2 . (37)
As Ḡ(S) ∞ = sup w>0 Ḡ(jw) 2 = sup w>0 Ḡn(w) Ḡd (w) , string stability criterion (32) is equivalent to Ḡd (w) -Ḡn (w) > 0, ∀w > 0.

(38)

Exploiting the definitions of G c and G o along with (31a) -(31e), after certain algebraic manipulations, (38) is recast as (39) Ḡd (w) -Ḡn (w) = 4w 8 + a 6 w 6 + a 4 w 4 + a 2 w 2 + a 0 , where coefficients a (.) are functions of γ and p c :

a 6 = 8γ 2 p 2 c , (40a)
a 4 = -p 4 c (-4γ 4 + 16γ 3 -44γ 2 + 24γ + 3), (40b) a 2 = 2p 6 c (2γ 4 -4γ 3 + 6γ 2 -3), (40c) a 0 = γ 2 p 8 c (3γ 2 -γ + 6). (40d)
After some algebraic and numeric manipulations, under condition (33), we find that all the coefficients a 6 -a 0 are positive definite independently from the value of p c , and hence, the string stability constraint (38) is satisfied.

IV. RESULTS AND DISCUSSION

A. Study Setup

The proposed control strategy is tested by implementing the vehicles platoon in a custom simulator called ICARS that is developed under the Robot Operating System (ROS). ICARS has features such as real vehicle dynamics and models, both low and high level measurement units with noises, etc. The platoon is tested in a virtual map of Centrale Nantes (ECN) campus as an example of urban environment (road with curvatures, bumps, humans, etc).

We consider a platoon composed of N follower vehicles (of type Renault ZOE) plus a manually driven leader (of type Renault FLUENCE). The physical parameters of the vehicles are given in Table I.

The vehicles are initially parked. While the leader moves to the destination points, its path is built online and sent via the radio communication module to the followers as the reference path to follow. The radio system receives the vehicle information in the platoon one by one.

The lateral controller presented in [START_REF] Thuilot | Accurate automatic guidance of an urban electric vehicle relying on a kinematic gps sensor[END_REF] is in charge of producing the steering angle of the vehicle to track the reference path of the leader (by minimizing both the lateral and angular deviations, r i and ψ i , respectively).

The longitudinal controller is implemented at a rate of 100 Hz. The control algorithm parameters are selected to guarantee the internal stability and string stability given in Theorem III.1, Corollary III.2 and Theorem III.4. Table II presented the control parameters. For safety and comfortability, the vehicle velocity and acceleration are constrained to certain levels as given in Table II.

B. Simulation Results

Three Renault ZOE are used as followers, i.e., N = 3. Lateral motion is illustrated in Fig. 5a -Fig. 5c. The reference and actual paths of the leader and followers respectively are shown in Fig. 5a. Fig. 5b and 5c present lateral and angular deviations of the followers with respect to the reference

i ∈ [-6, 1] m /s 2 v i ∈ [0, 8] m /s d r,i 10 m γ 6 pc 1 λ i 1
path generated form the leader. Initial high deviations due to the time taken by the followers to join the platoon and converge to the leader path. The deviations increase when the leader moves through a curved segment of the road/path. These figures enlighten the tracking capabilities of the lateral controller and the potential of the longitudinal one. Fig. 5d -5h show the longitudinal motion results. As shown in Fig. 5e, the leader moves with variable velocity depending on the path conditions (by a human driver). This study investigates also the effect of accelerations perturbations as shown in Fig. 5d. The vehicles eventually reach the consensus. At certain places in the path (e.g., road curvature), the position and velocity errors become higher which is desired to allow the follower to adapt its velocity based on the road conditions. The observer estimation error, z2,i = e q,ij -ẑ2,i , is given in Fig. 5h which shows a convergence of the estimated predecessor velocity.

To quantify our results, the Root Mean Squared Error (RMSE) of the estimation and tracking errors, calculated after the initialization period is passed, are reported in Table III which are in the acceptable ranges. Furthermore, this table shows that the position tracking errors are attenuated as the RMSE of e s,3 < e s,2 < e s,1 , which illustrate the string stability of the platoon. z2,i = e q,ij -ẑ2,i Fig. 5. Platoon simulation results. L: leader actual path, and F i : follower i actual path. The followers track the path perfectly V. CONCLUSIONS Platoon kinematic and dynamic models are introduced in both curvilinear and Cartesian coordinates with a framework that allows us to design the control law in the curvilinear coordinates. An observer-based control law is proposed to achieve both consensus and string stability of the platoon vehicles. This control scheme requires a minimum number of communication links and can still achieve a higher traffic flow. This strategy reduces the traffic in the communication channel which in turn can enhance the communication robustness, and enables us to use a low-cost communication module. Conditions for platoon internal stability and string stability are provided. Simulation studies are conducted and illustrate the relevance of our approach.

Fig. 1 .

 1 Fig. 1. Kinematic model of car-like vehicle.

Fig. 2 .

 2 Fig. 2. Block diagram of the vehicle longitudinal dynamic model, where S is the Laplace operator.

Fig. 3 .

 3 Fig. 3. Platoon of car-like vehicles.

Fig. 4 .

 4 Fig. 4. Information flow topology: Hybrid PLF through communication link between leader and followers and direct sensor-based measurements between followers.

  28) Exploiting the similarity transformation technique by multiplying the left and right sides of the matrix Āc by Q = I 0 -Γ I , and Q-1 = I 0 Γ I , respectively, and by conditions (24) and (25), one can obtain

TABLE III RMSE

 III OF TRACKING ERRORS IN SIMULATION.

	RMSE	Follower i	1	2	3
		e s,i [m]	0.2675	0.1141	0.0290
		e q,i [ m /s]	0.1199	0.0635	0.0480
		z2,i [ m /s]	-	0.0635	0.0480
		r i [m]	0.1324	0.1080	0.1104
		ψ i [rad]	0.0209	0.0196	0.0198

Demonstration of this study is available in this video 1 .

1 https://www.dropbox.com/s/te3p0dod0kjtye3/obsrv_ sim.mp4?dl=0

ACKNOWLEDGEMENTS

The work presented in this paper is carried out in the framework of the Valet project, reference ANR-15-CE22-0013-02.


Michael G B Blum

. The chapter presents the main concepts underlying regression adjustment. A theorem that compares theoretical properties of posterior distributions obtained with and without regression adjustment is presented. Last, a practical application of regression adjustment in population genetics shows that regression adjustment shrinks posterior distributions compared to rejection approaches, which is a solution to avoid inflated credibility intervals.

Posterior distribution

Posterior distribution

Non-linear and heteroscedastic adjustment

Posterior distribution s obs Model parameter

Introduction

In this chapter, we present regression approaches for Approximate Bayesian Computation (ABC). As for most methodological developments related to ABC, regression approaches originate with coalescent modeling in population genetics [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF]. After performing rejection sampling by accepting parameters that generate summary statistics close enough to those observed, parameters are adjusted to account for the discrepancy between simulated and observed summary statistics. Because adjustment is based on a regression model, such approaches are coined as regression adjustment in the following.

Regression adjustment is a peculiar approach in the landscape of Bayesian approaches where sampling techniques are usually proposed to account for mismatches between simulations and observations [START_REF] Marjoram | Markov chain Monte Carlo without likelihoods[END_REF][START_REF] Scott A Sisson | Sequential Monte Carlo without likelihoods[END_REF]. We suggest various reasons explaining why regression adjustment is now a common step in practical applications of ABC. First, it is convenient and generic because the simulation mechanism is used to generate simulated summary statistics as a first step and it is not used afterwards. For instance, the software ms is used to generate DNA sequences or genotypes when performing ABC inference in population genetics [START_REF] Richard R Hudson | Generating samples under a wright-fisher neutral model of genetic variation[END_REF]. Statistical routines, which account for mismatches, are completely separated from the simulation mechanism and are used in a second step. Regression adjustment can therefore be readily applied in a wide range of contexts without implementation efforts. By contrast, when considering sampling techniques, statistical operations and simulations are embedded within a single algorithm [START_REF] Scott A Sisson | Sequential Monte Carlo without likelihoods[END_REF][START_REF] Mark | Adaptive approximate Bayesian computation[END_REF], which may require new algorithmic devel-opment for each specific statistical problem. Second, regression approaches have been shown to produce reduced statistical errors compared to rejection algorithms in a quite diverse range of statistical problems [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF][START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF][START_REF] Saulnier | Inferring epidemiological parameters from phylogenies using regression-ABC: A comparative study[END_REF]. Last, regression approaches are implemented in different ABC software including DIYABC [START_REF] Cornuet | DIYABC v2. 0: a software to make approximate Bayesian computation inferences about population history using single nucleotide polymorphism, DNA sequence and microsatellite data[END_REF] and the R abc package [START_REF] Csilléry | abc: an R package for approximate Bayesian computation (abc)[END_REF].

