L Briñón-Arranz
email: lara.brinon-arranz@gipsa-lab.fr

Multi-Robot Symmetric Formations for Gradient and Hessian Estimation with Application to Source Seeking

Lara Briñón-Arranz, Alessandro Renzaglia, and Luca Schenato

Abstract-This paper deals with the problem of estimating in a collaborative way the gradient and the Hessian matrix of an unknown signal via noisy measurements collected by a group of robots. We propose symmetric formations with a reduced number of robots for both the two-dimensional (2-D) and the threedimensional (3-D) cases, such that the gradient and Hessian of the signal are estimated at the center of the formation via simple computation on local quantities independently of the orientation of the formation. If only gradient information is required, the proposed formations are suitable for mobile robots that need to move in circular motion. We also provide explicit bounds for the approximation error and for the noise perturbation that can be used to optimally scale the formation radius. Numerical simulations illustrate the performance of the proposed strategy for source seeking against alternative solutions available in the literature and show how Hessian estimation can provide faster convergence even in presence of noisy measurements.

Index Terms-Networked Robots, Sensor Networks, Cooperating Robots, Cooperative Estimation, Source seeking.

I. INTRODUCTION

S OURCE localization is a fundamental problem in nature which is relevant to many complex applications, such as environmental monitoring [START_REF] Ogren | Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment[END_REF], search and rescue operations [START_REF] Kumar | Robot and sensor networks for first responders[END_REF], odor source detection [START_REF] Marques | Particle swarm-based olfactory guided search[END_REF] and pollution sensing [START_REF] Li | Moth-inspired chemical plume tracing on an autonomous underwater vehicle[END_REF]. The problem of localizing the source of a signal has been approached in two distinct manners: deploying a fixed network of sensors that collect measurements to cooperatively estimate the source location or employing a mobile robot or a robot team equipped with appropriate sensors which is steered toward the source using collected signal measurements while moving (see [START_REF] Bayat | Environmental monitoring using autonomous vehicles: a survey of recent searching techniques[END_REF] and the references therein).

The majority of the source seeking strategies proposed in the current literature exploit the measurements collected by a group of mobile robots to estimate properties of the signal such that the model parameters of the scalar field of interest or its gradient. This estimated information can then be used to steer the robots toward the source location, for instance via gradientbased methods. There are several approaches to estimate the gradient of a signal, as extremum seeking techniques [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF], least-squares methods [START_REF] Ryan | Peak-seeking control using gradient and hessian estimates[END_REF], [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF], consensus-based parametric algorithms [START_REF] Li | Cooperative distributed source seeking by multiple robots: Algorithms and experiments[END_REF] and cooperative Kalman filters [START_REF] Wu | Robust cooperative exploration with a switching strategy[END_REF]. However, all these strategies do not explicitly quantify the quality of the estimation algorithm nor the impact of measurement noise and at best the performance is analyzed only in simulation. Such error analysis is of paramount importance in practical applications to properly design formation shapes or exploration strategies [START_REF] Leonard | Collective motion, sensor networks, and ocean sampling[END_REF]. Such analysis is often avoided in the literature since no specific formation patterns of the robots are enforced, thus resulting in algorithms which are highly nonlinear in the relative distance measures with high communication and computational complexity. Finally, only few works addressed the problem of multi-robot source seeking in 3-D scenarios [START_REF] Cochran | 3-D source seeking for underactuated vehicles without position measurement[END_REF], [START_REF] Lin | 3-D velocity regulation for nonholonomic source seeking without position measurement[END_REF] or proposed strategies to estimate the Hessian of the unknown signal from noisy measurements [START_REF] Ryan | Peak-seeking control using gradient and hessian estimates[END_REF], [START_REF] Fabbiano | Source localization by gradient estimation based on poisson integral[END_REF], [START_REF] Williams | Probabilistic spatial mapping and curve tracking in distributed multi-agent systems[END_REF].

Based on the ideas presented in [START_REF] Briñón-Arranz | Distributed source seeking via a circular formation of agents under communication constraints[END_REF], where a strategy to estimate only the gradient of 2-D signals is proposed, this paper aims to overcome the main drawbacks of related works presenting an alternative strategy to cooperatively estimate the gradient and the Hessian matrix of an unknown signal for both 2-D and 3-D scenarios. The major contribution is to show that under some specific symmetric formation patterns, the estimates can be obtained via simple averages that involve only the products of the measurements collected by the robots and their relative position with respect to the formation center. Differently, the aforementioned works typically require the solution of a least-squares problem of the size of the formation size, which then scales quadratically from both a computational and a memory/communication perspective. Besides reduced communication and computational complexity which is appropriate for implementation in small robotic platforms, this approach also provides explicit a priori computation of the approximation error and of the measurements noise error which are shown to be monotonically decreasing and increasing functions of the formation radius, respectively. As so, if the radius is not properly chosen, either the approximation error or the measurement noise can induce poor estimation quality.

To validate the quality of the proposed strategy, we numerically compare it with an alternative least-squares-based solution [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF] and we show how Hessian information can be used to drive the center of the multi-robot formation to the location of the source much faster than gradient-based algorithms only if the formation radius is properly chosen, thus stressing the value of theoretical analysis of this work. Additionally, the robustness of the estimations with respect to perturbed robots' positions in the formation is studied through numerical simulations.

The rest of the paper is organized as follows. First, Section II states the problem formulation and the paper contributions. Section III exposes the results dealing with the gradient and the Hessian estimation for the 2-D case. Section IV presents the contribution for the 3-D scenario. Section V characterizes the effect of measurement noise in 3-D scenario and discusses how to optimally determine the formation radius. The performance of the proposed estimation method is analyzed through numerical simulations in Section VI. Finally, we present our conclusions and future directions.

II. PROBLEM FORMULATION

The main purpose of the source seeking is to steer a group of robots to the source location of a signal distribution. In this setup, each robot represents a mobile sensor or an autonomous vehicle equipped with a sensor that is able to measure the signal strength emitted by the source. In mathematical terms, the signal distribution σ(r) is a spatial function representing the scalar field at location r achieving its maximum in the position r * where the source is located and smoothly decreasing to zero far from the source. Formally speaking, we will consider the following assumption on this signal, where we denote its gradient and Hessian at a location r as ∇σ(r) and H σ(r) respectively: Assumption 1 The function σ : R p → R + , where p = 2 (2-D) or p = 3 (3-D), is three times continuously differentiable, i.e. σ ∈ C 3 , and all its partial derivatives up to order three are globally bounded. Moreover, ∇σ(r * ) = 0, ∇σ(r) = 0, ∀r = r * , and H σ(r * ) < -aI p , where a > 0 and I p ∈ R p×p denotes the identity matrix.

This assumption implies that there exist scalars L and M such that

| σ(r) -σ(c) -∇σ(c) T (r -c) =:ϕ ∇ (r,c) | ≤ L r -c 2 | ϕ ∇ (r, c) - 1 2 (r -c) T H σ(c) (r -c) =:ϕ H (r,c) | ≤ M r -c 3
where ϕ ∇ (r, c) and ϕ H (r, c) correspond to the first and second order remainders of the Taylor expansion about the point c, respectively. The knowledge of the gradient and the Hessian can then be used to drive the group of robots towards the source via gradient-ascent (GA) or Newton-Raphson-like ascent (NRA) algorithms, for example by steering the center of robots' formation as follows

c k+1 =c k + ε∇σ(c k ), GA c k+1 =c k -ε([H σ(c k ) ] α ) -1 ∇σ(c k ), NRA
where ε > 0 is the step size and the operator [•] α is defined as

[A] α = A, if A ≤ -αI p , [A] α = -I p otherwise,
where 0 < α < a. The previous saturation operator is necessary when the formation center c is far from the source and the Hessian is likely to be positive semidefinite and therefore drives the robots away from the source. Moreover, when c ≈ r * , by assumption we have that A < -aI p and therefore the saturation operator is not active and we recover the standard Newton-Rapshon ascent. A similar approach has been followed in [START_REF] Ryan | Peak-seeking control using gradient and hessian estimates[END_REF]. The core of this manuscript is to provide a procedure to estimate the gradient and the Hessian, referred as ∇σ(c) and H σ(c) , based on local measurements by multiple robots in specific symmetric configurations for both 2-D and 3-D settings. Moreover, explicit bounds on the estimation error are provided based on the parameters L and M defined above.

