Ugo Boscain
email: ugo.boscain@polytechnique.edu

Francesca C Chittaro
email: francesca.chittaro@lss.supelec.fr

Paolo Mason
email: paolo.mason@lss.supelec.fr

Rémi Pacqueau
email: remi.pacqueau@polytechnique.edu

Mario Sigalotti
email: mario.sigalotti@inria.fr

Motion planning in quantum control via intersection of eigenvalues

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

In the simplest case, the problem of controlling a quantum mechanical system consists in controlling an equation of the form

i ψ = H 0 + m i=1 u i H i ψ, (1)
where ψ belongs to the unit sphere in a Hilbert space H, the self-adjoint operator H 0 describes the free evolution of the system, H i , i = 1, . . . , m are self-adjoint operators describing the coupling between the controls and the system, and u i : [0, T ] → R are the controls, usually representing an electric field, a magnetic field or a laser pulse. Nowadays quantum control is of crucial importance in Nuclear Magnetic Resonance (NMR), in photochemistry and 1 This research has been supported by the European Research Council, ERC StG 2009 "GeCoMethods", contract number 239748, by the ANR "GCM", program "Blanc -CSD", and by the DIGITEO project "CON-GEO".

for the design of quantum gates in quantum computing (see [START_REF] Hübler | Nuclear magnetic resonance quantum computing exploiting the pure spin state of para hydrogen[END_REF], [START_REF] Shapiro | Principles of the Quantum Control of Molecular Processes[END_REF] [START_REF] Nielsen | Quantum computation and quantum information[END_REF], and references therein).

Many results are available when the Hilbert space has finite dimension (see for instance [START_REF] Albertini | Notions of controllability for bilinear multilevel quantum systems[END_REF], [START_REF] Allen | Optical Resonance and Two-Level Atoms[END_REF], [START_REF] Altafini | Controllability of quantum mechanical systems by root space decomposition of su(n)[END_REF], [START_REF] Boscain | Optimal control in laser-induced population transfer for two-and three-level quantum systems[END_REF], [START_REF] Boscain | On the k+p problem for a three-level quantum system: Optimality implies resonance[END_REF], [START_REF] Boscain | Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimal energy[END_REF], [START_REF] Dahleh | Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications[END_REF], [START_REF] Dahleh | Optimal control of two-level quantum systems[END_REF], [START_REF] Gaubatz | Population switching between vibrational levels in molecular beams[END_REF], [START_REF] Khaneja | Time optimal control in spin systems[END_REF], [START_REF] Ramakrishna | Controllability of molecular systems[END_REF], and the monograph [START_REF]Introduction to quantum control and dynamics[END_REF]). On the other hand in the infinite dimensional case positive controllability results are extremely difficult to obtain. For exact controllability results for a one-dimensional well of potential see [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF]. For approximate controllability results for discrete spectrum, via Galerkin approximations, see [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF]. For approximate controllability results via Lyapunov methods, see [START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF], [START_REF] Mirrahimi | Lyapunov control of a quantum particle in a decaying potential[END_REF]. Notice that except for the results given in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF], most of the controllability results are obtained for systems in the form

i∂ t ψ(t, x) = -∆ + V 0 (x) + m i=1 u i V i (x) ψ(t, x) (2)
where x ∈ Ω ⊂ R n , ∆ is the Laplacian (with Dirichlet boundary conditions, when Ω is bounded), V 0 is the potential in absence of controls, and V i are suitable control potentials.

For infinite dimensional systems, the need of generating an infinite number of new directions requires highly oscillating controls, and usually the techniques mentioned above do not lead to explicit control strategies.

Another approach could be to design a slow control path that passes through eigenvalues intersections in the space of controls allowing exchanges of probability between the intersecting levels (in this case, we say that the wave function climbs the spectrum). This idea was presented for the first time in [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF] for a couple of specific systems. To prove that such a path indeed produces a transition, one should: i) take advantage of the adiabatic decoupling far from eigenvalues crossing in the space of controls; ii) use another nonadiabatic decoupling (that we call conical decoupling) close to eigenvalues intersections that permits to prove that, while passing through eigenvalue intersections, the system makes a transition from a level to another.

