Ludovic Sacchelli
email: ludovic.sacchelli@univ-lyon1.fr

Jean-Baptiste Caillau
email: jean-baptiste.caillau@univ-cotedazur.fr

Thierry Combot
email: thierry.combot@u-bourgogne.fr

Jean-Baptiste Pomet
email: jean-baptiste.pomet@inria.fr

Zermelo-Markov-Dubins with two trailers

Keywords: Zermelo navigation, minimum time, turnpike, integrability, Kovacic algorithm

We study the minimum time problem for a simplified model of a ship towing a long spread of cables. Constraints are on the curvature of the trajectory as well as on the shape of what represent the spread of cables here. This model turns out to be the same as a cart towing two trailers and rolling without sleeping on a plane in uniform translation. We analyse the Hamiltonian system describing the extremal flow given by Pontrjagin maximum principle. We detail the equilibria of the system and prove that, contrary to the case of one trailer studied previously by part of the authors, it is not solvable by quadratures. Preliminary numerical results are given.

INTRODUCTION

The present note takes place in a line of work motivated by the optimization of turns and maneuvers of marine vessels towing a set of long and fragile underwater cables. It is a follow-up to [START_REF] Caillau | Zermelo-Markov-Dubins problem and extensions in marine navigation[END_REF], where the interested reader can find many more details about motivations in terms of marine seismic acquisition. Each of these ships collects data from the ocean ground via sonic sources and sensors located in a spread of cables it is towing. This can be done only while the ship is sailing along a straight line. In a typical campaign, the ship runs on parallel pre-defined straight lines, and must perform a u-turn at the end of each straight line to position itself at the starting point of the next one, acquisition being stopped during this maneuver. This maneuver is not constrained to follow a specific path, and hence can be optimized. The objective is to perform it in minimum time with given starting and end points, while being gentle enough to preserve the integrity of the spread of streamers. Since a typical u-turn can take more than an hour, minimizing time is important. Integrity of the spread of streamers, a few kilometers long, is primarily a matter of bounding the curvature of the trajectory. The starting and end points are at the end of a straight line and the beginning of the next one respectively; however if the state of the model takes into account the shape of the towed spreader, it should also be specified that it has to be in the right position at the end of the u-turn, i.e. in the relative equilibrium that is asymptotically attained during a straight line. There are also motivations in terms of traffic near airports of unmanned aerial vehicles [START_REF] Techy | Minimum-time path planning for unmanned aerial vehicles in steady uniform winds[END_REF].

After describing the model in Section 2, we apply Pontrjagin maximum principle: the minimum time extremals of the problem are solutions of a Hamiltonian system and can be regular or singular, as detailed in Section 3. Then, equilibria of the singular extremals are studied in Section 4; among these, hyperbolic points play an important role and are related to the turnpike phenomenon. In Section 5, we give obstructions to solvability by quadratures (integrability) both for the singular and regular extremals thanks to differential Galois theory. Preliminary numerical simulations of the system with two trailers are provided in the final section.

MODEL

If one takes into account only the position and orientation of the ship and the constraint is a bound on the curvature of the trajectory, one gets the so called Dubin's problem [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF]: the magnitude of the speed being fixed, one seeks the shortest path from a point to another (including direction of the tangent) with a bound on curvature. The maximum curvature has to be small enough in order to preserve the integrity of the towed equipment during the turn. See [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF]; [START_REF] Sussmann | Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control[END_REF]; [START_REF] Boissonnat | Shortest paths of bounded curvature in the plane[END_REF], and textbooks for Dubin's problem. There are two drawbacks to this approach: it does not take into account possible sea currents, and it does not contain any description of the hydrodynamic behavior of the towed cables (the dynamic equations only contain a kinematic of the ship itself).

Adding the sea current into the problem, still without modelling the cables behavior, leads to a so-called Zermelo-Markov-Dubins problem (the term was apparently coined in Bakolas and Tsiotras ( 2013)) where it is well documented, see also [START_REF] Techy | Minimum-time path planning for unmanned aerial vehicles in steady uniform winds[END_REF]. In [START_REF] Caillau | Zermelo-Markov-Dubins problem and extensions in marine navigation[END_REF], we introduced a possible model for the towed cables, consisting in replacing them with a finite number of rigid links or "trailers", their dynamic (in fact kinematic) equations coming either from a simple punctual drag force applied by the ocean to the spread at each "joint", or from mimicking the equations of rolling without slipping, in the frame that moves with the fluid, as if each link was a trailer on wheels (see [START_REF] Jean | The car with N trailers: characterisation of the singular configurations[END_REF]). That are two types of models, that we call for short rolling without slipping models and drag models. Models also differ by the number of links, or number of trailers. For a single trailer, drag coincides with rolling without slipping. The state variables may be chosen as follows: two cartesian coordinates (x, y) and an angle θ for the position and orientation of the ship, or towing vehicle in the vehicle with trailers point of view, plus as many angles as trailers, α i being the angles between the (i-1) th and the i th trailer (where the towing vehicle is counted as the 0 th trailer).

The case of a single trailer (where, as we just mentioned, drag or rolling without slipping models coincide) was examined in [START_REF] Caillau | Zermelo-Markov-Dubins problem and extensions in marine navigation[END_REF]. There, we explain among other things that this optimal control problem is Liouville integrable. Here we investigate only the case of two trailers and "rolling without slipping"; the conclusions do not differ for the "drag" model but we do not present them due to space limitation. They enjoy interesting properties but we prove that they are not integrable, prohibiting an almost explicit resolution.

Since the model is a heuristically approached model for a ship towing a spread of streamers but an exact kinematic model for a cart towing two trailers all rolling without slipping for instance on a conveyor belt in translation, and despite the motivation for navigation, we now consider the latter, illustrated in Figure 2, rather than the ship.

Parameters. There are five scalar parameters to this minimum time problem: W ∈ R 2 , the current speed with respect to a frame fixed to the ocean's floor, supposed constant (depends neither on time nor on the point), the magnitude V 0 > 0 of the longitudinal speed of the towing vehicle with respect to the mobile frame, the minimum curvature radius R min > 0, and the length L > 0 of each link. Via a rescaling of time and space, and a rotation that brings W to the 0x semi-axis, the magnitude of the longitudinal velocity as well as the maximum curvature (or minimum curvature radius) of the trajectory in the ping on a conveyor belt. Using complex notations in the plane, z = x+i y is the middle of the axle of the cart, whose speed with respect to the conveyor belt is e iθ (magnitude normalized to 1). The control u is the angular velocity θ of the cart, after re-mormalization.

The first trailer is attached to the cart at point z, and its angle with respect to the axis of the cart is α 1 , so that the other end of the cart is at point z 1 = z + e iθ+α1 , rolling without sleeping means that the velocity of point z 1 with respect to the conveyor belt is along the axis of the trailer, i.e. has polar angle θ + α 1 . The second trailer is attached to the first one at z 1 , its angle with respect to first one is α 2 , the other end of that second cart is at θ+α1+α2) , and the velocity of z 2 with respect to the conveyor belt has to have polar angle θ + α 1 + α 2 .

z 2 = z 1 + e i(
mobile frame become equal to 1, and there remains only two scaled dimension-less parameters: = L/R min , w = W /V 0 .

(1) We will see in (8) that w and cannot be too large.

Equations. Variables x, y, θ, α 1 , α 2 , u being defined in Figure 2, the state is q = (x, y, θ, α 1 , α 2 ) ∈ R 2 × S 1 × (S 1 ) 2 (α 1 , α 2 are further restricted below), and the control is

u ∈ [-1, 1]. The equations read                    ẋ = cos θ + w , ẏ = sin θ , θ =u , α1 = -u - sin α 1 , α2 = sin α 1 - cos α 1 sin α 2 , |u| ≤ 1 , (2)
or q = F 0 (q) + uF 1 (q) , |u| ≤ 1 , with

F 0 = (cos θ + w) ∂ ∂x + sin θ ∂ ∂y - sin α 1 ∂ ∂α 1 + sin α 1 - cos α 1 sin α 2 ∂ ∂α 2
(3) and

F 1 = ∂ ∂θ - ∂ ∂α 1 • (4)
There are constraints on some parameters ans states. First, it is clear that ẋ is non negative if w ≥ 1, forbidding any kind of controllability. Hence we assume 0 ≤ w < 1 .