Remark 1 Although we propose a simple kinematic model for source seeking, it can be extended to non-holonomic dynamic models as in [START_REF] Fabbiano | Source localization by gradient estimation based on poisson integral[END_REF] and [START_REF] Briñón-Arranz | Distributed source seeking via a circular formation of agents under communication constraints[END_REF], but it is not included here for space limitations. The scope of this work is rather to focus on the gradient and Hessian estimation process in the presence of noisy measurements and to evaluate the benefit of estimating the Hessian to speed up the source seeking process. This direction has only been marginally explored in [START_REF] Ryan | Peak-seeking control using gradient and hessian estimates[END_REF] for 2-D scenarios. In 3-D scenarios, Hessian estimation has been proposed in [START_REF] Fabbiano | Source localization by gradient estimation based on poisson integral[END_REF] but only to create suitable damping in the gradient-descent process to guarantee stability rather than to speed up source seeking.

III. 2-D ESTIMATION

A. Robots' formation

Consider a team of N robots distributed uniformly along a circular formation, as shown in panel (a) of Fig. 1, described by a radius D and a central point c ∈ R 2 , such that

r i = c + DR φi e i = 1, . . . , N (1)
where

r i ∈ R 2 denotes the position of robot i, φ i = 2πi/N is the rotation angle, R φ = c φ -s φ s φ c φ
denotes the rotation matrix where c φ and s φ represent the cosine and sine of angle φ respectively, and e = [1, 0] T . For the sake of simplicity, the robots' dynamics are not considered in this paper. In a previous work of one of the authors, the stabilization problem of a group of vehicles with unicycle-like dynamics to the uniformly distributed circular formation defined by ( 1) is studied under a cooperative control approach, see [START_REF] Briñón-Arranz | Cooperative control design for time-varying formations of multi-agent systems[END_REF].

B. Gradient estimation

In the sake of clarity, we briefly recall in this section the results dealing with the gradient estimation of a 2-D signal analyzed in [START_REF] Briñón-Arranz | Distributed source seeking via a circular formation of agents under communication constraints[END_REF].

Theorem 1 Assume that σ(r) : R 2 → R satisfies Assumption 1 and the robots' formation is given by (1). Considering a team of N ≥ 3 robots uniformly distributed along the circle centered at c and defining

∇σ(c) := 2 N D 2 N i=1 σ(r i )(r i -c) (2)
then it holds ∇σ(c) -∇σ(c) ≤ LD.

(

) 3
The proof is based on the Taylor series expansion and uses the trigonometric properties of the uniformly distributed circular formation, see [START_REF] Briñón-Arranz | Distributed source seeking via a circular formation of agents under communication constraints[END_REF] for details. This theorem basically states that the estimation error decreases linearly with the formation radius D, and therefore can be made arbitrarily small. The following result presents the minimal number of robots required to exactly compute the gradient of a quadratic signal at the center of the circular formation which is relevant since it represents a good approximation of the true signal near the maximum.

Corollary 1 Let σ(r) : R 2 → R be a quadratic function, i.e. σ(r) = r T Sr + p T r + q, where S is negative definite, and consider the robots' formation given by (1), then for a team of N ≥ 4 robots we have ∇σ(c) = ∇σ(c).

Remark 2 Note that the previous results, as well as the results presented in the sequel, hold for any configuration defined by (1) with φ i = φ 0 + 2πi/N where φ 0 is an arbitrary rotation angle. As so, the proposed formation is suitable for robots that have to move in circular motion, i.e. φ 0 (t) = ω 0 t, as fixed-wing unmanned aerial vehicles.

C. Hessian matrix estimation

This section presents mathematical results dealing with the estimation of the Hessian matrix of a two-dimensional signal. For estimating the Hessian, an additional robot is placed at the formation center as shown in panel (b) of Fig. 1.

Theorem 2 Assume that σ(r) : R 2 → R satisfies Assumption 1. Let N robots be deployed as in (1) and 1 additional robot placed at the center c. If N ≥ 5 then the quantity:

K σ(c) := 16 N D 4 N i=1 (σ(r i ) -σ(c))(r i -c)(r i -c) T (4)
satisfies

K σ(c) -(3H σ(c) + R π 2 H σ(c) R T π 2 ) F ≤ 16M D (5)
where • F indicates the Frobenius norm. ∈ R 2×2 where stands for the symmetric terms of the matrix. Using the first-order Taylor expansion of each measurement σ(r i ) about the point c and recalling that r i = D where r i := r i -c, then the following equation holds for all i = 1, . . . , N :

σ(r i ) -σ(c) =∇σ(c) T r i + 1 2 r T i H σ(c) r i + ϕ H (r i , c)
where ϕ H (r i , c) denotes the remainder of the Taylor expansion. Pre-multiplying the previous equation by 16 r i /(N D 4 ) and post-multiplying it by r T i , and summing over i = 1, . . . , N , we obtain

16 N D 4 N i=1 (σ(r i ) -σ(c)) r i r T i = 16 N D 4 N i=1 r i ∇σ(c) T r i r T i + 8 N D 4 N i=1 r i r T i H σ(c) r i r T i + 16 N D 4 N i=1 r i ϕ H (r i , c) r T i .
We start by analyzing the first summatory of the right-side equation:

N i=1 r i ∇σ(c) T r i r T i = D 3 N i=1 R φi e∇σ(c) T R φi ee T R T φi =D 3 N i=1 ∇σ x (c) c 3 φ i c 2 φ i s φ i c 2 φ i s φ i c φ i s 2 φ i +∇σ y (c) c 2 φ i s φ i c φ i s 2 φ i c φ i s 2 φ i s 3 φ i = 0
where we used trigonometric properties that allow to write the powers c m φ , c m φ as linear combinations of s φ , c φ , . . . , s mφ , c mφ and the fact

8 N D 4 N i=1 r i r T i H σ(c) r i r T i = 8 N N i=1 R φi ee T R T φi H σ(c) R φi ee T R T φi = 8 N N i=1 b 11 b 12 b 12 b 22 where b 11 =H 11 c 4 φi + 2H 12 c 3 φi s φi + H 22 c 2 φi s 2 φi , b 12 =H 11 c 3 φi s φi + 2H 12 c 2 φi s 2 φi + H 22 c φi s 3 φi , b 22 =H 11 c 2 φi s 2 φi + 2H 12 c φi s 3 φi + H 22 s 4 φi .
Applying trigonometric properties, and observing that it includes fourth-power trigonometric terms, similarly as before we need to have N ≥ 5 for the following equation to hold:

8 N N i=1 b 11 b 12 b 12 b 22 =8 3 8 H 11 + 1 8 H 22 1 4 H 12 1 4 H 12 1 8 H 11 + 3 8 H 22 =3H σ(c) + R π 2 H σ(c) R T π 2 .
We define the approximation error as follows

Ψ H (D, c) = 16 N D 4 N i=1 (r i -c)ϕ H (r i , c)(r i -c) T
and by Assumption 1 this error can be bounded as

Ψ H (D, c) F ≤ 16 N D 4 N i=1 |ϕ H (r i , c)| r i -c 2 ≤ 16M D.
Note that once K σ(c) is computed, an approximation for the Hessian is obtained by the unique solution of the linear system

3 H σ(c) + R π 2 H σ(c) R T π 2 = K σ(c)
which can be easily computed.

The case of quadratic signals is also analyzed and the minimal number of robots required to exactly compute the Hessian matrix at the center of the circular formation is presented in the sequel:

Corollary 2 Let σ(r) : R 2 → R be a quadratic function, i.e. σ(r) = r T Sr + p T r + q, where S is negative definite, and let N robots be deployed as in (1) and 1 additional robot placed at the center c. Then for N ≥ 5, the matrix

K σ(c) defined in (4) satisfies K σ(c) = 3H σ(c) + R π 2 H σ(c) R T π 2 and therefore H σ(c) = H σ(c) .
The previous result follows directly from Theorem 2, since in the case of quadratic functions the Taylor's remainders are zero, i.e. ϕ H (r i , c) = 0, ∀i = 1, . . . , N .

