Hector Garcia De Marina
email: hgdemarina@ieee.org

Bayu Jayawardhana
email: b.jayawardhana@rug.nl

Ming Cao
email: m.cao@rug.nl

Distributed algorithm for controlling scale-free polygonal formations. ⋆

Keywords: Formation control, Distributed control, Multi-agent system

This paper presents a distributed algorithm for controlling the deployment of a team of mobile agents in formations whose shapes can be characterized by a broad class of polygons, including regular ones, where each agent occupies a corner of the polygon. The algorithm shares the appealing properties of the popular distance-based rigid formation control, but with the additional advantage of requiring the control of fewer pairs of neighboring agents. Furthermore, the scale of the polygon can be controlled by only one pair of neighboring agents. We also exploit the exponential stability of the controlled formation in order to steer the formation as a whole with translations and rotations in a prescribed way. We provide both theoretical analysis and illustrative simulations.

INTRODUCTION

Accomplishing the tasks of surveillance, exploration, and search & rescue, among others, often requires formation control of multi-agent systems (see for instance [START_REF] Oh | A survey of multi-agent formation control[END_REF]). In particular, an appealing formation control framework based on rigid frameworks for the above mentioned tasks has been discussed in [START_REF] Krick | Stabilization of infinitesimally rigid formations of multi-robot networks[END_REF]. In such setups the agents can form a desired shape by only controlling the distances between neighbors. It is worth mentioning as follows some of the properties of distance-based rigid formation control. Firstly, the agents do not need to share a common frame of coordinates. Secondly, the system is robust against biases when sensing the distances between of neighboring agents (see Garcia de [START_REF] Garcia De Marina | Controlling rigid formations of mobile agents under inconsistent measurements[END_REF]). Thirdly, the motion of the formation can be de-coupled into rotational, translational and scaling movements (we refer to Garcia de Marina et al. (2016a,b)). Forthly, the stability of the closed-loop system is exponentially stable for agents modelled by first or second-order integrators [START_REF] Sun | Exponential stability for formation control systems with generalized controllers: A unified approach[END_REF]. On the other hand, the main drawback of this distance-based approach is that to control the formation, one needs to control at least 2n-3 distances in 2D, in order to be able to achieve a desired shape. This is not required for other approaches such as the positionbased control [START_REF] Oh | A survey of multi-agent formation control[END_REF], but with the possible price of loosing many of the above listed advantages. This paper presents an algorithm for controlling formations in the shapes from a broad class of polygons, i.e., a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain, ⋆ The work of Hector Garcia de Marina was supported by Mistrale project, http://mistrale.eu. The work of Cao was supported in part by the European Research Council (ERC-StG-307207) and the Netherlands Organization for Scientific Research (NWO-vidi-14134).

where each agent occupies a corner of the polygon. We will show that the algorithm has all the advantages from distance-based rigid formation control and at the same time it only needs to control a smaller number of agent pairs. In particular, the assignment of neighboring agents, e.g., the sensing topology of the team, is based on a daisy chain configuration, i.e., a configuration where the agents are connected in series. We will also show that by controlling only the distance between the first and last agent in the sensing topology, one can scale the size of the whole formation up and down freely.

The algorithm is motivated by the distance-based control of non-rigid formations as recently studied in [START_REF] Dimarogonas | On the stability of distance-based formation control[END_REF]. In particular, we exploit the effect of having mismatches in the prescribed distances between neighboring agents. Although one cannot define a particular shape by controlling a non-rigid setup, it is reported in Garcia de Marina and [START_REF] Garcia De Marina | Controlling a triangular flexible formation of autonomous agents[END_REF] that biases in the range sensors of neighboring agents1 can be used to lead the formation to converge to a collinear configuration for a daisy chain network consisting of three agents. In this work we will employ a slightly different approach than in Garcia de [START_REF] Garcia De Marina | Controlling a triangular flexible formation of autonomous agents[END_REF]. In fact, we will strengthen the resultant mismatched control law by clearly identifying two terms. The first term is responsible for controlling distances and derived from the standard gradient descent technique for the chosen potential function. The second term involving the mismatches has a clear interpretation and is responsible for the steady-state collinear configuration. Furthermore, as hindsight, the first term is surprisingly in the same form as the control law presented in Kv-into and Parsegov (2012) and [START_REF] Proskurnikov | Problem of uniform deployment on a line segment for second-order agents[END_REF] for steering equally-spaced agents to a line. We will show that with the technique introduced in Garcia de [START_REF] Garcia De Marina | Controlling a triangular flexible formation of autonomous agents[END_REF], the second term responsible for the alignment of the formation can be modified in order to control a prescribed angle and a prescribed distance between two pairs of consecutive neighboring agents in the daisy chain. Furthermore, the scale of the whole formation can be set by one pair of neighboring agents. The proposed algorithm renders the controlled formation convergning to the prescribed shape exponentially stable. This property, combined with a non-fixed steady-state orientation, allows us to achieve translations and rotations of the desired shape by following the technique introduced in Garcia de Marina et al. (2016a). This paper is organized as follows. We introduce some notation and the notion of framework in Section 2. Then in Section 3 we introduce the daisy chain topology for distance-based control. The addition of distance mismatches between neighboring agents in the control terms leads to an algorithm for deploying agents in a collinear and equally (or relatively) spaced configuration. We modify this algorithm by the addition of rotational matrices in Section 4 in order to control the relative angle between two consecutive relative positions in the framework. We prove the exponential stability under the new algorithm for a broad class of polygons, including regular ones. At the end of Section 4 we exploit this exponential stability in order to control the scale of the desired shape by only controlling the distance between the first and the last agent of the framework, and to induce rigid body motions, i.e., rotations and translations, to the polygon. We present a numerical simulation in Section 5 in order to validate the theoretical findings and we finish the paper with some conclusions in Section 6.

NOTATIONS AND DEFINITIONS For a given matrix

A ∈ R n×p , define A ∆ = A⊗ I 2 ∈ R 2n×2p ,
where the symbol ⊗ denotes the Kronecker product and I 2 is the 2 × 2 identity matrix. We denote by |X | the cardinality of the set X .

Consider a formation of n ≥ 3 autonomous agents whose positions are denoted by p i ∈ R 2 with i ∈ {1, . . . , n}.

The agents are able to sense the relative positions of their neighboring agents. The neighbor relationships are described by an undirected graph G = (V, E) with the vertex set V = {1, . . . , n} and the ordered edge set E ⊆ V × V. The set N i of the neighbors of agent i is defined by

N i ∆ = {j ∈ V : (i, j) ∈ E}. We define the elements of the incidence matrix B ∈ R |V|×|E| for G by b ik ∆ =    +1 if i = E tail k -1 if i = E head k 0 otherwise , (1)
where E tail k and E head k denote the tail and head nodes, respectively, of the edge E k , i.e.,

E k = (E tail k , E head k ).
A framework is defined by the pair (G, p), where p is the stacked vector of the agents' positions p i with i ∈ {1, . . . , n}. The stacked vector of the sensed relative positions by the agents can then be described by z = B T p.

(2) Note that each vector z k = p ip j in z corresponds to the relative position associated with the edge E k = (i, j).

DISTANCE-BASED DAISY CHAIN FRAMEWORKS, MISMATCHES AND THE UNIFORM DEPLOYMENT ON A LINE PROBLEM

Assume that the agent's dynamics are described by the first-order model ṗ = u, (3) where u is the stacked vector of control inputs u i ∈ R 2 for i = {1, . . . , n}.

Consider the following incidence matrix defining a daisychain topology

B =       1 0 . . . 0 0 -1 1 . . . 0 0 . . . . . . . . . . . . . . . 0 0 . . . -1 1 0 0 . . . 0 -1       , (4)
where -1) . The incidence matrix that will help us to control the angles defined by the vectors z k and z k+1 also corresponds to a daisy chain topology but -2) , i.e., the first column of B θ is related to θ 1 as the angle between z 1 and z 2 and so on.