(5) Another point is that we want a model where, for any control in the prescribed bound (|u| ≤ 1), the angles α 1 , α 2 remain "small" if their initial condition is "small". Let us set u = ±1 and examine a solution α 1 (t), α 2 (t) with initial condition close to zero. If > 1, α 1 goes unbounded, but if < 1 it converges exponentially to ±α * 1 , with

α * 1 = arcsin . (6) We set < 1. Fixing α 1 = ±α * 1 , α 2 goes unbounded if tan α * 1 > 1, but converges exponentially, if tan α * 1 < 1, to ±α * 2 with α * 2 = arcsin(tan α * 1 ) = arcsin( / 1 -2 ). (7) Hence we assume that tan α * 1 < 1, i.e. 0 < < √ 2/2 . (8) Then, R 2 ×S 1 ×(-α * 1 , α * 1 )×(-α * 2 , α * 2 ) is an invariant subset of R 2 × S 1 × S 1 2
for any control; we chose this restricted state space and assume from now on that

q = (x, y, θ, α 1 , α 2 ) ∈ R 2 ×S 1 ×(-α * 1 , α * 1 )×(-α * 2 , α * 2 ) . (9)

EXTREMALS OF MINIMUM TIME PROBLEM

In order to apply Pontrjagin maximum principle (see e.g. the textbook Agrachev and Sachkov ( 2004)), we define the Hamiltonian of the problem:

H = p 0 + p x (cos θ + w) + p y sin θ + (p θ -p α1 )u - p α1 sin α 1 + p α2 (sin α 1 -cos α 1 sin α 2 ). ( 10
)
Any minimum time trajectory t → (q(t), u(t)) must be the projection of a curve t → (q(t), p(t), u(t)) in the cotangent bundle of the state space, solution of the Hamiltonian system associated with H, with p 0 a non positive constant, a control u(t) that maximizes u → H(q(t), p(t), u) for almost all time t; H must be identically zero along the solution.

In particular, the adjoint state p = (p x , p y , p θ , p α1 , p α2 ) is a solution of:

               ṗx = 0 ṗy = 0 ṗθ = p x sin θ -p y cos θ ṗα1 = p α1 cos α 1 - p α2 (cos α 1 + sin α 1 sin α 2 ) ṗα2 = p α2 cos α 1 cos α 2
The set {p α2 = 0} is invariant; the system restricted to this set is equivalent to the case of one trailer only, treated in [START_REF] Caillau | Zermelo-Markov-Dubins problem and extensions in marine navigation[END_REF]; p α2 never vanishes if it is not identically zero. Similarly, (p α1 , p α2 ) either does not vanish or is identically zero, and the latter case reflects the original problem with zero trailer (see introduction).

As is customary, we use the notations H i = p, F i , i = 0, 1 (Hamiltonian lifts), as well as H 01 = {H 0 , H 1 }, H 001 = {H 0 , H 01 }, and so on... where {., .} stands for the Poisson bracket. Then H = H 0 + u H 1 . Also, in order to replace p 0 p x , p y let us define γ, ρ, φ by

γ = -p 0 -p x w , p x p y = ρ cos φ sin φ , ρ ≥ 0. ( 11
)
These are constant on any extremal. Pontrjagin's maximum principle also implies (γ, ρ, p θ , p α1 , p α2 ) = 0.

(12) The following identities can be checked via elementary (long) computations:

H 0 = -γ + H 101 (13) H 1 =p θ -p α1 (14)
H 01 = p α2 (sin α 1 sin α 2 + cos α 1 ) - p α1 cos α 1 -ρ sin(φ -θ) (15) H 001 = -p α1 + p α2 (1 + cos α 2 ) 2 (16)
Solutions are concatenations of regular arcs, or bang arcs, on which H 1 is nonzero, hence u = signH 1 , for almost all time, and of singular arcs, on which H 1 is identically zero, and that we particularly study in the rest of this section.

Since Ḣ1 = H 01 and Ḣ01 = H 001 + uH 101 , one has H 1 = H 0 = H 01 = H 001 + u H 101 = 0 identically on a singular arc (H 0 = 0 because H = 0 on any minimum time solution). The three first equality restrict the flow to some co-dimension 3 subset and the last one yields the control if H 101 = 0.

Let us prove that this is always the case, i.e. that all singular arcs are of order one (see [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]). Indeed, if H 101 (equal to γ according to (13)) is zero, so must be H 001 . A computation yields Ḣ001 = H 0001 = cos α 1 3

(-p α1 + (2 + cos α 2 )p α2 ) ,
that must also be zero; this and ( 16) implies p α2 cos α 1 = 0. Since cos α 1 does not vanish in the state space, see ( 9), ( 6) and ( 8), p α2 must be identically zero, which yields a contradiction to (12).

Hence γ = 0 and the control is given by u = -H 001 /γ. The description of the extremals reduces to

                   α1 = - sin α 1 - p α1 γ 2 + p α2 γ 2 (1 + cos α 2 ), α2 = 1 (sin α 1 -cos α 1 sin α 2 ), ṗα1 = p α1 cos α 1 - p α2 (cos α 1 + sin α 1 sin α 2 ), ṗα2 = p α2 cos α 1 cos α 2 . ( 17
)
The six other variables are recovered from α 1 , α 2 , p α1 , p α2 either through H 0 = H 1 = H 01 = 0 or by quadrature.

ANALYSIS OF EQUILIBRIA OF SINGULAR FLOW

Lemma 1. System (17) has 8 equilibria (with angles taken modulo 2π) given by the 4-tuple A = {(0, 0, 0, 0), (π, 0, 0, 0), (0, π, 0, 0), (π, π, 0, 0)} and the 4-tuple

B = π 4 , π 2 , - √ 2 γ, - γ √ 2 , 3π 4 , - π 2 , - √ 2 γ, - γ √ 2 , - π 4 , π 2 , √ 2 γ, γ √ 2 , - 3π 4 , - π 2 , √ 2 γ, γ √ 2 .
Proof. Equilibria of the system are deduced by direct analysis as follows.

• From ṗα2 = 0, either p α2 = 0 or cos α 2 = 0. Having cos α 1 = 0 is forbidden by α2 = 0, however. • If p α2 = 0, then p α1 = 0 follows from ṗα1 = 0. α1 = 0 and α2 = 0 then yield sin α 1 = sin α 2 = 0. This implies equilibria in family A

• On the other hand, if cos α 2 = 0, then α 2 = ±π/2 + 2kπ, k ∈ Z. We now show these equilibria correspond to family B. • If α 2 = π/2 + 2kπ, k ∈ Z, then α2 = 0 implies that cos α 1 = sin α 1 , and ṗα1 = 0 implies that p α1 = 2p α2 . Finally, α1 = 0 implies that

p α2 = -γ cos α 1 = ± γ/ √ 2. • If α 2 = -π/2 + 2kπ, k ∈ Z, then α2 = 0 now
implies that cos α 1 = -sin α 1 , and ṗα1 = 0 implies again that p α1 = 2p α2 . Finally, α1 = 0 implies that

p α2 = γ cos α 1 = ± γ/ √ 2.
This covers all possible cases. 2

By direct evaluation, we can obtain more information on the vector field near these equilibria. Lemma 2. Equilibria in family A have a linear part with eigenvalues (with multiplicity)

- 1 , - 1 , 1 , 1
Equilibria in family B have a linear part with eigenvalues

- 1 , 1 , - i √ 2 , i
√ 2 (with i denoting the imaginary unit i 2 = -1).

INTEGRABILITY PROPERTIES

Singular flow. Here, we study integrability of ( 17), or more precisely of its restriction to the invariant hyperplane {p α2 = 0}, on which the system reduces to:

             α1 = - sin α 1 - p α1 γ 2 α2 = 1 (sin α 1 -cos α 1 sin α 2 ) ṗα1 = p α1 cos α 1 (18)
Theorem 3. The singular flow is not solvable by quadrature.

Proof. We remark that

H r = sin(α 1 ) + p α1 γ 2 + cos(α 1 ) 2
is a first integral of ( 18). On the level H r = 1, we can solve easily in p α1 p α1 = -2 sin(α 1 )γ or p α1 = 0.

This first solution gives the equation α1 = sin(α1) and we deduce the following solution α 1 (t) = 2arctan e t/ , p α1 (t) = -4γ e t/ + e -t/ • (19) We will now try to obtain the corresponding solution α 2 of system (18). We introduce a variable change α 2 (t) = i ln z(τ ), t = ln τ and the equation in z(τ ) becomes

z (τ ) = (τ 2 -1)z(τ ) 2 2τ (τ 2 + 1) - 2iz(τ ) τ 2 + 1 - τ 2 -1 2τ (τ 2 + 1) • (20)
This is a Riccati equation, the parameter disappears, and noting

z(τ ) = - 2τ (τ 2 + 1) τ 2 -1 φ (τ ) φ(τ )
this equation reduces to a second order linear ODE Kovacic algorithm Kovacic (1986), we find that this equation has Galois group SL 2 (C), and thus is not solvable by quadrature. Thus α 2 (t) cannot be obtained by quadrature in system (18) when α 1 , p α1 are given by ( 19).