Remark 3

The proposed strategy resembles a sampled version of the Poisson integral for harmonic functions if σ(c) is replaced with 1 N N i=1 σ(r i ) in (4) as proposed in [START_REF] Fabbiano | Source localization by gradient estimation based on poisson integral[END_REF]. However, such procedure is suitable only for harmonic signals σ(•) which are solutions of diffusion equations at steady state, while our proposed strategy can be applied to any signal which is twice differentiable. For example a quadratic function, which is a relevant approximation of the signal near its maximum, is not a harmonic function.

IV. 3-D ESTIMATION A. Robots' formation

Consider a team of N = 2n, n ∈ N robots forming a symmetric configuration composed of two parallel circular formations whose centers are aligned with the z-axis as shown in panel (a) of Fig. 2. The center point c ∈ R 3 is located between the two circles at distance D sin θ F from each one. One half of the robots is uniformly distributed in the upper circular formation, the other half is uniformly distributed in the lower circular formation and thus, their relative vectors with respect to the center point c are also evenly spaced, see Fig. 2. The robots' positions in such configuration are expressed in spherical coordinates as follows:

r i = c + D [s θi c φi , s θi s φi , c θi ] T θ i = θ F , if i = 2m -1 π -θ F , if i = 2m, m = 1, . . . , n (6)
where r i ∈ R 3 denotes the position of robot i, φ i = 2πi/N is the azimuthal angle, D the radial distance to the center c, θ i is the polar angle and θ F is defined such that s θ F = 2/3. Note that this symmetric formation has two interesting and convenient properties. First of all, it is easy to see that all the robots are placed at the same distance from the center, i.e., r i -c = D, ∀i. Moreover, the sum of the relative position vectors of all robots with respect to the center is equal to zero for N = 2n, n ≥ 2, as proven in the sequel:

N i=1 (r i -c) = D s θ F N i=1 c φ i s θ F N i=1 s φ i i=2m-1 c θ F -i=2m c θ F = 0 since N i=1 cos(2πi/N ) = N i=1 sin(2πi/N ) = 0.

B. Gradient estimation

Consider the symmetric formation of robots given by ( 6) collecting measurements of a 3-D signal distribution σ(r). The following theorem is proposed: Theorem 3 Assume that σ(r) : R 3 → R satisfies Assumption 1 and the robots' formation is given by [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF]. Considering a team of N = 2n robots with n ≥ 2 and defining

∇σ(c) := 3 N D 2 N i=1 σ(r i )(r i -c) (7)
then it holds

∇σ(c) -∇σ(c) ≤ 3LD. ( 8
)
Proof 2 Using the first-order Taylor expansion of each measurement σ(r i ) about the point c and recalling that r i = D where r i := r i -c, then the following equation holds for all i = 1, . . . , N :

σ(r i ) -σ(c) = ∇σ(c) T r i + ϕ ∇ (r i , c),
where ϕ ∇ (r i , c) denotes the remainder of the Taylor expansion. Multiplying the previous equation by 3 r i /(N D 2 ) and summing over i = 1, . . . , N , we obtain

3 N D 2 N i=1 σ(r i ) r i - 3σ(c) N D 2 N i=1 r i = 3 N D 2 N i=1 ∇σ(c) T r i r i + 3 N D 2 N i=1 ϕ ∇ (r i , c) r i .
Due to the symmetric properties of the formation, we have that N i=1 r i = 0 and thus

∇σ(c) = 3∇σ(c) N D 2 N i=1 r i r T i + Ψ ∇ (D, c), (9)
where

Ψ ∇ (D, c) = 3 N D 2 N i=1 ϕ ∇ (r i , c) r i .
We analyze the second term of the previous equation using [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF] to express the position of the robots r i to obtain

N i=1 r i r T i = D 2 N i=1 s 2 θ F c 2 φ i s 2 θ F c φ i s φ i s 2 θ F s 2 φ i s θ F c φ i c θ i s θ F s φ i c θ i c 2 θ i
.

The elements depending on the polar angle θ i must be decomposed to consider the robots with even index i = 2m and the rest with i = 2m -1, m = 1, . . . , n, then for n ≥ 2 we have

N i=1 c φi c θi = c θ F n i=2m-1 c φi - n i=2m c φi = 0
and the same results holds for the sine. Considering the properties of the proposed formation (6) for which s 2 θ F = 2/3, and applying trigonometric properties, the following equality holds:

N i=1 r i r T i = D 2 N 2 s 2 θ F 0 N 2 s 2 θ F 0 0 N c 2 θ F = N D 2 3 I 3 , consequently ∇σ(c) = ∇σ(c) + Ψ ∇ (D, c).
According to Assumption 1 the term Ψ ∇ (D, c) satisfies the inequality

Ψ ∇ (D, c) ≤ 3 N D 2 N i=1 |ϕ ∇ (r i , c)| r i -c ≤ 3LD,
which concludes the proof.

Following the same ideas of Theorem 3, it can be proven that the gradient of a 3-D quadratic signal can be exactly computed by the symmetric configuration defined by ( 6):

Corollary 3 Let σ(r) : R 3 → R be a quadratic function, i.e. σ(r) = r T Sr + p T r + q, where S is negative definite, and consider the robots' formation given by (6), then for a team of N = 2n robots with n ≥ 3 we have ∇σ(c) = ∇σ(c).

C. Hessian matrix estimation

Consider the symmetric formation of robots given by ( 6) taking measurements of a signal distribution σ(r). In order to estimate the Hessian matrix of the signal σ(r), three additional robots are required, two placed along the z-axis of the previous symmetric formation, such that the robots positions are given by ( 6) with θ N +1 = 0 and θ N +2 = π and another one placed at the center c of the formation, as shown in panel (b) of Fig. 2. The following theorem is proposed: Theorem 4 Assume that σ(r) : R 3 → R satisfies Assumption 1. Let N = 2n robots deployed as in [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF], two additional robots are defined by [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF] with θ N+1 = 0 and θ N+2 = π respectively, and the robot N + 3 is placed at the center c of the formation. If n ≥ 4 then the quantity

K σ(c) := 18 N D 4 N+2 i=1 (σ(r i )-σ(c))(r i -c)(r i -c) T (10) satisfies K σ(c) - d1 H12 d2 2H13 2H23 d3 =:L(H σ(c) ) F ≤ 18(N + 2)DM N ( 11
)
with d1 =3H11/2 + H22/2 + H33 d2 =H11/2 + 3H22/2 + H33 d3 =H11 + H22 + H33 + 18H33/N ( 12
)
where the Hessian matrix at the center of the formation is denoted by H σ(c) = H11 H12 H22 H13 H23 H33

.

