Valerio Modugno

Gabriele Nava

Daniele Pucci

Francesco Nori

Giuseppe Oriolo

Serena Ivaldi

Safe trajectory optimization for whole-body motion of humanoids

Multi-task prioritized controllers generate complex behaviors for humanoids that concurrently satisfy several tasks and constraints. In our previous work we automatically learned the task priorities that maximized the robot performance in whole-body reaching tasks, ensuring that the optimized priorities were leading to safe behaviors. Here, we take the opposite approach: we optimize the task trajectories for whole-body balancing tasks with switching contacts, ensuring that the optimized movements are safe and never violate any of the robot and problem constraints. We use (1+1)-CMA-ES with Constrained Covariance Adaptation as a constrained black box stochastic optimization algorithm, with an instance of (1+1)-CMA-ES for bootstrapping the search. We apply our learning framework to the prioritized whole-body torque controller of iCub, to optimize the robot's movement for standing up from a chair.

I. INTRODUCTION

The generation of complex whole-body movements for humanoid robots typically involves the definition of a set of tasks (i.e., tracking a desired trajectory for the joints or the end-effectors, either positions or forces), performance objectives (e.g., minimize the torques), under a set of constraints (e.g., joint limits, motors limits) that assure that the motion is physically feasible on the real robot. For example, let us consider the humanoid iCub (Fig. 1) that must stand up from a chair. This motion, trivial for a human, is very challenging to be realized for a humanoid. During the execution of the stand up we want to be sure that the robot produces the right acceleration of its Center of Mass (CoM) while balancing. This necessitates optimizing its posture at each time step. At the same time we want to guarantee the feasibility and the safety 1 of the generated motions, i.e., we must take into account torque and joint limits, collisions, external forces, balancing constraints to prevent falling etc.

To solve this multi-task, multi-constraint problem, the research community has converged to a consensus framework in the last years: that is by using prioritized multitask controllers, either with strict task priorities [START_REF] Prete | Prioritized motion force control of constrained fully-actuated robots: atask space inverse dynamics[END_REF] or soft task priorities (also called weights) [START_REF] Salini | Synthesis of complex humanoid whole-body behavior: A focus on sequencing and tasks transitions[END_REF]. Typically, they are formulated as Quadratic Programming (QP) problems [START_REF] Liu | Generalized hierarchical control[END_REF].

*This work was supported by the European Projects CoDyCo (FP7, n.600716), Comanoid (H2020, n.645097) and An.Dy (H2020, n.731540) 1 Sapienza Università di Roma, Italy.

{modugno@dis,oriolo@diag}.uniroma1.it 2 Inria Nancy -Grand Est, CNRS & Univ. Lorraine, Loria, France.

serena.ivaldi@inria.fr 3 Fondazione Istituto Italiano di Tecnologia, Italy.

firstname.lastname@iit.it 1 By safety, here we mean the satisfaction of all the mechanical constraints of the robot, task constraints that guarantee that motions are feasible and do not make the robot fall, and constraints deriving from the interaction with the environment. Fig. 1: The humanoid robot iCub performing a whole-body motion (stand-up from the chair), with several tasks and constraints. In this paper we optimize the desired task trajectory w.r.t. a fitness function, guaranteeing that the global robot behavior is safe: it never violates any of the constraints.

The main problem with these methods is that when the global movement to realize is complex, like in the stand-up case, a significant amount of manual tuning is required to optimize the motion and adapt it to the robot, as one cannot guarantee that the QP controller can successfully execute any desired trajectory while satisfying all the constraints, not to mention the possibility that the QP solver uses constraints relaxation to find a solution. Generally, one can optimize the task priorities/weights, their evolution in time (e.g., their relative priority, or scalar value, or the task transitions), and/or the task trajectories, by acting on the desired trajectories (e.g., desired CoM) or some via-points (e.g., to avoid collisions or potentially unstable configurations). In previous work [START_REF] Modugno | Learning soft task priorities for control of redundant robots[END_REF] we proposed to automatically learn the task priorities by stochastic optimisation. Our hypothesis was that the tasks were fixed, but not the task weights. In [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF] we proposed to use constrained optimization to find solutions that never violate constraints, guaranteeing a safe trajectory and control generation.

In this paper, we take the opposite approach: we consider a controller with a fixed strategy and task priorities, and optimize the task trajectories, i.e. the reference trajectories or desired trajectories of the tasks. We do not use a weighted QP controller but the prioritized whole-body controller proposed by Nava et al. [START_REF] Nava | Stability Analysis and Design of Momentum-based Controllers for Humanoid Robots[END_REF], which is capable of performing highly dynamic tasks [START_REF] Pucci | Highly dynamic balancing via force control[END_REF]. We parametrize the task trajectories, and use constrained black-box stochastic optimization to ensure that the optimized trajectories are "safe" and that they never violate any constraint. As in [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF], we rely on (1+1)-CMA-ES with CCA to find solutions that do not violate constraints, but we improve the search bootstrap phase. To show the effectiveness of our approach, we show how we can optimize a particular motion for iCub: standing-up from a chair.

This kind of motion is characterized by switching contacts (from the legs on the chair to the feet on the ground), physical interaction with the environment and several balance constraints to be verified. Executing this kind of motion with the previous approach (learning task priorities) is hardly possible, because it is not suited to segment a complex movement into phases characterized by different contacts configurations.

The paper is organized as follows. Section II presents the related research in safe trajectory optimization for humanoid robots. Section III formulates our learning problem and describes the framework for learning task trajectories for controlling redundant robots like humanoids, including the whole-body controller and the constrained optimization algorithms retained for the study. Section IV illustrates the experiments with the iCub humanoid robot.

II. RELATED WORK

The problem of whole-body control for redundant robots such as humanoids involves fulfilling multiple operational, posture and force tasks, while satisfying several constraints that guarantee the physical feasibility of the generated motions [START_REF] Ivaldi | Special issue on whole-body control of contacts and dynamics for humanoid robots[END_REF], [START_REF] Flacco | Prioritized multi-task motion control of redundant robots under hard joint constraints[END_REF]. The problem is classically solved by prioritized multi-task controllers with strict task priorities [START_REF] Prete | Prioritized motion force control of constrained fully-actuated robots: atask space inverse dynamics[END_REF], [START_REF] Escande | Hierarchical quadratic programming: Fast online humanoid-robot motion generation[END_REF] or soft task priorities (also called weights) [START_REF] Salini | Synthesis of complex humanoid whole-body behavior: A focus on sequencing and tasks transitions[END_REF], [START_REF] Bouyarmane | Using a multi-objective controller to synthesize simulated humanoid robot motion with changing contact configurations[END_REF]. It is formulated as a QP problem subject to constraints, that is solved at each time step of the control loop [START_REF] Liu | Generalized hierarchical control[END_REF], [START_REF] Vaillant | Multi-contact vertical ladder climbing with an hrp-2 humanoid[END_REF].

To optimize the global robot movement, one classic approach is to fix the desired task trajectories (e.g., they are known in advance, or demonstrated by a human) and optimize the task priorities/weights (including their value in time, their transitions, etc.). Very frequently, this optimization is done manually. In [START_REF] Modugno | Learning soft task priorities for control of redundant robots[END_REF], [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF] we parametrized the task priorities and optimized their parameters to obtain behaviors that were maximizing a fitness function while satisfying the problem constraints.

Another approach consists in fixing the task priorities and optimize the desired task trajectories (in the form of set-points, via-points or entire trajectories). For example, in [START_REF] Lober | Efficient reinforcement learning for humanoid whole-body control[END_REF] the authors parametrized the task trajectories of a QP controller with a set of waypoints, and used Bayesian Optimization to optimize the global robot's movement reducing a task cost.

Trajectory optimization has been a central area of research in robotics for decades, especially to generate highly dynamic motions. In some cases, the problem of trajectory optimisation has been decoupled from the controller used to execute the trajectory on the robot. For example, in [START_REF] Mordatch | Discovery of complex behaviors through contact-invariant optimization[END_REF] Mordatch et al. propose the Contact Invariant Optimization method to synthesize highly complex behaviours, decomposed in different phases with their own set of contacts and weight functions. In [START_REF] Park | Itomp: Incremental trajectory optimization for real-time replanning in dynamic environments[END_REF] Park et al. propose the Incremental Optimization for Real-Time Replanning (ITOMP), an optimal trajectory planning method that can avoid moving obstacle by interleaving planning and execution.

Both in [START_REF] Mordatch | Discovery of complex behaviors through contact-invariant optimization[END_REF] and [START_REF] Park | Itomp: Incremental trajectory optimization for real-time replanning in dynamic environments[END_REF], the problem's constraints are managed as an additional cost inside the objective function, which might lead to solutions that are not feasible.