φ (τ ) + τ 3 + iτ 2 -3τ + i τ (τ 2 -1)(τ -i) φ (τ ) - (τ 2 -1) 2 4(τ 2 + 1) 2 τ 2 φ(τ ) = 0 Using the
As it is impossible to obtain α 2 (t) by quadrature in the particular case ( 19), then it is not possible in general, and thus system (18) and then system (17) are not solvable by quadrature. 2 Regular flow. We fix u = -1 (u = 1 is similar). In restriction to the invariant hyperplane p α2 = 0, the equations are (the time has been rescaled according to t → t):

   α1 = -sin α 1 α2 = sin α 1 -cos α 1 sin α 2 ṗα1 = p α1 cos α 1 (21)
Theorem 4. For ∈ (0, √ 2/2), the regular flow is not solvable by quadrature.

Proof. Again this system admits a first integral p α1 (sin α 1 ), which allows to recover p α1 once α 1 is known. Solving the first equation in α 1 , we find the solution

α 1 (t) = 2 arctan 1 - √ 1 -2 tanh 1 2 t 1 -2 .
We substitute in the second equation, and making the variable change

α 2 (t) = i ln(z(τ )), tanh 1 2 t 1 -2 1 -2 = τ we have a Riccati equation z (τ ) = ( 2 -(τ -1) 2 )z(τ ) 2 -4i (τ -1)z(τ ) -2 + τ 2 -2τ + 1) ( 2 + τ 2 -1)( 2 + (τ -1) 2 )
We can now transform this Riccati equation to a second order linear equation by the variable change

z(τ ) = - ( 2 + τ 2 -1)( 2 + τ 2 -2τ + 1) ( -1 + τ )( + 1 -τ ) φ (τ ) φ(τ ) which gives the equation φ (τ )- (iτ 4 + ( -3i)τ 3 -(3i 2 + 4 -3i)τ 2 -(3 3 -3i 2 -5 + i)τ -2 3 + 2 ) (iτ + -i) ( -1 + τ ) ( + 1 -τ ) ( 2 + τ 2 -1) ×2φ (τ ) - ( -1 + τ ) 2 ( + 1 -τ ) 2 φ(τ ) ( 2 + τ 2 -1) 2 ( 2 + τ 2 -2τ + 1) 2 = 0. ( 22
)
For a generic , the Kovacic algorithm finds no solutions, and thus generically the system (21) is not integrable by quadrature. The system could possibly nevertheless be integrated for specific values of ; computing the possible confluences between singularities, we find = 0, ±1 that are outside the interval studied. Thus for any ∈ (0, √ 2/2), there are exactly 6 singularities (the point at infinity is regular) 22) for the corresponding has a resonance between its exponents.

1 -, 1 + , 1 -i , 1 + i , 1 -2 , -1 -2 , with respectively the local exponents (0, 2), (0, 2), 1 2 ± 1 2 √ 2, - 1 2 ± 1 2 √ 2,
i √ 1 -2 ± √ 2 4 -3 2 + 1 2(1 -2 ) , i √ 1 -2 ± √ 2 4 -3 2 + 1 2( 2 -1)
We will now follow the Kovacic algorithm [START_REF] Kovacic | An algorithm for solving second order linear homogeneous differential equations[END_REF] case by case. We see that not all exponents are rational, and thus not all solutions of ( 22) can be algebraic. Thus case III is not possible. For case I, we need a hyperexponential solution, which requires that a sum of exponents to be a non positive integer. We obtain the following equation

√ 2 4 -3 2 + 1 1 -2 + κ √ 2 = -n, n ∈ N (23)
where , κ ∈ {-1, 0, 1} depend on the choice of the exponents. This equation will give constraints to the possible . For case II, we need to consider the symmetric square of equation ( 22), which is a linear differential equation of order 3 with the same singularities. The computed exponents (3 for each singularity) are the following (0, 2, 4), (0, 2, 4), (1, ±

√ 2), (-1, ± √ 2), i √ 1 -2 1 -2 , i √ 1 -2 ± √ 2 4 -3 2 + 1 1 -2 , i √ 1 -2 2 -1 , i √ 1 -2 ± √ 2 4 -3 2 + 1 2 -1 .
We again need a hyperexponential solution, and thus a sum of exponents equal to a nonpositive integer. This gives again relation ( 23), but with , κ ∈ {-2, -1, 0, 1, 2}.

For ∈ (0, √ 2/2), we see in figure 3 that the possible non positive integer values in relation ( 23) are -4, -3, -2, -1, 0, and the corresponding values of in (0, √ 2/2) are Remark. The fact that systems ( 18) and ( 21), when reduced to p α2 = 0, can be solved through second order differential equations is not a generic situation, and thus further calculations could be possible through the use of special functions.

1 3 √ 3, 1 7 √ 21, 2 7 √ 7 -1 14 √ 14, 2 7 √ 14 -1 7 √ 7, 1 2 3 - √ 2,

NUMERICAL SIMULATIONS

We use the BOCOP software (python version from the ct project1 ) to provide preliminary numerical computations on the problem with two trailers. The algorithm uses the midpoint scheme to discretise then optimise the problem. Three simulations are presented Figure 4 (unconstrained final trailer angles-as no alignment is required, the problem is equivalent to the standard Zermelo-Markov-Dubins one) and Figure 5 (trailers must be aligned at final time). These simulations all share the same value of the current (w = 0.8), of ( = 0.6 < √ 2/2), the same initial conditions (x 0 , y 0 , θ 0 ) = (0, 0, π/7) and final conditions (x f , y f ) = (4, 7) (normalised values as explained in Section 2) while the final velocity angles differ (remember that θ is the argument of the velocity in a fixed frame, attached to the bottom of the sea): (i) θ = π/2, (ii-a) and (ii-b) θ = -π/2. The numerical simulations presented are reproducible online2 on the web site of the ct project.

In the first case (i), one can observe that the structure of the computed trajectories is B + SB -B + B -(B + = positive bang, that is left turn, B -= right turn, while S = singular) vs. B + SB -for the unconstrained case. In case (ii-a) the structure is again B + SB -B + B -(B + SB - for the unconstrained case) and clearly not globally minimising, while in case (ii-b) (same problem, but different solution) the structure is B + SB + B -B + (B + SB + in the unconstrained case) and gives a much better final time. In the three cases, as for one trailer [START_REF] Caillau | Zermelo-Markov-Dubins problem and extensions in marine navigation[END_REF] the main part of the trajectory is a singular arc. This illustrates the so-called turnpike phenomenon; for a distant enough target in the (x, y) plane, the requested minimum time is long enough to allow the singular arc of the extremal to come close to the hyperbolic equilibrium (0, 0, 0, 0) of family A described in Section 4. The longer the minimum time, the closer this singular part is to the straight line encountered in the Zermelo-Markov-Dubins problem without trailer. In contrast with the one trailer case, the trajectories are terminated not by two but three bang arcs that accommodate the alignment requirement of the two trailers at the end. These short bang arcs are much more time efficient than the "run-in" procedure (a final straight line) performed until now by ships at the end of the maneuver during real exploration campaigns. Such comparisons will be the topic of future work.

Fig. 1 .

 1 Fig. 1. dimensions of a seismic acquisition spread. Illustration from Caillau et al. (2019).

Fig. 2 .

 2 Fig.2. A cart with two trailers rolling without slipping on a conveyor belt. Using complex notations in the plane, z = x+i y is the middle of the axle of the cart, whose speed with respect to the conveyor belt is e iθ (magnitude normalized to 1). The control u is the angular velocity θ of the cart, after re-mormalization.The first trailer is attached to the cart at point z, and its angle with respect to the axis of the cart is α 1 , so that the other end of the cart is at point z 1 = z + e iθ+α1 , rolling without sleeping means that the velocity of point z 1 with respect to the conveyor belt is along the axis of the trailer, i.e. has polar angle θ + α 1 . The second trailer is attached to the first one at z 1 , its angle with respect to first one is α 2 , the other end of that second cart is at z 2 = z 1 + e i(θ+α1+α2) , and the velocity of z 2 with respect to the conveyor belt has to have polar angle θ + α 1 + α 2 .

Fig. 3 .

 3 Fig. 3. The sum of exponents in function of for all possible choices of , κ. When one of these curves intersects a non positive integer ordinate, equation (22) for the corresponding has a resonance between its exponents.

Fig. 4 .

 4 Fig. 4. Trajectories of a ship with two trailers in a constant current, unconstrained case (Zermelo-Markov-Dubins): cases (i), (ii-a) and (ii-b) from left to right. Respective minimum times are (i) t f = 7.901, (ii-a) t f = 22.63 and (ii-b) t f = 11.38. In particular, the extremal obtained case (ii-a) for the same problem as (ii-b) is a strict local minimum. Note that since the final trailer angles are free, no alignment is obtained. Compare with Figure 5 where the longer trajectories including two additional bang arcs allow to realign the two trailers.