In this chapter, I introduce regression adjustment using a comprehensive framework that includes linear adjustment [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF] as well as more flexible adjustments such as non-linear models [START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF]. The first section presents the main concepts underlying regression adjustment. The second section presents a theorem that compares theoretical properties of posterior distributions obtained with and without regression adjustment. The third section presents a practical application of regression adjustment in ABC. It shows that regression adjustment shrinks posterior distributions when compared to a standard rejection approach. The fourth section presents recent regression approaches for ABC that are not based on regression adjustment.

Principle of regression adjustment

Partial posterior distribution

Bayesian inference is based on the posterior distribution defined as π(θ|y obs ) ∝ p(y obs |θ)π(θ)

where θ ∈ R p is the vector of parameters, and y obs are the data. Up to a renormalizing constant, the posterior distribution depends on the prior π(θ) and on the likelihood function p(y obs |θ). In the context of ABC, inference is no longer based on the posterior distribution π(θ|y obs ) but on the partial posterior distribution π(θ|s obs ) where s obs is a q-dimensional vector of descriptive statistics. The partial posterior distribution is defined as follows

π(θ|s obs ) ∝ p(s obs |θ)π(θ). (2)
Obviously, the partial posterior is equal to the posterior if the descriptive statistics s obs are sufficient for the parameter θ.

Rejection algorithm followed by adjustment

To simulate a sample from the partial posterior p(θ|s obs ), the rejection algorithm followed by adjustment works as follows 1. Simulate n values θ (i) , i = 1, . . . , n, according to the prior distribution π.

2. Simulate descriptive statistics s (i) using the generative model p(s (i) |θ (i) ).

3. Associate with each pair (θ (i) , s (i) ) a weight w (i) ∝ K h ( s (i) -s obs ) where • -• is a distance function, h > 0 is the bandwidth parameter, and K is an univariate statistical kernel with

K h ( • ) = K( • /h).
4. Fit a regression model where the response is θ and the predictive variables are the summary statistics s (equations ( 3) or ( 5)). Use a regression model to adjust the θ (i) in order to produce a weighted sample of adjusted values. Homoscedastic adjustment (equation (4)) or heteroscedastic adjustment (equation ( 6)) can be used to produce a weighted sample (θ

(i)
c , w (i) ), i = 1, . . . , n, which approximates the posterior distribution.

To run the rejection algorithm followed by adjustment, there are several choices to make. The first choice concerns the kernel K. Usual choices for K encompass uniform kernels that give a weight of 1 to all accepted simulations and zero otherwise [START_REF] Pritchard | Population growth of human Y chromosomes: a study of y chromosome microsatellites chromosome microsatellites[END_REF] or the Epanechnikov kernel for a smoother version of the rejection algorithm [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF]. However, as for traditional density estimation, the choice of statistical kernel has a weak impact on estimated distribution [START_REF] Bernard W Silverman | Density estimation for statistics and data analysis[END_REF]. The second choice concerns the threshold parameter h. For kernels with a finite support, the threshold h corresponds to (half) the window size within which simulations are accepted. For the theorem presented in section 0.5, I assume that h is chosen without taking into account the simulations s (1) , . . . , s (n) . This technical assumption does not hold in practice where we generally choose to accept a given percentage p, typically 1% or 0.1%, of the simulations. This practice amounts at setting h to the first p-quantile of the distances s (i) -s obs . A theorem where the threshold depends on simulations has been provided [START_REF] Biau | New insights into approximate Bayesian computation[END_REF]. Choice of threshold h corresponds to bias-variance tradeoff. When choosing small values of h, the number of accepted simulations is small and estimators might have a large variance. By contrast, when choosing large values of h, the number of accepted simulations is large and estimators might be biased [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF].

Regression adjustment

The principle of regression adjustment is to adjust simulated parameters θ (i) with nonzero weights w (i) > 0 in order to account for the difference between the simulated statistics s (i) and the observed one s obs . To adjust parameter values, a regression model is fitted in the neighborhood of s obs

θ (i) = m(s (i) ) + ε, i = 1, • • • , n (3)
where m(s) is the conditional expectation of θ given s and ε is the residual. The regression model of equation (3) assumes homoscedasticity, i.e. it assumes that the variance of the residuals does not depend on s. To produce samples from the partial posterior distribution, the θ (i) 's are adjusted as follows

θ (i) c = m(s obs ) + ε(i) (4) = m(s obs ) + (θ (i) -m(s (i) )),
where m represents an estimator of the conditional expectation of θ given s, and ε(i) is the i th empirical residual. In its original formulation, regression adjustment assumes that m is a linear function [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF] and it was later extended to non-linear adjustments [START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF].

The homoscedastic assumption of equation ( 3) may not be always valid. When the number of simulations is not very large because of computational constraints, local approximations, such as the homoscedastic assumption, are no longer valid because the neighborhood corresponding to simulations for which w (i) = 0 is too large. Regression adjustment can account for heteroscedasticity that occurs when the variance of the residuals depend on the summary statistics. When accounting for heteroscedasticity, the regression equation can be written as follows [START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF]

θ (i) = m(s (i) ) + σ(s (i) )ζ, i = 1, • • • , n (5)
where σ(s) is the square root of the conditional variance of θ given s, and ζ is the residual. Heteroscedastic adjustment involves an additional scaling step in addition to homoscedastic adjustment (4) (Figure 1)

θ (i) c = m(s obs ) + σ(s obs ) ζ(i) = m(s obs ) + σ(s obs ) σ(s (i) ) (θ (i) -m(s (i) )), (6)
where m and σ are estimators of the conditional mean and of the conditional standard deviation.

Fitting regression models

Equations ( 4) and ( 6) of regression adjustment depend on the estimator of the conditional mean m and possibly of the conditional variance σ. Model fitting is performed using weighted least squares. The conditional mean is learned by minimizing the following weighted least square criterion

E(m) = n i=1 (θ (i) -m(s (i) )) 2 w (i) . (7)
For linear adjustment, we assume that m(s) = α + βs [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF]. The parameters α and β are inferred by minimizing the weighted least square criterion given in equation [START_REF] Michael | A comparative review of dimension reduction methods in approximate Bayesian computation[END_REF].

For heteroscedastic adjustment (equation ( 4)), the conditional variance should also be inferred. The conditional variance is learned after minimization of a least square criterion. It is obtained by fitting a regression model where the answer is the logarithm of the squared residuals. The weighted least squares criterion is given as follows

E(log σ 2 ) = n i=1 log((ε (i) ) 2 ) -log σ 2 (s) 2 w (i) .
Neural networks has been proposed to estimate m and σ 2 [START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF]. This choice was motivated by the possibility offered by neural networks to reduce the dimension of descriptive statistics via an internal projection on a space of lower dimension [START_REF] Ripley | Neural networks and related methods for classification[END_REF].