An alternative approach is to reframe the trajectory optimization problem as an optimal control one. For locally linearized systems with quadratic cost function and Gaussian noise, one of the most prominent methods is the iLQG [START_REF] Todorov | A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems[END_REF]: it iteratively improves an optimal controller that locally optimizes a linearisation of a general non-linear system around the current trajectory. In [START_REF] Tassa | Synthesis and stabilization of complex behaviors through online trajectory optimization[END_REF] ILQG was used as optimization routine inside a Model Predictive Control scheme. In [START_REF] Tassa | Control-limited differential dynamic programming[END_REF] Tassa et al. propose a generalized version of Differential Dynamic Programming to provide a second order approximation of the Bellman equation, (while ILQG employs a first order approximation of the same expression). These trajectory optimization methods provide explicit means to manage constraints on the control inputs with backtracking line search in [START_REF] Todorov | A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems[END_REF], [START_REF] Tassa | Synthesis and stabilization of complex behaviors through online trajectory optimization[END_REF], or by solving quadratic program with box constraints for each time step in [START_REF] Tassa | Control-limited differential dynamic programming[END_REF]. None of them combat the state space constraint issue in an explicit way, so there is no guarantee of constraint satisfaction in scenarios with real robots. Toussaint proposed a probabilistic extension of ILQG that it can deal with non-Gaussian noise models but relaxes more the constraints [START_REF] Toussaint | Robot trajectory optimization using approximate inference[END_REF]. More recently, Akrour et al. in [START_REF] Akrour | Model-free trajectory optimization for reinforcement learning[END_REF] extended [START_REF] Todorov | A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems[END_REF] by dropping the linear approximation of the model dynamics, and replacing it with a model free approach based on a quadratic Q-function. However, as the previous methods, it ignores the existence of constraints defined in the state and action spaces.

Reist et al. propose a framework that combines optimal control and trajectory planning to compute a set of policies that stabilize from any starting point a non-linear system towards a goal state, while taking into account the control-state constraints [START_REF] Reist | Feedback-motion-planning with simulation-based lqr-trees[END_REF]. The motion planning module generates an open loop trajectory for the system that satisfies the controlstate constraints. Then the feedback module ensures local stability around the nodes of the computed trajectory. The union of the controllable region around each ball produce the so called "funnel". All the states in the funnel under the optimal control policy reach the goal and satisfy all the problem's constraints. It has two main limitations: first, it only allows quadratic cost functions; second, it cannot be easily extended to control problems with switching contacts.

III. METHOD

The framework proposed in this paper addresses all the issues of the other methods presented in Section II. Our learning module relies on a derivative-free constrained optimization method that poses no restrictions on the structure of the problem and guarantees constraint satisfaction both in the state and in the control space. The integration of the learning module with an established feedback torque controller [START_REF] Nava | Stability Analysis and Design of Momentum-based Controllers for Humanoid Robots[END_REF], [START_REF] Pucci | Highly dynamic balancing via force control[END_REF] let us obtain solutions that are robust and that can be easily extended to multi-contact scenarios. Figure 2 outlines our method. The balance torque controller of [START_REF] Nava | Stability Analysis and Design of Momentum-based Controllers for Humanoid Robots[END_REF], [START_REF] Pucci | Highly dynamic balancing via force control[END_REF] tracks the desired task trajectory (i.e., the CoM) sending joint torque commands to the robot (III-B). The task trajectories p(π) are parametrized functions, where π are the free parameters to be optimized (III-C). At the end of a roll-out (i.e., execution of the stand-up from the chair) we evaluate the fitness of the movement and the constraint violation. A constrained stochastic optimisation algorithm updates the parameters of the task trajectories (III-D).

A. Notation

Throughout the paper we use the following definitions:

• a (•) b : point b expressed in frame a • I: inertial frame • I p B , I
R B ,: base position and orientation in I frame • q j : joint positions • M, C, G: dynamic matrices and vectors of the floating base robot

• J C k : k -th contact Jacobian • J (•) : robot Jacobian • f: external wrench • h: centroidal momentum • φ (•): objective function • n IC : # of inequality constraints g i (•) • n EC : # of equality constraints h i (•) • n C = n IC + n EC : total number of constraints • Π ⊆ R n P : parameter space of the RBFs network • π k ∈ Π: k-th

B. Whole-body torque controller

We hereby describe the balance torque controller. Let us consider a dynamic representation of a floating base robot. We represent the configuration space of a humanoid robot by a triplet q = ( I p B , I R B , q j ) ∈ R 3 × SO(3) × R n where I p B and I R B describe respectively the position and orientation of the base of the humanoid (usually located in the torso) respect of an inertial reference frame, while q j represents the vector of joint angles. The total Degrees of Freedom (DoF) of the robot are n + 6 because the free floating base adds 6 DoF. The generalized velocity associated to the system is ν = ( I ṗB , I ω B , qj ) = (v B , q j ) where v B describes the spatial velocities of the base frame with respect to the inertial frame; I ω B is defined as

[ I ω B ] × = I ṘB I R T B , with [•]
× being the screw symmetric matrix obtained from I ω B . With the hypothesis of a flat contact surface (reasonable hypothesis in our case, since our contacts are on a flat ground and flat chair) the constrained dynamic model describing the robot is:

0 = J C k ν (1)
Su = M(q) ν + C(q, ν)ν + G(q) -∑ k J C k f k (2)
where u ∈ R n denotes the generalized force at the joints, S is a selection matrix, M ∈ R n+6 × R n+6 is the generalized mass matrix, C(q, ν) and G(q) account respectively for the non inertial forces and the gravity torque. f k ∈ R 6 represents the external wrench and is related to the Ground Reaction Forces (GRF) and the Center of Pressure (CoP) while the Jacobian J C k maps the robot velocity to the linear and angular velocity of the frame where the external wrench is exerted:

J C k (q) = J b C k (q) J j C k (q) (3)
J b C k = 1 3 -[ I p C k -I p B ] × 0 3 1 3 (4)
The dimension of J C k depends on the number and the kind of contacts with the environment. For example a double stance contact that constraints both orientation and position of the contact point has 12 dimensions (6 for each foot in contact with the ground). Equation 1 is equal to zero because we make the hypothesis of no slippage on the contact surfaces. Due to the fact that a free floating model is underactuated, we can rewrite the equation 2 into two different parts:

0 = J C k ν + JC k ν (5) 0 = M b (q) M b j (q) ν + C b (q, ν)ν + G b (q) -∑ k J b C k f k ∈ R 6 (6) u = M b j (q) T M j (q) ν + C j (q, ν)ν + G j (q) -∑ k J j C k f k ∈ R n (7) with M b (q) ∈ R 6×6 M j (q) ∈ R n×n and M b j (q) ∈ R 6×n .
It is proven that it is possible to diagonalize the generalized mass matrix M leading to a decoupling of base and joint accelerations [START_REF] Traversaro | A unified view of the equations of motion used for control design of humanoid robots[END_REF]. We introduce the following change of coordinates q = q , ν = T(q)ν

T = c X B c X B M -1 b M b j 0 n×6 1 n (8) c X B = 1 3 -[ I p c -I p B ] × 0 3×3 1 3 ( 9
)
where the c superscript represents the frame centered in the center of mass of the floating base platform, oriented as the inertial frame I. Equation 2 becomes:

Su = M(q) ν + C(q, ν)ν + G -∑ i J T C i f i (10)
where

M = M b (q) 0 6×n 0 n×6 M j (q j ) (11) M b (q) = m1 3 0 3×3 0 3×3 Ĩ(q) (12)
G = mg e 3 (13)

J C i (q) = J b C i (q) J j C i (q) (14)
J b C i = 1 3 -[ I p C i -I p c ] × 0 3×3 1 3 ( 15
)
with m the mass of the robot and Ĩ the inertia of the robot defined in the CoM frame. Due to the change of coordinate, the floating base velocities now represent the linear and angular velocities of the CoM. In particular the angular velocity is the average angular velocity. Equation 10 unifies in one expression the centroidal dynamics and the dynamic description of the free floating base. From now on all the subsequent equations will refer to Equation 10 and to lighten the notation we will drop the overline sign. The change of coordinate allows for a simple computation of the robot momentum. From [START_REF] Lee | A momentum-based balance controller for humanoid robots on non-level and non-stationary ground[END_REF] and Equation 10,the robot momentum is defined as h = M b v b .The D'Alembert's principle states that the rate of change of the robot momentum depends on the external forces acting on the robot (force from the environment and gravity in the CoM). From Equation 10and d'Alembert's principle we can write:

ḣ = d dt (M b v B ) = J T B f -mg e 3 (16)
By inverting the previous relation we can find an expression that connects the desired rate of the change of the centroidal momentum and the external forces:

f = J -T B ( ḣ * + mg e 3 ) (17)
ḣ * = ḣdes + K p (h -h d ) + K i h -h d (18)
where ḣ * combines a feed-forward term and a Proportional Integral correction to ensure that the system achieves the desired centroidal momentum. The control torques that realize the instantaneous external forces and satisfy the holonomic constraints at the contact point are:

u = Λ † (J C M(F -J T C f) -JC ν)) + N Λ u 0 ( 19
)
with Λ = J j M j , (•) † the pseudo-inversion, F = C + G and u 0 = F -J j f + K j p (q j -q des j ) + K d qj the posture task projected in the null space of the primary task that is introduced for the stability of the zero dynamics. In [START_REF] Nava | Stability Analysis and Design of Momentum-based Controllers for Humanoid Robots[END_REF] the authors empirically proved that in a centroidal momentum controller the posture task do not always assure the internal stability. To overcome this issue in Equation 18 they replace the momentum error with

h err = J l G (q j ) J k G (q d j ) q j ( 20
)
where

[J l G (q j )J k G (q d j )
] comes from a reduced expression of the centroidal momentum matrix that maps the joints velocities to the velocities of the floating base and is defined as J G (q j ) = -M b J -1 b J j . This substitution is motivated by the Lyapunov stability of the linearized closed loop dynamics around the joint positions-velocities equilibrium point (q d j , 0). Equation 17is mathematically sound only if the robot is standing on one foot and we ignore all the balance related constraints (GRF contained inside the friction cone and CoP inside the support polygon). To resolve the redundancy in the external forces in a multi-contact scenario and to take into account all the conditions for a stable stance we can recast the computation of the external forces as an optimization problem. This is done in our case, since the robot transitions between 2 contact forces in the legs when seated (contacts between legs and chair) and 2 in the feet when it stands up (contacts between feet sole and ground).