Fig. 5 .

 5 Fig. 5. Trajectories of a ship with two trailers in a constant current, constrained case (alignment of the trailers at the end): cases (i), (ii-a) and (ii-b) from left to right. Respective minimum times are (i) t f = 8.897, (ii-a) t f = 31.82 and (ii-b) t f = 19.08. In particular, the extremal obtained case (ii-a) for the same problem as (ii-b) is a strict local minimum. Note that for case (iib) the curvature constraint is indeed fulfilled; as the trajectories are portrayed in a fixed frame (attached to the bottom of the sea) and not in moving frame (moving at the speed of the current, directed along the (Ox) axis), the turns in cases (ii-a) and (ii-b) apparently have different curvatures although these curvatures are indeed equal (with turns in opposite directions).

ct: Control Toolbox, see ct.gitlabpages.inria.fr/gallery

See ct.gitlabpages.inria.fr/gallery/nav/nav.html

ACKNOWLEDGEMENTS

The authors thank T. Mensch and T. Moulinier, formerly with CGG, for introducing them to the problem (see Caillau et al. (2019)). Thanks are also extended to Y. Du for her help with the numerical simulations.


M Cazes

N Franiatte

A Delmas

J-M André

M Rodier

I Chraibi Kaadoud

Evaluation of the sensitivity of cognitive biases in the design of artificial intelligence

Keywords: cognitive bias, artificial intelligence, decision making, human factors, human-IA interaction

The reduction of algorithmic biases is a major issue in the field of artificial intelligence. Despite the diversity of sources and temporalities at the origin of these biases, the current solutions to achieve ethical results are mostly technical. The consideration of human factors and more particularly cognitive biases remains incomplete. Nevertheless, the task of designing artificial intelligence systems is conducive to the emergence of cognitive biases. The aim of our study is to test the awareness of individuals who design artificial intelligence systems of the impact of their cognitive biases in their productions. The study focuses on conformity bias, confirmation bias and illusory correlation bias. The first results of this pre-experimentation show that these individuals believe that their decisions are subject to the cognitive biases of conformity, illusory correlation and confirmation.

Introduction

The question of biases, and their reduction, is a major issue in artificial intelligence (AI), especially since the raise of all the ethical issues related to deep learning algorithms application [START_REF] Eirini | Bias in data-driven artificial intelligence systems-An introductory survey[END_REF][START_REF] Susan | Gender bias in artificial intelligence : The need for diversity and gender theory in machine learning[END_REF][START_REF] Deborah | Actionable Auditing : Investigating the Impact of Publicly Naming Biased Performance Results of Commercial AI Products[END_REF][START_REF] Julia | The accuracy, fairness, and limits of predicting recidivism[END_REF]. These lead to more research on human-AI interaction field that studies human impact on AI systems [START_REF] Susan | Gender bias in artificial intelligence : The need for diversity and gender theory in machine learning[END_REF][START_REF] Aylin | Semantics derived automatically from language corpora contain human-like biases[END_REF], and inversely, on the question of the impact of those systems on human [START_REF] Diana | Evaluation and selection of biases in machine learning[END_REF][START_REF] Gregory | Bias in artificial intelligence[END_REF][START_REF] Bruno | Fair, transparent, and accountable algorithmic decision-making processes[END_REF]. Our work is in the field of human-AI interaction, not only machine learning, and questions the impact of human on AI systems during their design. We define an AI system as a computational software, an algorithm or a scientific methodology using a set of algorithms, that assists humans to make a decision based on data analysis and data processing (mining, clustering, classification, prediction, recommandations, object recognition, etc.) whether in industrial or scientific context. We consider as an * these authors contributed equally to this work AI system any system that involves at least one of the following elements: an AI symbolic system (e.g. decision trees, expert systems) or a machine learning or deep learning algorithm (e.g. shallow and deep neural networks). From a software management point of view, the conception of an AI system goes through different steps that can be summarized as follows [START_REF] Marco | Applications of AI in classical software engineering[END_REF]: design, implementation of the code, test and production. We will refer in our work to all these steps as AI design. By doing so, we aim to group all the actors that can impact the product life cycle of an AI system. We will refer to these actors as AI professionals: AI researchers, AI manager products, data scientists, data analysts, data architects, data engineers and AI developers. Thus, the purpose of the current work is to propose a pre-experiment in the shape of a survey submitted to AI professionals in order to identify whether they are aware of being influenced in their work by cognitive biases classically known in human factors: conformity bias, confirmation bias, and illusory correlation bias. We therefore propose to conduct a study to test the following hypotheses (Figure 5):

-H0: Actors involved in the design of AI systems are aware that the biases of illusory correlation, conformity and confirmation influence their task. -H1: Feedback collected during the qualitative interviews shows that AI systems design choices are influenced by these cognitive biases. We adopt a different approach for works in literature by using human factors knowledge to address the issue of the presence of biases in AI systems by focusing first on the AI professionals' sensitivity of the cognitive biases. We will test H0 in this pre-experiment. This work is thus a first step in a larger question of the evaluation of cognitive biases on AI systems from a human factor perspective. As the subject of biases in AI systems is much debated and studied within the AI community, we are concerned about clarifying the contribution and position of the current work: we do not aim to establish any causal effect of relation between cognitive biases and algorithmic biases. These concepts are clearly distincts. However there will be cited in our work since they both intervene in the larger research field of human-AI interaction. [START_REF] Collectif | Grand Dictionnaire de la Psychologie[END_REF][START_REF] Helen | How computer systems embody values[END_REF][START_REF] Haselton | The Evolution of Cognitive Bias[END_REF][START_REF] Eirini | Bias in data-driven artificial intelligence systems-An introductory survey[END_REF][START_REF] Wason | On the failure to eliminate hypotheses in a conceptual task[END_REF][START_REF] Banchet | Psychologie sociale : Nouveau cours de psychologie[END_REF][START_REF] Loren | Genesis of popular but erroneous psycho-diagnostic observations *[END_REF][START_REF] Patrice | Algorithmes : Biais , Discrimination et Équité[END_REF][START_REF] Institut | Algorithmes : contrôle des biais svp[END_REF] 1 The biases of artificial intelligence [START_REF] Deborah | Actionable Auditing : Investigating the Impact of Publicly Naming Biased Performance Results of Commercial AI Products[END_REF] showed that Amazon's facial recognition AI was very effective in performing facial recognition for light-skinned men (almost zero error rate) but had an error rate of 31.4% for dark-skinned women. This is one of the studies that highlights ethical issues related to discriminatory practices of AI. Their social consequences are concrete [START_REF] Eirini | Bias in data-driven artificial intelligence systems-An introductory survey[END_REF] as shown by the results of the NGO ProPublica's survey showing that the software used to predict crimes in the USA (The Correctional Offender Management Profiling for Alternative Sanctions, COMPAS) had a 'racist bias' [START_REF] Julia | The accuracy, fairness, and limits of predicting recidivism[END_REF]. The explanation of these discriminatory practices requires an understanding of the emergence of bias in the results of AI algorithms. We consider algorithmic bias, as defined in Table 1, including for example 'Problems related to the gathering or processing of data that might result in prejudiced decisions on the bases of demographic features such as race, sex, and so forth' [START_REF] Eirini | Bias in data-driven artificial intelligence systems-An introductory survey[END_REF] . The sources of algorithmic biases are of various natures and occur at different temporalities. Algorithmic biases may appear upstream (selection of learning data), during (inclusion of certain variables) and downstream (interpretation bias) of the learning phase [START_REF] Selena | Viewpoint algorithms, platforms, and ethnic bias[END_REF]. Moreover, it is essential to qualify the nature of the source of these biases: technical bias (omitted variable bias, database bias, selection bias...) or 'society bias' (emotional or cognitive biases) [START_REF] Institut | Algorithmes : contrôle des biais svp[END_REF]. However, despite the identification of the diversity of the sources of algorithmic biases, the correction of these biases is considered only from a statistical and algorithmic performance point of view i.e. in relation to certain metrics such as the percentage of prediction error or the accuracy in machine learning [START_REF] Eirini | Bias in data-driven artificial intelligence systems-An introductory survey[END_REF] (Figure 2). Indeed, [START_REF] Eirini | Bias in data-driven artificial intelligence systems-An introductory survey[END_REF] denounces the sole search for technical performance, which is neither relevant nor desirable to deal entirely with algorithmic biases. The influence of the human characteristics of designers on programmes is known [START_REF] Gregory | Bias in artificial intelligence[END_REF], as recommendations are made regarding the need to incorporate diversity in the profiles of AI professionals [START_REF] Bruno | Fair, transparent, and accountable algorithmic decision-making processes[END_REF][START_REF] Sarah | Discriminating systems[END_REF] (Figure 2, bottom right). However, there are no empirical studies investigating how human factors -and especially cognitive biases -can be concretely illustrated in the AI design process [START_REF] Tore | Studying the Transfer of Biases from Programmers to Programs[END_REF].