In general, the assumptions of homoscedasticity and linearity (equation ( 3)) are violated when the percentage of accepted simulation is large. By contrast, heteroscedastic and non-linear regression models (equation ( 5)) are more flexible. Because of this additional flexibility, the estimated posterior distributions obtained after heteroscedastic and non-linear adjustment is less sensitive to the percentage of accepted simulations [START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF]. In a coalescent model where the objective was to estimate the mutation rate, heteroscedastic adjustment with neural networks was found to be less sensitive to the percentage of accepted simulations than linear and homoscedastic adjustment [START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF]. In a model of phylodynamics, it was found again that statistical error obtained with neural networks decreases at first-because the regression method requires a large enough training dataset-and then reaches a plateau [START_REF] Saulnier | Inferring epidemiological parameters from phylogenies using regression-ABC: A comparative study[END_REF]. However for larger phylodynamics dataset, statistical error obtained with neural networks increases for higher tolerance values. Poor regularization or the limited size of neural networks were advanced as putative explanations [START_REF] Saulnier | Inferring epidemiological parameters from phylogenies using regression-ABC: A comparative study[END_REF].

In principle, estimation of the conditional mean m and of the conditional variance σ 2 can be performed with different regression approaches. For instance, the R abc package implements different regression models for regression adjustment including linear regression, ridge regression and neural networks [START_REF] Csilléry | abc: an R package for approximate Bayesian computation (abc)[END_REF]. Lasso regression is another regression approach that can be considered. Regression adjustment based on lasso was shown to provide smaller errors than neural network in a phylodynamic model [START_REF] Saulnier | Inferring epidemiological parameters from phylogenies using regression-ABC: A comparative study[END_REF]. An advantage of lasso, ridge regression and neural networks compared to standard multiple regression is that they account for the large dimension of the summary statistics using different regularization techniques. Instead of considering regularized regression, dimension reduction is an alternative where the initial summary statistics are replaced by a reduced set of summary statistics or a combination of the initial summary statistics [START_REF] Fearnhead | Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate bayesian computation[END_REF][START_REF] Michael | A comparative review of dimension reduction methods in approximate Bayesian computation[END_REF]. The key and practical advantage of regression approaches with regularization is that they implicitly account for the large number of summary statistics and the additional step of variable selection can be avoided.

Parameter transformations

When the parameters are bounded or positive, parameters can be transformed before regression adjustment. Transformations guarantee that the adjusted parameter values lie in the range of the prior distribution [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF]. An additional advantage of the log and logit transformations is that they stabilize the variance of the regression model and make regression model (3) more homoscedastic [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF].

Positive parameters are regressed on a logarithm scale φ = log(θ),

φ = m(s) + ε.
Parameters are then adjusted on the logarithm scale

φ (i) c = m(s obs ) + (φ (i) -m(s (i) )
). The final adjusted values are obtained by exponentiation of the adjusted parameter values

θ (i) c = exp(φ (i) c
). Instead of using a logarithm transformation, bounded parameters are adjusted using a logit transformation. Heteroscedastic adjustment can also be performed after log or logit transformations.

Shrinkage

An important property of regression adjustment concerns posterior shrinkage. When considering linear regression, the empirical variance of the residuals is smaller than the total variance. In addition, residuals are centered for linear regression. These two properties imply that for linear adjustment, the empirical variance of θ (i) c is smaller than the empirical variance of the non-adjusted values θ (i) obtained with the rejection algorithm. Following homoscedastic and linear adjustment, the posterior variance is consequently reduced. For non-linear adjustment, shrinkage property has also been reported and the additional step generated by heteroscedastic adjustment does not necessarily involve additional shrinkage when comparing θ

(i) c to θ (i) c [5].

Theoretical results about regression adjustment

The following theoretical section is technical and can be skipped by readers not interested by mathematical results about ABC estimators based on regression adjustment. In this section, we give the main theorem that describes

FIGURE 1

Posterior distributions obtained with and without regression adjustment. The shrinking effect of adjustment is visible since the posterior variance of the green and blue histograms is reduced compared to the posterior variance of the orange histogram. the statistical properties of posterior distributions obtained with or without regression adjustment. To this end, the estimators of the posterior distribution are defined as follows πj (θ|s obs ) = n i=1 Kh (θ

(i) j -θ)w (i) , j = 0, 1, 2, (8)
where θ

(i) 0 = θ (i) (no adjustment), θ (i) j = θ (i)
c for j = 1, 2 (homoscedastic adjustment), K is an univariate kernel, and Kh (•) = K(•)/h . Linear adjustment corresponds to j = 1 and quadratic adjustment corresponds to j = 2. In nonparametric statistics, estimators of the conditional density with adjustment have already been proposed [START_REF] Hyndman | Estimating and visualizing conditional densities[END_REF][START_REF] Bruce E Hansen | Nonparametric conditional density estimation[END_REF].

To present the main theorem, we introduce the following notations: if X n is a sequence of random variables and a n is a deterministic sequence, the notation X n = o P (a n ) means that X n /a n converges to zero in probability and X n = O P (a n ) means that the ratio X n /a n is bounded in probability when n goes to infinity. The technical assumptions of the theorem are given in the appendix of [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF].

Theorem 1 We assume that conditions (A1)-(A5) given in the appendix of [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF] hold. The bias and variance of the estimators πj (θ|s obs ), j = 0, 1, 2 are given by

E[π j (θ|s obs ) -π(θ|s obs )] = C 1 h 2 + C 2,j h 2 + O P ((h 2 + h 2 ) 2 ) + O P ( 1 nh q ), (9)
Var[π j (θ|s obs )] = C 3 nh q h

(1 + o P (1)), [START_REF] Csilléry | Approximate Bayesian computation (abc) in practice[END_REF] where q is the dimension of the vector of summary statistics and the constants C 1 , C 2,j et C 3 are given in [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF].

Proof: See [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF].

There are other theorems that provide asymptotic biases and variances of ABC estimators but they do not study the properties of estimators arising after regression adjustment. Considering posterior expectation (e.g. posterior moments) instead of the posterior density, [START_REF] Barber | The rate of convergence for Approximate Bayesian Computation[END_REF] provides asymptotic bias and variance of an estimator obtained with rejection algorithm. [START_REF] Biau | New insights into approximate Bayesian computation[END_REF] studied asymptotic properties when the window size h depends on the data instead of being fixed in advance. Another version of ABC called lazy ABC exists and provides a bias proportional to h instead of h 2 while keeping the same variance of the order of 1/(nh q ), which is inversely proportional to the acceptance probability in the rejection algorithm [START_REF] Fearnhead | Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate bayesian computation[END_REF].

Remark 1. Curse of dimensionality The mean square error of the estimators is the sum of the squared bias and of the variance. With elementary algebra, we can show that for the three estimators πj (θ|s obs ), j = 0, 1, 2, the mean square error is of the order of n -1/(q+5) for an optimal choice of h. The speed with which the error approaches 0 therefore decreases drastically when the dimension of the descriptive statistics increases. This theorem highlights (in an admittedly complicated manner) the importance of reducing the dimension of the statistics. However, the findings from these asymptotic theorems, which are classic in non-parametric statistics, are often much more pessimistic than the results observed in practice. It is especially true because asymptotic theorems in the vein of Theorem 1 do not take into account correlations between summary statistics [START_REF] Scott | Multivariate density estimation: theory, practice, and visualization[END_REF].

Remark 2. Comparing biases of estimators with and without adjustment It is not possible to compare biases (i.e. the constant C 2,j , j = 0, 1, 2) for any statistical model. However, if we assume that the residual distribution of ε in equation ( 3) does not depend on s, then the constant C 2,2 is 0. When assuming homoscedasticity, the estimator that achieves asymptotically the smallest mean square error is the estimator with quadratic adjustment p2 (θ|s obs ). Assuming additionally that the conditional expectation m is linear in s, then both p1 (θ|s obs ) and p2 (θ|s obs ) have a mean square error lower than the error obtained without adjustment.