C. Trajectories Parametrization

In our application, our desired task trajectory is the one of the CoM. The CoM Cartesian trajectory is modelled as a weighted sum of normalized Radial Basis Functions (RBFs):

p * ,i com (π i ,t) = ∑ n r k=1 π ik ψ k (µ k , σ k ,t) ∑ n r k=1 ψ k (µ k , σ k ,t) (21)
where p * ,i com is the desired profile in time of one of the component of the final CoM trajectory,

ψ k (µ k , σ k ,t) = exp -1/2[(t -µ k )/σ k ] 2 ,
with fixed mean µ k and variance σ k of the basis functions, n r is the number of RBFs and π i = (π i1 , . . . , π in r ) ⊆ R n P is the set of parameters for each trajectory dimension. Generally for the design of Cartesian trajectory some conditions must be verified, for example passing through a set of waypoints or imposing particular conditions on initial or final velocities and accelerations. In this work, due to the application scenario, we employed an extended version of the RBF that allows for waypoints. We augment the RBF model by adding a set of RBFs ψ j , one for each waypoint, located in correspondence of the time µ c j in which we impose the waypoint passage:

p * ,i com (π i ,t) = ∑ n wp j=1 π c i j ψ j (µ c j , σ j ,t) + ∑ n r k=1 π f ik ψ k (µ k , σ k ,t) ∑ n r +n wp k=1 ψ k (µ k , σ k ,t) (22)
Once we set the value of the free parameters π f i , we need to compute the value of the fixed parameters in order to match the waypoint passage conditions. Given the vector of values f wp = [f 1 , . . . , f n wp ] that defines the value of the function p * ,i com at each way-point, we can write:

π i c = A -1 b (23)
where

A =    ψ 1 (µ c 1 , σ 1 , µ c 1 ) • • • ψ n wp (µ c n wp , σ n wp , µ c 1 ) . . . . . . ψ 1 (µ c 1 , σ 1 , µ c n wp ) • • • ψ n wp (µ c n wp , σ n wp , µ c n wp )    (24) b =     f wp 1 ∑ nr k=1 π f ik ψ k (µ k , σ k , µ c 1 ) -∑ nr+nwp k=1 ψ k (µ k , σ k , µ c 1 ) . . . f wp nwp ∑ nr k=1 π f ik ψ k (µ k , σ k , µ c nwp ) -∑ nr+nwp k=1 ψ k (µ k , σ k , µ c nwp )     (25)
For the case of exponential radial basis functions the matrix A is always invertible due to its particular structure.

D. Constrained Stochastic Optimisation

In the following section we formalize and describe our method for learning the parameters of the task trajectory with constrained stochastic optimisation. Given a fitness function φ (π) : R n P → R and a set of equality and inequality constraints g, h, we want to find an optimal solution π ∈ Π ⊆ R n P to the problem:

π • = arg max π φ (π) (26)
s.t. g i (π) ≤ 0 i = 1, . . . , n IC (27) h i (π) = 0 i = 1, . . . , n EC (28)
In our previous work [START_REF] Modugno | Learning soft task priorities for control of redundant robots[END_REF], [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF], we solve the problem with derivative-free methods, imposing no constraints on the structure of the fitness function and of the constraints. We use the known extensions of CMA-ES [START_REF] Hansen | Completely derandomized selfadaptation in evolution strategies[END_REF] that we benchmarked in [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF] for learning task priorities, to harness the known advantage of having to tune few parameters.

For CMA-ES a single iteration (called generation) comprises several stages. From a multivariate Gaussian distribution N ( π, σ 2 Σ) K samples π k are drawn with a σ 2 step size; an evaluation of the objective function φ , called fitness, is performed for each sample π k . Based on the fitness, the samples are ranked and the Gaussian distribution is updated according to the best K e candidates π 1:K e , called elites.

The update changes the mean, the covariance and the step size of the search distribution. The update for σ 2 and Σ exploits information both from last generation and all the previous ones. The step size update plays a key role to avoid premature convergence of the algorithm around local minima. A set of probability weights influences the mean update with one P k associated to each elite candidate. a default value for each weight is P k = ln (0.5(K e + 1))-ln (k). The initialization of CMA-ES requires the user to specify the value of the exploration rate, a number between 0 and 1, that affects the first value of the covariance matrix.

The classical CMA-ES is not designed to solve constrained black-box optimization problems, but there exist several variants of it in the literature that handle constraints. In [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF] we benchmarked different ones and we showed that (1+1)-CMA-ES with Covariance Constrained Adaptation (CCA) [START_REF] Arnold | A (1+1)-CMA-ES for constrained optimisation[END_REF] was the most appropriate for our robotics problems with several parameters and constraints. So we adopt it here. The constraints satisfaction is ensured by combining (i) a rejection rule that prevents the mean update when at least one constraint is violated and (ii) a different exploration strategy that exploits the information from violated constraints to change the covariance and steer the optimization inside the feasible region. In its basic version (1+1)-CMA-ES shows small differences with CMA-ES. In every generation only one sample (π 1 , therefore K = 1) is drawn from the search distribution according to the following rule: π 1 = π + σ Dz, where D is the Cholesky factor of the covariance matrix Σ = D T D and z is a standard normal distributed vector z ∼ N (0, I). The information about the successful steps are stored in a search path s ∈ R n P . For a detailed description of the update procedure of the original algorithm please refer to [START_REF] Igel | A computational efficient covariance matrix update and a (1+1)-CMA for evolution strategies[END_REF]. In [START_REF] Arnold | A (1+1)-CMA-ES for constrained optimisation[END_REF] the authors kept in place the original update scheme when the candidate belongs to the free region while adding a new update procedure for the constraints violation case. To reduce the probability that the candidate may belong to the infeasible region, the authors propose to reduce the covariance matrix in the direction that is orthogonal to the constraint boundary. Each time the j-th constraint is violated, we update the corresponding constraint vector trace v j ∈ R n P and the matrix D according to:

v new j = (1 -c c )v j + c c Dz D new = D - β ∑ n C j=1 1 {g j (π 1 >0)} n C ∑ j=1 1 {g j (π 1 >0)} v j w T w T w
where c c and β are constants that control the update step respectively for v j and D, w j = D -1 v j and 1 {g j (π 1 >0)} is equal to one when g j (π 1 ) > 0 and zero otherwise.

To summarize, the method tests one sample every generation and extends the covariance update to take into account the localization of the infeasible region in a neighborhood of the search distribution. The algorithm proceeds to update the mean of the search distribution only if the current candidate have a better fitness and the new sample belongs to the feasible region; these two conditions guarantees that the solution of the optimization problem always stays outside the infeasible region.

This method requires the starting point to satisfy all constraints in order to function properly. It means that we cannot initialize the method from scratch with random values but we need to provide a feasible starting point. To overcome this known limitation in this work we provide an extension of the current method with the introduction of a new step in the algorithm. In order to find a feasible starting point we start the algorithm with an unconstrained optimization problem with the subsequent fitness:

φ bs (T, n c ) = -∑ T t ∑ n C i=1 | ê(t, i, π)|, if constraints violation -∑ T t ∑ n C i=1 | ŝ(t, i, π)|, if no constraints violation where T is the number of control steps, ê(t, i, π) = 1 {g i (π)>0} g i (π) and ê(t, i, π) = 1 {h i (π) =0} h i (π) respectively
for the inequality and equality constraints that are not satisfied at time t and ŝ(t, i, π) = 1 {g i (π)<=0} g i (π), ŝ(t, i, π) = 1 {h i (π)=0} h i (π) represent all the satisfied constraints at time t. To summarize during an experiment we sum all the constraints violations while ignoring all the satisfied one. Only if the candidate is feasible throughout the experiment, we sum the total amount of constraints satisfaction.

IV. EXPERIMENTS

In this section we present our experiments with the iCub humanoid performing a "stand-up from the chair" movement. All experiments are simulated in Gazebo. Our method is capable of optimizing the task trajectories to generate a safe q si -10, 0, 0 -20, 30, 0, 45 25.5, 0, 0, -18.5, -5.5, 0 TABLE I: The intermediate (q si ) and the final joint pose (q st ) that define the secondary task.

motion even when the robot is switching contacts in physical interaction with the environment.

Robotics setup -Our experiments were performed with the iCub humanoid robot. iCub is a 53-DOF humanoid robot [START_REF] Natale | The iCub Platform: A Tool for Studying Intrinsically Motivated Learning[END_REF], but only 23-DOF are torque-controlled and used for balancing tasks. It is equipped with 6 force/torque sensors, placed in the middle of arms, legs and feet, that are used to compute the whole-body dynamics [START_REF] Fumagalli | Force feedback exploiting tactile and proximal force/torque sensing[END_REF]. We developed our controller in Matlab/Simulink using the WBToolbox [START_REF] Romano | A whole-body software abstraction layer for control design of free-floating mechanical systems[END_REF], which can be applied to both the simulated and real robot, whereas the learning and trajectory optimisation is done with our framework for learning task priorities [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF]. Considering the high chance of breaking the real robot during learning, we perform all the learning procedure in simulation. The rollouts are performed in Gazebo.