The first step is quite standard in quantum mechanics (see for instance [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics[END_REF]), while the second one will be explained in full details below. The main feature that makes this method work is the fact that, as a consequence of the self-adjointness of the operators H 0 and H i , i = 1 . . . m, the codimension of the eigenvalues intersections is bigger than one. For instance in the case in which the Hamiltonian is real, the codimension of eigenvalues intersections is two and hence, in the presence of two controls, generically they occur at isolated points in the space of controls. Since climbing of levels can be realized only at eigenvalue crossings, thanks to the fact that the intersections have codimension two we can construct closed paths that pass through each crossing only once, and therefore realize the climbing; see the red path in Figure 1. Transversal eigenvalue crossings of a real Hamiltonian in the presence of two controls are called conical intersections (see [START_REF] De Verdière | Construction de laplaciens dont une partie finie (avec multiplicités) du spectre est donnée[END_REF]). A non-real Hamiltonian can have eigenvalue intersections which are stable under small perturbations of the Hamiltonian and are transversal at isolated points only if it depends on at least three controls. This approach has already been explored in [START_REF] Guérin | Control of quantum dynamics by laser pulses: Adiabatic floquet theory[END_REF], [START_REF] Yatsenko | Topology of adiabatic passage[END_REF] for the STIRAP process. From the controllability point of view, this approach has been applied in [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF] to a class of models generalizing the Eberly and Law system [START_REF] Bloch | The controllability of infinite quantum systems and closed subspace criteria[END_REF].

This method has two advantages. First it provides explicit expressions of controls (motion planning); second it may be applied without difficulties also in the case in which the dependence on controls is nonlinear. The drawback is that the method works only when one can produce eigenvalue crossings via the external fields.

Let u = (u 1 , . . . , u m ) and σ(u) be the spectrum of H(u).

In the following we do not need that the dependence on u is affine as in [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF]. Let σ * (u) ⊂ σ(u) be a band that is uniformly separated from the rest of the spectrum in a connected domain U in the space of controls (see Figure 2) and P * (u) the corresponding orthogonal projector. Roughly speaking, the Adiabatic Theorem (see [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics[END_REF] and references therein) states that if u : [0, T ] → R m is a smooth control and ψ ε u is a trajectory of system (1), corresponding to the control [0, T /ε] ∋ t → u(εt) and such that ψ ε u (0) = ψ 0 ∈ P * (u)H, then as ε gets small, ψ ε u on [0, T /ε] belongs approximately to P * (u)H and its evolution is not influenced by the rest of the spectrum. The quality of the approximation depends on the "gap" C between σ * (u) and the rest of the spectrum. Roughly speaking the error is proportional to 1/C. When P * (u)H is finite dimensional, the adiabatic theory permits also to build a finite dimensional Hamiltonian H eff , describing the approximate evolution inside P * (u)H. The precise construction of H eff as function of u will be presented in a forthcoming paper, and is based on general adiabatic theory ( [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics[END_REF]).

In this paper we are interested in controllability problems inside the finite dimensional space P * (u)H, for a system governed by an Hamiltonian H eff (u). We are then looking for control strategies that can be slowed down (in order to guarantee the adiabatic decoupling with arbitrary precision from the rest of the spectrum). Notice that we are not interested in controlling relative phases, since the adiabatic approximation loses any information about them.

II. SLOW CONTROLLABILITY IN FINITE DIMENSION

We are ready to state our definition of controllability inside the space

P * (u)H ≃ C N . Definition 1: Consider a Schrödinger equation in C N of the type i ψ = H(u)ψ where ψ(t) ∈ C N , u : [0, T ] → U ⊂ R m
, and H(u) is a self-adjoint operator on C N , which is smooth as a function of u. We say that the system is slowly controllable between probabilities (SCBP for short) in Ū ⊂ U if for every c, d ∈ [0, 1] N such that j c j = 1, j d j = 1, and for every ū ∈ Ū , there exists a control function u(t) = (u 1 (t), . . . , u m (t)) : [0, T ] → Ū , continuous and piecewise smooth with u(0) = u(T ) = ū, such that the following holds: for every ψ 0 such that (|ψ 0 1 | 2 , . . . , |ψ 0 N | 2 ) = c there exists a sequence of continuous, piecewise smooth monotone functions

µ i : [0, T i ] → [0, T ], T = µ i (T i ), with lim i→∞ μi (t) ∞ = 0 such that for every i ∈ N the control [0, T i ] ∋ t → u(µ i (t)) steers ψ 0 to some vector ψ(T i ) with (|ψ 1 1 | 2 , . . . , |ψ 1 N | 2 ) = d. Remark 1:
In the previous definition, µ i (t) are reparametrizations of the time. For i → ∞ the intervals [0, T i ] become larger and larger. In our construction the rescalings can be taken piecewise-affine (although in general not linear).