FIGURE 1 -Sources of algorithmic biases and solutions implemented [START_REF] Sarah | Discriminating systems[END_REF] [31] [START_REF] Selena | Viewpoint algorithms, platforms, and ethnic bias[END_REF] We note that the debates around algorithmic biases bring into play problems of taking into account human factors in the actors of the design of artificial intelligence systems. We propose to carry out a study on this topic and more particularly the issue of the cognitive biases of the actors of this design chain.

Artificial intelligence design and cognitive biases

The conception of an AI system is a complex task. [START_REF] Edgar | Le défi de la complexité[END_REF] develops the concept of complexity: a phenomenon is complex when 'the whole is more than the sum of its parts' i.e.

there is an emergence of information. The conception of an AI system is a question of manipulating phenomena whose complexity is not mastered [START_REF] Arrieta | Explainable Artificial Intelligence (XAI) : Concepts, taxonomies, opportunities and challenges toward responsible AI[END_REF][START_REF] Shane | Principles of Explanation in Human-AI Systems[END_REF]. In particular concerning the neural networks which make information emerge from which it is difficult to directly apprehend the chain of causality. In industry, designing AI systems is a task subject to constraints of time, means, objectives and expectations on the part of customers. These elements are part of a complex and uncertain context that is conducive to the use of heuristics on the part of designers and developers. [START_REF] Kahneman | Système 1 / Système 2 : Les deux vitesses de la pensée[END_REF] defined a heuristic as 'a simple procedure that makes it possible to find adequate, although often imperfect, answers to difficult questions'. These heuristics reduce the complexity of the task but can lead to the appearance of cognitive biases.

[23] wrote as early as 1974 : 'In general, these heuristics are very useful, but sometimes they lead to severe and systematic errors'. The 'severe and systematic errors' considered in the quotation from Kahneman and Tversky [START_REF] Daniel | Judgment under Uncertainty : Heuristics and Biases[END_REF] correspond to what will be referred to later as cognitive biases. It seems to us that they can be defined in this way as a systematic deviation of logical and rational thinking from reality. Some authors have sought to model the functioning of cognitive biases by considering the theory of dual processes [START_REF] Wason | Dual processes in reasoning ?* P. C. WASON[END_REF][START_REF] Kahneman | Système 1 / Système 2 : Les deux vitesses de la pensée[END_REF] (Figure 2). The latter is based on the dichotomy between two main modes of reasoning: unconscious, rapid and automatic processes on the one hand, and slow and deliberate processes on the other. These two modes are colloquially referred to as 'intuitive' and 'analytical' but are also known as System 1 (intuitive) and System 2 (analytical). According to [START_REF] Kahneman | Système 1 / Système 2 : Les deux vitesses de la pensée[END_REF], these two systems should be 'seen as agents with their own capabilities, limitations and functions'. Nevertheless, these 'agents' interact with each other and thus divide up certain tasks related to information processing. For example, while system 1 will handle automatic activities such as 'orienting towards the source of a sudden noise' or 'making a face of disgust at a horrible image', system 2 will handle tasks such as 'focusing on the voice of a particular person in a crowded and noisy room' or 'checking the validity of a complex logical argument'. Also, it appears that the notion of expertise is associated with the development of better performance of system 1. However, this 'intuitive expertise' is only valid when the task to be performed is part of a predictable environment in which the individual can learn from regularities [START_REF] Daniel | Conditions for Intuitive Expertise : A Failure to Disagree[END_REF]. In the case of developers of AI systems, the result of their actions on the neural network involves an element of uncertainty. In other words, their understanding of the impact of the modification of certain parameters is not complete. For example, [START_REF] Rob | Thinking critically about and researching algorithms[END_REF] emphasised the fact that it seems unlikely that a single programmer would have a complete understanding of a large complex system designed by many teams around the world. Thus, despite the sense of expertise and mastery, the system of intuitive thinking cannot be fully effective.

In the field of computer science, the concrete bias impacts on developers has already been proved. Studies have shown that developers' cognitive confirmation bias has an influence on the quality of their programs. Indeed, [START_REF] Gül | Influence of confirmation biases of developers on software quality : an empirical study[END_REF] demonstrate that developers do unit testing -a software testing method to determine whether the written code is fit for use [START_REF] Dorota | Automated defect prevention : best practices in software management[END_REF] -to show that their system works rather than testing it. More precisely, in the field of AI, research highlighted the concrete impacts of cognitive biases of the people involved in the design and implementation of AI programs. For example, the choice of the set of training variables for the creation of AI recruitment software has an important role in the quality and results of the program. However, this choice is made by managers and developers who are subject to their own cognitive biases such as generalization bias [START_REF] Melika | Cognitive biases in developing biased Artificial Intelligence recruitment system[END_REF].

Finally, the task of designing an AI system has characteristics conducive to the emergence of cognitive biases. There are a multitude of cognitive biases, and it will be a matter of identifying the most relevant ones in our context.

Impossible and irrelevant identification of all cognitive biases

There are multiple typologies that classify cognitive biases according to pre-established criteria [START_REF] Hogarth | Judgement and choice : The psychology of decision[END_REF][START_REF] Bazerman M | The Principles of Quantum Mechanics[END_REF][START_REF] Dobelli | The Art of Thinking Clearly[END_REF]. From the seminal work of Kahneman and Tversky [START_REF] Daniel | Judgment under Uncertainty : Heuristics and Biases[END_REF] to more recent research [START_REF] Olivier | Comprendre et prévenir l'erreur récurrente dans les processus de décision stratégique : l'apport de la Behavioral Strategy[END_REF], there does not seem to be a consensus on the comprehensiveness and classification of cognitive biases. Moreover, cognitive biases raise several epistemological questions, which are important to consider as they directly influence the applications of research results [START_REF] Lebraty | Biais cognitifs : quel statut dans la prise de décision assistée ?[END_REF]. In order to qualify 'diversion' i.e. a bias, a rational behaviour must be established, i.e. a norm from which the behaviour deviates [START_REF] Pascal | Les canons de la rationalité : essai de classification des points de vue dans le débat sur les biais cognitifs et la rationalité humaine[END_REF]. But how can behaviour be described as irrational ? Similarly, the nature of the task seems to influence the cognitive mechanism at work. [START_REF] Leda | The logic of social exchange : Has natural selection shaped how humans reason ? Studies with the Wason selection task[END_REF] was thus able to demonstrate the existence of content effects for Wason's task: human beings use reasoning strategies appropriate to the nature of the problem they are facing. Cognitive biases would therefore not be 'mysterious irrationalities' but adaptations of the mind [START_REF] Haselton | The Evolution of Cognitive Bias[END_REF]. But how can cognitive processing be described as biased if it is context-dependent ? These elements show the complexity surrounding the notion of cognitive bias, which often makes it difficult to identify and evaluate them for a specific task. [START_REF] Patrice | Algorithmes : Biais , Discrimination et Équité[END_REF] already identified cognitive biases that could have impacts during the design and implementation of an AI system. In accordance with this work, we have chosen to focus on the study of three cognitive biases: conformity bias, confirmation bias, and the bias of illusory correlation.

Conformity bias, confirmation bias, illusory correlation bias

According to the codex of cognitive bias [START_REF] John | Codex des biais cognitifs[END_REF], the conformity bias and illusory correlation are due to the problem of the lack of sense of the individual's environment. In order to create meaning, individuals extrapolate 'attributes on the basis of stereotypes, generalities or antecedents' (translation of [START_REF] John | Codex des biais cognitifs[END_REF]). This led, among other reasons, to the development of the explainable AI and interpretable AI fields. Indeed, the neural networks used create a combinatorial system can be difficult to understand for the human cognitive system [START_REF] Arrieta | Explainable Artificial Intelligence (XAI) : Concepts, taxonomies, opportunities and challenges toward responsible AI[END_REF] and by extent to understand how AI systems lead to the given result. The difficulty of apprehending the internal functioning of neural networks can thus lead to a lack of understanding on the part of the AI professionals [START_REF] Arrieta | Explainable Artificial Intelligence (XAI) : Concepts, taxonomies, opportunities and challenges toward responsible AI[END_REF][START_REF] Shane | Principles of Explanation in Human-AI Systems[END_REF] on the behavior or inner mecanisms of their AI systems. Consequently, it is possible to assume that this task takes place in an environment conducive to the expression of the biases mentioned above (Figure 3). Moreover, designing an AI program is mainly by selecting the data that will be used as a basis for learning the neural network. Data from various sources, types and formats must be chosen so that AI can produce quality information [START_REF] Usama | From data mining to knowledge discovery in databases[END_REF]. Therefore, it is ne-cessary to select information in an environment that is saturated with it. Indeed, according to the codex typology of cognitive biases, too much information leads to cognitive biases [START_REF] John | Codex des biais cognitifs[END_REF]. Faced with the need to filter this information, individuals are attracted by what confirms their own convictions. This mechanism is at the origin of confirmation bias. The design of AI systems would therefore be an environment conducive to the emergence of confirmation bias (Figure 3). Finally, the characteristics of the AI systems design task seem to create an environment conducive to the emergence of the following cognitive biases: illusory correlation, conformity and confirmation bias. In order to understand more precisely how these biases operate, the following section will detail the cognitive mechanisms underlying their appearance.