Stand-up problem -We apply our method to solve the problem of finding a safe trajectory for the robot to stand up from a chair, where "safe" means physically feasible and that it does not violate any problem/robot constraint. The goal of the experiment is to learn an optimal CoM trajectory to move the robot from a sitting position to a double support stance that guarantees constraints satisfaction and minimizes the risk of falling. The desired trajectory is executed by the centroidal momentum controller described in Section III-B. For this experiment we constrain the desired CoM trajectory to lay on the sagittal plane of the robot (x, z plane in the robot world frame). For each trajectory component, we associate a RBFs network with n f = 5 and we add two fixed viapoints for the starting and the final CoM positions. Then we use a third RBFs network to model the time law. To allow for the stand up movements in the controller, we need to switch from the lower back/legs contact points (when the robot is sitting on the chair) to the feet contact points. Therefore we introduce a switching time t switch . This induces a natural segmentation of the movement in two distinct phases: the first is when the robot is sitting, the second is when the robot breaks the contacts with the chair and starts to stand up to reach the standing position. For the posture task we set two different keyframe joint postures: one at the end of sitting phase (q si ) and one a the end of the sit up movement (q st ).

To ensure a smooth transition in the joint space, we used the method from [START_REF] Pattacini | An experimental evaluation of a novel minimum-jerk cartesian controller for humanoid robots[END_REF]. To assure the feasibility of the optimal solution and to lighten the burden of the deployment on the real robot we introduce a set of constraints. For this experiment we set n C = n IC = 93 inequality constraints: joint position limits, joint torque limits and a dynamic balance constraint to assure that after the t switch the Zero Moment Point (ZMP) is located inside the support polygon of the robot.

A. Optimization for the simulated iCub in Gazebo

To obtain an optimal CoM trajectory that never violates the constraints we apply (1+1)-CMA-ES with CCA to our framework. As described in Section III-D, to avoid stalling, the algorithm requires a starting point that verifies the constraints, but for our application scenario it is not straightforward to set the right parameters that verify all the constraints at once.

Therefore we split the optimisation problem in two. In the first, we run an unconstrained optimisation problem, where we want to minimize the constraints violation to 0 (no constraint violation). In the second, we run a constrained optimization problem to find a solution that satisfies our "true" cost function. The two problems have similar settings (t switch = 1.5s, 3 RBFs networks with 5 RBFs each, for a total number of parameters equal to 15, T = 4.5s with a control step of 10ms) but different fitness functions.

The fitness function for the first problem, or bootstrap problem, (unconstrained optimization) is:

φ uc = l, if fallen φ bs (T, n c ) otherwise
where φ bs is the fitness function from Section III-D, and l is a penalty that we apply when the robot falls. The fitness function for the second problem, (constrained optimization) is:

φ co = l, if fallen -1 ∑ 3 i w i w 1 ∑ T i |p com (i)-p * com | εmax + w 2 ∑ T i u 2 (i) umax + w 3 ∑ T i b(i) bmax
, otherwise where φ co ∈ [-1, 0], w i are fixed weights, ε max , u max , b max are normalization factors for, respectively, the CoM position error (where p com (i) is the current CoM position and p * com is the desired CoM), the effort penalty (u(i) is the torque at time i) and the backward penalty. The latter is defined as

b(i) =1 {p x com (i-1)>p x com (i)} (p x com (i -1) -p x com (i))+ (29)
1 {p z com (i-1)>p z com (i)} (p z com (i -1) -p z com (i)) (30)
where p x com , p z com are the x and z coordinate of the CoM position. This term penalizes all the trajectories that produce backward movements along the x or z direction. We use it to minimize undesirable oscillations of the robot along the sagittal plane.

To find a solution for the first problem (bootstrap) we performed 500 rollouts of (1+1)-CMA-ES (without CCA) with an exploration rate equal to 0.1. The solution is used as the starting point for the second problem, which is solved with (1+1)-CMA-ES with CCA, with the same exploration rate. A typical fitness profile for φ co after 500 rollouts is shown in Figure 4. For this experiment, the weights for the penalty terms are respectively w 1 = 1.3 for the CoM error, w 2 = 1 for the effort and w 3 = 3 for the backward penalty.

We compare the optimized solution with one that was hand-tuned by an expert. The hand-tuned solution lasts for 11s, with t switch = 4.53s. Even if the hand-tuned solution has a smoother profile for the CoM, it makes the robot move faster during the stand-up and with higher torques (with a peak of torques at t switch ). The optimized trajectory, on the contrary, is faster (less than 5s), has lower energy consumption and lower CoM tracking error, as shown in Table II. By design, our solution satisfies all the problem constraints: as shown in Figure 3, it satisfies a very conservative constraint on the ZMP, that the hand-tuned solution was violating even if it was generating a physically feasible movement.

V. CONCLUSIONS

In this paper we propose a method to compute safe task trajectories for whole-body control of humanoid robots, where "safe" means that we ensure that the optimized trajectories lead to behaviors that never violate any of the problem/robot constraints. We propose to use (1+1)-CMA-ES with CCA, that we previously benchmarked in [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF], to optimize the parameters of the task trajectories, while for controlling the robot we rely on a whole-body multi-task centroidal momentum controller from [START_REF] Nava | Stability Analysis and Design of Momentum-based Controllers for Humanoid Robots[END_REF], [START_REF] Pucci | Highly dynamic balancing via force control[END_REF].

We demonstrate our method on the iCub humanoid, to find a safe trajectory for the motion "stand-up from the chair", with multiple switching contacts and several constraints. By comparing the optimal solution with an hand-tuned one we showed that our method improves the task trajectories satis- fying all constraints and reducing the energetic consumption. Realizing this kind of motions was not possible in our previous framework where task priorities were optimized [START_REF] Modugno | Learning soft task priorities for safe control of humanoid robots with constrained stochastic optimization[END_REF].

Currently, the solution is optimized in simulation, as the number of roll-outs is too high for executing the algorithm directly on the real robot. However, the feedback controller and the good dynamics model make it applicable with minor adaptations to the real robot. In the future, we plan to leverage transferability approaches for deploying and refining the solution directly on the real robot 2 We plan to take inspiration from the "Intelligent Trial and Error" approach, which essentially suggests to generate a large diversity of high-performing trajectories offline and select the best one to test on the robot online by trial and error [START_REF] Cully | Robots that can adapt like animals[END_REF].

Fig. 2 :

 2 Fig. 2: Overview of the proposed method. The balance controller executes a desired task trajectory, represented by a parametrized function. An outer learning loop enables the optimization of the parameters of the task trajectory, taking into account the constraint violations in an explicit way.

Fig. 3 :

 3 Fig.3: Comparison of three typical different solutions. Rows: in the first row we show the results for a trajectory that is manually hand-tuned by an expert, while in second and third rows we show the final solutions after the bootstrapping and the final trajectory obtained by the constrained optimization. Columns: A plots the joint torques, B plots the joints positions during the experiment, C shows the CoM trajectory of the robot on the sagittal plane, D shows the evolution of the ZMP on the ground plane. In D, the square is a conservative estimation of the support polygon of the robot: for this reason, even if the ZMP of the hand-tuned experiment moves outside the support polygon, the robot does not fall. However, our optimization algorithm finds solutions that satisfy the conservative ZMP constraint.

Fig. 4 :

 4 Fig. 4: Evolution of a typical fitness value for φ co .

  candidate at the current generation • K: total number of candidates for each generation • K e : number of best candidates or elites • π 1:K e : best candidates of the current generation • N ( π, Σ): Gaussian distribution with mean π and covariance Σ • σ 2 : step size • l(π k ): penalty factor • 1 {•} indicator function

This means that our algorithm will be kept offline for the initialization of the starting solution, and will not run online on the robot: for that part, we will turn to data-efficient / micro-data learning algorithms.


Laure Ambroise

Nathalie Claveau

Muriel Perez

Isabelle Prim Allaz

Martine Seville

Christine Teyssier
email: christine.teyssier@univ-st-etienne.fr

FIRM'S GROWTH PROFILES AND CEO'S ATTITUDES: THE MODERATING ROLE OF GROWTH INTENTION ON FIRM'S GROWTH?

Keywords: high growth, growth profiles, CEO's growth intentions, CEO'

The question of SME growth has been a concern of scholars, managers, practitioners and governments for a long time. The reality of growth appears to be multi-dimensional (Davidsson and Wiklund, 2006) and its determinants and measurements, especially in small businesses, have always presented a significant challenge for researchers. Hindering growth requires a good knowledge of the causes, the effects and the processes of development.

. This research is an attempt to make some progress in the understanding of the SME's growth by using an integrative model of influences that allows envisaging the simultaneous influences of the firm's growth profiles (the firm's affiliation to one of the growth paths obtained in a statistical typology) and the CEO's behavior on the growth rate (in terms of

FIRM'S GROWTH PROFILES AND CEO'

variation of sales in [2009][2010]. The aim of this research model is then to question the factors or contingency of the SME's growth: factors of inertia or factors of impulse.

After the presentation of our literature review, we present the empirical part of the work. A typology of SMEs growth paths, based on a large sample of firms (17 000), is presented: six groups, composed by firms that have experienced differentiated growth trajectories over a period of eight years, are identified. Then a sizable survey among 427 French SME CEOs is set up to access qualitatively the strategic orientations of CEOs: their perception of the market growth; their strategic style; their attitudes toward growth; their perception of abilities (self-capabilities) to manage their firm. CEOs growth intentions (at 1 and 3 years) are introduced as moderating variables. Structural equation modeling analyses are conducted.