Application of regression adjustment to estimate admixture proportions using polymorphism data

To illustrate regression adjustment, I consider an example of parameter inference in population genetics. Description of coalescent modeling in population genetics is out of the scope of this chapter and we refer interested readers to dedicated reviews [START_REF] Richard R Hudson | Gene genealogies and the coalescent process[END_REF][START_REF] Noah | Genealogical trees, coalescent theory and the analysis of genetic polymorphisms[END_REF]. This example illustrates that ABC can be used to infer evolutionary events such as admixture between sister species. I assume that two populations (A and B) diverged in the past and admixed with admixture proportions p and 1 -p to form a new hybrid species C that subsequently split to form two sister species C 1 and C 2 (Figure 2). Simulations are performed using the software DIYABC [START_REF] Cornuet | DIYABC v2. 0: a software to make approximate Bayesian computation inferences about population history using single nucleotide polymorphism, DNA sequence and microsatellite data[END_REF]. The model of Figure 2 corresponds to a model of divergence and admixture between species of a complex of species from the butterfly gender Coenonympha. We assume that 2 populations of the Darwin's Heath (Coenonympha darwiniana) originated through hybridization between the Pearly Heath (Coenonympha arcania) and the Alpine Heath (Coenonympha gardetta) [START_REF] Capblancq | Hybridization promotes speciation in coenonympha butterflies[END_REF]. A total of 16 summary statistics based on Single Nucleotide Polymorphisms (SNPs) are used for parameter inference [START_REF] Capblancq | Hybridization promotes speciation in coenonympha butterflies[END_REF]. A total of 10 6 simulations are performed and the percentage of accepted simulations is of 0.5%. I consider four different forms of regression adjustment: linear and homoscedastic adjustment, non-linear (neural networks) and homoscedastic adjustment, linear and heteroscedastic adjustment, non-linear and heteroscedastic adjustment. All adjustments were performed with the R package abc

Time A B C C 1 C 2 p 1-p FIGURE 2
Graphical description of the model of admixture between sister species. Two populations (A and B) diverge in the past and admixed with admixture proportions p and 1 -p to form a new hybrid species C that subsequently diverge to form two sister species (C 1 and C 2 ). [START_REF] Csilléry | abc: an R package for approximate Bayesian computation (abc)[END_REF][START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF]. I evaluate parameter inference using a cross-validation criterion [START_REF] Csilléry | abc: an R package for approximate Bayesian computation (abc)[END_REF].

The cross-validation error decreases when considering linear adjustment (Figure 3). However, considering heteroscedastic instead of homoscedastic adjustment does not provide an additional decrease of the cross-validation error (Figure 3). Then, using real data from a butterfly species complex, we compare the posterior distribution of the admixture proportion p obtained without adjustment, with linear and homoscedastic adjustment, and with non-linear and homoscedastic adjustment (Figure 4). For this example, considering regression adjustment considerably changes the shape of the posterior distribution. The posterior mean without adjustment is of p = 0.51 (95% C.I. = (0.12, 0.88)). By contrast, when considering linear and homoscedastic adjustment, the posterior mean is of 0.93 (95% C.I. = (0.86, 0.98). When considering non-linear and homoscedastic adjustment, the posterior mean is 0.84 (95% C.I. = (0.69, 0.93). Regression adjustment confirms a larger contribution of C. arcania to the genetic composition of the ancestral C. darwiniana population [START_REF] Capblancq | Hybridization promotes speciation in coenonympha butterflies[END_REF]. This example shows that regression adjustment not only shrinks credibility intervals but can also shift posterior estimates. Compared to rejection, the posterior shift observed with regression adjustments provides a result that is more consistent with published results [START_REF] Capblancq | Hybridization promotes speciation in coenonympha butterflies[END_REF].

FIGURE 4

Posterior distribution of the admixture coefficient p obtained using real data from a butterfly species complex to compute observed summary statistics. This example shows that regression adjustment not only shrinks credibility intervals but can also shift posterior estimates. The admixture coefficient p measures the relative contribution of Coenonympha arcania to the ancestral C. darwiniana population.

Regression methods besides regression adjustment

There are other regression methods besides regression adjustment that have been proposed to estimate E[θ|s obs ] and π(θ|s obs ) using the simulations as a training set. A first set of methods consider kernel methods to perform regression [START_REF] Shigeki Nakagome | Kernel approximate Bayesian computation in population genetic inferences[END_REF]. The principle is to define a kernel function K to compare observed statistics to simulated summary statistics. Because of the so-called kernel trick, regression with kernel methods amounts at regressing θ with Φ(s) where < Φ(s), Φ(s ) >= K(s, s ) for two vector of summary statistics s and s . Then, an estimate of the posterior mean is obtained as follows

E[θ|s obs ] = n i=1 w i θ (i) , (11)
where w i depends on the inverse the Gram matrix containing the values K(s (i) , s (j) ) for i, j = 1 . . . , n. A formula to estimate posterior density can also be obtained in the same lines as formula [START_REF] Csilléry | abc: an R package for approximate Bayesian computation (abc)[END_REF]. Simulations suggest that kernel ABC gives better performance than regression adjustment when highdimensional summary statistics are used. For a given statistical error, it was reported that fewer simulations should be performed when using kernel ABC instead of regression adjustment [START_REF] Shigeki Nakagome | Kernel approximate Bayesian computation in population genetic inferences[END_REF]. Other kernel approaches have been proposed for ABC where simulated and observed samples or summary statistics are directly compared through a distance measure between empirical probability distributions [START_REF] Mitrovic | DR-ABC: approximate bayesian computation with kernel-based distribution regression[END_REF][START_REF] Park | K2-ABC: Approximate bayesian computation with kernel embeddings[END_REF]. Another regression method in ABC that does not use regression adjustment considers quantile regression forest [START_REF] Marin | ABC random forests for bayesian parameter inference[END_REF]. Generally used to estimate conditional mean, random forests also provide information about the full conditional distribution of the response variable [START_REF] Meinshausen | Quantile regression forests[END_REF]. By inverting the estimated conditional cumulative distribution function of the response variable, quantiles can be inferred [START_REF] Meinshausen | Quantile regression forests[END_REF]. The principle of quantile regression forest is to use random forests in order to give a weight w i to each simulation (θ (i) , s (i) ). These weights are then used to estimate the conditional cumulative posterior distribution function F (θ|s obs ) and to provide posterior quantiles by inversion. An advantage of quantile regression forest is that tolerance rate should not be specified and standard parameters of random forest can be considered instead. A simulation study of coalescent models shows that regression adjustment can shrink posterior excessively by contrast to quantile regression forest [START_REF] Marin | ABC random forests for bayesian parameter inference[END_REF].

Conclusion

This chapter introduces regression adjustment for Approximate Bayesian Computation [START_REF] Mark | Approximate Bayesian computation in population genetics[END_REF][START_REF] Michael | Non-linear regression models for Approximate Bayesian Computation[END_REF]. We explain why regression adjustment shrinks posterior distribution which is a desirable feature because credibility intervals obtained with rejection methods can be too wide [START_REF] Michael | Approximate Bayesian Computation: A nonparametric perspective[END_REF]. When inferring admixture with SNP data in a complex of butterfly species, the posterior distribution obtained with regression adjustment was not only shrunk when compared to standard rejection but also shifted to larger values, which confirm results obtained for this species complex with other statistical approaches [START_REF] Capblancq | Hybridization promotes speciation in coenonympha butterflies[END_REF]. We have introduced different variants of regression adjustment and it might be difficult for ABC users to choose which adjustment is appropriate in their context. We argue that there is no best strategy in general. In the admixture example, we found, based on a cross-validation error criterion, that homoscedastic linear adjustment provides considerable improvement compared to rejection. More advanced adjustments provide almost negligible improvements if no improvement at all. However, in a model of phylodynamics, non-linear adjustment was reported to achieve considerable improvement compared to linear adjustment [START_REF] Saulnier | Inferring epidemiological parameters from phylogenies using regression-ABC: A comparative study[END_REF]. In practical applications of ABC, we suggest to compute errors such as cross-validation estimation errors to choose a particular method for regression adjustment.

With the rapid development of complex machine learning approaches, we envision that regression approaches for Approximate Bayesian Computation can be further improved to provide more reliable inference for complex models in biology and ecology.