Proof 3 Using the first-order Taylor expansion of each measurement σ(r i ) about the point c and recalling that r i = D, where r i := r i -c then the following equation holds for all i = 1, . . . , N + 2:

σ(r i ) -σ(c) =∇σ(c) T r i + 1 2 r T i H σ(c) r i + ϕ H (r i , c),
where ϕ H (r i , c) denotes the remainder of the Taylor expansion. Pre-multiplying the previous equation by 18 r i /(N D 4 ) and post-multiplying by r T i , and then summing over i = 1, . . . ,N +2, we obtain

K σ(c) = 18 N D 4 N +2 i=1 r i ∇σ(c) T r i r T i + + 9 N D 4 N +2 i=1 r i r T i H σ(c) r i r T i + 18 N D 4 N +2 i=1 r i ϕ H (r i , c) r T i .
To analyze this equation, we decompose the summation in two parts: first, consider the N robots of the cylindrical formation and second, the two additional ones. Following the same steps that in previous cases, applying trigonometric properties and considering robots with even and odd index, we can show that the first term of the right-side of the previous equation is equal to zero for n ≥ 4 and the second term is equal to

3 2 H11+ 1 2 H22+H33 H12 1 2 H11+ 3 2 H22+H33 2H13 2H23 H11+H22+H33+ 18 N H33
.

Additional details on the mathematical derivations can be found in [START_REF] Briñón-Arranz | Technical report on: Multi-robot symmetric formations for gradient and Hessian estimation with application to source seeking[END_REF]. The error term defined by

Ψ H (D, c) = 18 N D 4 N +2 i=1 r i ϕ H (r i , c) r T i satisfies Ψ H (D, c) F ≤ 18 ND 4 N+2 i=1 |ϕ H (r i , c)| r i 2 ≤ 18(N +2)DM N .
Note that once K σ(c) is computed, an estimate H σ(c) for the true Hessian H σ(c) is given by the unique solution of the linear system

L( H σ(c) ) = K σ(c) (13)
which can be obtained by inverting the operator L.

Similarly to the 2-D setting, the case of a quadratic signal is also analyzed in the sequel: Corollary 4 Let σ(r) : R 3 → R be a quadratic function, i.e. σ(r) = r T Sr + p T r + q, where S is negative definite. Considering N = 2n robots deployed as in [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF], two additional robots are defined by [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF] with θ N+1 = 0 and θ N+2 = π respectively, and the robot N + 3 is placed at the center c of the formation. Then for n ≥ 4 the matrix K σ(c) defined in [START_REF] Wu | Robust cooperative exploration with a switching strategy[END_REF]

satisfies K σ(c) = L(H σ(c) ) and therefore H σ(c) = H σ(c) .
Remark 4 Both Corollaries 3 and 4 present the minimal number of robots required to exactly compute the gradient and the Hessian matrix of quadratic signals at the center of the 3-D symmetric formation, respectively. Details of the proofs can be found in [START_REF] Briñón-Arranz | Technical report on: Multi-robot symmetric formations for gradient and Hessian estimation with application to source seeking[END_REF].

V. NOISE AND APPROXIMATION ERROR ANALYSIS

In this section we address the problem of noisy measurements and their effect on the gradient and Hessian estimation, and we compare it with the approximation errors of equations ( 8) and [START_REF] Leonard | Collective motion, sensor networks, and ocean sampling[END_REF]. We limit the analysis to the 3-D scenario, being the 2-D scenario very similar. We assume that the measurements are corrupted by Gaussian zero-mean white noise which models possible small scale spatial variations due to turbulence or local perturbation:

y(r i ) = σ(r i ) + v i (r i ), v i (r i ) ∼ N (0, υ 2 )
where υ 2 represents the variance of the noise. We further assume that the noise is independent in each robot measurement, i.e. E[v i (r i )v j (r j )] = 0, i = j which is a realistic assumption if r i -r j > D min where D min represents the spatial correlation distance of the disturbance. We start by considering the effect on the gradient estimation. Since the noise is additive in the measurement, the gradient estimate defined in [START_REF] Ryan | Peak-seeking control using gradient and hessian estimates[END_REF] becomes

∇σ(c) := 3 N D 2 N i=1 σ(r i )(r i -c) + 3 N D 2 N i=1 v i (r i -c) Ψ ∇ v (D,c)
.

The term Ψ ∇ v (D, c) represents the effect of noise. It has clearly zero mean, i.e. E[Ψ ∇ v (D, c)] = 0 and its expected standard deviation is given by:

s ∇ (D) := E[ Ψ ∇ v (D, c) 2 ] = 3 N D 2 (E[ N i=1 N j=1 υ i υ j r T i r j ]) 1 2 = 3υ N D 2 N i=1 r T i r i 1 2 = 3υ N 1 2 D
, where we used the results from the proof of Theorem 3. Note that this term is a monotonically decreasing function of the formation radius and the number of robots. This term has to be compared with the approximation error provided in [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF] which is monotonically increasing with the formation radius. This observation shows a clear trade-off when deciding the formation radius in order to balance the effect of approximation error and the noise error. As a rule-of-thumb we can try to design the formation radius D * ∇ to exactly balance these two errors (see [START_REF] Briñón-Arranz | Technical report on: Multi-robot symmetric formations for gradient and Hessian estimation with application to source seeking[END_REF] for more details):

s ∇ (D * ∇ ) = 3LD * ∇ =⇒ D * ∇ = υ 1 2 L 1 2 N 1 4 , s ∇ (D * ∇ ) = 3(υL) 1 2 N 1 4
which provides a rule-of-thumb to chose the formation radius if L, υ and N are known. The previous expression shows that both the minimal estimation error and the optimal formation radius decrease as the number of robots increases, however it is important to keep in mind that we must have r i -r j > D min for the previous equation to be realistic.

We now consider the effect on the Hessian estimation. Since even in this case the noise is additive in the estimate of the Hessian, it is straightforward to see that to the estimate H σ(c) there is the additional term:

Ψ H v (D, c) = 18 N D 4 N+2 i=1 (v i -v c )(r i -c)(r i -c) T
Such matrix has clearly zero mean, while the expected standard deviation in terms of the Frobenius norm is given by:

s H (D) :=(E[ Ψ H v (D, c) 2 F ]) 1 2 =E trace(Ψ H v (D, c)(Ψ H v (D, c)) T ) = 18 N D 4 (E[ N+2 i=1 N+2 j=1 (v i -v c )(v j -v c )( r T i r j ) 2 ]) 1 2 = 18 N D 4 ( N+2 i=1 E[v 2 i ] r i 4 + N+2 i=1 N+2 j=1 E[v 2 c ]( r T i r j ) 2 ) 1 2 = 18υ N D 2 N + 2 + ( N 3 + 2) 2 1 2 ≈ 6υ D 2
, where the last equation has been obtained via tedious but straightforward calculations which are reported in [START_REF] Briñón-Arranz | Technical report on: Multi-robot symmetric formations for gradient and Hessian estimation with application to source seeking[END_REF] in the interest of space, and the last approximation is obtained for N 1. As compared to the effect of noise in the gradient estimation s ∇ (D), the standard deviation s H (D) does not decrease to zero as the number of robots increases, and it is also more sensitive to the formation radius since it is inversely proportional to the square of the radius. This last fact is to be expected since we are estimating second order derivatives via finite differences and therefore measurement noise is amplified more than when computing first derivatives.

Similar consideration can be derived for the Hessian in terms of finding the optimal radius D * H to balance the approximation error and the noise error in the scenario N 1:

s H (D * ∇ ) = 18M D * H ⇒ D * H = υ 3M 1 3
, s H (D * H ) = 18(υM )

1 3 3 1 3
.