If the selected spectrum σ * (u) ⊂ σ(u) is discrete and non-degenerate, then SCBP never holds unless we are in the trivial case N = 1. Indeed an application of the adiabatic theorem inside P * (u)H does not permit any probability transfer.

We then consider the case in which there are eigenvalue intersections in the space U : as we will show, every intersection allows transfers of probability between the corresponding two energy levels. That is why we focus on a neighborhood of a conical eigenvalue crossing: the basic block of the algorithm consists in studying the controllability of a twolevel system driven by two controls (see Figure 1).

We prove that, close to a conical intersection, a two level system is SCBP with explicit computations, without any adiabatic approximation.

The general model for such a system with two controls is

i ψ = H(u 1 (t), u 2 (t))ψ (3)
where

ψ = (ψ 1 , ψ 2 ) : [0, T ] → C 2 with |ψ 1 | 2 + |ψ 2 | 2 = 1, u 1 , u 2 : [0, T ] → R and H(u 1 , u 2 ) is a real-valued matrix of the form H(u 1 , u 2 ) = f 1 (u 1 , u 2 ) f 2 (u 1 , u 2 ) f 2 (u 1 , u 2 ) -f 1 (u 1 , u 2 ) . (4)
In the following we assume that f 1 and f 2 are smooth.

Remark 2: In formula (4) we have assumed that the trace is zero. This is not restrictive, since a nonvanishing trace gives rise only to a common factor of phase.

The eigenvalues of the matrix H(u 1 , u 2 ) are

λ ± = ± f 1 (u 1 , u 2 ) 2 + f 2 (u 1 , u 2 ) 2 .
(

) 5
Hence a degeneration occurs at (ū 1 , ū2 ) if and only if

f 1 (ū 1 , ū2 ) = f 2 (ū 1 , ū2 ) = 0.
Notice that for a generic choice of f 1 and f 2 , the points at which the eigenvalues lose simplicity are isolated in the plane (u 1 , u 2 ). Theorem 1: Consider a two-level quantum system

i ψ = H(u 1 (t), u 2 (t))ψ (6)
where

ψ = (ψ 1 , ψ 2 ) : [0, T ] → C 2 with |ψ 1 | 2 + |ψ 2 | 2 = 1, u 1 , u 2 : [0, T ] → U ⊂ R and H(u 1 , u 2
) is a real-valued matrix of the form (4). Assume that det(H(ū 1 , ū2 )) = 0 and that the differential of the function

(u 1 , u 2 ) → (f 1 (u 1 , u 2 ), f 2 (u 1 , u 2 )
) is invertible at (ū 1 , ū2 ). Then the system is SCBP in a neighborhood of (ū 1 , ū2 ).

Here we sketch the explicit strategy we intend to use. We perform the change of coordinates

(v 1 , v 2 ) = F (u 1 , u 2 ) := ρ (f 1 (u 1 , u 2 ), f 2 (u 1 , u 2 )), (7)
ρ > 0, from a neighborhood of (ū 1 , ū2 ) to an open set containing the unit ball; we then construct a closed oriented path in the space of controls (v 1 , v 2 ) that is piecewise smooth and passes through the conical intersection (the origin in these coordinates); we choose a path that possesses the graph shown in Figure 3: we go straight from the starting point (-1, 0) to the singularity, then we make an angle α and continue with another segment, and finally we come back to the starting point with an arc of circle. Assuming without loss of generality that ρ = 1 (this corresponds to a simple time reparametrization), this path can be realized as the graph of the following function:

(v 1 , v 2 ) : t →      ( t L -1, 0) t ∈ [0, L] (cos α, sin α)( t L -1) t ∈ [L, 2L] (cos(θ(t)), sin(θ(t))) t ∈ [2L, T ], (8)
where

L = T 2 + π -α , θ(t) = α + t -2L L .
This function is indeed continuous and piecewise smooth, and satisfies the condition (v 1 (0), v 2 (0)) = (v 1 (T ), v 2 (T )).