Dr Johannes Karges

Dr Olivier Blacque

Dr Philippe Goldner

Hui Chao

Gilles Gasser
email: gilles.gasser@chimieparistech.psl.eu@www.gassergroup.com

Prof H Chao

Towards Long Wavelength Absorbing Photodynamic Therapy Photosensitizers via the Extension of a [Ru(bipy) 3 ] 2+ Core

Complementary to classical treatment methods used against cancer, photodynamic therapy (PDT) has received increased attention over the last years. PDT relies on the generation of reactive oxygen species (ROS) upon light irradiation to trigger cell death. As the wavelength employed during such treatments directly influences the light penetration depth and therefore the possibility to treat deep seated tumours or large tumours, research efforts have been made towards the development of photosensitizers (PS) with an absorption in the phototherapeutic window (600-900 nm). To tackle this drawback, we report herein the preparation and characterisation of new Ru(II)containing PDT PSs, that are based on a [Ru(bipy) 3 ] 2+ core (1; bipy:

2,2'-bipyridine) and that are extended with methyl groups (2) or vinyl dimethylamino groups (3). As anticipated with our design, we found a red-shift of 65 nm of the maximum absorption of complex 3 in comparison to complex 1. In addition, we report on the in-depth photophysical properties as well as (photo-)cytotoxicity against cervical cancerous HeLa cells of the investigated compounds.

Introduction

Over the last decades, cancer has emerged to be one of the deadliest diseases worldwide. [1] Next to the classical treatments (e.g., chemotherapy, surgery and radiotherapy), the use of Photodynamic Therapy (PDT) has received increased attention as a complementary medical technique to these blockbusters.

PDT is based on the combination of light, a photoactive compound called a Photosensitizer (PS) and oxygen. Ideally, the PS should be nontoxic in the absence of light and generate highly toxic species upon light irradiation. The mechanism of action of PDT is based on the generation of reactive oxygen species (ROS).

More specifically, upon light irradiation, the PS is excited to a singlet state, which can be transformed into an excited triplet state by an intersystem crossing (ISC) process. From there, the PS is able to influence its biological environment by two pathways, namely Type I and Type II. During a Type I reaction, an electron or proton is transferred to/from the PS from/to its biological surrounding. This leads to the generation of radicals and ROS like superoxides or hydroxyl radicals. In a Type II reaction, the energy of the exited triplet state of the PS is transferred to molecular oxygen ( 3 O 2 ) to produce singlet oxygen ( 1 O 2 ). This highly energetic form of oxygen is highly reactive. Consequently, during both pathways, ROS or 1 O 2 react with its biological surrounding, generating cellular damages and therefore ultimately trigger cell death. [2] The most commonly used PS for PDT treatments is Photofrin (Figure 1), which is approved for the treatment of bladder cancer, early stage lung cancer, oesophageal cancer and early non-small cell lung cancer in various countries. To date, the majority of approved PSs are based on a tetrapyrrolic scaffold. Due to their relatively similar structures, the majority of these compounds have a tendency to share the same disadvantages, which are 1) poor water solubility; 2) tedious synthesis; 3) photobleaching effect and 4) slow clearance from the body causing photosensitivity. At this stage, it is important to mention that some compounds based on a tetrapyrrolic scaffold do not share these limitations. [2c, 3] To tackle these drawbacks, new classes of PDT PSs are currently being developed. Among the different classes, Ru(II) polypyridyl complexes seem to be excellent candidates. The majority of these compounds have generally a high water solubility, long luminescence decay, high 1 O 2 production as well as a high chemical and photophysical stability. [4] Therefore, it is not surprising that the complex TLD-1433 (Figure 1, λ ex = 525 nm, ε max = 2000 M -1 cm -1 , Φ ~ 0.99 in CH 3 CN) has just completed phase I clinical trial as a novel PDT PS for the treatment of bladder cancer. [4h, 5] To date, most studied Ru(II) polypyridyl complexes lack significant absorption in the phototherapeutic window (600-900 nm). [6] It is well-established that the wavelength used during treatments directly correlates with the tissue penetration depth.

Longer wavelengths are able to penetrate deeper in the tissue and are hence potentially able to treat deeper-seated tumours or larger tumours. Additionally, as longer wavelengths are less energetic, less photodamage caused by the light source has been associated with treatments at longer wavelengths. Based on this, PSs with an absorption at wavelengths in the phototherapeutic window are sought after. [7] With the aim to develop Ru(II) polypyridyl complexes with a redshift absorption in view of applications as PDT PSs, we have extended the parent complex [Ru(bipy) 3 ] 2+ (1) (bipy = 2,2′bipyridine) with methyl groups (2) and the conjugated system with vinyl dimethylamino groups (3) (Figure 2). Recent investigations have shown that the π-extension of the bipy core of the Ru(II) polypyridine complex caused a red shift of the absorption of the resulting complex. [8] In addition, previous systematic studies of differently substituted Ru(II) polypyridine complex have indicated that dialkylamino substituents strongly promote a desired red shift in absorption. [9] Combining these concepts, we report here compound 3, with an extended π-system and dimethylamine groups, which was indeed found to have a red-shifted absorption of 65 nm in comparison to complex 1 and an absorption in the phototherapeutic window as well as highly increased extinction coefficients. Importantly, we could show the ability of this compound to produce 1 O 2 at longer wavelengths as well as to cause phototoxicity at this wavelength in cancerous cells while having no observed dark toxicity.

Results and Discussion

Synthesis and Characterisation

The complexes synthesised in this work are shown in Figure 2.

Compound 1 was obtained from a commercial source whereas 2 was synthesised as previously reported by complexation of RuCl 2 dmso 4 with 4,4´-dimethyl-2,2´-bipyridine. [10] To the best of our knowledge, the synthesis the complex 3 has not been yet

reported. The ligand (E,E')-4,4'-bis(N,N-dimethylaminovinyl)-2,2'bipyridine present in complex 3 was synthesised as previously reported. [11] However, complexation attempts of this ligand using

X-ray crystallography

The crystal structures of 4,4'-bis(N,N-dimethylaminovinyl)-2,2'bipyridine and 2 have been determined by single crystal X-ray diffraction studies. Crystal data, structure refinement parameters and molecular structures are presented in Table S1 as well as reported by Viau et al. in 2003. [12] The asymmetric unit in a of diethyl ether in a ration 1/2/1. It is interesting to note that the crystal structure of 2 is isostructural with the previously reported crystal structure of the iron complex [Fe(4,4′-dimethyl-2,2′dipyridyl) 3 ][PF 6 ] 2 (Figure S9). [13] In the crystal, ions and solvent molecules are linked together through C-H⋯F and C-H⋯O interactions. Despite the numerous aromatic rings of complex 2 no π…π or C-H…π interactions are observed.

Photophysical properties

The photophysical properties of the complexes were then evaluated to assess their potential as PDT PSs. To investigate this, we have measured the absorption of complexes 1-3 in CH 3 CN (Figure 4, Table 1). Typically, Ru(II) polypyridyl complexes have, as the lowest energy absorption band, a spinallowed metal-to-ligand charge transfer (MLCT) transition, which occurs for the prototype complex [Ru(bpy) 3 ] 2+ 1 at 450 nm. The band at 285 nm was assigned to spin-allowed ligand-centered (LC) transition and the shoulders around 350 nm to metalcentered (MC) transitions. [6f] The comparison between complexes 1 and 2 shows only small differences indicating that the additional methyl groups in 2 do not significantly influence the absorption properties. On the contrary, the absorption of complex 3 was highly modified with a strong increase of the extinction coefficient as well as a strong red-shift of 65 nm of the maximum of the MLCT transition caused by the extension of the π-system as well as the insertion of dimethylamine groups at the terminal end. Importantly, the absorption tail of the compound is in the desired phototherapeutic window (600-900 nm). For further investigation

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of the excited state, the emission properties of the complexes were investigated upon excitation of the compounds in CH 3 CN at 355 nm. The emission signal was measurable between 550-850 nm (Figure S10) with a maximum at 622 nm for complex 2 and 621 nm for complex 3. Comparison to the standard compound 1

(Φ em = 0.059) shows that complex 2 has an increased emission (Φ em = 0.083) whereas complex 3 is only weakly emitting and could only be detected at the detection limit of our used setup.