Conformity Bias

Conformity bias can be defined as the 'modification of an individual's behavior or judgment to bring it into harmony with the behavior or judgment of the majority' (translation of [START_REF] Banchet | Psychologie sociale : Nouveau cours de psychologie[END_REF]). As Asch's pioneering experience shows, it is a powerful mechanism in decision-making [START_REF] Asch | Effects of group pressure upon the modification and distortion of judgments[END_REF]. The consequences of conformity bias are ambivalent and contextual. It is, for example, one of the factors at the origin of collective intelligence phenomena [START_REF] Joaquin | Aggregated knowledge from a small number of debates outperforms the wisdom of large crowds[END_REF] but also of 'collective conservatism' [START_REF] Thaler R | Nudge[END_REF].

Confirmation bias

[49] was one of the first to point out the existence of a confirmation bias. [START_REF] Bazerman M | The Principles of Quantum Mechanics[END_REF] state that in forming a judgment, a large majority of individuals reason by trial and error on the basis of previous judgments, and tend to confirm the accuracy of their initial hypotheses. Thus, confirmation bias can be defined as the tendency to explain facts with consistent stories and neglect facts that contradict them [START_REF] Raymond | Confirmation Bias : A Ubiquitous Phenomenon in Many Guises[END_REF][START_REF] Erica | Motivated Reasoning and Performance on the Wason Selection Task[END_REF]. Confirmation bias can have different consequences depending on the context in which it is embodied. For example, in the run-up to the US presidential election, [START_REF] Krebs | Political Polarization[END_REF] studied the evolution of political books, highlighting a confirmation bias that shows that people who buy these books do so not for information, but for confirmation of their opinion.

The illusory correlation bias

The first descriptions and explanations of the phenomenon of illusory correlation come from social psychology, and more particularly from the studies of the researchers Chapman and Chapman [START_REF] Loren | Genesis of popular but erroneous psycho-diagnostic observations *[END_REF]. When two events are correlated, individuals attribute cause and effect relationships to one of them. The meaning of this correlation may be erroneous, or even totally illusory [START_REF] Arun | Can executive information systems reinforce biases ?[END_REF].

The three biases of conformity, confirmation and illusory correlation are therefore the focus of our study. After having identified them, we will transpose them into the context that actors in AI systems development may encounter.

5 Experimentation

Purpose and hypothesis

We focus our study on the impact of cognitive biases in the AI systems design process. Indeed, the nature and conditions of the AI systems design task are, in our opinion, clues to suppose that the biases of illusory correlation, conformity and confirmation would be at the origin of weaknesses in AI systems creation. There is a lack of empirical studies on how cognitive biases act precisely on decision making in an AI system design context. We therefore propose to conduct a study to test the following hypotheses (Figure 4). As a reminder, our assumptions are as follows:

-H0: Actors involved in the design of AI systems are aware that the biases of illusory correlation, conformity and confirmation influence their task. -H1: Feedback collected during the qualitative interviews shows that AI systems design choices are influenced by these cognitive biases.

Methodology for testing the H0 hypothesis

To test the H0 hypothesis, an online questionnaire was carried out. The purpose of this questionnaire is to perform a preliminary study to obtain the self-assessments of AI professionals of the impact of cognitive biases in their work, and to recruit target people to have more accurate feedback. The online tool FramaForms was used, which anonymizes the data and removes it permanently after six months. The questionnaire was sent via the bull-i3 mailing list on January 4, 2021, in the BAIA newsletter on January 10, 2021 and in the RISC newsletter on January 26, 2021. We have blocked the recording of the answers to the questionnaire on February 17, 2021. Bull-i3 is a mailing list of the IRIT (Toulouse Institute of Computer Science Research) which gathers all the members (industrialists, researchers, professors, PhD students,...) of the Information, Intelligence and Interaction communities concerned by the issues of these research fields. BAIA is the newsletter of the Bordeaux Artificial Intelligence Alliance and RISC is the information relay on cognitive sciences. These three mailing lists are French. This is a methodological choice. We wanted to avoid bias related to the translation of the questionnaire or cultural bias. The questionnaire is addressed to what we call AI professionals as defined above. The exclusion criteria set concern people whose expertise is not related to AI e.g. the answer of a journalist was discarded because his profile did not correspond to an AI professional.

The questionnaire is structured in four main parts, each of which includes several multiple-choice questions: I) Sensitivity to cognitive bias ; II) Cognitive biases and your profession ; III) Your profile. The questions were originally in French (Figure 5).

Professional Profile

36 out of 39 people surveyed (92.3%) are in either the digital or artificial intelligence sector, or both. Four people combine one of these sectors with a specialty or do not belong to either one (i.e. 'Administration, data mining project', 'Social Robotics', 'Geomatics' and one person looking for work). The respondents are equally divided between the private sector (19 people) and the public sector (19 people), with one person not belonging to either of these two sectors ('In search of employment'). Of the 19 people working in the private sector, 9 (47.4%) belong to a large group, 6 (31.2%) to a start-up, 3 (15.8%) to a PME (Petite ou Moyenne entreprise in french, Small or Medium Factory) and 1 (5.3%) to neither of these categories. Finally, as regards the professions practiced by the respondents: 35.9% of them qualify as researchers/professors, the remainder are divided between student/doctoral students (17.9%), data related professions (25.6%) and other professions (including for example an application architect, a research engineer and an innovation manager) (Figure 6).

Sensitivity to cognitive biases

In the questionnaire sent out, we briefly defined the biases of conformity, confirmation and illusory correlation. 66.7% of the people surveyed told us that they were aware of at least one of these three cognitive biases. Among the latter, this knowledge comes in 100% of cases from personal interest and/or professional training.

Potential impacts of cognitive biases on tasks in the work environment

Finally, all respondents believe that some decisions in their professional environments have already been influenced by at least one of the three cognitive biases. Among the answers obtained concerning the frequency of this influence : 56.4% answered 'Yes, regularly' and 43.6% 'Yes, rarely'.

According to the people surveyed, conformity bias seems to be the most frequent cognitive bias, followed by confirmation bias and then illusory correlation bias (Figure 7).

Discussion

First and foremost, the gender figures (28.2% women and 71.8% men) seem to reflect the current context, as women are still in the minority in the digital and artificial intelligence sectors. Indeed, according to INSEE, in 2017, the share of women in IT professions was 28%. Also, a study by the World Economic Forum and LinkedIn (2018) reports that only 22% of jobs in the field of artificial intelligence are held by women, and even less by senior managers. This survey is a pre-experiment that establishes that all respondents believe that the cognitive biases studied in this article have (or have already had) an impact on their professional decision-making (39/39 or 100% of them). It confirms the need to take the human factors into account in the development of AI systems and to underline the relevance of the three cognitive biases mentioned in the article by Bertail et al [START_REF] Patrice | Algorithmes : Biais , Discrimination et Équité[END_REF] from which we drew inspiration. Also, these results allow us to note that the conformity bias is considered by respondents to be more influential in their professional tasks than the other two proposed biases (of confirmation and illusory correlation). It would therefore be possible to envisage that certain cognitive biases are more likely to influence decision making than others. However, this finding should be qualified in that the conformity bias may be better known to respondents than the other two biases. Also, another point to consider is the fact that it is generally easier for human beings to think that cognitive biases influence others before themselves (which would also bias our own analysis). This is an important point raised notably in the work of Fabrizio Butera [START_REF] Hernandez | Biais de confirmation : pouvezvous le contrer ?[END_REF]. We can summarize it here with the example of confirmation bias: 'The problem with confirmation bias is that it is itself subject to bias: people agree that confirmation bias exists in others, but find it difficult to admit that it exists in them as well. A meta-bias if you like.' This notion of 'meta-bias' should therefore be taken into account in our research and should lead us to interpret any results very cautiously.