We will prove that the controls (u 1 (t), u 2 (t)) = F -1 (v 1 (t), v 2 (t)) guarantee SCBP, in the following sense: for every c, d probabilistic weights there is an α ∈ [0, π] (depending only on c, d) such that for any

ψ 0 = (ψ 0 1 , ψ 0 2 ) with (|ψ 0 1 | 2 , |ψ 0 2 | 2 ) = (c 1 , c 2 )
we can find a sequence of time rescalings µ α i that satisfy the conditions in the definition of SCBP and such that the control (v 1 (µ α i (t)), v 2 (µ α i (t))) defined on the interval [0, T i ] steers ψ 0 to a state The angle α can be explicitly computed (see ( 10)); for instance, if ψ 0 = e iφ (1, 0), then we have the following values (see Figure 4):

ψ 1 = (ψ 1 1 , ψ 1 2 ) such that (|ψ 1 1 | 2 , |ψ 1 2 | 2 ) = (d 1 , d 2 ).
(|ψ 1 1 | 2 , |ψ 1 2 | 2 ) = (1, 0) ⇒ α = π (|ψ 1 1 | 2 , |ψ 1 2 | 2 ) = (0, 1) ⇒ α = 0 (|ψ 1 1 | 2 , |ψ 1 2 | 2 ) = 1 2 , 1 2 ⇒ α = π 2 .
The above control satisfies u(0) = u(T ) = ū := F -1 (-1, 0). Control strategies for different ū can be obtained from the one above, applying a rotation and a dilation in the plane (v 1 , v 2 ).

III. SKETCH OF THE PROOF

A. Normal form at an eigenvalue intersection

Assume that f 1 (ū 1 , ū2 ) = f 2 (ū 1 , ū2 ) = 0 and that the differential of the function

(u 1 , u 2 ) → (f 1 (u 1 , u 2 ), f 2 (u 1 , u 2 )
) is invertible at (ū 1 , ū2 ). Then, locally near (ū 1 , ū2 ) we can perform the change of variables [START_REF] Boscain | Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimal energy[END_REF]; up to a simple reparametrization of time, from now on we assume that ρ = 1. We thus obtain the normal form

i ψ = v 1 v 2 v 2 -v 1 ψ, (9)
whose eigenvalues are

λ ± = ± v 2 1 + v 2 2 .

B. The control procedure

In this subsection we solve explicitly the differential system (9) associated with the control (8) in the case in which the initial condition is an eigenvector of the Hamiltonian; we show how the choice of the angle α and of the (α-dependent) time reparametrization µ α k let us control the probability weights of the wave function at the final time (µ α k ) -1 (T ). The choice of such an initial condition gives rise to a family of reparametrizations µ α k that is particularly simple (multiplication by a constant factor); for initial conditions that are nontrivial combinations of the two eigenvectors, to obtain SCBP one may still apply controls of the form (8), up to a suitable, possibly piecewise-linear, time reparametrization.

Let us now analyze the evolution of the system (9). We assume then that our initial state is ψ 0 = (e iφ , 0) (i.e. c = (1, 0)) and that we are given a pair

(d 1 , d 2 ) ∈ [0, 1] 2 , d 1 + d 2 = 1. We choose α = 2 arcsin( d 1 ). ( 10
)
First segment: Putting ψ = (ψ 1 , ψ 2 ), the system (9) evolves accordingly to the differential equation

ψ1 (t) = -i (t/L -1) ψ 1 (t), ψ2 (t) = i (t/L -1) ψ 2 (t),
therefore the solution at the time t is

ψ(t) = e -i t 2 2L -t ψ 1 (0), e i t 2 2L -t ψ 2 (0) . If ψ(0) = ψ 0 , we get ψ(L) = (e iφ1 , 0), φ 1 = φ + L/2.
This means that the evolution only adds a (relative) phase factor, but does not change the probabilities.

Second segment: In this case we have to solve the Schrödinger equation

i d dt ψ 1 ψ 2 = cos(α)(t/L -1) sin(α)(t/L -1) sin(α)(t/L -1) -cos(α)(t/L -1) ψ 1 ψ 2 t ∈ [L, 2L],
with initial condition ψ(L) = (e iφ1 , 0). The solution of the Cauchy problem is

ψ 1 (t) ψ 2 (t) = e iφ1 cos(φ 2 (t)) -i sin(φ 2 (t)) cos(α) -i sin(φ 2 (t)) sin(α) , where φ 2 (t) is the dynamical phase φ 2 (t) = t 2 2L -t + L 2 .
In particular, we notice that for (v 1 , v 2 ) = (cos(α), sin(α)), the eigenvectors of the Hamiltonian are ϕ + = (cos(α/2), sin(α/2)), ϕ -= (-sin(α/2), cos(α/2)), respectively associated with the eigenvalues λ = +1 and -1. Therefore, we can write the wave function at the time t = 2L as ψ(2L) = e iφ1 e -iφ2(2L) cos(α/2)ϕ + + e iφ2(2L) sin(α/2)ϕ -.