B ∈ R |V|×(|V|
B θ ∈ R (|V|-1)×(|V|

Distance-based mismatched gradient-descent control

For illustration, let us consider the case when our daisy chain framework consists of three agents. We then choose for the distance-based control the following potential function

V (z) = 1 4 (||z 1 || 2 -d 2 1 ) 2 + 1 4 (||z 2 || 2 -d 2 2 ) 2 , (5)
where d 1 and d 2 are the desired distances between the corresponding neighboring agents. Taking the gradientdescent of (5) (as used in Garcia de Marina et al. ( 2016a)) we arrive at the following system

   ṗ1 = -z 1 e 1 ṗ2 = z 1 e 1 -z 2 e 2 ṗ3 = z 2 e 2 , (6)
where are the different desired distances that the agents i and j respectively in E k = (i, j) want to maintain for the same edge. We consider that the mismatches are assigned to the second agent in ( 6) such that we can arrive at the following expression

e k = ||z k || 2 -d 2 k , k ∈ {1, 2} are the distance error signals. Inspired by Garcia de Marina et al. (2016a), let us now include a distance mismatch µ k ∈ R in the edge E k = (i, j), namely d 2 tail k = d 2 head k -µ k , (7
   ṗ1 = -z 1 e 1 ṗ2 = z 1 e 1 -z 2 e 2 + µ 1 z 1 -µ 2 z 2 ṗ3 = z 2 e 2 . ( 8
)
One can identify that system (8) can be derived from a potential function as it has been done for system (6) with the exception of the term µ 1 z 1µ 2 z 2 . In fact, the gradient-descent-derived terms are responsible for the distance control between neighboring agents. If one drops all the terms in (8) involving the control of the d k 's, then one gets

   ṗ1 = 0 ṗ2 = µ 1 z 1 -µ 2 z 2 ṗ3 = 0. (9)
If one considers µ 1 = µ 2 = c then system (9) follows precisely from the algorithm presented in [START_REF] Kvinto | Equidistant arrangement of agents on line: Analysis of the algorithm and its generalization[END_REF] and in [START_REF] Proskurnikov | Problem of uniform deployment on a line segment for second-order agents[END_REF] for solving the problem of deployment on a line, i.e., two fixed points p 1 and p n defining a segment and the rest of agents will be deployed on such a segment at spots equally separated. In particular, as one will see in the following section, the algorithm is stable for c ∈ R + and its compatibility with the distance-based gradient-descent control and its relation with biases in range sensors have been studied in Garcia de Marina and Sun (2017).

Deployment on a line problem

In this section we will prove the stability of the algorithm introduced in system (9) for µ 1 = µ 2 = c but for a general daisy chain topology. The stability analysis in this section is different from the one presented in [START_REF] Kvinto | Equidistant arrangement of agents on line: Analysis of the algorithm and its generalization[END_REF] and in [START_REF] Proskurnikov | Problem of uniform deployment on a line segment for second-order agents[END_REF].

In particular, in this paper we analyze the derived error signals from the algorithm. This approach serves as a starting point for controlling polygons in the plane, not only collinear configurations. Let us define the following error vector e θ = B T θ z, (10) where e θ ∈ R (|V|-2) . Then the extension to n agents from system (9) can be generalized as

               ṗ1 = 0 ṗ2 = ce θ1 . . . ṗn-1 = ce θn-2 ṗn = 0, (11)
where c ∈ R + is a constant gain. Let us write the dynamics of the signal e θ (t). We first derive the dynamics of z from system (11), namely

ż = B T ṗ = -cB θ B T θ z = -cB θ e θ , (12)
and noting that ėθ = B T θ ż we have that ėθ = -cB T θ B θ e θ .

(13) Proposition 1. The origin of system ( 13) is globally exponentially stable. That is, all the agents from system (11) will converge to a fixed point, namely p(t) → p * as t → ∞, where all the agents are equally spaced with respect to each other in a collinear fashion.

Proof. Consider the following Lyapunov function

V = 1 2 ||e θ || 2 , whose time derivative is given by dV dt = e T θ ėT θ = -ce T θ B T θ B θ e θ . ( 14
)
We know that B θ defines a daisy chain topology, i.e., it does not contain any cycles, therefore the matrix B T θ B θ is positive definite [START_REF] Dimarogonas | On the stability of distance-based formation control[END_REF]).

Hence the exponential stability of the origin of e θ follows. Since the signal e θ (t) converges exponentially fast to zero, we know B T θ z(t) → 0 as t → ∞, i.e., z k (t)z k+1 (t) → 0 as t → ∞. Thus, by observing system (11), we have that ṗ(t) also converges exponentially fast to zero. So one can conclude that p(t) converges to a fixed point p * where all the agents are equally spaced and collinearly positioned. Remark 1. Note that for the case p 1 (0) = p n (0), all the agents will converge to the same point, i.e., z(t) → 0 as t → ∞.

Controlling relative magnitudes between relative positions

The relative magnitude between two consecutive relative positions z k and z k+1 can be trivially defined as r k z k = r k+1 z k+1 , where r k , r k+1 ∈ R + are the scaling factors that determine how the magnitude of one relative position with respect to its next neighboring one. This case encompasses, as in ( 9), the particular case of having all the agents equally spaced in the steady state, e.g., r k = 1, ∀k{1, . . . , |E|}. In particular, we have that

z = D r z, (15)
where

D r ∆ = diag ([r 1 . . . r k ]) , with k ∈ {1, . . . , |E|}. So by redefining e θ = B T θ z,
(16) we have that the error dynamics derived from ( 11), as we have done before in Proposition 1, are given by ėθ

= -cB T θ D r B θ e θ , (17)
where the matrix -B T θ D r B θ is Hurwitz since D r is a diagonal positive definite matrix. Therefore, the set defined by the vanishing of the signal ( 16) is globally exponentially stable for system (11).

CONTROLLING POLYGONAL FORMATIONS IN THE PLANE

It is possible to extend the results of Proposition 1 in order to deploy the team of agents on the plane in a more general way. We are going to show that by following the technique introduced in Garcia de Marina and [START_REF] Garcia De Marina | Controlling a triangular flexible formation of autonomous agents[END_REF] one is able to control the relative angle θ k between two consecutive vectors z k and z k+1 . For formations where all these consecutive angles are equal to θ * , we provide a bound to such an angle in order to assert the (exponential) stability of the system. In particular, we will show that such a bound applies to the particular case of controlling regular polygons in the plane.

We introduce the consecutive angles θ k to be controlled in the redefinition of the error signal e θ as follows

e θ k = W θ k 2 z k -W θ k 2 T z k+1 , ∀k ∈ {1, . . . , |V|-2}, (18)
where θ k ∈ (-π, π] and W (α) is the rotational matrix

W (α) = cos(α) -sin(α) sin(α) cos(α) . ( 19
)
Note that in (18) we are comparing the clockwise rotated z k with the counterclockwise rotated z k+1 .

We now write in a compact form the stacked vector of errors in ( 18) as e θ = B T W z, (20) where

B W =             W θ 1 2 0 . . . 0 0 -W θ 1 2 T W θ 2 2 . . . 0 0 . . . . . . . . . . . . . . . 0 0 . . . -W θ n-1 2 T W θ n-2 2 0 0 . . . 0 -W θ n-2 2 T             . ( 21
)
Note that trivially B W is equal to B θ , as defined in the end of Section 2, if we set θ k = 0, ∀k ∈ {1, . . . , |V| -2}. Therefore to deploy on a line is a particular case of the problem considered in this section.

Let us now write the dynamics of z derived from system (11) by employing the error signal ( 18)

ż = -cB θ B T W z, (22)
so it allows us to derive the new error (linear) system dynamics given by ėθ

= -cB T W B θ e θ = -A(θ)e θ , (23)
where θ ∈ R (|V|-2) is the stacked vector of all θ k and A(θ) is shown in (24) in the next page. Note that now A(θ) is not positive definite in general. Therefore in order to check the stability of the origin of e θ in ( 23), one has to do an eigenvalue analysis for A(θ).

Our numerical simulations have shown that not for all the values of θ the origin of ( 23) is stable. In fact, the team of agents might converge to a different shape at the same time when they perform a steady-state motion. This effect has been shown not only for rigid formations with distance mismatches (see [START_REF] Mou | Undirected rigid formations are problematic[END_REF]), but also for flexible formations (as in Garcia de Marina and Sun (2017)) as the daisy chain setup described in this paper. Nevertheless, we can provide an analytical result for the stability of a broad class of polygons where θ k = θ * . In particular we provide a bound for θ * such that the formation is stable.

Fortunately, this bound also applies to the set of regular polygons, which can be of interest in the field of formation control [START_REF] Krick | Stabilization of infinitesimally rigid formations of multi-robot networks[END_REF]. Theorem 2. Consider the n ≥ 3 agent system ( 11) with e θ defined as in ( 18) and θ k = θ * , ∀k ∈ {1, . . . , n -2} . Then, the origin of e θ (t) in system ( 23) is exponentially stable if and only if |θ * | ≤ 2π n-1 .