From a practical perspective, the optimal radius formation should be chosen to balance all approximation errors and noise effects in both the gradient and the Hessian. Since the noise error increases rapidly for decreasing formation radius a Fig. 3: Evolution of the distance between the center of the formation and the source location for the GA algorithm when the gradient is estimated via the least-squares method in [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF] (dashed lines) and using our proposed approach (solid lines).

sensible choice for the radius is the largest between D * ∇ and D * H .

VI. NUMERICAL SIMULATIONS

In this section, we present numerical simulations to illustrate the performance of the proposed cooperative estimation algorithms applied to the source seeking problem. The signal strength considered is a 3-D Gaussian function whose surface sets are ellipsoids given by σ(r) = 100e -r T Sr , where S = 10 -4 100 1 1 1 1 0 1 0 10

. The maximum corresponding to the source is located at r * = [0, 0, 0] T denoted by the black ×.

We start by first comparing the quality of the proposed approximation for the gradient in a 3-D scenario with an alternative solution presented in [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF], which is representative of many other algorithms based on least-squares solutions as in [START_REF] Ryan | Peak-seeking control using gradient and hessian estimates[END_REF], [START_REF] Li | Cooperative distributed source seeking by multiple robots: Algorithms and experiments[END_REF]. Fig. 3 compares the results of the implementation of the GA algorithm using the gradient estimation proposed in Theorem 3 with respect to the gradient estimation proposed in [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF]. Both algorithms were implemented using the same stepsize ε = 0.1, the same formation of N = 8 robots as defined in [START_REF] Khong | Multi-agent source seeking via discrete-time extremum seeking control[END_REF] and no noise was injected in order to evaluate only the approximation error. The evolution of the distance between the formation center and the source location, i.e., c k -r * , at each time step for three different values of the formation radius is shown. The geometrical properties of our proposed symmetric formation lead to a much better estimation of the gradient and thus the center of the formation is driven much closer to the proximity of the source location. It can be seen that the error of our proposed estimation strategy considerably outperforms the least-squares based method from [START_REF] Zhu | Cooperative control of multiple uavs for source seeking[END_REF]. Similar strategies presented in literature which do not exploit the geometric properties of the robots' formation exhibit analogous limitations.

In the next simulation we want to evaluate the benefit of using Hessian information to improve rate of convergence for source seeking applications, and to evaluate the impact of noise and approximation error. Fig. 4 shows a simulation 7), [START_REF] Wu | Robust cooperative exploration with a switching strategy[END_REF] and [START_REF] Lin | 3-D velocity regulation for nonholonomic source seeking without position measurement[END_REF].

In order to evaluate the performance of our proposed strategy we also simulate two ideal algorithms implementing the GA and NRA source seeking defined in Section II assuming exact computation of the gradient and the Hessian at the center of the formation, thus providing a baseline for performance evaluation in a more realistic multi-robot scenario with noisy measurements. The evolution of these two algorithms is shown in Fig. 5 in black pointed and dashed lines, respectively. Fig. 5a also displays the evolution of the NRA algorithm implemented with three different values for the symmetric formation radius with ε = 0.1 and α = 0.001 when the signal measurements are corrupted by white noise with υ = 2. For each iteration of the algorithm, the average over 100 trials of the distance between the center of the formation and the source is provided and the standard deviation is reported each 1000 iterations to make figures easier to read. As discussed in Section V, these curves clearly show the trade-off between approximation error and noise error. If the formation radius is chosen too small (D = 1m), then the noise error dominates and the estimation of the Hessian is so poor that it is not even negative definite and therefore never used in the actual algorithm. As so, the NRA algorithm becomes a simple GA and no benefit in estimating the Hessian appears. If the formation radius is instead chosen too large (D = 10m), then the approximation error dominates and the center of the formation converges to a point which can be far from the true source. Differently, if the radius is properly chosen to balance both measurement noise error and approximation error, then

Fig. 1 :

 1 Fig. 1: Minimal uniformly spaced circular formations for 2-D estimation: (a) formation with N = 3 robots to estimate the gradient and (b) formation with N = 5 robots and a green additional robot at the center to estimate the Hessian matrix.

Proof 1

 1 Let us preliminary define the notation ∇σ(c) = [∇σ x (c) ∇σ y (c)] T and H σ(c) = H11 H12 H22

N

  i=1 cos(m2πi/N ) = N i=1 sin(m2πi/N ) = 0 for N > m (see Appendix for details). Since the first element includes terms with m = 3 we need N ≥ 4 to satisfy the previous property. Let us now consider the second term of the right-side equation containing the Hessian matrix:

Fig. 2 :

 2 Fig. 2: Minimal symmetric cylindrical formations for 3-D estimation: (a) formation with N = 4 robots to estimate the gradient and (b) formation with N = 8 robots and three additional robots in green to estimate the Hessian matrix.

Fig. 4 :

 4 Fig. 4: 3D symmetric formation of 11 robots performing a source seeking mission based on the gradient and Hessian cooperative estimation of the signal at the formation center.

convergence behavior is very close to the ideal NRA with no noise and no approximation error.

With a view to study the robustness of our proposed estimation strategy with respect to formation errors, i.e., when the robots' positions do not exactly fit with the desired symmetric formation, we considered a perturbation vector δ i = [δ xi , δ yi , δ zi ] T where δ xi , δ yi , δ zi , ∼ N (0, δ 2 ) such that the perturbed position of each robot i is defined as r p i = r i +δ i , where r i is the desired robot's position in the symmetric formation. Fig. 5b shows the average over 100 trials of the distance between the center of the formation and the source for each iteration of the NRA algorithm using the same 3-D 11robots formation as in the previous case, with three different values of the radius formation D = 1, 5, and 10m, considering now misalignment in the robots' positions with δ = 0.1m. The standard deviation is reported each 1000 iterations. As shown in the figure, the gradient and Hessian estimation errors due to the perturbed robots' positions increase for smaller values of the formation radius. This behavior is similar to the noisy measurements case and again the formation radius plays an important role to balance approximation error and formation error. It is however important to remark that, even if our strategy strongly relies on the symmetric geometrical properties of the formation, the estimations computed in presence of errors in the robots' positions can be still exploited to locate the source and the estimation errors remain bounded.

VII. CONCLUSION

This paper deals with the cooperative estimation of the gradient and Hessian matrix of a signal in the context of source seeking missions. We showed that the signal measurements collected by a planar circular formation of robots allow estimating both quantities at the center of the formation for the 2-D case. Moreover, another symmetric formation is proposed to deal with 3-D scenarios. In both cases, a limited number of robots and low computations are required to compute the estimates whose error is bounded and depends on the radius of the formation. Additionally, we mathematically proved that both the gradient and the Hessian can be exactly calculated for quadratic signals. Thanks to these results, the convergence rate of Newton-Raphson based source seeking algorithms is improved as shown in numerical simulations. A detailed analysis of the effect of noisy measurements in both the gradient and the Hessian estimations is provided. A further analysis of the effect of formation errors has been carried out in simulation, showing the robustness of our approach also with respect to errors in the robots' positions.

The design of an adaptive formation radius to increase the convergence rate of the Newton-Rhapson-based source seeking algorithm will be a key point in future research.


Professor Catherine Viot Full
email: catherine.viot@univ-lyon1.fr

Caroline Bayart
email: caroline.bayart@univ-lyon1.fr

Agnès Lancini
email: agnes.lancini@univ-lyon1.fr

The Consumer Intention to Adopt Smart Connected-Products: Does the Category Matter?