Results highlight the influence of the various determinants of SME's growth. They confirm the idea that small business managers can choose many different ways to achieve growth and that the goals of growing are interdependent with other strategic objectives [START_REF] Delmar | The effect of small business managers' growth motivation on firm growth: a longitudinal study[END_REF]. Regarding growth experience, the results underline that, even if growth deserves to be considered as crucial for SMEs, it should not be taken for granted.

CEOs' perceptions and attitudes have more influence on SME's growth rate when CEOs have low short or mid-term intention to develop. Besides, the analyses show the interest of a focus on managers' growth intention to study the SMEs growth process (Sadler- [START_REF] Sadler-Smith | Managerial Behavior, Entrepreneurial Style, and Small Firm Performance[END_REF][START_REF] Janssen | Do Managers' Characteristics Influence the Employment Growth of SMEs?[END_REF][START_REF] Wiklund | Building an integrative model of small business growth[END_REF]. The five selected variables chosen to explain firm growth are relevant. But their relevance depends on the moderating variable "growth intention" that emphasizes the extent to which a manager articulates growth as an objective, as well as its commitment to the execution of an ongoing growth strategy [START_REF] Barringer | A quantitative content analysis of the characteristics of rapid-growth firms and their founders[END_REF] or the entrepreneur's goals or aspirations for its firm growth trajectory [START_REF] Dutta | The evolution of growth intentions: toward a cognition-based model[END_REF]. Indeed, CEO's strategic styles have a significant and positive impact on the future growth rate. Finally, CEO's attitudes toward growth have a notable impact only in the short term for CEOs who have a low intention to grow. Regarding CEO's perception of the market growth, it has obviously a significant and important influence of SME's growth rate.

INTRODUCTION

The question of SME growth has been a concern of scholars, managers, practitioners and governments for a long time. Entrepreneurship researchers have pointed out growth as a crucial indicator of venture success [START_REF] Baum | A multidimensional model of venture growth[END_REF]. The economic crisis faced by most European countries since 2008 has revived the debate about the significant contribution of the small enterprises growth to the dynamism of our economies. In France, the new definition of the "intermediary size firms" category, enacted by the Law of Modernization of the Economy (August 2008), is no stranger to this general trend. The implementation of policy measures is designed to promote businesses growth, particularly in fostering the transition of SMEs in the larger size class. By their size and flexibility, intermediary size firms have competitive advantages in conquering niche or innovative markets [START_REF] Retailleau | Les entreprises de taille intermédiaire au coeur d'une nouvelle dynamique de croissance[END_REF]. Clearly, growth in SMEs has always been an essential driving force in the process of economic growth and job creation. This explains the number of studies which have been reviewed on the subject of small firm growth (Davidsson and Wiklund, 2006;[START_REF] Shepherd | Are we comparing apples with apples or apples with oranges? Appropriateness of knowledge accumulation across growth studies[END_REF].

But the reality of growth is complex and the determinants of business growth and its measurements, especially in small businesses, have always presented a significant challenge for researchers. Hindering growth requires a good knowledge of the causes, the effects and the processes of development.

Economic literature (based on economic motivations) and stages of development theories of growth generally assume that growth is a "natural and imperative phenomenon" and that has nothing to do with the entrepreneur's priorities [START_REF] Greiner | Evolution and revolution as organizations grow[END_REF][START_REF] Churchill | The five stages of small business growth[END_REF]. In this approach, each of development stages follows the previous one in an incremental and progressive way [START_REF] Greiner | Evolution and revolution as organizations grow[END_REF][START_REF] Churchill | The five stages of small business growth[END_REF]. Therefore only the development profile counts.

The reality of growth appears rather multi-dimensional (Davidsson and Wiklund, 2006). Current studies on the firm's growth suggest that it is fundamental to develop integrative and comprehensive approaches of growth. More precisely, research on business growth tend to focus on entrepreneurial orientation, environmental characteristics, firm's resources and manager's personal attitudes towards growth [START_REF] Wiklund | Building an integrative model of small business growth[END_REF]. It also points out a need to study the influence of entrepreneurial decision making and entrepreneurial growth intention on the business growth [START_REF] Achtenhagen | Business growth -Do practitioners and scholars really talk about the same thing ?[END_REF]. This research is an attempt to make some progress in the understanding of the SME's growth by using an integrative approach. Our integrative model of influences allows envisaging the simultaneous influences of the firm's growth profiles (the firm's affiliation to one of the growth paths obtained in a statistical typology) and the CEO's behavior on the growth rate (in terms of variation of sales in [2009][2010]. The aim of this research model is then to question the factors or contingency of the SME's growth: factors of inertia or factors of impulse.

The current trend in growth studies considers that other types of research (conceptual, process) and methods (qualitative) are needed (cf. the 2010 special edition of ET&P). Even though our approach is quantitative, it partly answers to these concerns as it takes into account CEOs intentions and perceptions of their environment with a quite important sample.

These perceptions can be regarded as qualitative information.

After the presentation of our literature review, we present the empirical part of the work. A typology of SMEs growth paths, based on a large sample of firms, is presented: six groups of firms, which have experienced differentiated growth trajectories over a period of eight years, have been identified. Then a sizable survey among 427 French SME CEOs has been set up to access qualitatively the strategic orientations of CEOs (their perception of the market growth; their strategic objectives; their attitudes toward growth; their abilities (selfcapabilities) to manage their firm. And finally, their growth intentions are introduced as moderating variables.

1. SMALL BUSINESSES GROWTH: A LITERATURE REVIEW 1.1.Small businesses growth: a complex and multidimensional phenomenon Business growth is a complex and multidimensional phenomenon [START_REF] Birley | Growth and performance contrasts between types of small firms[END_REF][START_REF] Delmar | Arriving at the high-growth firm[END_REF]. Studying the SMEs growth therefore implies to deal with a real conceptual complexity. Factors influencing the growth in small firms are numerous and interconnected. [START_REF] Storey | Understanding the small sector business, Routledge, chapter 5, The growth of small firm[END_REF] shows that growth is a combination of factors related to three dimensions: (1) the entrepreneur's resources, (2) the firm and (3) the strategy.

In each dimension, a great number of components interact with each other. As an example, prior self-employment is expected to influence ownership, itself having an impact on the external equity.

As a consequence, it would be unreasonable to investigate growth issues without previously defining what growth means [START_REF] Delmar | Measuring growth: methodological considerations and empirical results[END_REF]. In fact, it appears that there is a gap between what business growth means for practioners and how it is defined and measured by academic researchers [START_REF] Achtenhagen | Business growth -Do practitioners and scholars really talk about the same thing ?[END_REF]. Moreover, the type of formula and indicators used by academic researchers for measuring growth has an impact on the nature of the relationship between independent variables and growth [START_REF] Weinzimmer | Measuring organizational growth: issues, consequences and guidelines[END_REF][START_REF] Shepherd | Are we comparing apples with apples or apples with oranges? Appropriateness of knowledge accumulation across growth studies[END_REF]. It is a problem, for at least two reasons. First, we deplore a lack of theoretical progress due to contradictory findings and weaknesses in comparing and accumulating knowledge across studies (Davidsson and al., 2006). Secondly, there are some influences of academic results on policy measures to foster growth, and they could be inadequate with businesses real priorities. Moreover, researchers have commonly adopted growth as a measure of success in entrepreneurial firms, but this variable may not completely reflect a given manager's definition of what constitutes successful efforts [START_REF] Dutta | The evolution of growth intentions: toward a cognition-based model[END_REF]. More precisely, firms' managers consider growth as a much more complex phenomenon than scholars do. For example, manager's perceived growth can be translated in terms of increase in the company value compared to its competitors. It can also be formulated as quality management or social climate improvement in the firm. In others words, growth deserves to be analysed as a result of a "process of development in which an interacting series of internal changes leads to increase in size accompanied by changes in the characteristics of the growing object" [START_REF] Penrose | The theory of the growth of the firm[END_REF]. The reality of growth is complex (Davidsson and Wiklund, 2006). Current studies on the firm's growth suggest that it's fundamental to develop integrative and comprehensive approaches of growth. [START_REF] Wiklund | Building an integrative model of small business growth[END_REF] found little overlap between the different theoretical perspectives developed in the literature on growth, and the final use of a limited number of variables. This is the reason why they include entrepreneurial orientation, environmental characteristics, firm's resources and manager's personal attitudes towards growth. Another crucial challenge for the future study of growth lies in the focus on what happens in practice. More precisely, research on business growth needs to focus on the influence of entrepreneurial decision making and entrepreneurial growth intention on the business growth [START_REF] Achtenhagen | Business growth -Do practitioners and scholars really talk about the same thing ?[END_REF].

1.2.Impact of managerial styles, attitudes and abilities on growth paths [START_REF] Penrose | The theory of the growth of the firm[END_REF] showed that managers of growing companies are characterized by specific systems of values, largely oriented towards imagination, long-term vision and naturally in favour of ambitious development projects. Her work has inspired many extensions. In their Upper Echelons Theory, [START_REF] Hambrick | Upper echelons: the organization as a reflection of its top managers[END_REF] showed that cognitive characteristics of the management team define both the strategic direction of the firm (product innovation and diversification, terms of internal and external growth, choices of financing), but also its growth and profitability. Following them, many authors use psychological variables related to managers' values, goals and personal satisfaction [START_REF] Smith | Top management team demography and process: the role of social integration and communication[END_REF][START_REF] Kilduff | Top management team diversity and firm performance: examining the role of cognitions[END_REF]. As a consequence, small business managers can choose many different ways to achieve growth and the goal of growing can be interdependent with other strategic objectives [START_REF] Delmar | The effect of small business managers' growth motivation on firm growth: a longitudinal study[END_REF]. For example, managers may want to expand their firm by acquiring another firm or growing organically; increasing sales while outsourcing the production; expanding sales but not employment. If there is a contradiction between the goal of expanding the firm and the others individuals goals, managers may choose actions that do not completely fulfill the expansion goal. The theory of planned behavior [START_REF] Azjen | The Theory of Planned Behavior[END_REF] provides a recognized explanatory model of behavior and intention. This theory postulates that human behavior, to be effective, must be decided and planned.