Proof. Since all the 2D rotational matrices W (α) as in ( 19) commute, then A(θ) is unitarily similar to diag{C(θ), C(θ) † }, where

C(θ) =         2 cos θ * 2 -w(θ * ) -w(θ * ) † . . . . . . . . . . . . -w(θ * ) -w(θ * ) † 2 cos θ * 2         , ( 25
)
with C(θ) ∈ R (n-2)×(n-2) , w(θ) = e j θ 2 , j is the imaginary unit, and the symbol † denotes for the complex conjugate transpose. The matrix C is tridiagonal and Toeplitz, so its eigenvalues have the following analytical expression [START_REF] Noschese | Tridiagonal toeplitz matrices: properties and novel applications[END_REF]):

λ k (θ * ) = 2 cos θ * 2 + 2 cos kπ n -1 , k ∈ {1, . . . , n -2}
(26) hence C is positive definite (so the eigenvalues of A(θ) are positive) if and only if |θ * | ≤ 2π n-1 . Therefore the origin of e θ (t) in system ( 23) is exponentially stable. This fact implies the (exponential) convergence of system (11) to the set given by e θ = 0 with e θ k as in (18). Remark 2. For the particular case of n = 3, we have that for all the values of θ 1 ∈ (-π, π] the system is stable, i.e., we are defining a triangle where its scale is determined by the constant positions p 1 (0) and p 3 (0). Remark 3. Note that for regular polygons we have that θ * = π -π(n-2) n which satisfies the bound in Theorem 2. The angle θ k is not an inner angle of the polygon, but the angle between two consecutives z k and z k+1 .

Note that the algorithm presented in ( 11) with e θ as in ( 20) is able to control 2D shapes employing (n -2) edges, which requires fewer than the (2n -3) edges as in the gradient-descent control of the distance-based rigid setup. This is also the case if one employs position-based control [START_REF] Oh | A survey of multi-agent formation control[END_REF]). However, the algorithm presented in this paper has two important features that are not present in the position-based approach. First, the agents can work employing their own local frames of coordinates. Second, the steady-state orientation of the shape is not fixed, therefore allowing for rotational motion as we will show. These two advantages come from the fact that the presented algorithm is an extension of the mismatched distance-based formation control.

Controlling the scale of the prescribed shape

Consider the example where six agents want to form an hexagon, so the four agents in the middle of the chain would control the inter-angles θ 1 , . . . , θ 4 = π -2 3 π and by looking at (11) we notice that agents 1 and 6 are stationary. The idea is to apply the distance-based control to these two agents at the ends of the daisy chain, and therefore closing the chain. For example, if we are controlling a regular polygon, then all the side-lengths are equal, i.e., D r is the identity matrix in (15). Therefore, if we control the distance d between the first and the last agent, then the rest of distances between neighboring agents will also be equal to d.

For controlling the scale we assign the following control law, derived from a potential function like in (5), to agents 1 and n

ṗ1 = -(p n -p 1 )(||p n -p 1 || 2 -d 2 ) ṗn = (p n -p 1 )(||p n -p 1 || 2 -d 2 ) . (27)
We have already noted that the convergence to the desired distance between agents 1 and n in the nonlinear system (27) is exponential [START_REF] Sun | Exponential stability for formation control systems with generalized controllers: A unified approach[END_REF]). One can treat the terms (27) as a disturbance that vanishes exponentially fast when they are into the dynamics of (11), and hence the stability result in Theorem 2 is not compromised and the scale of the formation can be controlled by only two

          W ( θ 1 2 ) + W ( θ 1 2 ) T -W ( θ 2 2 ) 0 . . . 0 0 0 -W ( θ 2 2 ) T W ( θ 2 2 ) + W ( θ 2 2 ) T -W ( θ 3 2 ) . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . -W ( θ n-3 2 ) T W ( θ n-3 2 ) + W ( θ n-3 2 ) T -W ( θ n-2 2 ) 0 0 0 . . . 0 -W ( θ n-2 2 ) T W ( θ n-2 2 ) + W ( θ n-2 2 ) T           . (24)
agents. In fact, only one agent is needed if we set ṗn = 0 in (27).

In case that one also desires to control the steady-state orientation of the formation, then a position-based control (with an exponential equilibrium) can be applied to agents 1 and n. Remark 4. For the three agents example, the main difference of system (11) with respect to the one presented in Garcia de Marina and Sun ( 2017) is that in the latter the sensing of p 3 -p 1 is not necessary for determining the scale of the triangular formation. In Garcia de Marina and Sun (2017), the agents are also controlling the size of z k as in system (8) where the gradient descent terms have not been dropped out.

Steering the prescribed shape in the plane

The exponential stability in Theorem 2 can be further exploited. For example, one can employ the technique in Garcia de Marina et al. (2016a) in order to steer the whole group with rotational and translational motions. It is obvious that a vector in the plane can be constructed as a linear combination of two non-parallel vectors. Therefore, agent i can construct a velocity vector ṗ * i by just combining two relative positions (available for the agent) from the formation. We note that this is indeed possible for agent i from the system (11) in combination with (27). The main idea is to design a collection of steady-state velocities ṗ * i by employing the relative positions such that p ∈ {z : (e θ = 0) ∧ (||p 1p n || = d)} such that the desired shape is not destroyed, i.e., rigid body motions. For example, the control law introducing such an idea is given by

                         ṗ1 = -(p n -p 1 )(||p n -p 1 || 2 -d 2 ) +µ 11 (p n -p 1 ) + µ 12 z 1 . . . ṗi = ce θi+1 + µ i1 z i-1 + µ i2 z i . . . p n = (p n -p 1 )(||p n -p 1 || 2 -d 2 ) +µ n1 p n -p 1 + µ n2 z n-1 , (28)
where i ∈ {2, . . . , (n -1)} and µ n {1,2} are the motion parameters responsible for the design of the velocities ṗ * i . We illustrate the physical meaning of (28) in Figure 1.

An algorithm describing how to compute these motion parameters such that they define rigid motions for generic shapes can be found in Garcia de Marina et al. (2016a). In fact, these motion parameters can be considered as a parametric disturbances for system (11) considered in Theorem 2. In particular, it has been introduced in Garcia de Marina and [START_REF] Garcia De Marina | Controlling a triangular flexible formation of autonomous agents[END_REF], inspired by the work in 1. Explanation of control law (28) for a square as a prescribed shape. The shape is achieved by the control of the relative angles θ 1 and θ 2 by agents 2 and 3 respectively to π 2 < 2π 3 rads, satisfying the bound in Theorem 2. Note that θ 1 and θ 2 do not define the inner angles of the polygon, but the angles between two consecutive z k and z k+1 . An inner angle is simply πθ k . We set D r in (15) to the identity matrix, so all the norms ||z * k || will be equal at the steady-state. The scale of the square is determined by the control of ||p 1p 4 || (black solid). The velocity of an agent ṗ * i , at the desired shape, is the linear combination of the vectors from its associated relative positions. This velocity ṗ * i can be decomposed into both translational (blue vectors) and rotational (red vectors) components. Note that these velocities are constant with respect to a frame of coordinates O b attached to the desired (body) shape. [START_REF] Mou | Undirected rigid formations are problematic[END_REF], that the error-distance system defined by a rigid framework, whose equilibrium is the desired shape described by θ, is autonomous and exponentially stable. Therefore, the stability of the error-distance system will not be compromised for small µ n {1,2} 's [START_REF] Mou | Undirected rigid formations are problematic[END_REF]), or for big control gains (Garcia de Marina et al. (2016a)). This fact can be employed for giving bounds to the parameters µ n {1,2} 's and the gain c in (28) in order to guarantee the exponential stability of the system for a set of desired velocities ṗ * i (Garcia de Marina et al. (2016a)). Remark 5. It is important to note that by the addition of the motion parameters in (28) we might add undesired equilibria in the system.

O b Og z * 1 ||p 1 -p 4 || * z * 3 z * 2 θ * 1 θ * 2 Fig.

SIMULATIONS

In this section we are going to validate the result of Theorem 2 together with system (28). We consider a team of six agents for achieving a regular hexagon, so D r is the identity matrix in (15). Since the inner angles of the hexagon are 2π 3 , we then set θ * = π -2π 3 = π 3 which is smaller than the bound 2π 5 given in Theorem 2. We define the error distance to be controlled by the agents 1 and 6 as e d The red and magenta agents are the agents 1 and 6 respectively. The red dashed line corresponds to the controlled inter-distance between these two ending point agents in the daisy chain. Note how these two agents converge to the desired inter-distance d (the side of the hexagon) describing almost a straight line before the rotational terms are dominant. The rest of the agents control the angles θ k in order to achieve a regular hexagon. All the agents converge to a rotational motion about the centroid of the formed hexagon. This motion is given by setting all the motion parameters in (28) to same constant. At time t = 150 the agents 1 and 6 change the distance d to be controlled to three times the starting one. This change results in a rescaling of the whole formation.

to induce a spinning motion of the hexagon around its centroid. Such a motion can be accomplished by setting all the motion parameters in (28) equal to µ = 0.025. This can be checked by simple geometrical arguments or by employing the algorithm given in Garcia de Marina et al. (2016a). The simulation results are described in Figure 2.