Keywords: Brand Trust, Smart connected products, Theory of planned behavior, Insurance

The development of the Internet Of Things (IOT) could be considered as the third period of the digital transformation. More and more Smart Connected-Products (SCPs) are used, whether in the private or public domain. And many different sectors, as automotive, health or insurance develop such products, which can gather, analyze and generate data thanks to sensors and connection. If companies look like interested by this business opportunity, the adoption of SCP is not obvious to consumers.

Drawing on the Theory of Planned Behavior (TPB) and on the concept of Consumer innovativeness, this study proposes a model to better understand the intention to adopt SCP. To test our proposed research model, we used an online survey to collect data from a convenient sample of young people from generation Z.

According to the results, it seems that intention to adopt SCPs depends on feelings about this decision (subjective norms), personal attitudes towards SCPs and cognitive innovativeness (intellectually interesting and challenging products). But as far as the intention to adopt SCPs linked to insurer is concerned, the subjective norms and the cognitive innovativeness are identified as two main variables.

Introduction

As underlined by Hoffman and Novak (2015, p. 11),"We are now entering the third phase of the Internet, the Internet of Things phase, where consumers can not only interact directly with the devices themselves, but in addition the devices can interact with each other". The first phase of internet was essentially technological and informational-oriented, while the second one was mainly social-oriented.

Smart-connected products (SCPs), although highly varied1 , generally share the same components: chips, sensors, and software programs connected to the Internet. They are able to collect, analyze, and diffuse data [START_REF] Rijsdijk | How today's consumers perceive tomorrow's smart products[END_REF].

SCPs incorporate new technologies and thus are innovative products. So, it seems logical to wonder if their adoption by the consumer follows the same model as that brought to light for the adoption of new products. For example, it has been shown that the adoption of new products relies on individual traits such as innovativeness [START_REF] Roehrich | Innovativités hédoniste et sociale: Proposition d'une échelle de mesure[END_REF]. According to the Theory of Planed Behavior (TPB) [START_REF] Fishbein | Belief, attitude, intention and behavior: an introduction to theory and research[END_REF], the intention to adopt a given behavior -adoption of SCPs -depends on the attitude towards this behavior (am I favorable/unfavorable to the adoption of SCPs), the perceived control towards this behavior (would I be able to operate SCPs) and the subjective norms (what do people think about using SCPs).

This paper aims to examine the following research questions.

1) Can the adoption of SCPs be explained by the consumer innovativeness?

2) Does the TBP constitute a relevant theoretical framework to explain the adoption of SCPs?

3) Does the product category matter: does the adoption of SCPs linked to the insurer differ from the adoption of others SCPs?

The consumer attitude towards SCPs is ambivalent. SCPs can generate resistance (Chouk and Many, 2016). On the one hand, three consumers in four consider that SCPs tend to improve their everyday life. On the other one, two in three consider these products as pointless2 .

3 An undeniable enthusiasm for SCPs can be observed, both among practitioners and researchers. Indeed, a better understanding of factors enabling or inhibiting the adoption of such products is needed. This research makes an important theoretical contribution by revisiting the concept of innovativeness and a well-established theoretical framework: the TPB.

This paper is structured as follow. The conceptual and theoretical frameworks are first presented. Then, an integrative model -including the consumer innovativeness and the TPBis proposed and tested. The results are then discussed.

Conceptual and theoretical framework

Smart Connected Products (SCP): Definition and Issues

Definition of smart-connected products

Digital technologies know a new transformation since the early 2010s with the emergence of smart-connected products [START_REF] Barlatier | Management de l'innovation et nouvelle ère numérique -enjeux et perspectives[END_REF] and the internet of things. [START_REF] Porter | How smart-connected products are transforming companies[END_REF] speak about a '3rd wave of competition based on the IT' which is characterized by the fact that information technologies are an integral part of products, through the presence of electronic chips, sensors, and software programs connected to the Internet. They so distinguish three components characterizing SCP. "All smart, connected products, from home appliances to industrial equipment, share three core elements: physical components (such as mechanical and electrical parts); smart components (sensors, microprocessors, data storage, controls, software, an embedded operating system, and a digital user interface); and connectivity components (ports, antennae, protocols, and networks that enable communication between the product and the product cloud, which runs on remote servers and contains the product's external operating system)" (Porter and Heppelmann, 2015, p. 98).

Interest of smart-connected products for companies and brands

This new digital era opens a range of possibilities which comes to revolutionize as well the social sphere as the organizational one. According to Gartner, there will be more than 20 billion of SCPs in the world, in 2020. The progress of crossed analysis with Big Data could leverage their utility for companies. Thanks to their connectivity component, SCPs produce a huge quantity of data relative to uses and behaviors, so allowing a continuous improvement of their features and performance. Smart-connected devices produce an important quantity of data which might be of interest to companies and brands. These data, if analyzed, could potentially constitute a competitive advantage because they allow a better understanding of the consumer behavior, and improve the customer knowledge.

Certain services could know a real revolution thanks to SCPs. This is, for example, the case of the insurance industry. The SCPs will most probably have a radical impact in this sector, because they could question its core business, namely the mutualization of risks.

Theory of Planned Behaviour

The development of integrated models of behavior has received considerable attention in the literature [START_REF] Armitage | Efficacy of the Theory of Planned Behaviour: A metaanalytic review[END_REF]. The Theory of Planned Behavior [START_REF] Ajzen | From intentions to actions: A theory of planned behaviour[END_REF][START_REF] Ajzen | The theory of planned behavior[END_REF] is essentially an extension of the previous Theory of Reasoned Action [START_REF] Fishbein | Belief, attitude, intention and behavior: an introduction to theory and research[END_REF], commonly known TRA. This theory is based on the premise that individuals consider available information to assess the impact of their choices (expected advantages vs. perceived disadvantages). Behavior is a direct function of behavioral intention. TRA provides a conceptual framework for explaining behavior intention by two major components: attitude towards desirable behavior and subjective norms.

However, the TRA has to address a few limitations. This theory doesn't consider factors which constrain individuals and may have negative impact on their intention to engage in a behavior (skills, abilities, available time) [START_REF] Kefi | Mesures perceptuelles de l'usage des systèmes d'information : Application de la théorie du comportement planifié[END_REF]. [START_REF] Ajzen | From intentions to actions: A theory of planned behaviour[END_REF] has proposed an extension of the theory by including perceptions of behavioral control as an additional predictor of intentions and behavior. He developed the TPB, widely cited in international scientific literature [START_REF] Eagly | The psychology of attitudes[END_REF]. This theory was successfully used to explain and predict individual decision Pavlou to follow a particular behavior in many different contexts [START_REF] Sheppard | The Theory of Reasoned Action: A Meta-Analysis of Past Research with Recommendations for Modifications and Future Research[END_REF][START_REF] Pavlou | Understanding and predicting electronic commerce adoption: An extension of the theory of planned behavior[END_REF]. Several academic journals proposed meta analyses which highlight its predictive power on the intention to engage in a particular behavior [START_REF] Godin | The theory of planned behavior: A review of its applications in health-related behaviors[END_REF][START_REF] Notani | Moderators of Perceived Behavioral Control's Predictiveness in the Theory of Planned Behavior: A Meta-Analysis[END_REF][START_REF] Armitage | Efficacy of the Theory of Planned Behaviour: A metaanalytic review[END_REF]. According to this social cognitive theory, proximal determinants of behavior are divided in two main categories:

-emotional and affective factors: attitude (or overall evaluation of the behavior), and subjective norms (or individual's beliefs about significant others' evaluation of this behavior) ; -cognitive factors: perceived behavioral control or expectancy that performance of the behavior is under control.