Azjen assumes that behavior is determined by intention, defined itself by attitudes, social norms and perceived control of the situation. More precisely, attitudes reflect the manager's judgment -positive or negative -about the desirability of his behavior and its consequences.

Social norms concern the considerations about influence and opinion of close people regarding managers' behaviors. Finally, perceived control of the situation refers managers' beliefs on their own ability to conduct actions successfully, ie a kind of self-efficacy.

1.3.Role of growth intention on growth trajectory

Even if growth deserves to be considered as crucial for SMEs, it should not be taken for granted. In fact, most firms are born small to stay small [START_REF] Kolvereid | Growth aspirations among Norwegian entrepreneurs[END_REF], because many small business managers are not specially willing to pursue the expansion of their firm [START_REF] Storey | Understanding the small sector business, Routledge, chapter 5, The growth of small firm[END_REF]Wiklund and al., 2003). The reasons involved by managers are numerous, but generally based on the negative consequences of growth on quality of products and services, managers workload and work tasks, employees well-being, or more economic factors like financial outcomes and impacts on independence toward external stakeholders and management control [START_REF] Wiklund | Expected consequences of growth willingness in different samples of small firms[END_REF]Wiklund and al., 2003). For example, the risk of deterioration of the working atmosphere and the difficulties of monitoring are the main criteria of dissuasion for growth pointed out by [START_REF] Davidsson | Entrepreneurship -And after? A study of growth willingness in small firms[END_REF], while Wiklund and al. (2003) underline recurrent conflicts between managers regarding their growth strategy.

However, the authors refute the influence of personal financial goals on managers' behavior towards growth. In the field of entrepreneurship, research conducted on the motivations of entrepreneurs show that the motivation to create a firm meets other requirements related to the environment, such as autonomy, recognition, a sense of belonging and security [START_REF] Boissin | Profils de dirigeant et croissance des jeunes entreprises innovantes[END_REF]. For these reasons, focus on managers' growth intention is essential in the study of the growth process in SMEs (Sadler- [START_REF] Sadler-Smith | Managerial Behavior, Entrepreneurial Style, and Small Firm Performance[END_REF][START_REF] Janssen | Do Managers' Characteristics Influence the Employment Growth of SMEs?[END_REF][START_REF] Delmar | The effect of small business managers' growth motivation on firm growth: a longitudinal study[END_REF][START_REF] Wiklund | Building an integrative model of small business growth[END_REF]. The variable "growth intention" emphasizes the extent to which a manager articulates growth as an objective, as well as its commitment to the execution of an ongoing growth strategy [START_REF] Barringer | A quantitative content analysis of the characteristics of rapid-growth firms and their founders[END_REF] or the entrepreneur's goals or aspirations for its firm growth trajectory [START_REF] Dutta | The evolution of growth intentions: toward a cognition-based model[END_REF]. [START_REF] Hambrick | Upper echelons: the organization as a reflection of its top managers[END_REF] have stressed that leaders must necessarily consider the growth of their business as a deliberate choice and priority, if they want to meet the challenge of sustained long-term growth. [START_REF] Wiklund | Aspiring for and achieving growth: the moderating role of resources and opportunities[END_REF] show that growth intentions are much higher when the growth rate is high. It is not a coincidence that one of the first criteria considered by the authors who have begun to study specifically the phenomenon of highgrowth in the SME is the "growth intention", ie aspirations of the leader to see its business growing [START_REF] Davidsson | Entrepreneurship -And after? A study of growth willingness in small firms[END_REF]. Growth is hardly due to chance; indeed, involvement to growth is the starting point not only to access growth, but also to maintain it [START_REF] Wiklund | Expected consequences of growth willingness in different samples of small firms[END_REF][START_REF] Barringer | A quantitative content analysis of the characteristics of rapid-growth firms and their founders[END_REF][START_REF] Wiklund | Building an integrative model of small business growth[END_REF]. This impact of the growth intentions is enhanced in the SME context where the interpenetration between managers and their firm explains that strategic decision-making power is frequently concentrated in their hands [START_REF] Janssen | L'influence de l'interpénétration du dirigeant et de son entreprise sur l'endettement bancaire des PME : état de la question[END_REF]. Determinants related to leadership, and specifically managers' growth intentions, are more likely to have an impact on SMEs growth than for larger businesses. Moreover, growth intentions are non-linear over time, but they evolve as the firm develops [START_REF] Dutta | The evolution of growth intentions: toward a cognition-based model[END_REF]. Initial growth intentions are themselves modified or heavily transformed, according to the growth success (earlier performance) and the learning process (feedback from earlier growth processes and performance) that managers implement to meet the development requirements [START_REF] Delmar | The effect of small business managers' growth motivation on firm growth: a longitudinal study[END_REF].

That being said, the motivation towards growth is not sufficient to achieve a high level of growth, if it is not accompanied by the abilities of managers to manage growth and to search and secure resources needed by the development path of the firm. For example, these abilities are linked to the leaders' education and experience, and to the environmental dynamism of the company [START_REF] Wiklund | Aspiring for and achieving growth: the moderating role of resources and opportunities[END_REF]. Some authors found that the positive relationship between growth intentions and growth rate is stronger when the level of education and the experience of the manager are high, and when the environment is dynamic [START_REF] Orser | On the Growth of Small Enterprises: The Role of Intentions, Gender and Experience[END_REF][START_REF] Wiklund | Aspiring for and achieving growth: the moderating role of resources and opportunities[END_REF][START_REF] Barringer | A quantitative content analysis of the characteristics of rapid-growth firms and their founders[END_REF]. Therefore, growth intentions are likely to be heterogeneous and deeply influenced by a wide range of individual, organizational and environmental factors. In this research, we believe it is impossible to consider growth intentions independently from other variables which characterize the manager's behavior: What does growth represent for him? Which goals does he look for while its firm is growing? Which abilities does he implement to manage growth?

RESEARCH MODEL AND METHODOLOGY

Data collection and measures

On a methodological point of view, a joint research approach is deliberately implemented. In a first time, a quantitative approach was undertaken to develop a typology of SMEs' growth paths. In a second time, we have conducted a survey, interviewing more than 400 CEOs on their intentions, objectives, attitudes and abilities towards growth.

Approach by clustering: a typology of SMEs' growth paths

More precisely, we conducted a non-hierarchical classification approach with dynamic clustering (SPSS 18.00). This method of classification, which allows using large volumes of data while optimizing the classification criteria, is now widely accepted as analytical method that perfectly suits for the study of complex phenomenon of business growth [START_REF] Delmar | A taxonomy of high-growth firms[END_REF][START_REF] Delmar | Arriving at the high-growth firm[END_REF]. This cluster analysis was used to classify a large number of businesses into relatively homogeneous groups, according to their profile and growth dynamics, by crossing several multidimensional criteria [START_REF] Ambroise | Des paliers de croissance au potentiel de développement de la PME[END_REF]. The initial sample consists of 17,404 French companies located in the Rhône-Alpes region (France) and created before 2004, whose turnover was more than € 500,000 in 2008.

These initial criteria were used to eliminate the very small and start-up firms because we considered them as subject to significant failure rates in their first years of existence. The data come from the DIANE-NEO database 1 (Van Dijk Editor). The period of study runs from 2000 to 2008. A combination of five discrimination variables of growth's trajectory was chosen. These variables allow us to consider the phenomenon of growth in a multidimensional way: in terms of intensity, volatility, positioning relative to average performance of the industry, recurrence and sustainability of the phenomenon over time (Appendix 1). This first step in our methodology allows capturing a potential heterogeneity in the growth screening. Six groups of firms with differentiated growth trajectories over a period of eight years could be expected to represent a maximum variation regarding their experience with growth, while allowing for the recognition of common patterns within and between groups. Due to its small size, the 6 th group is not considered in further analyzes.