Besides, during the study, we realized the semantic -even conceptual -ambiguity that can exist around the term 'bias', whether cognitive or algorithmic. Initially, the connotation of the latter seemed to us rather negative, often leading us to wonder about the potential means of 'debiaising' the human being. Nevertheless, in the course of our work, we found it interesting to also question their possible usefulness. Consequently, the term 'bias' semantically poses a confusion, in the sense that it always refers to a distortion, often interpreted as an error, whereas this distortion is also an essential means of survival for the human being. Moreover, it is difficult to find appropriate solutions to these biases (which would imply that it would be a problem A second element that can be discussed is the potential social desirability bias, which results from a tendency of the individual to want to present himself favourably in the eyes of society [START_REF] Douglas | A new scale of social desirability independent of psychopathology[END_REF]. Indeed, although our survey was anonymized, this psychological mechanism is sometimes implicitly exercised without the subject being aware of it. It would therefore be relevant to conduct another study, not related to AI systems design, in order to verify whether the awareness of cognitive biases by individuals in their professional environments shows a trend similar to the one found in this article.

Finally, the interest of this article is to question the taking into account the human factor -and more particularly cognitive biases -in the whole AI system design cycle. The questionnaire provided initial insight into the place of cognitive biases among AI designers and developers. The study of the potential impact of cognitive biases on the actors of AI system development, and their incarnations in the resulting work, represents a problem that could also be raised in future work. This is what we would like to do next, by testing hypothesis H1 through individual interviews with people who have completed our questionnaire and agreed to be contacted. The objective is to have more qualitative and targeted feedback on the cognitive biases that can occur according to the different expertises related to the AI domain.

Conclusion

The aim of this work is to raise the human factors and, more particularly, cognitive biases issues in the task of designing AI systems. Indeed, the appearance of biases from different sources [START_REF] Institut | Algorithmes : contrôle des biais svp[END_REF] and at all stages of designing AI systems [START_REF] Selena | Viewpoint algorithms, platforms, and ethnic bias[END_REF] raises the question of the impact of cognitive biases during the design of an AI system. In other words, we wish to raise the question of the impact of cognitive bias in individuals at the origin of the realisation of intelligent systems. The AI system and cognitive bias relationship is an open scientific question which tends to grow in importance, particularly thanks to the growing concern about explainable AI systems intrinsically linked to the question of responsibility and ethics in AI [START_REF] Arrieta | Explainable Artificial Intelligence (XAI) : Concepts, taxonomies, opportunities and challenges toward responsible AI[END_REF].

Our objective is to test the awareness of those involved in the design of AI systems of the influence that their cognitive biases may have in their professional decision-making.

Our work suggests that people involved in AI system design processes believe that their decisions are subject to the cognitive biases of conformity, illusory correlation and confirmation. Future work should try to understand whether there is an embodiment of these cognitive biases in the AI system, and if so, how they are embodied. This involves testing our second hypothesis (H1, Figure 4). Indeed, it is not so much a question of the cognitive biases of the individuals involved in the design of AI systems, which are natural and useful [START_REF] Haselton | The Evolution of Cognitive Bias[END_REF], but of their incarnations and therefore their consequences in artificial intelligence. Thus, it would be relevant to carry out a study using feedback, in order to evaluate how these cognitive biases are embodied, or not, in the work of individuals responsible for designing AI systems.

FIGURE 2 -

 2 FIGURE 2 -Functions and processes associated with Systems 1 and 2, based on the work of Daniel Kahneman and Amos Tversky [23] [21]

FIGURE 3 -

 3 FIGURE 3 -Schematic representation of our hypothesis: link between elements of the context of the task and the cognitive biases they could encourage to appear

FIGURE 4 -

 4 FIGURE 4 -Hypothesis on the impact of cognitive biases in AI systems design

FIGURE 5 -

 5 FIGURE 5 -Questions in the questionnaire sent

FIGURE 6 -

 6 FIGURE 6 -Respondents' responses regarding the title of their professional designation.

FIGURE 7 -

 7 FIGURE 7 -Biases chosen by respondents who answered positively to the question 'Do you think that some decision making in your professional environment has already been influenced by one of the previously mentioned cognitive biases' (Question originally in French)

Table

  1 provides definitions and non-exhaustive examples of the two distinct concepts according to the literature.

TABLE 1 -

 1 Differences between cognitive and algorithmic bias: extraction of definitions from scientific studies and non-exhaustive examples

  were in the 35-55 age category. Next, in order of importance, come the 18-25 age group (10 people or 25.6%), then the 25-35 age group (7 people or 17.9%) and finally the 55+ age group (4 people or 10.3%).

5.3.1 Profile of sampled respondents

39 people responded to our questionnaire. Among them, 11 women (28.2%) and 28 men (71.8%). Regarding age (in years), 18 people surveyed (46.2% of the people surveyed)

  ). For Fabrizio Butera, cited by Hernandez[START_REF] Hernandez | Biais de confirmation : pouvezvous le contrer ?[END_REF]: 'The term bias is confusing. It refers to a tunnel of the mind. It implies that one cannot get out of it. If you are in a tunnel, you have to go from the entrance to the exit without making a detour. This implies that we are all helpless in the face of the slippery slope of our biases. Yet this is not what we observe socially.' Thus, in this article we use the terms 'cognitive bias' and 'algorithmic bias' while trying to define as well as possible what they represent for us, in order to avoid any confusion.

Authors contributions

*MC and NF contributed equally to this work. All authors contributed to manuscript revision, read and approved the submitted version.


Alexandre Hannud Abdo
email: abdo@member.fsf.org

Keywords: knowledge society, open science, research evaluation, responsible research and innovation, sociology of science

Following an invitation by the editors of Social Science Information to react to an article by Olof Hallonsten, this article joins a debate about ways of evaluating science in our current context. This article presents an argument in support of the following four assertions and their importance to properly approach today the transformations of science evaluation and governance in the last decades: (a) scientific communities have failed to update their self-governance as societies transitioned from 'rural-labor societies' to 'urban-knowledge societies'; (b) the ensuing discrepancy from expectations contributed to the economization of science; (c) we must consider two distinct processes of democratization; and (d) geopolitics plays an important role in the establishment of commodification in wealthy nations.

Résumé

A la suite de l'invitation des éditeurs d'Information sur les sciences sociales à réagir à un article d'Olof Hallonsten, cet article prend part au débat portant sur l'évaluation de la science à notre époque. Cet article défend les quatre constats qui suivent, ainsi que leur importance pour appréhender aujourd'hui correctement les transformations de l'évaluation et de la gouvernance qu'a connu la science au cours des dernières décennies : (a) les communautés scientifiques ont échoué à renouveler leur propre manière de se gouverner alors que les sociétés passaient de sociétés rurales fondées sur le travail à des sociétés urbaines fondées sur le savoir ; (b) l'écart qui en résulta par rapport aux attentes contribua à l'économisation de la science ; (c) nous devons envisager deux processus de démocratisation distincts ; et (d) la géopolitique joue un rôle important dans l'avènement de la marchandisation dans les pays riches.

Mots-clés

évaluation de la recherche scientifique, recherche et innovation responsables, science ouverte, société du savoir, sociologie des sciences Following an invitation by the editors of Social Science Information to react to an article by Olof Hallonsten, this article joins a debate about ways of evaluating science in our current context. I will start by providing my own synthesis of Hallonsten's argument [START_REF] Hallonsten | Stop evaluating science: A historical-sociological argument[END_REF], in order to position my response.

Through an historical examination of trends of 'economization', 'distrust', 'democratization', and 'commodification' of science, Hallonsten claims that the narrative asserting that science is somehow poorly governed or needs stricter management is absurd. He suggests that considering in some depth the world in which we live should suffice to reveal how preposterous it is to presume that counting citations or anything that can be inventoried could somehow reflect the pervasive and nonlinear role of scientific knowledge in contemporary life. Scientific communities are diverse and self-organizing systems; if we want 'better science', regardless of what we mean by that, then we must engage with it by embracing its complexity and stop the make-believe tale that it is a commodity to be produced, traded, and priced.

Let me say that I agree with this reading of the text. Left without shallow quantitative managerial devices, scientific communities would have kept improving our knowledge of the world and delivering useful innovation at a pace no slower than what is promoted by current trends of economization and commodification, with the caveat that more effort would have been accorded to knowledge and innovation that are less readily measured. Having said that, I consider that the argument, as presented, misses the mark about the transformation of science in four important ways that limit its analytical power and its effectiveness as a call to action. They correspond to the four assertions of this contribution: (a) scientific communities have failed to update their self-governance as societies transitioned from 'rural-labor societies' to 'urban-knowledge societies'; (b) the ensuing discrepancy from expectations contributed to the economization of science; (c) we must consider two distinct processes of democratization; and (d) geopolitics plays an important role in the establishment of commodification in wealthy nations.