CONCLUSIONS

In this paper we have presented a distributed algorithm for controlling formations of a broad class of polygonal shapes, including regular ones, defined by a daisy chain topology. The first step is to construct an algorithm for deploying agents equally spaced on a line, which is derived from adding distance mismatches to a standard distance-based controller in the literature. Consequently, a number of properties, such as having agents working using their own local frames of coordinates and non-controlled steady-state orientation, are preserved. In the second step we show that with the addition of rotational matrices one can control the relative angle between two consecutive relative positions in the framework. It turns out that the desired shape is then exponentially stable for a broad class of polygons including regular ones. By exploiting this stability property, one can add a series of useful properties to the formation. Firstly, one can control the relative size of consecutive relative positions in the framework. Secondly, the scale of the whole shape can be achieved by only controlling the distance between the first and the last agent of the framework. Thirdly, motion parameters can be employed in order to steer the formation as a combination of translations and rotations. We are currently working to implement the proposed algorithm in a robotic testbed Garcia de Marina et al. (2017).

∆=Fig. 2 .

 2 Fig.2. The figure on top shows the trajectories of the agents in solid color, where the crosses are the initial conditions. The red and magenta agents are the agents 1 and 6 respectively. The red dashed line corresponds to the controlled inter-distance between these two ending point agents in the daisy chain. Note how these two agents converge to the desired inter-distance d (the side of the hexagon) describing almost a straight line before the rotational terms are dominant. The rest of the agents control the angles θ k in order to achieve a regular hexagon. All the agents converge to a rotational motion about the centroid of the formed hexagon. This motion is given by setting all the motion parameters in (28) to same constant. At time t = 150 the agents 1 and 6 change the distance d to be controlled to three times the starting one. This change results in a rescaling of the whole formation.

In the cited paper, the mismatches have been addressed as a biases. Nevertheless, mathematically speaking in the cited paper both concepts are equivalent.


Dr Susan D Shenkin
email: susan.shenkin@ed.ac.uk

Prof Cyril Pernet

Prof Thomas E Nichols

Dr Jean-Baptiste Poline

Prof Paul M Matthews

Aad Van Der Lugt

Dr Clare Mackay

Prof Linda Lanyon

Prof Bernard Mazoyer

Prof James P Boardman

Dr Paul M Thompson

Dr Nick Fox

Daniel S Marcus

Dr Aziz Sheikh

Simon R Cox

Dr Devasuda Anblagan

Dr Dominic E Job

Prof David Alexander Dickie

David Rodriguez

Dr Joanna M Wardlaw

Henry H Wheeler

Dr Trevor Ahearn

Dr John Ashburner

Ellen Backhouse

Prof Marcelo Barria

Dr Richard Baldock

Dr Mark Bastin

Dr Monique Breteler

Prof Albert Burger

Dr Jonathan Cavanagh

James Cheney

Prof James Cole

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Dr Olimpia Curran

Prof Klaus Ebmeier

Prof Alan Evans

Lorna Gibson

Dr Ludavica Griffanti

Laura Gui

Dr Lewis Hou

Dr Neda Jahanshad

Dr Rajeev Krishnadas

Prof Graeme Laurie

Prof Steve Lawrie

Prof Jennifer Macfarlane

Dr Andrew Mcintosh

Prof John Mclean

Prof Shadia Mikhael

Prof Karla Miller

Prof Giovanni Montana

Dr Alison Murray

Dr Katie Nicol

Dr Wiro Niessen

Prof Mario Parra

Dr Mark Parsons

Dr Steve Pavis

Prof Douglas Potter

Prof Rebecca Reynolds

Neil Roberts

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Dr Dave Robertson

Dr Mark Rodrigues

Prof David Rodriguez Gonzalez

Prof Burkhard Schafer

Dr Ruth Sibbet

Prof Colin Smith

Prof Leslie Smith

Dr Steve Smith

Prof Roger Staff

Dr John Starr

Amos Storkey

Prof Daniel Taylor

Dr Tony Stöcker

Dr Maria Valdes Hernandez

Prof Lana Vasung

Improving data availability for brain image biobanking in healthy subjects: Practice-based suggestions from an international multidisciplinary working group

Keywords: neuroimaging, brain image biobank, data sharing

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

Introduction

Neuroimaging has become embedded in substantial research endeavours to understand normal brain function and effects of disease (e.g. Thompson et al., 2003;[START_REF] Fox | Imaging cerebral atrophy: normal ageing to Alzheimer's disease[END_REF][START_REF] Lemaitre | Age-and sex-related effects on the neuroanatomy of healthy elderly[END_REF][START_REF] Marcus | Open Access Series of Imaging Studies (OASIS): Longitudinal MRI Data in Nondemented and Demented Older Adults[END_REF]Wardlaw et al., 2011;Weiner et al., 2016).

Until recently, many neuroimaging studies were in single centres and, inevitably, of modest size [START_REF] Dickie | Do brain image databanks support understanding of normal ageing brain structure? A systematic review[END_REF]. Many much larger population scanning initiatives are now ongoing [START_REF] Jr | The Alzheimer's disease neuroimaging initiative (ADNI): MRI methods[END_REF], and many multicentre clinical trials routinely include imaging as part of inclusion criteria and as outcome measures [START_REF] Cash | Imaging endpoints for clinical trials in Alzheimer's disease[END_REF], providing the potential for large multicentre collections capturing the range of brain structure in the population. The importance of maximising the value captured in this large amount of imaging data -to detect how differences in brain structure and function relate to behavioural or clinical outcomes -is now widely recognised (Toga, 2002;[START_REF] Barkhof | Making better use of our brain MRI research data[END_REF]Poline, 2012). The value of data for answering new questions can grow with sample size, e.g. for replication, increasing population representativeness, and increasing study power. To address this issue, a growing number of electronic databanks including brain imaging are available, either from dedicated cohorts (e.g. Alzheimer's Disease Neuroimaging Initiative, UK Biobank, IMAGEN), or collections of studies (e.g. Brain Imaging in Normal Subjects, Dementia Platform UK, Open Access Series of Imaging Studies): see Table 1.

Brain images from 'healthy' subjects are important: The wide variation in brain structure and function both within and between individuals at different ages has long been recognised (Wardlaw et al., 2011;[START_REF] Dickie | Variance in Brain Volume with Advancing Age: Implications for Defining the Limits of Normality[END_REF]. Methodologies that use appropriately representative populations are needed to provide normative populations, particularly for healthy subjects (i.e. those without neurological diseases such as stroke or dementia). They can provide informative reports for users (e.g. 'brain on 5 th percentile for volume at age 70' for a specified population) and simultaneously embrace the spectrum of individual variation 3 (Dickie et al., 2015a(Dickie et al., , 2015b)). Brain imaging is increasingly used in the diagnosis of neurological diseases, and mental health disorders [START_REF] Fox | Imaging cerebral atrophy: normal ageing to Alzheimer's disease[END_REF]. Data from existing cohort or population studies (e.g. [START_REF] Marcus | Open Access Series of Imaging Studies (OASIS): Longitudinal MRI Data in Nondemented and Demented Older Adults[END_REF], can help define boundaries between health and disease, to aid diagnosis and trial inclusion, to provide effect size estimates for planning trials, and, where relevant, controls for case-control studies (e.g. Dickie et al., 2015a;ADNI: Potvin et al., 2016).

Current status of brain imaging banks: Large repositories of brain imaging data from wellcharacterised subjects in accessible databanks are required to achieve this, while ensuring that data protection concerns are also addressed. These comprise data initiatives that are planned around harmonised protocols, such as ADNI (Alzheimer's Disease Neuroimaging Initiative) [START_REF] Weiner | Alzheimer's Disease Neuroimaging Initiative. Impact of the Alzheimer's Disease Neuroimaging Initiative, 2004 to 2014[END_REF], UK Biobank (Matthews & Sudlow, 2016), Human Connectome Project [START_REF] Van Essen | The WU-Minn Human Connectome Project: An overview[END_REF], OASIS (Open Access Series of Imaging Studies) (Marcus et al., 2007a(Marcus et al., , 2007b[START_REF] Marcus | Open Access Series of Imaging Studies (OASIS): Longitudinal MRI Data in Nondemented and Demented Older Adults[END_REF], and those that represent data aggregation without initial harmonisation e.g. ENIGMA (Enhancing Neuro Imaging Genetics through Meta- Analysis -Thompson et al., 2014, 2015).

The value of brain images is hugely enhanced by the information on the characteristics of individual subjects and the study in which they participated, but at present studies vary widely in what data they present on the study, subject or image data, and how these data are presented [START_REF] Dickie | Do brain image databanks support understanding of normal ageing brain structure? A systematic review[END_REF].

