Andrea Cherubini

Robin Passama

Philippe Fraisse

André Crosnier

A unified multimodal control

Keywords: Reactive and Sensor-based Control, Human-Robot Interaction, Visual Servoing

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Introduction

Recently, the attention of robotics researchers worldwide has turned towards the field of human-robot interaction (HRI [START_REF] Bicchi | Safety for physical human-robot interaction[END_REF][START_REF] Santis | An atlas of physical human-robot interaction[END_REF][START_REF] De Luca | Integrated control for pHRI: Collision avoidance, detection, reaction and collaboration[END_REF][START_REF] Sentis | A Whole-Body Control Framework for Humanoids Operating in Human Environments[END_REF][START_REF] Erden | End-point Impedance Measurements at Human Hand during Interactive Manual Welding with Robot[END_REF]), to enable close collaboration between human and robot [START_REF] Li | Human-Guided Robotic Manipulation: Theory and Experiments[END_REF][START_REF] Erden | Assisting manual welding with robot[END_REF]. In this context, the robot must infer the user intention, to interact more naturally, from the human perspective [START_REF] Grunwald | Touch: The direct type of human interaction with a redundant service robot[END_REF][START_REF] Erden | Human intent detection and physicallyinteractive control of a robot without force sensors[END_REF][START_REF] Khatib | Robotics-based Synthesis of Human Motion[END_REF]. To this end, both visual (e.g., based on Microsoft Kinect TM [START_REF]Microsoft Kinect homepage[END_REF]) and force feedback, have been used [START_REF] Maeda | Human-robot cooperative manipulation with motion estimation[END_REF][START_REF] Villani | Force Control[END_REF][START_REF] Lippiello | Interaction Control of Robot Manipulators Using Force and Vision[END_REF][START_REF] Flacco | A Depth Space Approach to Human-Robot Collision Avoidance[END_REF][START_REF] Corrales | Cooperative Tasks between Humans and Robots in Industrial Environments[END_REF]. Generally, we believe that direct sensor-based methods, such as visual servoing [START_REF] Chaumette | Visual servo control[END_REF], provide better solutions, for intuitive HRI, than planning techniques, requiring a priori models of the environment and agents [START_REF] Valle | Planning Algorithms[END_REF]. Moreover, force and vision should be used concurrently, since the information they provide is complementary. One pioneer work in this sense is [START_REF] Santis | Human-Robot Interaction Control Using Force and Vision[END_REF], where force and visual control are used to avoid collisions, while tracking human motion during interaction.

However, the authors do not provide a unified solution for integrating the two sensing modalities. Instead, since the vision and force sensors often measure different physical phenomena, it is preferable to directly combine their data at the control level, rather than to apply multi-sensory fusion, or to design complex state machines. This idea has been initially proposed in [START_REF] Nelson | Improved force control through visual servoing[END_REF][START_REF] Hosoda | Adaptive hybrid visual servoing/force control in unknown environment[END_REF], by adapting the hybrid position-force control paradigm [START_REF] Raibert | Hybrid position/force control of manipulators[END_REF]: force constrains some motion directions, while vision drives the remaining degrees of freedom. Later, the authors of [START_REF] Baeten | Integrated vision/force robotic servoing in the task space formalism[END_REF] have presented a list of hybrid control configurations, and divided the degrees of freedom to be controlled by vision and force. An alternative is impedance/admittance control [START_REF] Hogan | Impedance control: an approach to manipulation: parts I-III[END_REF], which has been integrated with visual [START_REF] Morel | Impedance based combination of visual and force control[END_REF] and even tactile [START_REF] Prats | Vision-tactile-force integration and robot physical interaction[END_REF] control, to account for external forces. Although many techniques for merging vision, force and position control have been designed, the presence of the human in the robot control loop is rarely accounted for.