Consumer innovativeness

Researchers in marketing are paying attention to consumer innovativeness since the 1960s.

Innovativeness as a concept

Innovativeness is defined as "the degree to which an individual makes innovation decisions independently of the communicated experience of others" (Midgley and Dowling, 1978, p. 235), or as "the degree to which an individual is relatively earlier in adopting new ideas than the average member of his social system" (Rogers and Shoemaker, 1971, p. 27).

For some authors, consumer's innovativeness is domain-or product-specific (Goldsmith and Hofaker, 1991). For others researchers, innovativeness is considered as a personality trait generalizable through several categories of products [START_REF] Midgley | Innovativeness: The concept and its measurement[END_REF]. Shared by all the individuals at different degrees, innovativeness is considered as a personality's trait dominating among pioneers, those consumers who tend to adopt new products in an early way and independently of their circle of acquaintances [START_REF] Rogers | Diffusion of innovations[END_REF]. This personality's trait must be considered as a latent construct that can be analyzed and measured through its demonstrations (Le [START_REF] Louarn | La tendance à innover des consommateurs: Analyse conceptuelle et proposition d'une échelle de mesure[END_REF]. [START_REF] Rogers | Diffusion of innovations[END_REF] initiated a new way in measuring consumer's innovativeness, based on the time of adoption of a new product that is the time elapsed between the product launch and its effective adoption by a given individual. [START_REF] Midgley | Innovativeness: The concept and its measurement[END_REF] criticized this measurement method for theoretical and methodological reasons. From a theoretical point of view, the main criticism concerns the fact that the time of adoption is a temporal construct which really has no relation with the latent construct of innovativeness. From a methodological point of view, measuring consumer's innovativeness by the time of adoption raise problems of validity and reliability: first, it is difficult to make comparisons between studies; second, the time of adoption cannot be used to predict behavior, and third, the measurement is biased, as it depends on the capacity of the consumer to remember the time elapsed between the launch and the adoption of the new product.

Consumer innovativeness measurement

In response to these critics, a new kind of measure was proposed: it consists in selfassessment scales describing typical behaviors of innovative consumers. Some of these scales are domain-or product-specific (Godsmith and Hofaker, 1991), while others are more universal, and are either unidimensional [START_REF] Bruner | Gadget lovers[END_REF], or multidimensional (Roehrich, 1994).

More recently, Vandecateele and Geuens (2010) underline some weaknesses of consumers' innovativeness scales: first, they do not take into account the entire motivations to buy novelties, and second, they present a poor predictive power of new products purchase. In order to improve the measure of consumer' innovativeness and its predictive power, they develop the Motivated Consumer Innovativeness Scale (MCI). This 20-item scale includes four motivations: functional innovativeness (product performance), hedonic innovativeness (pleasure), cognitive innovativeness (mental stimulation), and finally, social innovativeness (differentiation) (table 2).

Research hypothesis and model

Theory o Planned Behavior and intention to adopt SCPs

Attitude depends on perceived consequences of the behavior. Individuals estimate the likelihood that an outcome probably arises as a result of the behavior and try to evaluate it.

The more perceived advantages (or consequences seen as positive) are evident, the more attitude toward behavior will be positive. Attitude refers to stable judgments, which encourage individuals to engage in a particular behavior. According to [START_REF] Gao | A unified perspective on the factors influencing consumer acceptance of internet of things technology[END_REF] perceived utility and enjoyment impact positively the intention to adopt SCPs (hypothesis H1).

H1: Attitude towards the adoption of SCPs is positively related to the intention to adopt such products.

H1 is declined in two sub-hypothesis depending on considering SCPs in general "generic

SCPs" (H1a) or smart products connected with an insurer "SCPs linked to an insurer" (H1b).

Subjective norms represents not only perception of significant others' preferences about the adoption of new behavior by an individual, but also his/her motivation to comply with these expectations. It deals with the evaluation that behavior is approved or not by most people who are important to him/her [START_REF] Ramayah | A decomposed theory of reasoned action to explain intention to use Internet stock trading among Malaysian investors[END_REF]. If an individual thinks that his/her family members or close friends expect him/her to engage in a certain behavior, it will be his/her intention to do so. As SCPs are relatively new products and potential users don't have enough feed back regarding their use [START_REF] Gao | A unified perspective on the factors influencing consumer acceptance of internet of things technology[END_REF]). That's why significant others' preferences play a major role in SCPs adoption. We can state the following hypothesis (H2): Perceived behavioral control reflects the ability to act on one's environment and influence what will happen [START_REF] White | Motivation reconsidered: The concept of competence[END_REF]. Control factors could be both internal (personal deficiencies, skills…) and external (opportunities, barriers…). The more individuals believe they have control over their behavior, the more they are likely to engage in it. On the other hand, if individuals anticipate potential difficulties, they will be unwilling to adopt the behavior. By nature, SCPs are ubiquitous (inside and outside households), but also versatile products. So, many consumers are not able to understand how data are generated, shared and gathered. If perceived behavioral control is too weak, there will be sometimes a tendency not to buy these products. On the basis of these findings, we can state the following hypothesis (H3).

H3: Perceived control concerning adoption of SCPs is positively related to the intention to adopt such products.

H3 is declined in two sub-hypothesis depending on considering SCPs in general "generic

SCPs" (H1a) or smart products connected with an insurer "SCPs linked to an insurer" (H1b).

Consumer innovativeness and intention to adopt SCPs

The four motivations identified by Vandecateele and Geuens ( 2010) are proved to be relevant predictors of the adoption of new products.

Hedonic innovativeness, "the self-reported consumer innovativeness motivated by affective and sensory stimulation and gratification", explains the intention to adopt new products" [START_REF] Baumgartner | Exploratory consumer buying behavior: Conceptualization and measuremen[END_REF]Roehrich, 1994[START_REF] Roehrich | Innovativités hédoniste et sociale: Proposition d'une échelle de mesure[END_REF][START_REF] Venkatraman | Differentiating between cognitive and sensory innovativeness: Concepts, measurement, and implications[END_REF]. (Roehrich, 1994). Given that SCPs can be considered as new products, hypothesis H7 is postulated as follow.

H7: Social innovativeness is positively related to the intention to adopt SCPs.

H7a: Social innovativeness is positively related to the intention to adopt generic SCPs.

H7b: Social innovativeness is positively related to the intention to adopt SCPs connected with an insurer. To test the scales reliability, Principal Component Analyses were carried out 3 . Then, the model and hypothesis were tested using structural equation modeling with AMOS. A bootstrap was performed in order to obtain a P value for each estimated parameter.

Findings

TPB and intention to adopt SCPs

Hypotheses H1a, H2a, H2b, and H3a are validated, while H1b and H3b are rejected (Figure 2). The TPB is relevant in the case of generic SCPs. The adoption of such products depends on attitude, subjective norms, and perceived control. On the contrary, the TPB is not a relevant framework when it comes to explain the adoption of SCPs connected with an insurer.

Consumer innovativeness and intention to adopt SCPs

Only hypotheses H6b and H7b are validated, while H4a, H4b, H5a, H5b, H6a, and H5a are rejected (Figure 2). The consumer innovativeness is not relevant when it comes to explain the adoption of generic SCPs. On the contrary, two dimensions of the consumer innovativenessfunctional and social innovativeness -are predictors of the intention to adopt SCPs connected with an insurer. 3 Results are not presented here. As far as the intention to adopt SCPs linked to an insurer is concerned, the explanatory power of subjective norms, perceived control and attitude is not significant. The social and functional innovativeness are identified as the two main predictors of the intention to adopt such SCPs.

The consumer innovativeness framework seems to be more relevant than the TPB.