Only a small proportion of the images performed for research are included in biobanks, and in existing structural brain image biobanks, normal subjects over 60 years of age are relatively under-represented, with limited cognitive and medical metadata to support their classification as "normal" [START_REF] Dickie | Do brain image databanks support understanding of normal ageing brain structure? A systematic review[END_REF], and available with a limited range of neuroimaging sequences. For example, fluid attenuated inversion recovery (FLAIR) and T2* volumes are often not available, although they are essential for sensitively identifying and quantifying white matter hyper-intensities (WMH) and microbleeds respectively, neuropathologies present in normal ageing but associated with vascular cognitive impairment [START_REF] Wardlaw | Neuroimaging standards for research into small vessel disease and its contribution to ageing and neurodegeneration[END_REF][START_REF] Cox | Associations between education and brain structure at age 73 years, adjusted for age 11 IQ[END_REF]. Newer initiatives like BRAINS [START_REF] Job | A brain imaging repository of normal structural MRI across the life course: Brain Images of Normal Subjects[END_REF] provide a range of sequences (e.g., T1, T2, T2*, and FLAIR) for most subjects plus cognitive and medical information. Future data sharing will be facilitated by influencing how new data are collected in terms of core imaging sequences and meta-data variables.

Standards for sharing: The INCF (International Neuroinformatics Coordinating Facility)

Standards for Data Sharing Neuroimaging Task Force the Brain Imaging Data Structure (http://bids.neuroimaging.io/) to advance standard organisation and descriptions of data files, and the Neuroimaging Data Model (http://nidm.nidash.org/) for data provenance tracking, but ongoing work is needed around developing community consensus and adoption of standards [START_REF] Bjaalie | Global Neuroinformatics: The International Neuroinformatics Coordinating Facility[END_REF]. Issues such as privacy, de-identification, quality control, provenance, avoiding including the same subjects in multiple databases, ethics (historical and future), consent, essential components of 'good guardianship', costs, sustainability, software version control, , definitions of 'normality', and international variations in ethical and legal frameworks, also need further consideration [START_REF] González | An open source toolkit for medical imaging de-identification[END_REF]. The European Society of Radiology (ESR) published a position paper on Imaging Biobanks (European Society of Radiology, 2015) defining imaging biobanks, outlining their purpose, and advocating the creation of a network/federation of such repositories with existing biobanks.

Many funders advocate or mandate that data generated by studies they fund are made public and the International Committee of Medical Journal Editors (ICMJE) has proposed that deidentified patient information is shared before research manuscripts of randomised controlled trials will be considered for publication (Taichman et al., 2016). While this data sharing may be relatively straightforward for tabular demographic data (i.e. the types of alphanumeric data that can be held in traditional databases), the situation is much more complex for brain image data (Toga, 2002, Marcus et al., 2007a). Factors like the size of imaging files and the possibility of identifying subjects from images impose non-trivial technological challenges. While initiatives such as NeuroVault (www.neurovault.org - [START_REF] Gorgolewski | NeuroVault.org: a web-based repository for collecting and sharing unthresholded statistical maps of the human brain[END_REF] avoid the problem by publicly sharing statistical maps for data aggregation it does not include whole datasets. By contrast, a repository like OpenfMRI (www.openfmri.org) includes raw-data, with some subject-level variables, which allows newer analyses to be performed. Even when there is a desire to share imaging data, there are a number of technical, legal and practical problems to be overcome: (Poline 2012;Poldrack andGorgolewski, 2014, Pernet &Poline, 2015).

Learning from existing databanks and population studies

Against this background, a group of experts, including specialists in image acquisition and analysis, clinical disciplines, epidemiology, legal, ethics, and data science, met to discuss and debate conceptual, legal, ethical and technical issues around creating brain image banks. We aimed to highlight the issues that need to be addressed, from the ethical to the practical, achieve some consensus, promote best practice and provide useful advice for ongoing and planned studies. The primary aim of the meeting was to encourage data sharing, construct pan-institutional brain image databank consensus, and facilitate linking between databanks.

Here, we describe lessons learned from existing image databanks, provide advice on technical, epidemiological and legal challenges and identify areas where agreement was not reached that should be addressed as the field evolves.

Representatives of major groups involved in neuroimaging databases were present at the meeting (Table 1). We recognise that there are other imaging databases, many summarised in recent publications (e.g. Sharing the Wealth: Brain Imaging Repositories in 2015;

Neuroimage 124 Part B), but here we provide the lessons learned by this international group with experience in building databases and sharing data under various schemas, for healthy participants and people with neurological disorders, from prenatal to old-age. We noted that the information collected in each databank was very different depending on the perspective and expertise of the individual, e.g. computer scientists or data analysts versus clinicians or epidemiologists. 1). Here, we discuss each of these aspects, presenting lessons learned and recommendations.

Data collection

There is a great willingness from many people across the life course to volunteer for brain imaging studies: even when the participants are in their nineties and the study includes prolonged imaging [START_REF] Deary | Cohort profile: the Lothian Birth Cohorts of 1921 and 1936[END_REF]. However, such willing individuals -irrespective of age -tend to be fitter, better educated and less socially deprived than the general population (e.g. [START_REF] Deary | Cohort profile: the Lothian Birth Cohorts of 1921 and 1936[END_REF]Stafford et al., 2013). Extra effort is therefore needed to encourage more representative population sampling, or the consideration of statistical approaches to account for known bias [START_REF] Ganguli | Who wants a free brain scan? Assessing and correcting for recruitment biases in a population-based sMRI pilot study[END_REF].

A. Subject sampling

Several aspects of study design, fundamental to good epidemiology, are currently not prominent among brain image banks. The use of guidelines on study reporting should improve quality (http://equator-ntework.org). The method of subject selection (random, sequential, particular characteristics, etc.), population from which they were selected (e.g.

hospital clinic attendees, primary care attendees, general population) and method (e.g. direct mailing, adverts, any compensation or payment) should be described clearly. In general, research participation varies with social class, education, health status, by ethnic and other minority groups [START_REF] Deary | Cohort profile: the Lothian Birth Cohorts of 1921 and 1936[END_REF]Stafford et al., 2013), hence documenting the original study aims when data are collated for secondary use is important to describe the population.

This can be achieved by citing a paper, 'read me' descriptors, or website that describes the original study, linked from the database.

A subject should only contribute once to a particular analysis. However, several open datasets include the same individuals [START_REF] Dickie | Do brain image databanks support understanding of normal ageing brain structure? A systematic review[END_REF] available -however this approach may not be reliable, and also implies that even deidentified data about an individual cannot be considered truly anonymous, as data may be susceptible to data linkage attacks [START_REF] Fung | Privacy-preserving data publishing: a survey of recent developments[END_REF]. Simply asking participants about inclusion in other studies is unlikely to be reliable. Preferably, subjects would be assigned a unique study identifier linked to a unique study registration, but this is difficult to implement in practice, as there is no central database of studies. However, tools such as The Global Unique Identifier (GUID) may perform this function (https://dataarchive.nimh.nih.gov/rdocdb/s/guid/nda-guid.html). GUID is a universal subject ID which allows participants to be matched across labs and research data repositories, as well as allowing researchers to share data specific to a study participant without disclosing identifiable information. In genetic studies, it is sometimes possible to use a "checksum" method based on a person's genomic data, to test whether any individuals took part in two studies being compared, as this may be important to avoid reporting spurious associations (Franke et al., 2016).

Methods for long term follow up of healthy volunteers and adding linked data including health outcomes many years later, is potentially very valuable and should also be considered wherever possible (e.g. [START_REF] Deary | Cohort profile: the Lothian Birth Cohorts of 1921 and 1936[END_REF][START_REF] Matthews | The UK Biobank[END_REF].

B. Metadata collection

Some studies include extensive socio-demographic data, others have detailed clinical characteristics and vascular risk factors, others focus on cognitive testing, and several have extensive blood or urine biomarkers including genetics. A key aspect to consider is to define the minimum subject metadata to ensure maximum use of the data world-wide.

Age, sex (self-report), handedness (self-report and/or Edinburgh inventory test) and education (total years of education and highest qualification) are important variables that should always be reported, because they are associated with brain structure and function: age [START_REF] Dickie | Do brain image databanks support understanding of normal ageing brain structure? A systematic review[END_REF]; sex and handedness [START_REF] Good | Cerebral asymmetry and the effects of sex and handedness on brain structure: a voxelbased morphometric analysis of 465 normal adult human brains[END_REF]; and education [START_REF] Cox | Associations between education and brain structure at age 73 years, adjusted for age 11 IQ[END_REF].

The chromosomal identification) will depend on the purpose and location of the original study, and description of the population is encouraged (e.g. educational system to allow international comparison).

Further discussions are required to agree a minimum normative subject dataset, but this is likely to include some measure of sociodemographic variables, clinical features (main comorbidities, medications), cognitive ability (at least a general cognitive test).