In our previous work [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF], we have started the design of a multimodal framework for human-robot cooperation. The approach is marker-less, and has been validated in a mock-up industrial scenario. However, the following contributions are brought here, with regards to that work:

• a unified formalism, inspired by inverse kinematics [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF][START_REF] Mansard | A versatile generalized inverted kinematics implementation for collaborative working humanoid robots: The stack of tasks[END_REF] guarantees the controller stability, independently from the sensor modality;

• the use of smooth transitions (homotopies) between the sensor-based tasks, and of self-adapting gains, limits the robot accelerations, thus guaranteeing safer operation;

• in contrast with hybrid vision-force control, it is possible to control a same task direction using weighted combinations of different sensors, and sensor-based tasks can be expressed in different reference frames;

Other, minor improvements, with regards to [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF], include the introduction of force-based control, guaranteeing safety of HRI, and better accuracy, velocity, and control smoothness. Moreover, our task-oriented approach, in contrast with similar ones, such as the stack-of-tasks [START_REF] Mansard | A versatile generalized inverted kinematics implementation for collaborative working humanoid robots: The stack of tasks[END_REF] and constraint-based programming [START_REF] Schutter | Constraint-based task specification and estimation for sensor-based robot systems in the presence of geometric uncertainty[END_REF][START_REF] Ceriani | A constraint-based strategy for task-consistent safe human-robot interaction[END_REF], is directly usable in real HRI scenarios (to our knowledge, the method presented in [START_REF] Ceriani | A constraint-based strategy for task-consistent safe human-robot interaction[END_REF] has been used, for now, only for human collision avoidance). The article is organized as follows. In Section 2, we present our general framework for multimodal control for HRI. In Section 3, relevant variables and sensor-based tasks are defined. Based on these preliminaries, Section 4 shows how the general framework can be instantiated for an industrial case study. Experimental results are reported in Section 5, and summarized in the Conclusion.

Control framework

To safely interact with the human, the designed controller must rely on the various sensing modalities present on the robot. These may include cameras (for vision), force/torque sensors, skin (for tact), or proprioception (e.g., for positioning).

A commonly used approach to merge the various sensing modalities directly at the control level is hybrid sensors control, e.g. hybrid force/position [START_REF] Raibert | Hybrid position/force control of manipulators[END_REF] or hybrid force/vision [START_REF] Nelson | Improved force control through visual servoing[END_REF] control. This approach was recently extended in a framework integrating vision, force and tact to realize physical interaction tasks [START_REF] Prats | Reliable non-prehensile door opening through the combination of vision, tactile and force feedback[END_REF]. We hereby recall the formulation of that approach, and propose a more generic one, based on classic inverse kinematics control [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF].

Let k be the dimension of the operational space associated with the end effector (e.g., k = 3 in the case of a planar manipulator). Consider n senses and, for each sense, the task vector s m ∈ R k , with m = 1, . . . , n. For example, a task associated with the sense of vision could consist in controlling an on-board camera to make it look at a target point, and a task associated with the sense of force could consist in applying a desired wrench with the end effector (e.g., s f = [f x , f y , m z ] for a planar manipulator). In this work, since position, vision and force are used, n = 3. All n tasks have the same size k, and, if the sensor provides less than k measures, it will be sufficient to select the task components corresponding to the actual measures, as will be explained later.

Each task is related to the Cartesian velocity of the end effector, v ∈ R k by the k × k matrix L m (called interaction matrix in the case of visual servoing):

ṡm = L m v. (1)
Stacking the n tasks yields:

ṡ = Lv, with s =    s 1 . . . s n    ∈ R kn and L =    L 1 . . . L n    ∈ R kn × R k . (2)
As aforementioned, a combination of tasks defined by different senses (i.e., by components of the different s m ) is realizable, as long as its size is also k. The tasks are selected thanks to n positive definite square diagonal selection matrices of size k, denoted S m , that activate or deactivate a given task component. Then, the k-dimensional hybrid task to be realized, is a linear mapping of the complete s:

ṡ = S ṡ, with S = [S 1 . . . S n ] ∈ R k × R kn . (3)
Note that, as outlined above, if the m-th sensor provides less than k measures, the missing components can be deselected by simply setting to zero the corresponding row in S m . The selection matrices can also be used, as will be shown later, to weigh outputs from different sensors and combine them into a single task. Merging (3) and ( 2) gives the open-loop behaviour of the task in function of the end effector velocity: ṡ = SLv.