The notion of high -and hyper -growth refers to a form of sustained development of the company, characterized by a high growth rate over a period of time, and potentially a great and positive influence on job creation. Since the work of [START_REF] Birch | The job generation process[END_REF] on "gazelles", the study of the phenomenon has led to a multiplicity of definitions and methods of identification [START_REF] Mustar | Les PME à forte croissance et l'emploi[END_REF][START_REF] Almus | What characterizes a fast-growing firm?[END_REF][START_REF] Julien | Les PME à forte croissance -L'exemple de 17 gazelles dans 8 régions du Québec[END_REF][START_REF] Barringer | A quantitative content analysis of the characteristics of rapid-growth firms and their founders[END_REF][START_REF] Moreno | High-growth SMEs versus non high-growth SMEs: a discriminant analysis[END_REF][START_REF] Shepherd | Are we comparing apples with apples or apples with oranges? Appropriateness of knowledge accumulation across growth studies[END_REF][START_REF] Parker | What happens to gazelles? The importance of dynamic management strategy[END_REF]. Such diversity affects variables for measuring growth, which widely vary from researcher to another [START_REF] Weinzimmer | Measuring organizational growth: issues, consequences and guidelines[END_REF][START_REF] Delmar | Measuring growth: methodological considerations and empirical results[END_REF]. Commonly used indicators are given in terms of inputs (staff, funds committed), outputs (turnover, profits) or valuation (total assets, market capitalization) [START_REF] Garnsey | Growth setbacks in new firms[END_REF]. Nevertheless, the authors often prefer the growth rate of sales at the expense of the growth rate of employment or assets, because these criteria are too dependent on the choice of productive business (use of sub-contracting or external labor) or changes in accounting principles [START_REF] Delmar | Arriving at the high-growth firm[END_REF]Davidsson and al., 2006;[START_REF] Moreno | High-growth SMEs versus non high-growth SMEs: a discriminant analysis[END_REF][START_REF] Barbero | Managerial capabilities and paths to growth as determinants of high-growth small and medium-sized enterprises[END_REF]. A diversity of definitions is also observed regarding the mathematical methods of measure. The phenomenon can be calculated in absolute or in relative terms. But the approach in relative terms appears to be more frequent. Similarly, the positioning within the sector (measure of outperformance compared with the average or median development companies in the same sector) should be preferred. Finally, the high or hyper-growth cannot be an instantaneous event. But again, the period of observation of the phenomenon differs across studies, even if the authors agree to consider a period of 3 or 4 successive years of strong growth is symptomatic of a sustained and sustainable development. This diversity of definitions has led, in this quantitative step of 1 DIANE-NEO compiles financial information that French firms have to publish. our methodology, to the adoption of a multidimensional definition of hyper-growth: a firm is an hyper-growing firm if the growth rate of its annual turnover is more than 20%, while in the top quartile of the growth rate of turnover of its industry sector (based on the 2-digit French code NAF) and is observed on three or four successive years.

CEOs perceptions survey

The first step of our approach is essential for a better understanding of the growth phenomenon. But it is not sufficient to put into perspective the influence of CEO's intentions, strategic styles, attitudes and abilities to growth. So, a sizable survey among French SME CEOs has been set up to access more qualitatively the strategic orientations of CEOs. This survey was conducted in 2009-2010 on 427 SME, in the French Rhône-Alpes region. The questionnaire was administered during face-to-face interviews with CEOs. Among a large set of questions; they were asked to give: a/ their growth intentions; b/ their perceptions of the market growth; c/ their strategic style (referring to the CEOs general objective as to regard to their firms, aiming independence, profitability, growth, sustainability, or patrimonial concern); d/ their attitudes toward growth; e/ their perception of their abilities (selfcapabilities) to manage their firm (Appendix 2). Four of five dimensions are introduced in the integrative model of influences presented below, after being associated with the SME's growth profile (ie the firm's affiliation to one of the growth paths in the typology). The fift one, growth intentions at 1 and 3 years, are then introduced as moderating variables.

Integrative model of influences

The integrative model of influences presented below allows analyzing the simultaneous influences of both the firms growth profiles (the firm's affiliation to one of the growth paths obtained in the typology) and CEOs behaviors (the four dimensions mentioned above) on growth rates (in terms of variation of sales between 2009 and 2010). The aim of this research model (Figure 1) is then to question the factors of contingency of the SMEs growth: factors of inertia or factors of impulse. Growth intentions are considered as moderators.

Growth profiles: results of a clustering approach

As said before, a non-hierarchical classification on 17,404 French companies was used to categorize SMEs into homogeneous groups, according to their growth profile and dynamics (Table 1). This non-hierarchical classification approach with dynamic clustering was based on a combination of five multidimensional variables representing the growth trajectory: intensity, volatility, positioning relative to average performance of the industry, recurrence and sustainability of the phenomenon of fast growth over time. The final classification was stabilized in five groups: "decreasing", "sluggish", "growing", "fast growing", and "hyper-growing" firms. The first group, the "decreasing" firms, represents 9.1 per cent of the sample and the second one, the "sluggish" firms 29.8 per cent. The third group, the "growing" firms, rises 36.5 per cent, while the fourth group, the "fast growing" firms represents 18.7 per cent. Finally, the fifth group, the "hyper-growth" firms, reaches only 5 per cent of the initial population. A sixth group, the "mega-growing" firms, representing 0.5 per cent of the sample, was removed because of the extremely volatile financial data of firms and the very low number of companies belonging to this group. Firms growth profiles were then saved in a qualitative variable with five modalities.

(0) 0 (0) 1,09 (1) 2,12 (2) 3,43 (3)
(1) A ratio > 1 means the company is one of the 25% best performing companies in its industry.

CEOs perceptions as determinants of firms' growth

The aim of this research is to modelize different determinants of firm's growth, especially linked to CEO's attitudes and perceptions on several variables. On this fact, CEO's perception on the market growth, strategic style, attitude towards growth, perception of his abilities and growth intention were extracted from the early mentioned questionnaire.

In a first step, exploratory factor component analysis were conducted on the "strategic style", "attitudes towards growth" and "perception of abilities" items, in order to identify the different dimensions of these concepts and to reduce the number of variables. Then a structural model is proposed including the previous variables as well as the "perception of the market growth", directly asked to CEOs and measured as a quantitative variable (an expected growth percentage), based on two temporal horizons (one and three years). Finally, "growth intentions" (also measured on one and three years) are introduced as moderating variables in the model designed to explain firms' growth. The intention was classified as low when under the median value of the sample and high when above.

Exploratory factor analyses

Regarding CEO's strategic style, two dimensions are identified (Table 2): strategic style oriented towards 'Efficiency' (profitability, perennity and growth objectives) and strategic style oriented towards the firm 'Value and Independence'. The CEOs perceptions of the market growth go along three dimensions: 'Stimulation', 'Control' and 'No Risk2 ' (Table 3). CEOs perception of abilities is summarized in three principal area of expertise: ability to 'manage', 'develop' and 'maintain' the activity (Table 4). Considering the relative complexity of the model, we used confirmatory factor analyses within a PLS approach. [START_REF] Bagozzi | On the use of structural equation models in experimental designs[END_REF] have already pointed out that this approach bypasses problems of traditional covariance structure analyses linked to small sample sizes and the need for multivariate normal distribution. Another advantage of analyzing variance at a latent level using a structural equations model is to measure the overall effect, but also to break it down according to each modality of the explanatory variables selected (in this case the five determinants of firms' growth).

Moreover, the objective of this research is to test the moderating effect of CEOs intention to develop their firm. To this end, the methodology used is in line with multi-group analyses of variance at the latent level [START_REF] Yi | On the Use of Structural Equation Models in Experimental Designs: Two Extensions[END_REF]. This method offers the advantage of providing, for each measurement variable and each structural coefficient, a test of the difference between the value obtained for a given group and the value resulting from an analysis of the rest of the sample. So we performed multi-group analyses to directly specify the influence of a strong versus low CEOs' intention to develop their firm 3 on the latent variables concerned (in our case the firm's growth and some of its determinants).

Table 5: Analysis of firms' growth determinants using multi-group analysis within a PLS approach 4

On a statistical point of view (Table 5), all indices confirm the stability of the model 5 just like the reliability and internal validity of the first and second order latent variables (all Alpha's values are above 0,67 et all Rho's values above 0,86).

3 The sample is divided on two groups, the High growth intention group and the Low growth intention group with a one year horizon and a three year horizon. The Intention Growth median is the frontier between groups. 4 ***, p<0,01; **, 0,01<p<0,05, *, 0,05<p<0,12. 5 The global model presents the following GoF: Absolute GoF: 0,686; Relative GoF: 0,848; External GoF: 0,991; Internal GoF: 0,856.

General model of firms' growth determinants

Intention to develop at 1 year Intention to develop at 3 years

Moderating influence of CEOs intention to develop on firms' growth determinants

Even though models explain a small part of the variance of firm growth (between 7 and 15%), structural coefficients linked to determinants are significant.

The results demonstrate that determinants of growth are not the same according to the temporal horizon in which the CEO positions. Moreover, the intention to develop the firm (low versus strong) has a moderating effect.

In other words, if we consider the CEO's intention to grow at 1 year, his perception of his abilities as well as the firm's growth profile seem to have a relative weak influence on the firm's growth. On the other hand, the moderating effect of the level of CEO's intention to develop his firm is very important. When the CEO has a low intention to develop his firm, his strategic style and attitudes towards growth have a positive and significant impact on the firm's growth while his perception of the market growth has a negative effect. The influences of these determinants are exactly reverse when the CEO has a strong intention to develop. In this case, his previsions of the market growth and the firm's growth profile become salient while the influence of the other CEO's perceptions is really weak.

If we consider the CEO's intention to grow at 3 years, the firm's growth profile as well as the CEO's market growth previsions appear as the most striking determinants and have a positive impact on firms' growth. The results also highlight that the CEO's perceptions of his abilities can become a brake for the growth of his firm, in particular when he has a low intention to develop. These results highlight the risk of CEOs' over confidence in their abilities to manage specially when they have a low intention to grow. In this case, two factors negatively influence the firm's difficulty to grow: the weakness of the CEOs growth intention and their false perceptions of their own abilities to conduct their firms' evolution.

Considering the second order latent variables, the difference between CEOs who have strong versus low intention to grow at one year echoes essentially in their attitudes towards growth. In the case of a strong intention to develop, growth is perceived as a stimulating situation and CEOs think to have the capacity and abilities to face this challenge that is to say to manage, maintain and develop their firm's activity. On the other hand, CEOs who have low intention to develop perceive growth more as a source of risk but they are also confident in their abilities to control this situation. In a strange manner, they appear to be more confident in their abilities to develop their firm's activities. These conclusions join the above discussion on the risk of over confidence. This is even truer as the risky vision of the growth becomes more strengthen in the case of low intention to grow at 3 years (Table 5). Indeed, the contribution of the "risk" dimension in the CEO's attitudes toward growth is statistically more important when he has a low intention to develop.