The year we are having this conversation in is the year Wikipedia grew out of its teens. It is also the year when a mutation that turns humans into telepaths with shared eidetic memory became prevalent in more than half the world's population 1 . Those 'several decades [since] science is no longer viewed as a public good but as a financial good, and no longer expected to advance civilization or culture in a wider sense but to first and foremost drive economic growth' that Hallonsten mentions (2021: 12-13) have also transformed the relationship every single human being entertains with knowledge. The stakeholders of science have not expanded, they have exploded. Everywhere on earth, people constantly interact with new knowledge, and when given the chance of a conducive environment, inconspicuous techies make contributions to widely used free and open-source software and hardware that elsewhere could be awarded a master's degree, high-schools command reagents and join cutting-edge drug discovery and synthetic biology research, math enthusiasts collaborate across the globe with professional mathematicians in solving some of the toughest problems, indigenous peasants document and share knowledge and biological material 2 [START_REF] Albagli | Open Science, Open Issues. Rio de Janeiro: IBICT and Unirio[END_REF]Kothari et al., 2019), and the social sciences equally face a turn towards situated and participatory research [START_REF] Albert | Citizen social science: New and established approaches to participation in social research[END_REF]. Science communication broadcasts proliferate, with massive followings, and recent US opinion polls show decent levels of trust in science, which may vary significantly across issues, but where a majority of people align with scientific standards of trust when it comes to funding sources, peer review and transparency [START_REF] Funk | Key findings about Americans' confidence in science and their views on scientists' role in society[END_REF]. Furthermore, it should be keenly noted that almost nowhere in these phenomena do people articulate or justify science in terms of metrics and indicators.

The knowledge society, the one where communities engage with knowledge, is a fulfilled promise, yet for a number of reasons science did not react accordingly. Facing a world where knowledge justice becomes a popular demand (Kapczynski and Krikorian, 2010), universities that 1 A more revealing description of humans carrying Internet-connected smartphones; along the lines of cyberfeminism [START_REF] Haraway | Simians, Cyborgs, and Women: The Reinvention of Nature[END_REF] were conceived to train elites have done very little to adapt, let alone live up to the challenge, when they're the ones positioned to take it to the next level. In that, economization and commodification have conveniently served the corporatism of scientific institutions by providing an alibi and a distraction to shield them from updating their governance to account for contemporary realities of the knowledge public good [START_REF] Lafuente | La lengua de la ciencia y su (inaplazable) conexión con la sociedad[END_REF]. To this day, progress towards a scientific culture more fairly distributed throughout society, from access to publications, to open educational resources, to commitment with the territory, and engagement with pre-tertiary education, have come mostly as a result of hard fought, sometimes heroic, battles, despite the technological affordances and the increasing numbers of higher-education alumni favoring these ends. The advent of a free, multilingual and universal encyclopedia had to wait for the initiative of a pornography entrepreneur; and despite the fact that Wikipedia established itself as one of the main learning resources used by citizens and scientists alike, scientific institutions rarely do better than ignore it. Universal access to the scientific literature has to be 'illegally' provided by a grassroots movement on the backs of a suicide and a forced exile3 , while some major scientific professional associations are still actively fighting against it4 . Community-engaging institutional initiatives have seen an expansion5 [START_REF] De Filippo | A bridge between society and universities: A documentary analysis of science shops[END_REF] under recent trends towards Responsible Research and Innovation (RRI; [START_REF] Robinson | RRI legacies: Co-creation for responsible, equitable and fair innovation in Horizon Europe[END_REF], but they will remain a distant reality for most science if they are not radically scaled-up. At the same time, both private and public universities in many countries have raised costs for diplomas whose value has entered a regime of diminishing returns. And it took a world-stopping pandemic for most of them to make a move towards more accessible digital learning, but with prices and practices still seeing little change. In short, over the last decades, scientific institutions -faced with popular demand and disposing of the means to respond to it -have for the most part slipped away from the challenges and responsibilities associated with a mission to 'advance civilization or culture in a wider sense' instead of cooperating for the public good. As this gap grew with each techno-social shift during the period, embracing economization and the game of metrics has been the path of least resistance for science. By yielding control to bureaucrats that have an interest in controlling science for strictly economic ends, science as an institution has managed to avoid facing a world that is no longer satisfied by the producer-consumer game. Ironically, this is a world that, much like science itself, wants to know 'why?' [START_REF] Benkler | The Wealth of Networks: How Social Production Transforms Markets and Freedom[END_REF][START_REF] Kotler | The prosumer movement[END_REF].

The term democratization therefore can be attributed to two distinct but related processes. One is a deep democratization that, as we've just seen, still paralyzes a majority of scientists and institutions. The other is the one Hallonsten discusses, which comforts science and, as long as science submits itself, promises to fund and shield it from actually engaging with its expanded stakeholders. The forces supporting this latter alternative are, as he explains, connected to the broader economization of life and the political instrumentalization of science. In this context, behind the facade of efficiency and performance evaluation, the enthronement of metrics acts as a performative instrument to designate an elite, of what is in reality a self-organizing system, providing other elites with a lever to influence and direct scientific efforts [START_REF] Lebaron | Injonction comptable et révolution culturelle à l'Université[END_REF]. That scientific communities have significantly expanded over the years, and became more international, only makes such a 'need for elite' more pressing. These observations lead me to suggest that we'll communicate better if we refer to this second process by the more appropriate term 'elitization'6 .

The performative power yielded by elitization, as just noted, is particularly relevant in the international arena. By numerologically construing a mostly Anglo-Saxon elite as the global reference for performance evaluation, wealthy nations manage to command the academic efforts and investments of poor countries towards goals that serve them, in place of situated goals that might serve these countries better [START_REF] Acharya | Applied research in low-income countries: Why and how[END_REF]. Interestingly, such arrangements get deployed with little effort, given the cross-border nature of science, the shallow but deceiving 'neutrality' cover of quantitative performance metrics, and above all the globally dominant developed-developing ideology whereby the poor must overcome poverty by imitating the rich, or rather by following the path the rich tell them is right [START_REF] Sachs | Foreword[END_REF]. This, in turn, creates a strong motivation for decision makers in rich countries to push for elitization, even if they were to acknowledge that it might hurt their own scientific communities. In a globalized world, just like monopolist corporations get overlooked by antitrust agencies because their monopoly power allows them to take over foreign markets, bringing profits home, the elitization of science may be seen by wealthy countries as a strategy to frame the world's research and teaching around issues and perspectives to their advantage.

Fortunately, none of these forces are absolute, and none of the actors are trivial. Research communities have often organized and reacted to commodification by adapting publication and citation practices, and there are clear and steady advances towards a more positively democratized research, in the sense of an open one, as shown by the very examples listed above. Yet there's only so much researchers can do on their own, and research institutions are still shy. They are only recently adopting bolder requirements for research transparency, which are already obvious to society, and for the most part still remain removed from more participatory paradigms. For instance, we see this with the underwhelming outcome of the overwhelming societal response to the pandemic regarding innovation in medical equipment, where institutional support was required and failed [START_REF] Stirling | The COVID-19 pandemic highlights the need for open design not just open hardware[END_REF]. But insofar as research institutions are governed by researchers, among whom I figure, it is our responsibility to keep pushing for reforms. We do need to invest more -in terms of thought, time and money -in experimentation and significant deployment for bringing in the public, of which we're all a part. Finding the right mix of disciplined and mutually interesting ways to do this is imperative lest we spend our lives hiding behind economization, for this is no longer the 20th century.

  //www.breakinggoodproject.com/ ) , the Polymath Projects (https://polymathprojects.org/ ) , and Via Campesina (https://viacampesina.org/ ) .

				.
	2	Referring	to	initiatives	such	as	iGem	(https://igem.org/ ) ,	Breaking	Good
	(https:

I refer, of course, to Aaron Swartz and Alexandra Elbakyan.

For instance, the American Chemical Society (ACS) recently joined forces with editorial corporations to take down access to Sci-Hub, in the peak of the coronavirus pandemic, for the whole of India. See: https://torrentfreak.com/sci-hub-scientists-academics-teachers-and-students-protest-blocking-lawsuit-050121/

Such as university makerspaces and science shops, following an earlier expansion of autonomous hackerspaces.

Which has the added benefits of (1) not overloading the frequently employed term 'democratization of science' with a notion that is, in a sense, its opposite; and (2) not overlapping with terms used to describe other, non-commodified, processes of scientific elitism[START_REF] Bourdieu | Homo Academicus[END_REF].

Acknowledgement

The author would like to thank Allison-Marie Loconto for commenting and reviewing the manuscript.

Funding

The author would like to thank the LabEx SITES -IFRIS for their financial support.

Author biography

Alexandre Hannud Abdo is a researcher at the Laboratoire Interdisciplinaire Sciences Innovations Sociétés (LISIS, Gustave Eiffel University, France) and a member of Garoa Hacker Clube (Brazil). He is active in movements and organizations dedicated to cognitive justice, and his research intersects computational social science, 'science, technology and society', and participatory action research.