Inverse kinematics control relies on the assumption that matrix SL is in-vertible 1 . Then, the optimal2 solution of (4) ensuring exponential convergence of s to the desired constant task s * is:

v = (SL) -1 (s * -s) . (5)
Indeed, replacing this into (4) yields:

ṡ = s * -s, (6)
guaranteeing that s = s * is a stable equilibrium for the closed-loop system.

Let us now compare [START_REF] Erden | End-point Impedance Measurements at Human Hand during Interactive Manual Welding with Robot[END_REF] with the hybrid sensors control used in numerous works [START_REF] Nelson | Improved force control through visual servoing[END_REF][START_REF] Raibert | Hybrid position/force control of manipulators[END_REF][START_REF] Baeten | Integrated vision/force robotic servoing in the task space formalism[END_REF][START_REF] Prats | Reliable non-prehensile door opening through the combination of vision, tactile and force feedback[END_REF]. This approach consists in assigning each sensing modality to a Cartesian direction in the operational space, and then summing the velocities associated with the selected sensors:

v = m S m v m , (7)
with some assumption on the selection matrices, e.g., that they are orthogonal, as in [START_REF] Prats | Reliable non-prehensile door opening through the combination of vision, tactile and force feedback[END_REF].

Assuming each L m is invertible, exponential convergence of s m to s * m , according to [START_REF] Bicchi | Safety for physical human-robot interaction[END_REF], is guaranteed by applying:

v m = L -1 m (s * m -s m ) . (8)
Plugging ( 8) into [START_REF] Erden | Assisting manual welding with robot[END_REF], we obtain the hybrid sensors control expression:

v = S L (s * -s) with L =    L -1 1 . . . 0 0 . . . 0 0 . . . L -1 n    ∈ R kn × R kn . ( 9
)
This controller is optimal for (4), if and only if (9) coincides with (5):

S L (s * -s) = (SL) -1 (s * -s) ∀ s * , ṡ * ∈ R 2 . ( 10
)
This is equivalent, considering (3), to:

S L = (SL) -1 S. (11)
In general, this is not the case, but we hereby provide two necessary conditions for it to be true.

Property. Hybrid sensors control ( 9) is optimal if the diagonal selection matrices S m are all binary and orthogonal, and if the sensor matrices L m are all diagonal.

Proof. Since all S m are binary (hence, idempotent) and orthogonal:

n m=1 S m = I. (12)
Moreover, binary S m imply that S has full rank, so its right pseudoinverse can be derived to show, using [START_REF] Maeda | Human-robot cooperative manipulation with motion estimation[END_REF], that it coincides with its transpose:

S † = S SS -1 = S m S 2 m -1 = S m S m = S . (13)
Then, post-multiplying condition [START_REF]Microsoft Kinect homepage[END_REF] by S † = S , we obtain:

S LS † = (SL) -1 SS † , (14)
which leads to:

S LS = (SL) -1 . (15)
• The first member of (15) becomes:

S LS = m S m L -1 m S m . (16)
By commuting the matrix product (since all L m and S m are diagonal, and have the same size), and taking advantage of the idempotency of the S m , we obtain:

S LS = m S 2 m L -1 m = m S m L -1 m . (17)
• The second member of (15) becomes:

(SL) -1 = m S m L m -1 . (18)
Noting s im and l im the i-th elements of S m and L m , respectively:

(SL) -1 = diag m s 1m l 1m , . . . , m s km l km -1 = = diag 1 m s1ml1m , . . . , 1 m s km l km . ( 19
)
Since for each i, exactly one s i is non-null and equal to 1, this equation can be rewritten:

(SL) -1 = m S m L -1 m . (20)
Equations ( 17) and [START_REF] Nelson | Improved force control through visual servoing[END_REF] demonstrate that the first and second members of (15) coincide, and that the property is therefore valid.