DISCUSSION

Even if the issue of this article is essentially descriptive, its objective is to progress towards a better understanding of the phenomenon of growth. It also aims to describe factors that are likely to stimulate or to slow down the firm's growth in order to help the economic and political players who try to improve their politics of support for the economic development of the SMEs. In that perspective, there are several contributions for this research. Results allow highlighting the influences of the various determinants of SME growth with the prospect of a general model integrating simultaneously various variables.

Results confirm the idea that small business managers can choose many different ways to achieve growth and that the goals of growing are interdependent with other strategic objectives [START_REF] Delmar | The effect of small business managers' growth motivation on firm growth: a longitudinal study[END_REF].

Concerning growth experience, the results underline that, even if growth deserves to be considered as crucial for SMEs, it should not be taken for granted. If growth experience (approximated here by the growth profile variable) obviously contributes to firms' future growth rate, results demonstrate that CEOs' perceptions and attitudes have also a significant influence on the trajectory of the firm development. So our results appear to converge with recent works which criticize the "life cycle theory" approach of growth (Levie and Lichstenstein, 2010). For all that, we can underline that the impact of SME past growth experience is higher when considering CEOs having a strong short-or mid-term intention to growth compared to other independent variables. It remains to investigate whether the past growth experience influences CEO's intention to develop.

Conversely, we can point out that CEOs' perceptions and attitudes have more influence on SMEs growth rate when CEOs have low short or mid-term intentions to develop. In that perspective, analyzing CEOs perceptions of abilities and competencies seems to be an interesting avenue for research. [START_REF] Penrose | The theory of the growth of the firm[END_REF] identified two specific types of competencies that are able to influence the growth process: entrepreneurial and managerial competencies. In this research, we show that managers distinguish three field of expertise: managing, developing and maintaining firms activity. However, we underline that the more managers believe they have competencies, the less their firm grows. This paradoxical situation is especially true for CEOs with low intention to grow. Further research should explore this negative influence. It could be explained by a problem of manager's capabilities myopia. Indeed, managers probably tend to reproduce the same behavior based on their capabilities, while growth is precisely a capabilities challenge. Growth and particularly high-growth require a permanent renewal of capabilities (Hambrick and al., 2005) or the detention of scarce resources [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF]. It suggests that managerial competencies could be, in some cases, a factor of inertia. This could have, in return, some impact on management training programs.

Besides, the analysis presented above show the interest of a focus on managers' growth intention to study the SMEs growth processes (Sadler- [START_REF] Sadler-Smith | Managerial Behavior, Entrepreneurial Style, and Small Firm Performance[END_REF][START_REF] Janssen | Do Managers' Characteristics Influence the Employment Growth of SMEs?[END_REF][START_REF] Delmar | The effect of small business managers' growth motivation on firm growth: a longitudinal study[END_REF][START_REF] Wiklund | Building an integrative model of small business growth[END_REF]. The five selected variables chosen to explain firm growth are found to be relevant. But their relevance depends on the moderating variable "growth intention" that emphasizes the extent to which a manager articulates growth as an objective, as well as its commitment to the execution of an ongoing growth strategy [START_REF] Barringer | A quantitative content analysis of the characteristics of rapid-growth firms and their founders[END_REF] or the entrepreneur's goals or aspirations for its firm growth trajectory [START_REF] Dutta | The evolution of growth intentions: toward a cognition-based model[END_REF]. Indeed, CEO's strategic style has a significant and positive impact on the future growth rate. Growth is not only due to companies whose managers seek primarily growth. Our results demonstrate that CEO's strategic style has a significant influence on the firm's growth rate independently of his personal goal-seeking objective. This contribution joins those of [START_REF] Steffens | Performance Configurations over Time: Implications for Growth-and Profit-oriented Strategies[END_REF] which underline that the willingness of profitability is often the guarantee of a future growth.

Finally, CEO's attitude toward growth has a notable impact only in the short term for CEOs who have a low intention to grow. The influence of CEO's attitude toward growth on the growth rate disappears with a mid-term perspective. These results are somewhat contradictory to the existing literature on the subject. The time horizon of our study is probably too small for meaningful relationships between CEO's attitude toward growth and the development of the company. Another explanation can also be discussed: CEOs can clearly imagine the organizational opportunities, constraints and difficulties linked to growth in the short term but they are not able to fix them on a mid or long-term basis. In order to improve this research, we could work on a mid and long-term basis. Moreover it could be interesting to explore more qualitatively CEOs perceptions that echo to the three dimensions of growth vision (a source of stimulation, a situation that must be controlled and a source of risk).

Regarding CEO's market growth previsions, they have obviously a significant and important influence of SME's growth rate. It then appears remarkable that positive market growth previsions are negatively linked to growth when growth intention is low at short term.

These results can participate to literature under CEO's over optimism and confidence (Hambrick et Crozier, 1985).

LIMITATIONS OF THE STUDY

This study contains some limitations that should be pointed out. They concern, at least, the way of measuring (1) the CEO's intention to develop the firm and (2) his characteristics in terms of strategic style, attitudes towards growth, perception of his abilities.

More generally, this limitation reveals the problem faced by researchers willing to measure intentions or attitudes towards growth [START_REF] Achtenhagen | Business growth -Do practitioners and scholars really talk about the same thing ?[END_REF]. The manager's willingness and his visions of growth change over time, depending on the results and growth already achieved. Researchers frequently asks CEOs if they want to grow and how much they are willing to grow, and then conclude, if the business has grown, that the aspirations for growth are explaining this phenomenon. However, this means forgetting that these aspirations have their own internal dynamics, that they evolve over time: past growth is changing aspirations, sometimes to continue growing (because the CEO manages the situation with growth limitation) and sometimes to limit it (e.g. create a new company to test new ideas). In others words, growth deserves to be analyzed as a result of a "process of development in which an interacting series of internal changes leads to increase in size accompanied by changes in the characteristics of the growing object" [START_REF] Penrose | The theory of the growth of the firm[END_REF]. The use of cross-sectional data is a serious methodological limitation in this study. And growth profiles included in the model probably do not take into account all the dynamic impacts. Nevertheless, the sample size is a strength.

Moreover, firms' managers consider growth as a much more complex phenomenon than scholars do. For example, manager's perceived growth can be translated in terms of increase in the company value compared to his competitors. It can also be formulated as quality management or social climate improvement in the firm. Furthermore, growth must be considered as a multidimensional and complex phenomenon that can come in several interrelated dimensions: the manager's characteristics, the strategic choices, the characteristics of the firm and its environment.

The work presented here has tried to see how the growth was due to a combination of factors focused on the manager (growth intention, reading of the market, objectives, relationship to growth). We cannot pretend that explains the whole growth phenomenon. It would be interesting to analyze whether the observed relationships persist while taking into account the organization characteristics and its environment.

Figure 1 :

 1 Figure 1: Integrative model of influences

Table 1 -

 1 Growth characteristics by group

Table 2 :

 2 Component Analysis: CEO's strategic style

		CEO's strategic style
			Value &
		Efficiency	Independence
	Profitability	,892
	Firm perennity	,770
	Growth	,631
	Independence		,863
	Value		,821
	Cronbach's alpha	,610
	R²		,436**
	% of variance	42,112	22,45

Table 3 :

 3 Component Analysis: CEO's perception of the market growth

	CEOs perception of the market growth
	Stimulation	Control	No Risk

Table 4 :

 4 Component Analysis: CEO's perception of his abilities

	CEOs perception of his abilities
	Managing	Developing Maintaining

The "no risk" dimension means that the higher the score the less the CEO perceives a growing situation as risky.

For many companies, the variance ranges from 0 to 1, since we are working on growth rates. Thus, the variance enables a robust and realistic scaling with respect to the volatility of growth rates. The standard deviation would tend to homogenize volatilities. Indeed, for the variances range from 0 to 1, the standard deviations will be greater than the associated variances, while for variances strictly greater than 1, the standard deviations associated with them will be lower than variances.

Median

-0,056 (-0,038) 0,037 (0,033) 0,058 (0,057) 0,176 (0,163) 0,477 (0,433) V2. Rate variance of annual average rate of growth Mean Median 0,058 (0,013) 0,008 (0,005) 5,821 (0,032) 251,223 (0,076) 860,418 (0,367) V3. Relative performance (1) Mean Median -0,483 (-0,334) 0,303 (0,265) 0,459 (0,456) 1,355 (1,252) 3,602 (3,371) V4. Number of years of hyper-growth

CEO's market growth previsions:

Indicate the forecasted percentage of growth on your market:

-at 1 year ? -at 3 years ? .

CEO's strategic style:

What are your main objectives for your company?

CEO's attitudes towards growth:

For you, a strong period of growth of your company corresponds in:

Totally disagree

Totally agree

A rewarding challenge A source of danger and disturbance A source of personal satisfaction and bonus A possibility of improving the employees' wellbeing A source of financial enrichment An increase of the capacity to check and to control the operations A way to increase the independence of the company towards his(her) customers, suppliers and financiers An improvement of the chances of sustainability of the company A situation which the company can easily manage CEO's perception of his abilities:

Among the following activities, in which do you feel the most comfortable?