Availability of biomarkers (such as DNA) may also be useful. The potential to link to data collected for other reasons, such as participation in nationwide Biobanks or data collected during clinical care, may reduce the burden on participants if appropriate consent is provided and relevant reliable data are available.

C. Consent

Individuals must be informed and provide informed consent where possible. One issue is that it may be impossible to re-contact previous participants who have already provided consent for further use of their data, where imaging databanks were not specifically mentioned. In such case, retrospective approval might be given by an ethics committee. Individual projects should also decide whether there is a need to link subjects back to their anonymised data, and consent therefore must include information on being re-contacted.

Individuals and researchers should be informed of who 'owns' the data (often the host institution: principal investigators should be aware of any restrictions if they move between institutions), if there are copyrights on images, and consent could be indicated in a metatag. It is also important to consider what should be done if data are collected from people without capacity to consent (e.g., children: if consent is provided by a parent on their behalf, can the child rescind this on reaching adulthood, and when should a young person be asked to give consent to ongoing participation in a longitudinal study that his / her parent consented to during the child's infancy).

D. Practical consideration in collecting MRI scans across the lifespan

Perinatal: Data from healthy 'normal' fetuses throughout gestation and neonates born at term are currently sparse. Single centres have generally developed repositories to study specific cohorts (e.g. congenital heart disease or preterm neonates), using bespoke processing pipelines or pipelines developed using adult data but optimised for neonates [START_REF] Boardman | Abnormal deep grey matter development following preterm birth detected using deformation-based morphometry[END_REF][START_REF] Miller | Abnormal brain development in newborns with congenital heart disease[END_REF][START_REF] Ball | An optimised tract-based spatial statistics protocol for neonates: applications to prematurity and chronic lung disease[END_REF][START_REF] Blesa | Parcellation of the Healthy Neonatal Brain into 107 Regions Using Atlas Propagation through Intermediate Time Points in Childhood[END_REF]. There are some fetal and neonatal structural MRI atlases (for example: [START_REF] Gousias | Magnetic resonance imaging of the newborn brain: manual segmentation of labelled atlases in term-born and preterm infants[END_REF]Shi et al., 2011;[START_REF] Blesa | Parcellation of the Healthy Neonatal Brain into 107 Regions Using Atlas Propagation through Intermediate Time Points in Childhood[END_REF][START_REF] Makropoulos | Regional growth and atlasing of the developing human brain[END_REF][START_REF] Kabdebon | Anatomical correlations of the international 10-20 sensor placement system in infants[END_REF][START_REF] Oishi | Multi-contrast human neonatal brain atlas: application to normal neonate development analysis[END_REF], but to our knowledge there are no perinatal image banks hosting normal data acquired from multiple studies and sites. A perinatal sub-section of the BRAINS database is under development, and the developing Human Connectome Project (http://wp.doc.ic.ac.uk/dhcp/) aims to make data available from 1500 fetuses and newborns between 20-44 weeks' post-menstrual age.

Experience from perinatal brain repositories is that newborn neuroimaging research is acceptable to parents and carers, especially those with children at risk of long term impairment. Newborns, including those who require intensive care, can be looked after safely in the MRI environment [START_REF] Merchant | A patient care system for early 3.0 Tesla magnetic resonance imaging of very low birth weight infants[END_REF]. Research quality data can be acquired from infants without the use of sedation by using feed-and-wrap techniques, and by allowing flexibility within in the scan schedule to allow for coaxing to sleep and managing wakefulness. Perinatal image data are readily contextualised by maternal, pregnancy and birth information, and it can be analysed with information from infant biosamples [START_REF] Boardman | Common genetic variants and risk of brain injury after preterm birth[END_REF][START_REF] Blesa | Parcellation of the Healthy Neonatal Brain into 107 Regions Using Atlas Propagation through Intermediate Time Points in Childhood[END_REF] and standardised childhood neurodevelopmental / cognitive outcome tests [START_REF] Woodward | Neonatal MRI to predict neurodevelopmental outcomes in preterm infants[END_REF]: achieving prospective consensus over minimum datasets in these domains will maximise opportunities of perinatal data-sharing in future initiatives.

Older age: Normal ageing is associated with increasing variability in brain structure, such as brain tissue loss (atrophy) and the accumulation of white matter hyper-intensities, which may or may not have functional impact on cognition, mood, gait etc. (Dickie et al., 2015a). Many brain image biobanks were created to study pathological ageing, e.g. dementia, and only small numbers of 'healthy' older people were previously included in accessible databanks [START_REF] Dickie | Do brain image databanks support understanding of normal ageing brain structure? A systematic review[END_REF], making it difficult to define 'normal ageing' or pathological ageing.

The availability of data from UK Biobank (http://www.ukbiobank.ac.uk/) and other ongoing population imaging initiatives in North America and mainland Europe will change this. One conclusion from the workshop was that, for databanks to accurately represent the spectrum of health, a large number of participants, sampled in equal numbers across appropriate age bands are required, accompanied by detailed descriptions of the demographic and clinical characteristics of individuals.

Addressing Data heterogeneity

Where more than one study is included in a brain image bank, like 3-CITIES (Alperovitch et al., 2002) or BRAINS [START_REF] Job | A brain imaging repository of normal structural MRI across the life course: Brain Images of Normal Subjects[END_REF], there is usually substantial heterogeneity of the acquired demographic/clinical and imaging data. This can be addressed either by describing each variable (3-CITIES), or by harmonising metadata (BRAINS). Having many variables makes the database large and difficult to search, while transforming variables to agreed standards, which is simpler for the end user, is resource intensive and requires additional data provenance documentation. Alternatively, in approaches such as ENIGMA (Thompson et al., 2015) and UK Biobank [START_REF] Matthews | The UK Biobank[END_REF], both raw and derived data from each of the participating centres can be used for large scale analyses: either in meta-analyses, which can circumvent issues of data sharing and transfer between countries, but may restrict the analyses that can be performed; or in 'mega'-analyses using original data, which may be difficult to access due to data access controls, and/or too different to combine. Choosing the right framework clearly depends on the question(s) the database aims to address, and requires collaboration between local data providers and the databank.

A. Minimum provenance of study data

Brain imaging uses indirect measures to make inferences about brain structures. The meaningfulness of these measures will vary with the amount and heterogeneity of the data.

The version of scanner and software used for data collection and analysis should be clearly documented. Decisions need to be made about the inclusion of raw or processed data in the database, recording of processing steps, and whether outputs of imaging data will be raw, pre-processed or fully processed. The amount of data storage required for imaging (and e.g. genetic) data may require the use of high performance networking, storage and computing, which can be upgraded without compromising the database, adequate bandwidth, and/or the use of cloud computing, taking account of privacy and security issues [START_REF] Poldrack | Making big data open: data sharing in neuroimaging[END_REF].

Data provenance (study, aims, date performed, funders, principal investigator, recruitment method, publications) is important for appropriate citation and recognition of data sources, encouraging reproducibility, avoiding duplicates, and data versioning. This information can easily be documented on a per study basis without complex tools, though formal representation will maximise the potential for reuse. For example, the W3C-PROV specifications is a framework to interchange provenance information (http://www.w3.org/TR/prov-dm/), and its extension for neuroimaging, NIDM (http://nidm.nidash.org - [START_REF] Keator | Towards structured sharing of raw and derived neuroimaging data across existing resources[END_REF]

B. Role of Ontologies

To allow comparisons between banks, many groups are now working on methods to compare clinical and imaging variables, and using appropriate ontologies. 2012) or the Cognitive Atlas [START_REF] Poldrack | The Cognitive Atlas: Towards a knowledge foundation for cognitive neuroscience[END_REF]. It is important that the databank clearly describes which ontology was used, how decisions were made, and that all metadata variables are clearly defined.

Database infrastructure

Many of the studies that led to the creation of imaging databanks started over a decade ago, and reported issues relating to changing technology [START_REF] Mazziotta | A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM)[END_REF]. For example, technical staff need to consider the impact of hardware changes (e.g. upgrading or changing scanner software or hardware; changes in data storage solutions and formats) and software evolution, which can make keeping track of multiple analyses of the database challenging [START_REF] Poldrack | Making big data open: data sharing in neuroimaging[END_REF]. Such changes in technology have, for instance, been shown to impact on local brain volume [START_REF] Lorio | Neurobiological origin of spurious brain morphological changes: A quantitative MRI study[END_REF] and atrophy measurements [START_REF] Leung | Effects of changing from non-accelerated to accelerated MRI for follow-up in brain atrophy measurement[END_REF].