We notice that the probability of being in an eigenstate relative to the eigenvalue λ = +1 is (cos(α/2)) 2 ; in other words, we spread the probability weights between the two eigenstates.

Arc of circumference: The last part of the path comes back to the initial point (v 1 , v 2 ) = (-1, 0). The final state is then the solution at the time T of the equation

i d dt ψ 1 ψ 2 = cos(θ(t)) sin(θ(t)) sin(θ(t)) -cos(θ(t)) ψ 1 ψ 2 , (11)
with initial condition ψ(2L) = e iφ+ cos(α/2)ϕ + + e iφ-sin(α/2)ϕ -. We claim that there is a suitable choice of time rescalings µ α k such that the controls (v 1 (µ α k (t)), v 2 (µ α k (t)) defined on [0, (µ α k ) -1 (T )] steer the initial vector (e iφ , 0) to the vector of the form

ψ 1 = e iβ+ sin(α/2) 1 0 + e iβ-cos(α/2) 0 1 , (12)
for suitable phases β ± .

In order to do that, we study the system

i d dt x 1 x 2 = cos(α + t/τ ) sin(α + t/τ ) sin(α + t/τ ) -cos(α + t/τ ) x 1 x 2 , for t ∈ [0, (π -α)τ ],
where τ is a free parameter, with initial condition x(0) = (x 1 (0), x 2 (0)) = ϕ + . We notice that this system is completely analogous to [START_REF] De Verdière | Construction de laplaciens dont une partie finie (avec multiplicités) du spectre est donnée[END_REF], up to an affine time transformation.

We evaluate |x i (t)| 2 , i = 1, 2, at the final time t = (πα)τ , as a function of τ ; explicit computations lead to

|x 2 (t)| 2 = 1 4τ 2 + 1 4τ 2 + cos π -α 2 4τ 2 + 1 2 .
Therefore there is a sequence of times τ α k , going to infinity, such that for τ = τ α k and t α k = (π -α)τ α k , we have |x 2 (t α k )| 2 = 1. Hence, from the unitarity of the evolution,

|x 1 (t α k )| 2 = 0; analogously, for the initial condition x(0) = ϕ -we have |x 2 (t α k )| 2 = 0 and |x 1 (t α k )| 2 = 1. For α ∈ [0, π) the explicit form of τ α k is τ α k = π π -α 2 k 2 - 1 4 ,
while for α = π the equation is satisfied for any value of τ α k . This means that after an evolution of time t α k the eigenvector of the Hamiltonian evaluated at (v 1 , v 2 ) = (cos(α), sin(α)) relative to the eigenvalue λ = +1 has moved to the vector ψ = (0, 1), which is in fact the eigenvector of the Hamiltonian at (v 1 , v 2 ) = (-1, 0) relative to the same eigenvalue. Analogously, at the final time ϕ - has moved to ψ = (1, 0).

Coming back to the initial system, we have proved that if we choose the constant time rescaling

µ α k : t → L τ α k t, t ∈ 0, τ α k L T , then the chosen control (v 1 (µ α k (t)), v 2 (µ α k (t))), defined on [0, (µ α k ) -1 (T )],
steers the initial state ψ 0 = (e iφ , 0) to the final state [START_REF] Dahleh | Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications[END_REF], which means that we steered the vector with probabilities

(|ψ 0 1 | 2 , |ψ 0 2 | 2 ) = (1, 0) to a vector ψ 1 with probabilities (|ψ 1 1 | 2 , |ψ 1 2 | 2 ) = (sin 2 (α/2), cos 2 (α/2))
. By our choice of the angle α (10), we have

(|ψ 1 1 | 2 , |ψ 1 2 | 2 ) = (d 1 , d 2 )
. Remark 3: Notice that the above procedure also includes the special cases α = 0, π. If α = 0 the control goes straight from (-1, 0) to (1, 0), and then comes back with a semicircle; the wave function at the final time (τ 0 k /L)T is ψ = (0, 1) (modulo a phase factor), which means that the state has completely flipped to the eigenstate relative to λ = +1, in accord with the adiabatic theory. On the contrary, if α = π then the path reduces to the segment connecting (-1, 0) to the origin followed by the segment that comes back to the initial point. In this case, the final state is ψ = (1, 0) (with a phase factor), that is the trajectory comes back to the original state. Notice that in this case any reparametrization µ π leads to the same result.