Managerial Contributions

The present study yields some interesting insights for practice. For the manufacturers who target the young consumers with SCP's connected with an insurer, it is crucial to develop useful and practical SCPs, because the intention to adopt them depends partly on the functional innovativeness. As far as marketing business and communication is concerned, managers should underline this argument while considering as well the weight of subjective norms. Messages showing the interaction with other people and the positive reactions of friends or members family regarding the adoption of SCP's should be put forward. Insurers' communication should also put forward the relevance of the SCP, insisting on their power of differentiation (social innovativeness).

The recommendations for generic SCPs should differ. The adoption of generic SCPs is mainly explained by the TPB. None of the four dimensions of the consumer innovativeness is significantly linked to the intention to adopt such products. The attitude towards generic SCPs is the stronger predictor of intention to adopt them, although the perceived control and subjective norms are also significantly and positively related to this intention. Insurers' communication should put forward the relevance of generic SCP's, insisting on the opinion of the close friends -or relatives -of the consumer (subjective norms). They also have to "educate" the consumer to by-pass restraint for purchase SCPs due to their perceived complexity (perceived control).

Limitations and future research

As far as we know, this research is one of the first exploratory studies in the early stage of understanding the SCPs adoption phenomenon. The results must be confirmed with more representative and broader samples. The model can be refined and completed by the introduction of new mediating and/or moderating variables. For example, it is likely that variable such as familiarity and/or subjective knowledge towards SCPs play an important role in their adoption.

Future studies need more contextualization. They should focus on specifics SCPs and describe their features as well as the nature of the collected and transferred data, and the actors to whom these data can be transferred. It would be also interesting to survey Gen-Xers to highlight possible effects of generation.

Figure

  Figure 1 -Model and research Hypothesis

Figure 2 .

 2 Figure 2. Estimation of parameters 4

Table 2 : Dimensions of the MCI scale and comparisons with other scales

 2

	Dimensions		Definitions		Presence in other scales
		(Vandecateele and Geuens, 2010, p. 310)
	Hedonic	"The	self-reported	consumer	Venkatraman and Price
	innovativeness	innovativeness motivated by affective and	(1990)
		sensory stimulation and gratification"	Roehrich (1994, 2004)
					Baumgartner and Steenkamp
					(1996)

Toothbrush, shoes, cars, bathroom scale…

IFOP, 2014, France.

By analogy, it can be stated that hedonic innovativeness is a predictor of intention to adopt SCPs.

H4: Hedonic innovativeness is positively related to the intention to adopt SCPS. H4a: Hedonic innovativeness is positively related to the intention to adopt generic SCPs.

H4b: Hedonic innovativeness is positively related to the intention to adopt SCPs connected with an insurer.

It has been shown that the cognitive innovation, defined as "the self-reported consumer innovativeness motivated by the need for mental stimulation", is positively linked to the adoption of new products [START_REF] Venkatraman | Differentiating between cognitive and sensory innovativeness: Concepts, measurement, and implications[END_REF][START_REF] Baumgartner | Exploratory consumer buying behavior: Conceptualization and measuremen[END_REF].

Thus, hypothesis H5 can be stated. Functional innovativeness, the "self-reported consumer innovativeness motivated by the functional performance of innovations", predicts the adoption of new products [START_REF] Venkatraman | The impact of innovativeness and innovation type on adoption[END_REF]. By analogy, and given that SCPs are new products, hypothesis H6 can be stated. Social innovation, defined as the "self-reported consumer innovativeness motivated by the self-assertive social need for differentiation", predicts the intention to adopt new products

Empirical Study and Findings

Research methodology

The data were collected on a convenient sample of young people from generations Y and Z (N = 266; average age = 18.98 years, standard deviation = 1.197; age between 17 and 25).

Measurements

The TPB and consumer innovativeness concepts were measured with already validated scales.

Consumer innovativeness

Unidimensional scales of innovativeness, if they count a limited number of items (six items for the Goldsmith and Hofaker's scale (1991) do not allow the identification of specific motivations which can incite the consumer to adopt a new product. For that reason, the MCI scale (Vandecateele and Geuens, 2010) was chosen to measure consumer innovativeness. This scale was recently reused in recent researches and tends to show a universal character [START_REF] Li | The role of product originality, usefulness and motivated consumer innovativeness in new product adoption intentions[END_REF]. This scale is composed of 20, seven-point Lickert-type statements (Appendix 1).

Theory of Planned Behavior

The subjective norms concept was measured by a scale composed of six/seven-point Lickerttype statements (Appendix 1). The attitude toward smart connected products (SCP) was measured by a scale composed of three/seven-point Lickert-type statements (Appendix 1).

And finally, the perceived control was measured by a scale composed of four/seven-point Lickert-type statements (Appendix 1). These scales are adapted from Shih and Fang (2004) who conduct a survey on the adoption of internet banking. This scale was chosen because of the closeness between internet and the Internet of Things.

Intention to adopt SCPs

The intention to adopt generic SCPs is measured by two items (Appendix 1), while the intention to adopt SCPs in the insurance domain is measured by four items (Appendix 1).

Appendix 1 -Measurement scales

Theory of Planned Behavior (TPB) scale (Shih and Fang (2004) Subjective norms SUBNORM1 Most people who are important to me would think that using SCP is a foolish/wise idea. SUBNORM2 Most people who are important to me would think that using SCP is a good idea. SUBNORM3 Most people who are important to me would think I should use SCP. SUBNORM4 My family who are important to me would think that using SCP is a foolish/wise idea. SUBNORM5 My family who are important to me would think that using SCP is a good idea. SUBNORM6 My family who are important to me would think I should use SCP.

Perceived control

PER_CO1 I would be able to operate SCP. PER_CO2 I have the resources to use SCP. PER_CO3 I have the knowledge to use SCP. PER_CO4 I have the ability to SCP.

Attitude towards Smart Connected Products (SCP)

ATT-SCP1 I feel using SCP is a foolish/wise idea. ATT-SCP2 I feel using SCP is a good idea. ATT-SCP3 I like to use SCP.

Intention to adopt SCP

INTEN1 I plan to use SCP. INTEN2 I intend to use SCP within the next 3 months.

New items

INTEN3 I intend to use a bracelet or a smart connected-watch which will allow my insurer to obtain information on my physical activity (distance, steps…). INTEN4 In the medium term, it is likely that I subscribe to a connected insurance which will allow my insurer to obtain information on kilometers traveled with my car (pay how you drive insurance). INTEN5 In the medium term, it is likely that I subscribe to a connected insurance which will allow my insurer to obtain information on my way of driving (pays as you drive insurance).

Motivated Consumer Innovativeness (MCI) scale [START_REF] Vandecasteele | Motivated Consumer innovativeness: concept, measurement, and validation[END_REF] Social innovativeness SOC1 I love to use innovations that impress others. SOC2 I like to own a new product that distinguishes me from others who do not own this new product. SOC3 I prefer to try new products with which I can present myself to my friends and neighbors. SOC 4 I like to outdo others, and I prefer to do this by buying new products which my friends do not have. SOC 5 I deliberately buy novelties that are visible to others and which command respect from others.

Functional innovativeness

FUNC1 If a new time-saving product is launched, I will buy it right away. FUNC2 If a new product gives me more comfort than my current product, I would not hesitate to buy it. FUNC3 If an innovation is more functional, then I usually buy it. FUNC4 If I discover a new product in a more convenient size, I am very inclined to buy this. FUNC5 If a new product makes my work easier, then this new product is a "must" for me.

Hedonic innovativeness

HEDO1 Using novelties gives me a sense of personal enjoyment. HEDO2 It gives me a good feeling to acquire new products. HEDO3 Innovations make my life exciting and stimulating.