Changing requirements for data governance and data management also need to be considered: file names and structures may need to be updated to newer recommendations such as BIDS (Brain Imaging Data Structure) [START_REF] Gorgolewski | NeuroVault.org: a web-based repository for collecting and sharing unthresholded statistical maps of the human brain[END_REF]. Similarly, better understanding of disease aetiology, or changes in taxonomy, may affect how clinical characteristics are coded,

and therefore what a database entry means, e.g. changing definitions of dementia subtypes.

Studies may be cross-sectional or longitudinal. In studies of development or ageing, longitudinal data are particularly valuable [START_REF] Mills | Methods and considerations for longitudinal structural brain imaging analysis across development[END_REF], and systems need to be in place to ensure that future data acquisition can be matched to the correct subject, and that imaging parameters are similar enough to allow comparison.

A. Technical infrastructure

To maximize usage and usability, any planned databank should make use of a formal imaging database tools. We considered five software tools that create sharable, searchable databases and offer maximum flexibility: COINS (http://coins.mrn.org/) [START_REF] Landis | COINS Data Exchange: An open platform for compiling, curating, and disseminating neuroimaging data[END_REF] (Marcus et al, 2007a). The ability to link and search make those applications different from repositories; however each has strengths and limitations, some significant. Information reported below reflect user experience and discussion with software developers.

Each of these different software tools has different strengths, discussed at the meeting. allowing direct transfer between the database and the health care service. As imaging is just one facet of any research study, it is vital that all these tools also allow storing of non-MRI data such as demographic, clinical, behavioural, and genetic data. Other types of data such as electroencephalography can be stored but may not be viewable using imaging analysis/display interfaces, and may require additional software.

Another important aspect to consider relates to the software: (1) How are MRI data linked with other types of data? ( 2) How can the data be visualised?; and (3) How can the data be explored? Linkage (bringing together datasets) is an important aspect to consider regarding the size of the dataset and its utility. For instance, LORIS is built with two distinct databases (imaging vs non-imaging) that allows the user to interrogate, process and retrieve information separately or together. XNAT uses an XML-defined schema for searching data, and also uses and underlying object-relational database PostgreSQL. Like LORIS, this allows searching and accessing data separately. Considering the end-user perspective of the database one wishes to build, all five tools allow one to search for image data within and across projects and to visualise data, though with some restrictions depending on the software.

The final important aspects to consider are the overall maturity, usability, maintainability, extensibility and support of the software and access control. LORIS and XNAT, for instance, are well established and maintained, while newcomers (COINS, Scitran, NiDB) are more adapted to new data types and use more recent software technology. Mostly, the software can be installed with minimal programming expertise -but this differs between tools. Setting up multiple access control levels, linking or adding new functionalities, can however require much more expertise. For instance, XNAT, the most widely used platform, requires programming expertise to enable its extensible XML structure. In contrast, simpler tools with extensive search solutions exist (e.g. LORIS, NiDB), but modifying the search tool to suit dedicated needs also requires dedicated programming expertise. In terms of usability, it is worth considering data visualization, and whether the pipeline includes data quality control or basic data pre-processing. All the platforms considered include such options but various levels of programming expertise are required. It is important to also consider 'futureproofing' the technology, at a minimum ensuring accurate version control, and direct access to code. Another essential aspect to consider is access control: COINS, LORIS, Scitran,

XNAT have extensive security levels to create, read, and edit data, while NIDB is limited to user/project management. Thus, at present, the consensus was that there is no 'best' software, the field is moving rapidly, and it is worth considering every aspect discussed here with expert advice to choose a software tool that best suits specific needs.

B. Security

A brain scan is unique, and could allow identification of the individual (e.g. by someone who already holds a copy of their brain image). There is a trade-off between removing all potential identifiers and retaining the scientific value of the data. Clear processes are required regarding sensitive, or potentially identifiable, data to ensure that all reasonable safeguards are put in place, e.g. the DICOM Confidential software; completely removing all textual personally identifiable information; generating new "anonymous" identification numbers;

"defacing" brain MRI (Marcus et al., 2007b, Milchenko and[START_REF] Milchenko | Obscuring surface anatomy in volumetric imaging data[END_REF]Rodriguez, 2010).

A balance between accessibility and security is important, ensuring that potentially identifiable data (text or images) are protected from remote access in the database and if released for external use. It should be clear to the intended users what the data are, and what will be the safeguards to access. To test whether data can be hacked, a mock database can be developed, released and attempts made to infiltrate the security systems.

C. User interface

The user interface should suit the main proposed users. The requirements of clinicians, researchers, or industry are however likely to be very different (for instance searching per pathology vs. scan feature, looking for raw vs. processed data). It is worth considering the user interface and its design. In XNAT for example, the level of detail displayed to users can be made dependent on both their relationship with the databank (external user, contributor, etc.) and/or the intended use of the data of interest (Marcus et al., 2007a). The use of an Application Program Interface (API) allows for easier creation and customisation of user interfaces for different user groups, but can bring new security concerns.

D. Data release

In ongoing/longitudinal studies, there is a wide variation in when data are released: as collected, in batches, on completion, or after all analyses performed by initial research team.

There is currently no consensus on how to release data, but it is important to make that decision clear and have mechanisms in place to identify releases (data version control).

Primary researchers who developed the database should apply through the same mechanism once the databank is public.

E. Quality assurance and control of data

Quality assurance (QA) of all data is key to providing high quality and robust research findings [START_REF] Ducharme | Trajectories of cortical thickness maturation in normal brain development--The importance of quality control procedures[END_REF]

Database management

The legal and ethical framework of individual countries, and agreements reached between them, may affect how and where data are or can be stored. Systems are required to ensure data security, but allow appropriate access. Relevant approvals should be transparent, e.g. in publications and on websites.

During the meeting it was recognized that brain image databanks should have a Steering Committee, including independent and lay representatives, to monitor and review progress.

This has the advantage of providing oversight of data usage and the opportunity to review data requests, but can be time consuming, it may be difficult to get agreement among stakeholders, and therefore delay access to data, and it can be difficult to identify lay representatives. Consideration should be given to how 'sleuthing' of individual's data can be minimised or prevented, and to the legal and ethical aspects of data collection, storage and sharing. This is relevant both for adult and perinatal studies (for both mother and baby).

Decisions need to be made in advance about what will happen to data if an individual decides to opt out; loses capacity in the future (e.g. develops dementia); or gains capacity to make his or her own decisions (e.g. a child growing up). Decisions should be able to be reviewed by the steering committee if legal or ethical frameworks change.

A. Legal and ethical issues

The legal framework may vary between countries, and this should be considered in international collaboration. The general principle is that researchers should satisfy the governing body, e.g. ethics committee, which they are processing and dealing with people's data responsibly. It is useful to consider the concept of a 'motivated intruder': have reasonable steps been taken to protect data from someone making attempts to access the data.

Factors such as what will be done to divulge findings of abnormalities found on neuroimaging (e.g. tumours, or features of cerebrovascular disease, or multiple sclerosis) should be considered (Wardlaw et al., 2011;Wardlaw et al., 2015). Some primary studies have all scans evaluated by a neuroradiologist, others explicitly state that no feedback will be given on any tests. The issue for databanks is what to do if an abnormality of potential health significance is noted, or new health implications for existing findings come to light during secondary use -should the information be fed back to the participant, or the researchers that produced the data? These issues should be considered in the data donation agreement. If data are fully anonymised such feedback would not be possible.

The UK Health Research Authority (http://www.hra.nhs.uk/) is one example of a body that promotes research to improve clinical care of patients. The sharing and use of already collected data with appropriate safeguards fits this duty. The Royal Society, Science as an Open Enterprise, (2012) report, promotes Intelligent Openness -intelligibility, verifiability, accessibility if robust, if there is commercial confidence and personal privacy.

B. Publication

To maximise use, databanks should be registered on a publicly accessible registration website. Currently there is no international registry for neuroimaging studies or databanks. A general platform such as ClinicalTrials.gov could be used in the interim. [START_REF] Bandrowski | A cultural shift in publishing[END_REF]. The consensus from our meeting was to discourage authorship, but to have the data paper cited along with the inclusion of the databank acronym with a web-reference or, better still, a DOI.

A standardised template for reporting these papers (such as CONSORT for randomised controlled trials, see http://www.equator-network.org/) would be useful. This would allow an emphasis on clinical and epidemiological as well as technical perspectives.

C. Funding

Increasingly, funders are keen to encourage data sharing; indeed some go so far as to refuse future funding unless the results of prior funded studies have been published open source (http://www.wellcome.ac.uk/About-us/Policy/Policy-and-positionstatements/WTD002766.htm). While most brain imaging studies are individually funded through grants to acquire data and undertake initial analyses, it has previously been difficult

to obtain funding to create a brain image bank, or if initial funding was secured, then it may be insufficient to maintain the database long term, and deal with data requests. Storage and back up is, however, necessary, and inclusion for provision of this is encouraged. Researchers at ADNI estimate that 10-15% of funding, and 15% of time has been spent on data sharing [START_REF] Wilhelm | Approaches and costs for sharing clinical research data[END_REF]. The use of large-scale distributed computation can make the work more efficient, but users need to be aware of the heterogeneity of the constituent datasets e.g., in ENIGMA no one national government had to finance all the capital infrastructure.