These results fit with those of a recent study which compared different 4,4´-π-conjugated[2,2´]-bipyridines and which found that (E,E')-4,4'-bis(N,N'-dimethylaminovinyl)-2,2'-bipyridine itself already had a low fluorescence quantum yield of 0.015 in dichloromethane. [14] As an additional characterisation of the excited state, the luminescence lifetimes in degassed and air saturated CH 3 CN upon excitation at 355 nm were determined (Figure S11-S13). The measured lifetimes (Table 1) were found to be in the same range as for other investigated Ru(II) poylpyridine complexes. [9] Importantly, the lifetimes of the excited state is strongly decreasing in the presence of air.

Table 1. Spectroscopic properties of the investigated complexes 1-3 in CH3CN

Singlet oxygen generation

As discussed in the photophysical evaluation section, the lifetimes molecule. [15] The results presented in Table 2 show that compounds 1 and 2 are generating 1 O 2 decently whereas 3 generates 1 O 2 only poorly.

(Photo-)stability

An important property of a molecule, which is envisioned to be used as a PDT PS, is its (photo-)stability. To investigate this, we have assessed the stability in organic solvents of our complexes as it was shown in previous works that this could already be problematic. [16] For this purpose, the compounds were incubated in CH 3 CN and, in time intervals (0, 1, 4, 8, 12, 24, 48 h) As an internal standard, caffeine was used, which has already been shown to be suitable for these experiments. [17] After 48 h incubation, the compounds were extracted from the plasma and the HPLC chromatogram before and after the incubation compared. The analysis showed that no decomposition for the

Cellular Uptake

A crucial parameter for the bioactivity of a molecule is its cellular uptake. To investigate this, compounds 1-3 were incubated for 4 h in the dark in human cervical carcinoma (HeLa) cells. The amount of the metal Ru inside the cells was then determined using inductively coupled plasma mass spectrometry (ICP-MS).

The results (Figure 5) show that complex 3 has a much higher uptake than 1 or 2.

Cytotoxicity and Phototoxicity

After having assessed the chemical and photophysical properties of compounds 1-3, their influence on cell viability in the dark and upon light irradiation was investigated. For this purpose, the compounds were incubated in non-cancerous retinal pigment epithelium (RPE-1) and HeLa cells in the dark as well as upon light irradiation at 480 nm (10 min, 3.1 J/cm 2 ) and 540 nm (40 min, 9.5 J/cm 2 ). The obtained IC 50 values were further compared with the chemotherapeutic drug cisplatin and the PS Protoporphyrin IX (PpIX) (Table 3). For all investigated complexes, no toxicity in the dark could be observed (IC 50 > 200 μM), which is a desired characteristic for a potential PDT PS. Disappointingly, the exposure to light had only a small effect on the cell viability for the three compounds. While no toxicity was observed for compounds 1 and 2, compound 3 showed some phototoxicity. Of note, these findings are in agreement with a study of the [Ru(bipy) 3 ] 2+ complex which showed no dark and phototoxic effect in the high micromolar range. [18] To evaluate the ability of a compound to act as a PS, the phototoxic index (PI) is calculated as the ratio between the IC 50 values in the dark and upon light exposure. For compound 3, a PI value of 1.3 at 480 nm and 1.4 at 540 nm for HeLa cells and 1.3 at 480 nm and 1.2 at 540 nm for RPE-1 cells was determined. These results demonstrate that 3 is able to have a slight phototoxic effect upon exposure to higher wavelength which is a desired characteristic for a PS. However, the obtained PI values are quite low in comparison to established PSs like PpIX.

The results can be rationalised by the rather poor generation of singlet oxygen of this complex. One has also to highlight that the instability of this compound in human plasma and upon irradiation is extremely problematic and could also explain these poor biological results. Overall, the biological results obtained in this section are fitting with the ICP-MS experiments carried out which showed a much better cellular accumulation of complex 3 compared to complexes 1 and 2.

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Conclusions

In this study, we aimed to shift the absorption wavelength of the RuCl2dmso4 was synthesised as previously reported. [19] The ligand (E,E')-4,4'-bis(N,N-dimethylaminovinyl)-2,2'-bipyridine present in complex 3 was synthesised as previously reported. [11] Instrumentation and methods single wavelength X-ray source from a micro-focus sealed X-ray tube were used with the Cu Kα radiation (λ = 1.54184 Å) [20] for both analyses. The selected single crystals were mounted using polybutene oil on a flexible loop fixed on a goniometer head and transferred to the diffractometer. Preexperiments, data collections, data reductions and analytical absorption corrections [21] were performed with the program suite CrysAlisPro. [22] Using Olex2, [23] the structures were solved with the SHELXT [24] small molecule structure solution program and refined with the SHELXL2018/3

program package [25] by full-matrix least-squares minimization on F 2 .

Molecular graphics were created using Mercury 4.0. [26] The crystal data collections and structure refinement parameters are summarized in Table S1. CCDC 1914096 (for 2) and CCDC 1914097 (for 4,4'-bis(N,Ndimethylaminovinyl)-2,2'-bipyridine) contain the supplementary crystallographic data for these compounds, and can be obtained free of charge from the Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

Spectroscopic measurements

The [Ru(bipy)3]Cl2 in CH3CN (Φem=5.9%) [27] applying the following formula: bengal (Φ = 76 %) [28] was used and the singlet oxygen quantum yields were calculated using the following formula:

Stability in CH3CN and PBS

The stability of the compound in CH3CN and PBS was determined by UV/Vis spectroscopy. The compound was dissolved and stored at room temperature in the dark. The absorption spectrum from 250-700 nm was recorded with a SpectraMax M2 Microplate Reader (Molecular Devices)

after each time interval (0, 1, 4, 8, 12, 24, 48 h) and compared.

Stability in human plasma

The stability of the complexes was evaluated with caffeine as an internal standard, which has already shown to be suitable for these experiments. [17] The pooled human plasma was obtained from Biowest and caffeine from

Photostability

The samples were prepared in an air saturated CH3CN solution. To measure the photostability, the samples were irradiated at 450 nm in 96 well plates with an Atlas Photonics LUMOS BIO irradiator during time intervals from 0-10 min. The absorbance spectrum from 350-700 nm was recorded with a SpectraMax M2 Microplate Reader (Molecular Devices)

after each time interval and compared.

Cell culture

Human cervical carcinoma (HeLa) cells were cultured using DMEM media and retinal pigment epithelium (RPE-1) cells using DMEM/F-12 with addition of 10% FBS and 1% penstrep. The cells were cultivated and maintained at 37 °C in a cell culture incubator at 37 °C with 5% CO2

atmosphere. Before an experiment, the cells were passaged three times.

Cellular uptake

The cellular uptake of the complex was investigated by the determination of the Ru content inside the cells. The complex with a final concentration of 25 μM (1% DMSO, v%) was incubated for 4 h at 37 °C in the dark on a cell culture dish with a density of ca. 6 . 10 6 cells in 10 mL of media. After this time, the media was removed and the cells were washed with cell media. The cells were trypsinised, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted.

Each sample was the digested using a 60% HNO3 solution overnight. Each sample was diluted to solution of 2% HNO3 in water. The Ru content was determined using an ICP-MS apparatus and comparing the results with the Ru references. The Ru content was then associated with the number of cells.

(Photo-)cytotoxicity

The cytotoxicity of the compounds was accessed by measuring the cell viability using a fluorometric resazurin assay. The cultivated cells were seeded in triplicates in 96 well plates with a density of 4000 cells per well in 100 μL of media. After 24 h the medium was removed and the cells were During this time, the temperature was maintained constantly at 37 °C. The cells were grown in the incubator for additional 44 h. For the determination of the dark cytotoxicity, the cells were not irradiated and after the medium exchange directly incubated for 44 h. After this time, the medium was replaced with fresh medium containing resazurin with a final concentration of 0.2 mg/mL. After 4 h incubation, the amount of the fluorescent product resorufin was determined upon excitation at 540 nm and measurement its emission at 590 nm using a SpectraMax M2 Microplate Reader (Molecular Devices). The obtained data was analysed with the GraphPad Prism software.