With our choice of the initial condition the time reparametrization of the control functions involved in the definition of SCBP takes a particularly simple form. However, if the initial condition ψ 0 is a superposition of states, we can prove that the control functions of the form (8), suitably rescaled in time, still provide an explicit strategy for SCBP. The explicit computations are cumbersome, and we therefore postpone them to a forthcoming paper.

IV. CONCLUSION

In this paper, we provided a control strategy that permits to steer exactly a two level quantum system presenting a conical singularity to a final state with given probabilities. This strategy can be arbitrarily slowed down, which is important in order to reduce an infinite dimensional quantum system to a two level system as the one considered in this paper, that is then decoupled from the other levels. The tool permitting such a decoupling is the adiabatic theory. The aim of the further research in this direction is to get an approximate controllability result for the general system.

The main advantages of the proposed strategy are that it does not require the Hamiltonian to depend linearly on the controls, and that it provides an explicit expression of the controls.

Fig. 1 .

 1 Fig. 1. Climbing path u.

Fig. 2 .

 2 Fig. 2. Spectrum in function of u.

Fig. 3 .

 3 Fig. 3. The path in the plane (v 1 , v 2 ).

Fig. 4 .

 4 Fig. 4. Paths in the space of controls.


P Andry
email: andry@ensea.fr

P Gaussier
email: gaussier@ensea.fr

M Lagarde
email: lagarde@ensea.fr

S Boucenna
email: boucenna@ensea.fr

L Hafemeister
email: hafemeister@ensea.fr

Proprioception and imitation: on the road to agent individuation

Since Piaget, it is well accepded that higher level cognitive functions are settled on low level sensori-motor associations. In this paper, we will show that different kinds of interactive behaviors can emerge according to the kind of proprioceptive function available in a given sensori-motor system. We will study two different examples. In the first one, an internal proprioceptive signal is avaible for the learning of the visuo-motor coordination between an arm and a camera. An imitation behavior can emerge when the robot 's eye focuses on the hand of the experimenter instead of its own hand. The imitative behavior results from the error minimization between the visual signal and the proprioceptive signal. Here, the imitation results from the perception ambiguity: the robot mistakes its hand with the experimenter hand! In the second example, a robot head has to recognize the facial expression of the human caregiver. Yet, the robot has no visual feedback of its own facial expression. The human expressive resonance will allows the robot to select the visual features relevant for a particual facial expression. As a result, after few minutes of interactions, the robot can imitates the facial expression of the human partner. We will show the different proprioceptive signals used in both examples can be seen as bootstrap mechanisms for more complex interactions. Applied as a crude model of the human, we will propose that these mechanisms play an important role in the process of individuation.

Introduction

With the recent development of robotics, an important effort has been made to build robots able to interact easily with humans. This effort mainly result in [START_REF] Amari | Dynamic of pattern formation in lateral-inhibition type by neural fields[END_REF] researches trying to enhance the robot's control architecture in order to detect and take into account the other (from speech recognition, to the addition of special primitives processing face/body detection and framing and/or emotion recognition) and (2) technics used to enhance the expressiveness and the aspect of the robot. ( 1) is dedicated to enhance the robot interaction and (2) is dedicated to enhance the human acceptation of the robot, and both approaches intend to facilitate the exchanges between Human and robots. Nevertheless, as underlined by decades of studies in developmental psychology, understanding the roots of communication is still a major issue, and we still not know what are the right "properties" that a system must embed in order to detect, exchange, take turn easily as the young infants do, even at a pre-verbal stage.

In this paper, we wish to show that simple perception-action systems initially designed as homeostatic machines, without embedding any "interactive functions", have shown to be the basis of the emergence of higher level behaviors when placed in interactive situations. Interestingly, these emergent behaviors constitute new building blocks favoring at the same time higher level learning and interaction.

We will give two examples, where the architectures are not designed to interact with others. They are only designed to learn new sensory-motor associations when detecting changes in self dynamics (imbalances). When acting in front of the robot, the human is actively modifying the behavioral dynamics of the robot provoking new situations which are learned by the system.