Similarly, a distributed image databank could be supported by funding from individual countries, much as are some multinational clinical trials. There is a tension between the desire to share the data and the feasibility of actually affording to do so, and a range of models exist. [START_REF] Gorgolewski | The brain imaging data structure, a format for organizing and describing outputs of neuroimaging experiments[END_REF].

A federated international network of normative brain image biobanks is achievable (see table 2) and would have many advantages, including: 1) facilitating large scale meta-analyses of brain structure and function, in health and disease, following successful precedents (e.g.

genetics [START_REF] Hibar | Common genetic variants influence human subcortical brain structures[END_REF]depression -Schmaal et al., 2016;schizophrenia -van Erp et al., 2016;bipolar disorder -Hibar et al., 2016); 2) avoiding duplication of effort by data reuse, as occurs widely in physics; 3) providing population controls; 4) increasing research efficiency where research questions could be answered using existing data; and 5) providing a mechanism to replicate results in different demographic populations. The barriers to achieving this vision are political, ethical, technical, and financial, but a federated international group could work with funders, legal and ethical experts and industry along the lines of the solutions proposed here. Most large scale recent initiatives are still within geographical regions (e.g. the Obama BRAIN initiative; the European Human Brain Project), with some exceptions, such as the ENIGMA initiative spanning 35 countries (Thompson et al., 2015). Truly global, inter-regional initiatives are needed, to make full use of neuroimaging to understand the brain across the life-course.

Figure 1 .

 1 Figure 1. Major issues identified in building a brain image biobank (arrows indicate which aspects constrain each other)

  Comparing first the aspects related to the nature and format of imaging data, all of the above software tools support the Digital Imaging and Communications in Medicine (DICOMhttp://dicom.nema.org/) format and can interface directly with scanners. Some software tools support almost any other image formats (COINS, NiDB, XNAT), while others are restricted to specific ones (LORIS, Scitran). Only XNAT has an explicit tool to link the data from the database directly to clinical PACS (Picture Archiving and Communication System), i.e.

Table 1 : Databases presented at brain image bank meeting with relevant references, website, and data access policy

 1

	Data Base	Website	Data access policy
	ADNI (Weiner, 2015)	http://adni.loni.usc.edu/	ADNI: registration and
	ENIGMA (Thompson et	http://enigma.ini.usc.edu/	application for approval by
	al., 2015)		steering committee for data
			access
			ENIGMA: data is shared
			between members on an ad
			hoc basis
	BRAINS (Job et al.,	http://www.brainsimagebank.a	Open access search of
	2016)	c.uk/	available data; registration and

  potentially distorting any results if several such databanks are used in one analysis. Methods are needed to identify if a subject is included in multiple studies. Identification of the uniqueness of a subject's inclusion in a database is a significant problem, with few effective solutions at present. 'Pseudo-

anonymised' identifiers can identify individuals in longitudinal studies. Probabilistic matching of clinical or imaging data could identify repeat subjects if enough data were

  detail in which variables are collected, and what other meta-information is required depends on the setting. In a clinical setting, 'normal' or healthy is often defined as above or below a cut-off, the definition may be somewhat arbitrary (e.g. blood pressure of 140/90;

cognitive test score above a threshold; blood test value). In a research setting, the concept of 'normal' may refer to healthy controls, who do not have the disease of interest (but may have other conditions) and meet several other criteria defined within the study. People recruited to a study because they are 'healthy' or 'normal' may have occult or undeclared disease, have borderline values (e.g. blood pressure of 138/88) or develop disease in the future. It is essential that those participating in a study are well-described if their data are to be used for other purposes. A balance needs to be struck between overburdening subjects, versus inadequate information. The method of collection of data (e.g. self-report of sex vs.

  More importantlyfor new studies, consent must include information on data reuse and sharing. The Open Brain Consent initiative (https://open-brain-consent.readthedocs.io/en/latest/) provides sample consent forms to allow deposition of anonymised data to public data archives, and a collection of tools to facilitate anonymization of neuroimaging data to enable sharing. This initiative aims to facilitate neuroimaging data sharing. It is based on the legal and ethical framework of the USA, but is adaptable to other countries.

  . All QA steps of data collection (blinding of researchers, checking of data entry, standard operating procedures and calibration of equipment, particularly methods such as phantom scanning to describe scanner stability) should be recorded and provided with the data. Quality control should be implemented at all stages of the database from provenance to visualization. Data aggregation centres could provide a useful service with a common QC procedure across all datasets included. An important step is to be transparent on how this is implemented and what is tested on the data. If it is planned to incorporate processed data from external groups (e.g. templates, feature measures, etc.), how will the quality control be implemented? With increasing open access and secure webbased repositories, one option is to link to such repositories rather than incorporating secondary data into the database. This encourages early sharing of secondary (summary) data without the need to request access to the original databases (the model being pursued by UK Biobank) that have more stringent safeguards and comprehensive data access agreement to control research usage.

  records how the databank was established, and the decisions made at each stage (e.g. for UK Biobank at http://www.ukbiobank.ac.uk/wp-content/uploads/2011/11/UK-Biobank-Protocol.pdf) and a 'data paper' that can include technical data and then be cited in the methods section of future results papers (e.g. for the imaging in UK Biobank

	http://www.nature.com/neuro/journal/v19/n11/full/nn.4393.html). The wording on citation
	and authorship should not contravene ICJME (International Committee of Medical Journal
	Editors) rules http://www.icmje.org/recommendations/browse/roles-and-
	responsibilities/defining-the-role-of-authors-and-contributors.html. There is also the
	possibility of having a Research Resource Identifier (RRID

One option would be a data posting website for imaging, like dbGaP for genetics (http://www.ncbi.nlm.nih.gov/gap). Researchers should be encouraged to publish a 'protocol paper' which

  Some organisations, such as DataLad (http://datalad.org/) have developed infrastructure to provide access to scientific data available from various sources (e.g. lab or consortium websites or data sharing portals) through a single interface and integrated with software package University on January 27-30th 2015, which led to the development of the Brain Imaging Data Structure (BIDS -http://bids.neuroimaging.io/,

	managers
	Conclusions
	Brain image biobanking is a rapidly evolving field. Several related and relevant projects will
	complement our recommendations, such as the International Neuroinformatics Coordinating
	Facility (INCF) Neuroimaging Data Sharing Task Force
	(wiki.incf.org/mediawiki/index.php/Neuroimaging_Task_Force) meeting held at Stanford

One model of cost-recovery used in UKBiobank (http:www.ukbiobank.ac.uk) is a fixed charge for reviewing the application, and a variable component depending on the data requested.

Table 2 . Action points for global data sharing via brain image databanks Key issue Reason Suggested action

 2 Whalley, Dr Rebecca Woodfield, Prof David Wyper, Kaiming Yin, Dr May Yong, Dr Gabriel Ziegler.

	Data provenance unclear	Establish if results relevant	Clear data provenance tagged
		for user's population	to each data item e.g. W3C-
			PROV
	Acknowledge role of	Ensure clear recognition	Data use agreement templates
	original studies, and	and ownership of data	should be published, and
	databanks		shared between databanks
	Ensure appropriate	Define whether data	Ethical committee review of
	informed consent	storage, sharing and future	applications, share consent
		use is permitted	templates between databanks
	Oversight of databank	Appropriate use of data,	Studies should have a steering
		data sharing	committee, with independent
			oversight, and consider
			legal/ethical/lay members
	Variation in data collected Difficult to share data,	Agree minimum dataset (e.g.

Acknowledgements

Thanks to Guarantors of Brain, British Geriatrics Society (Scottish Branch), Royal Society of Edinburgh, SINAPSE (Scottish Imaging Network: a Platform for Scientific Excellence) SPIRIT, International Neuroinformatics Coordinating Facility and Nuffield Foundation for contributions towards funding the meeting which formed the basis of this paper. TEN is supported by the Wellcome Trust (100309/Z/12/Z).

Funding sources

The writing of this paper did not receive any specific grant from funding agencies in the public, commercial or non-for-profit sectors. Guarantors of Brain, British Geriatrics Society (Scottish Branch), Royal Society of Edinburgh, SINAPSE (Scottish Imaging Network: a Platform for Scientific Excellence) SPIRIT, International Neuroinformatics Coordinating Facility and Nuffield Foundation made contributions towards funding the meeting which formed the basis of this paper

Authorship

Papers that use data from