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The preparation and characterisation of new Ru(II)-containing photodynamic therapy, that are based on a [Ru(bipy) 3 ] 2+ core (bipy: 2,2'-bipyridine) and that are extended with methyl groups or vinyl dimethylamino groups is described. Their in-depth photophysical properties as well as (photo-)cytotoxicity against cervical cancerous HeLa cells is also reported.

Figure 1 .

 1 Figure 1. Structure of a) Photofrin and b) TLD-1433. [Ru(dmb)2(IP-TT)] 2+ (dmb=4,4′-dimethyl-2,2′-bipyridine, IP-TT=2-(2′,2″:5″,2′ ′′-terthiophene)imidazol[4,5-f][1,10]phenanthroline).

Figure 2 .

 2 Figure 2. Chemical structures of the investigated compounds in this study. The complexes were isolated as PF6 salts.

  Scheme 1. Synthesis of complexes 2 and 3. a) 4,4´-dimethyl-2,2´-bipyridine, DMF, reflux, 12 h, nitrogen atmosphere, 92%; b) tert-butoxy bis(dimethylamino)methane, DMF, 140°C, 40 h, nitrogen atmosphere, 85%.

Figures 3

 3 Figures 3 and S7. The crystal structure of 4,4'-bis(N,Ndimethylaminovinyl)-2,2'-bipyridine presented in the manuscript is a new monoclinic polymorph (a) of the previous structure (b)

  Figure S8). In our crystals the molecules are linked by C-H⋯N and C-H⋯π interactions. In the structure of the trisbipyridyl ruthenium(II) complex 2 the central Ru atom is expectedly coordinated to the six nitrogen atoms of the three substituted bipyridines in a distorted octahedral geometry. The complex cations crystallized with PF 6 -counter-ions and solvent molecules

Figure 3 .

 3 Figure 3. Molecular structure of compound 2. The thermal ellipsoids are drawn at the 30 % probability level and all H atoms and the solvent molecule of diethyl ether are omitted for clarity.

Figure 4 .

 4 Figure 4. Absorption spectra of complexes 1-3 in CH3CN.

  of the excited state of the investigated compounds are drastically decreasing in the presence of air indicating that the triplet state of the compound ( 3 PS) is able to interact with molecular oxygen ( 3 O 2 ). As the active species for most applied PSs in PDT, the production of singlet oxygen ( 1 O 2 ) is responsible for most PDT effects. To investigate the ability of our compounds to generate 1 O 2 , we used two different methods, namely 1) direct by measurement of the phosphorescence of 1 O 2 or 2) indirect by measurement of the change in absorbance of a reporter

  n.d. a) = not detectable due to missing absorbance at this wavelength.

  , their UV/Vis spectra measured. During the incubation in CH 3 CN, no change in the spectra for all compounds (Figure S14-S16) could be detected indicating their stability in this solvent. As a second experiment, the stability in an aqueous PBS solution was investigated. Also, here, no decomposition was observed (Figure S17-S19), proving the stability of these compounds. Finally, as a third experiment, the complexes were incubated in human plasma and the stability of the complexes identified by an HPLC analysis.

  compounds 1 and 2 (Figure S20-S21) occurred. On the contrary, FULL PAPER complex 3 was clearly transformed to a mixture of different kinds of unidentified products (Figure S22), clearly proving its decomposition. After investigation of the stability of the compounds in a biological environment, we have tested their stability upon irradiation in an air saturated CH 3 CN solution while monitoring their UV/Vis spectra in constant time intervals. The stability was compared to the PS Protoporphyrin IX (PpIX), which is well known to be photodegradating. The comparison between the spectra (Figure S23-S26) shows only a small decrease in absorption for 1 and 2 indicating only a small photobleaching effect. On the contrary, the bands of 3 strongly decrease and shift, proving its decomposition upon light exposure. Consequently, the modification caused by the irradiation was investigated by NMR spectroscopy. For this purpose, a solution of 3 in CD 3 CN was irradiated at 450 nm (30 min, 36.0 J/cm 2 ) and the change in the 1 H-NMR spectrum monitored. The compound was transformed in an unidentified mixture of different compounds (see Figure S27).

Figure 5 .

 5 Figure 5. Comparison of the cellular uptake of complexes 1-3 after 4 h incubation in HeLa cells.

Table 3 .

 3 IC50 values in the dark and upon irradiation at 480 (10 min, 3.1 J/cm 2 ) and 540 nm (40 min, 9.5 J/cm 2 ) for complexes 1-3 in comparison to cisplatin and Protoporphyrin IX (PpIX) on non-cancerous retinal pigment epithelium (RPE-1) and human cervical carcinoma (HeLa) cells. Average of three independent measurements. n.d. = not determinable.

  photophysical properties revealed that this compound was weakly emissive and has short excited state lifetimes. We assume that these properties are limiting the necessary energy transfer from the excited state 3 PS to molecular oxygen ( 3 O 2 ) to ultimately produce singlet oxygen ( 1 O 2 ). This probably explains the poor singlet oxygen quantum yield determined in this study for complex 3. Investigation of the stability of the compounds revealed that complexes 1 and 2 are stable in CH 3 CN, PBS and human plasma whereas 3 decomposes in human plasma as well as upon light irradiation over time. Biological evaluation on the cancerous cell line HeLa and the non-cancerous cell line RPE-1 revealed no dark toxicity for any of the investigated complexes in this study. While no toxicity upon light exposure for compounds 1 and 2 could be observed, complex 3 showed some slight phototoxicity in the high micromolar range against cervical cancerous HeLa cells and importantly no measurable dark cytotoxicity. Despite unfavourable photophysical properties, 3 showed a stronger cytotoxic effect than the two other complexes 1 and 2.Quantification of the cellular uptake of the complexes by ICP-MS experiments rationalized this observation with complex 3 being much better taken up by HeLa cells than compounds 1 and 2.