From visuo-motor learning to sequence learning by imitation

Our model is a perception-action architecture designed to equilibrate 2 kinds of information : Visual information (V), about the external environment and Proprioception information (Prop), i.e information about the self movements that are being done (angle of joints φ 1 ,φ 2 ,...,φ n , speed of the joints d dφ , etc.. of the devices). In order to compute basic behaviors, like tracking, pointing, reaching objects, the architecture is designed like an homeostatic machine (Figure 1 a.). It tries to minimize the differences between the inputs V and Prop, and the imbalances generate the movements of the system in order to reduce the corresponding error. To do so, V excites a Neural Field (NF) that compute a dynamical attractors centered on the input stimuli [START_REF] Amari | Dynamic of pattern formation in lateral-inhibition type by neural fields[END_REF]. If Prop can be expressed in the same referential as the NF, then it is easy to extract the command to give to the joints of the arm in order to move this latter (and therefore the value of Prop) toward the maximum of the attractor (Figure 1 b.). This read-out [START_REF] Schöner | Dynamics of behavior: theory and applications for autonomous robot architectures[END_REF] mechanism is the value of the derivative of the attractor at the position of prop. Hence, behaviors of the robot are NF attractors triggered by V information (representing sensorymotor "target"). Obviously, the read-out mechanisms is only possible if V and Prop can be computed in the same space. In previous work, we have shown that all information can be computed in a simple 2D space (corresponding to the vision produced by a sole CDD camera). To do so, we used a visuo-motor map of neurons, that associate multi-dimensional and redundant information about the joints into a simple 2D "visual" response. The associations are formed during a learning phase: the system produces initial random arm movements in order to learn to equilibrate input V and Prop information projected on the Visuomotor map. The result is a system that is able to transform motor information (φ 1 ,φ 2 ,...,φ n ) into a simpler 2D space corresponding to the visual one. Then, the movements to reach visual goals is computed in the visual space thanks to the read-out mechanism. The association of an adaptive visuo-motor map with two 1D neural fields can be seen as a simple and global dynamical representation of the working space controlling efficiently an arbitrary number of degrees of freedom according to 2D information coming from the visual system.

Moreover, having a system that is not able to discriminate visual information about self movements with the movements of others can be, for example, useful to trigger an imitative behavior. Based on movement detection, the system can't differentiate it's extremity from another moving target. As a consequence, moving in front of the robot induces visual changes (movement detection) that the robot interprets as an unforeseen self movement. The robot acts as an homeostat, it tends to correct by producing the opposite movements, inducing the following of the demonstrator gesture (numerous psychological works show comparable human behaviors when visual perception is ambiguous [START_REF] Nielsen | Volition: A new experimental approach[END_REF][START_REF] Fourneret | Limited conscious monitoring of motor performance in normal subjects[END_REF][START_REF] Jeannerod | To act or not to act. perspectives on the representation of actions[END_REF]). Applied to an eyearm robotic system, the generated error will induce movements of the robotic arm reproducing the moving path of the human hand: an imitative behavior emerges. Using this setup, we showed that our robot can imitate several kind of movements (square or circle trajectories, up and down movements. Hence, the imitative behavior emerges as a side effect of the perception ambiguity (and limitations). From this low-level imitative behavior, our robot is able to learn sequences of movements from others. To do this, we use a second sensori-motor loop (Figure 3 a.) based on a neurobiological model [START_REF] Gaussier | From perception-action loops to imitation processes[END_REF] inspired from some of the properties of the cerebellum and the hippocampus. This model uses associative learning rules between past inputs memorized as a STM and present inputs in order to learn the timing of simple sequences. In order to learn complex sequences in which the same state is repeated several times, we have added a mechanism that generates internal dynamics and that can be associated with the repeated inputs of the sequence. The association between the repeated inputs and different activities of the oscillator allows to code hidden states with different and un-ambiguous patterns of activities. As a result, our architecture manages to learn/predict and reproduce complex temporal sequences. Moreover, recent experiments shows that this architecture can be transposed to learning by demonstration [START_REF] Lagarde | The role of internal oscillators for the oneshot learning of complex temporal sequences[END_REF], and also to the learning of complex sequences of displacements with a navigating robot [START_REF] Lagarde | Learning new behaviors : Toward a control architecture merging spatial and temporal modalities[END_REF]. The coupling between the experimenter and the robot is performed via the visual environment (see also [START_REF] Cheng | Complex continuous meaningful humanoid interaction: A multi sensory-cue based approach[END_REF] for a similar dynamical coupling)but the visuo-motor coupling has been learned thanks to the proprioceptive signal.