1H

  and 13 C NMR spectra were recorded on a Bruker 500 MHz NMR spectrometer. Chemical shifts (δ) are reported in parts per million (ppm) referenced to tetramethylsilane (δ 0.00) ppm using the residual proton solvent peaks as internal standards. Coupling constants (J) are reported in Hertz (Hz) and the multiplicity is abbreviated as follows: s (singlet), d (doublet), dd (doublet of doublet). ESI-MS experiments were carried out using a LTQ-Orbitrap XL from Thermo Scientific and operated in positive ionization mode, with a spray voltage at 3.6 kV. No Sheath and auxiliary gas was used. Applied voltages were 40 and 100 V for the ion transfer capillary and the tube lens, respectively. The ion transfer capillary was held at 275°C. Detection was achieved in the Orbitrap with a resolution set to 100,000 (at m/z 400) and a m/z range between 150-2000 in profile mode. Spectrum was analyzed using the acquisition software XCalibur 2.1 (Thermo Fisher Scientific). The automatic gain control (AGC) allowed accumulation of up to 2*10 5 ions for FTMS scans, Maximum injection time was set to 300 ms and 1 µscan was acquired. 10 µL was injected using a Thermo Finnigan Surveyor HPLC system (Thermo Fisher Scientific) with a continuous infusion of methanol at 100 µL.min -1 . Elemental microanalyses were performed on a Thermo Flash 2000 elemental analyser. For analytic HPLC the following system has been used: 2 x Agilent G1361 1260 Prep Pump system with Agilent G7115A 1260 DAD WR Detector equipped with an Agilent Pursuit XRs 5C18 (100Å, C18 5 μm 250 x 4.6 mm) Column. The solvents (HPLC grade) were millipore water (0.1% TFA, solvent A) and acetonitrile (0.1% TFA, solvent B). Inductively coupled plasma mass spectrometry (ICP-MS) experiments were carried out on an iCAP RQ ICP-MS instrument (Thermo Fisher). Synthesis [Ru(4,4′-Dimethyl-2,2′-dipyridyl)3][PF6]2 (2)FULL PAPER[Ru(4,4′-Dimethyl-2,2′-dipyridyl)3][PF6]2 (2) was synthesized as previously published[8] using RuCl2dmso4. Experimental data fits with the literature.Purity of the sample was assessed by HPLC and elemental analysis.Elemental analysis calcd for C36H36F12N6P2Ru (%): C 45.82, H 3.85, N 8.91; found: C 45.71, H 3.69, N 8.83. .) was dissolved in dry DMF (12 mL) under nitrogen atmosphere and tert-Butoxy bis(dimethylamino)methane (0.9 mL, 4.36 mmol, 21.8 equiv.) was added slowly. The mixture was heated at 140 °C for 40 h. The solution was then cooled down and a sat. aqueous solution of NH4PF6 was added. The crude product, which precipitated as a PF6 salt was collected by filtration and washed with H2O and Et2O. The product was isolated via fractionated precipitation from CH3CN by adding dropwise Et2O. 215 mg of 3 (0.17 mmol, 85 %) were yielded as a dark red solid. 1 H-NMR (CD3CN, 500 MHz): 7.98(d, 4 J = 2.0 Hz, 6H, H3), 7.45(d, 3 J = 13.4 Hz, 6H, H8), 7.24 (d, 3 J = 6.2 Hz, 6H, H6), 6.90 (dd, 3,4 J = 6.2, 2.0 Hz, 6H, H5), 5.09(d, 3 J = 13.4 Hz, 6H, H7), 2.94 (s, 36H, H10). 13 C-NMR (CD3CN, 125 MHz): δ = 157.8 (C2), 150.2 (C4), 150.0 (C6), 147.0 (C8), 120.2 (C5), 116.7 (C3), 93.0 (C7), 40.9 (C10). ESI-HRMS (pos. detection mode): calcd for C54H66N12Ru m/z [M] 2+ 492.2283; found: 492.2284. Elemental analysis calcd for C54H66F12N12P2Ru (%): C 50.90, H 5.22, N 13.19; found: C 50.64, H 4.96, N 12.90. X-ray crystallography X-ray single-crystal data were collected at low temperatures, 160(1) K for 4,4'-bis(N,N-dimethylaminovinyl)-2,2'-bipyridine and at 183(1) K for compound 2, with an Oxford liquid-nitrogen Cryostream cooler on a Rigaku OD XtaLAB Synergy Dualflex (Pilatus 200K detector) diffractometer. A

  absorption of the samples has been measured with a SpectraMax M2 Spectrometer (Molecular Devices). The emission was measured by irradiation of the sample in fluorescence quartz cuvettes (width 1 cm) using a NT342B Nd-YAG pumped optical parametric oscillator (Ekspla) at 355 nm. Luminescence was focused and collected at right angle to the excitation pathway and directed to an Acton SP-2300i monochromator (Princeton Instruments). As a detector a PI-Max 4 CCD camera (Princeton Instruments) has been used. Luminescence quantum yield measurements For the determination of the luminescence quantum yield, the samples were prepared in a not degassed CH3CN solution with an absorbance of 0.1 at 355 nm. This solution was irradiated in fluorescence quartz cuvettes (width 1 cm) using a NT342B Nd-YAG pumped optical parametric oscillator (Ekspla) at 355 nm. The emission signal was focused and collected at right angle to the excitation pathway and directed to an Acton SP-2300i monochromator (Princeton Instruments). As a detector a XPI-Max 4 CCD camera (Princeton Instruments) has been used. The luminescence quantum yields were determined by comparison with the reference

  photomultiplier tube (Hamamatsu) has been used.

Φsample

  oxygen quantum yield, S = slope of the linear regression of the plot of the areas of the singlet oxygen luminescence peaks against the irradiation intensity, I = absorbance correction factor, I0 = light intensity of the irradiation source, A = absorbance of the sample at irradiation wavelength. -Indirect evaluation For the measurement in CH3CN: The samples were prepared in an airsaturated CH3CN solution containing the complex with an absorbance of 0.2 at the irradiation wavelength, N,N-dimethyl-4-nitrosoaniline aniline (RNO, 24 µM) and imidazole (12 mM). For the measurement in PBS buffer: The samples were prepared in an air-saturated PBS solution containing the complex with an absorbance of 0.1 at the irradiation wavelength, N,Ndimethyl-4-nitrosoaniline aniline (RNO, 20 µM) and histidine (10 mM). The samples were irradiated on 96 well plates with an Atlas Photonics LUMOS BIO irradiator for different times. The absorbance of the samples was measured during these time intervals with a SpectraMax M2 Microplate Reader (Molecular Devices). The difference in absorbance (A0-A) at 420 nm for the CH3CN solution or at 440 nm a PBS buffer solution was calculated and plotted against the irradiation times. From the plot the slope of the linear regression was calculated as well as the absorbance correction factor determined. The singlet oxygen quantum yields were calculated using the same formulas as used for the direct evaluation.

  TCI Chemicals. Stock Solutions of the compounds (40 μM) and caffeine (20 μM) were prepared in DMSO. One aliquot of the solutions was added to 975 μL of human plasma to a total volume of 1000 μL. Final concentrations of the compounds of 0.5 μM n-and caffeine of 0.25 μM were achieved. The resulting solution was incubated for 48 h at 37 °C with continuous gentle shaking (ca. 300 rpm). The reaction was stopped after the incubation time by addition of 2 mL of methanol. The mixture was centrifuged for 45 min at 650 g at 4 °C. The methanolic solution was filtered through a 0.2 μm membrane filter. The solvent was evaporated under reduced pressure and the residue was dissolved in 1:1 (v/v) CH3CN/ H2O 0.1% TFA solution. The solution was filtered through a 0.2 μm membrane filter and analysed using an HPLC System. Based on the big differences in lipophilicity, two different HPLC methods have been used. The solvents (HPLC grade) were millipore water (0.1% TFA, solvent A) and acetonitrile (solvent B). Method M1: 0-3 minutes: isocratic 95% A (5% B); 3-17 minutes: linear gradient from 95% A (5% B) to 0% A (100% B); 17-23 minutes: isocratic 0% A (100% B). Method M2: 0-3 minutes: isocratic 80% A (20% B); 3-17 minutes: linear gradient from 80% A (20% B) to 0% A (100% B); 17-23 minutes: isocratic 0% A (100% B). The flow rate was 1 mL/min and the chromatogram was detected at 250 nm.

  concentrations of the compound diluted in cell media achieving a total volume of 200 μL. The cells were incubated with the compound for 4 h. After this time, the media was removed and replaced with 200 μL of fresh medium. For the phototoxicity studies, the cells were exposed to light with an Atlas Photonics LUMOS BIO irradiator. Each well was constantly illuminated with either a 480 nm or 540 nm irradiation.

  Photodynamic TherapyPhotosensitizers via the Extension of a [Ru(bipy)3] 2+ Core

Table 2 .

 2 Singlet oxygen quantum yields in CH3CN and aqueous solution.

		direct	direct	indirect	indirect	indirect	Indirect
		450 nm	450 nm	450 nm	450 nm	540 nm	540 nm
		CH3CN	D2O	CH3CN	PBS	CH3CN	PBS
	1	54%	21%	57%	20%	n.d. a)	n . d . a)
	2	66%	25%	64%	27%	n.d. a)	n . d . a)
	3	n.d.	n.d.	21%	3%	18%	2%
	n.d. = not detectable.

Average of three independent measurements, ±10%.

Acknowledgments

Le Conseil Régional d'Ile de France is acknowledged for providing financial support for the purchase of a 500 MHz NMR spectrometer. This work was financially supported by an ERC Consolidator Grant PhotoMedMet to G.G. (GA 681679) and has received support under the program "Investissements d' Avenir" launched by the French Government and implemented by the ANR with the reference ANR-10-IDEX-0001-02 PSL (G.G.), the National Science Foundation of China (Nos. 21525105 and 21778079 for H.C.) and the 973 Program (No. 2015CB856301 for H.C.). Keywords: Anticancer • Bioinorganic Chemistry • Medicinal Inorganic Chemistry • Metals in Medicine • Photodynamic Therapy