Robot response to an expressive human

This second work is motivated by the question of how a robotic system (Figure 4, right), able to exhibit a set of emotional expressions can learn autonomously to a associate theses expressions with those of others. Here, "autonomously" refers to the ability to learn without the use of any external supervision. A robot with this property could therefore be able to associate its expressions with those of others, linking intuitively its behaviors with the responses of the others. This question is close to the understanding of how babies learn to recognize the facial expressions of their caregivers without having any explicit teaching signal allowing to associate for instance an "happy face" with their own internal emotional state of happiness. Using the cognitive system algebra [START_REF] Gaussier | Toward a cognitive system algebra: A perception/action perspective[END_REF], we showed that a simple sensori-motor architecture (see Figure 4, left) using a classical conditioning paradigm could solve the task if and only if we suppose that the baby produces first facial expressions according to his/her internal emotional state and that next the parents imitate the facial expression of their baby allowing in return the baby to associate these expressions with his/her internal state [START_REF] Gaussier | Toward a cognitive system algebra: Application to facial expression learning and imitation[END_REF]. If the adult facial expressions are not synchronized with the baby facial expression, the task cannot be learned. Moreover, recent psychological experiments [START_REF] Nadel | Human responses to an expressive robot[END_REF] have shown that humans "reproduce" involuntary a facial expression when observing and trying to recognize it. Interestingly, this facial response has also been observed in presence of our robotic head. This low level resonance to the facial expression of the other can be considered as a natural bootstrap for the baby learning ("empathy" from the parents). Because, the agent representing the baby must not be explicitly supervised, a simple solution is to suppose the agent representing the parent is nothing more than a mirror. We obtain an architecture allowing the robot to learn the "internal state"-"facial expression" associations. We showed that from our departure control architecture, learning is only possible if the parent agent (supposed to be the teacher) imitates the baby agent. The roles are switched according to the classical point of view of AI and learning theory. This advocates that taking account of the interaction dynamics between two agents can change our way of thinking learning and more generally cognition problems.

Discussion

Both examples show the importance of the interaction with a partner for the robot development and learning. Imitation appears as an important mechanism both for learning and communication [START_REF] Nadel | Imiter être imité dans le développement de l'intentionnalité[END_REF]. Interestingly, in our developmental procedure there is no need at first to suppose the robot is able to recognize the human partner as a model. At the opposite, the robot development is based on the perception ambiguity. The robot considers the human partner as a part of itself. The interaction and the associated correlations extend the proprioception of the robot to the human partner. They become a single dynamical system able to learn things that the robot could not learn by itself. Hence imitative behaviours would be the result at first of the agent unability to perceive something as being different from itself. We can imagine the capability to predict the feedback signal can be used to verify if there is some novelty in the sensory feedback related to the current robot action. Bounded novelty may result from the interaction with a partner hence allowing to recognize him/her at such. Hence we can propose a scheme in which the individuation comes after a phase of fusion/merging with the environement. The proprioceptive signal is necessary to close the loop and to build a global system composed of the agent and his/her caregivers allowing simple learning mechanisms to be sufficient for the learning of more and more complex behaviors (autopoietic loop). First correlations detections allow to build new predictive signals allowing then to differentiate the agent "inner part" from the outside and latter the other agents. We have shown for instance that the capability to discreminate faces from non faces could be the result of the emotional interaction and not one of its preriquisites as usually supposed in classical image processing works. In the same vein, joint attention to a given object could be the result of the emotional association between the facial expression of the experimenter and an object (social referencing) and not the opposite as it is classically supposed. Social interactions can be seen in the first periods of the development as a way to increase the potential of the proprioceptive function allowing to maintain more complex and stable dynamics that can be used for the learning of more complex tasks. Hence, in an epigenitic approach, the sensori-motor control appears as an essential element to avoid the symbol grounding problem and to build autonomous robots able to develop more and more cognitive capabilities thanks to the social interactions.

Fig. 1 .

 1 Fig. 1. a. The homeostatic controller trying to equilibrate visual and proprioceptive informations. The motor command is computed by the neural field and the readout groups whose activities are illustrated in b. (example in 1 dimension). In this case Visual and proprioceptive information are different. This unbalance will trigger the learning of a new association on the visuo-motor map.

Fig. 2 .

 2 Fig. 2. The proto-imitation principle applied to a robotic setup.

Fig. 3 .

 3 Fig. 3. a. : Complex sequences learning model b. : We manipulate Aibo passively. It learns the succession of orientations of the movement from these front left leg motor information. c. : Aibo reproduces the learnt sequence.

Fig. 4 .

 4 Fig. 4. The architecture used to associate a collection of feature points extracted from the visual flow with the expressed emotion by the robot. If a human come in front of the robot an start to imitate the expression, (s)he will closes the loop between vision and proprioception and allow the system to learn to recognize emotions of others.

Acknowledgments

This work was supported by the French Region Ile de France, the Institut Universitaire de France (IUF) and the FEELIX GROWING european project. (FP6 IST-045169)