To summarize, hybrid sensors control provides an optimal solution for (4) under two strong assumptions.

1. All the sensor tasks s m must be expressed in the same reference frame.

This can be stated from in [START_REF] Bicchi | Safety for physical human-robot interaction[END_REF], subject to the condition that the L m matrices are diagonal. 2. Only one sensor can be used to control each end effector direction. This can be stated from [START_REF] Erden | Assisting manual welding with robot[END_REF], subject to the condition that the S m are binary and orthogonal.

These assumptions are mentioned in all works that apply hybrid sensors control. However, they limit its use in practical applications. For instance, merging image-based visual servoing [START_REF] Chaumette | Visual servo control[END_REF], which defines the visual task in the image frame, with force control, usually implemented in the force sensor frame, would infringe the first assumption.

On the other hand, to guarantee stability of the closed-loop system, the classical inverse control scheme (5) only requires that SL is invertible (a weaker assumption, that is always true if the S m are binary, and the L m diagonal). Controller ( 5) can be applied even if the task frames associated with each sensor are different, and even if a task is defined for multiple robots [START_REF] Antonelli | The NSB control: a behavior-based approach for multi-robot systems[END_REF], or as a combination of heterogeneous sensor data (as shown in many recent works by Mansard et al. [START_REF] Mansard | A versatile generalized inverted kinematics implementation for collaborative working humanoid robots: The stack of tasks[END_REF], [START_REF] Mansard | A Unified Approach to Integrate Unilateral Constraints in the Stack of Tasks[END_REF], [START_REF] Park | Intermediate Desired Value Approach for Task Transition of Robots in Kinematic Control[END_REF]).

Let us now apply the previous result, by expressing the problem in the joint space, rather than in the operational space. The robot joint velocity is denoted q ∈ R j , with j the number of degrees of freedom. We assume that j ≥ k, so that s can be realized. If j > k, redundancy exists, and one can also minimize a scalar cost function h (q) ∈ R, while realizing the task s.

Each task is related to the joint velocity by: ṡm = J m (q, s m ) q,

where

J m (q, s m ) = ∂s m ∂q (22)
is the corresponding task Jacobian, of dimension k × j, that depends on both the robot configuration and on the task. By stacking the n tasks, and using (3), we obtain:

ṡ = S ṡ = SJ (q, s) q, ( 23
)
where:

J =    J 1 . . . J n    ∈ R kn × R j . ( 24
)
The multimodal controller that we propose, for driving s to s * is given by:

q = (SJ) † Λ (s * -s) + I -(SJ) † (SJ) h (25)
In the above equation:

• (SJ) † is the j × k right pseudoinverse of SJ.
We assume that SJ is full rank during operation, so that the pseudoinverse can be calculated. This was the case throughout the experiments and is a common assumption in inverse kinematics control [START_REF] Mansard | A Unified Approach to Integrate Unilateral Constraints in the Stack of Tasks[END_REF].

• Λ is a positive definite square diagonal matrix of dimension k that determines the convergence rate of s to s * ;

• the term h = ∂h ∂q (i.e., h = 0 when j = k) is introduced in order to minimize cost function h in case of redundancy. System [START_REF] Baeten | Integrated vision/force robotic servoing in the task space formalism[END_REF], controlled by [START_REF] Morel | Impedance based combination of visual and force control[END_REF], is globally asymptotically stable with respect to the k selected tasks. Indeed, plugging ( 25) into ( 23) yields:

ṡ = Λ (s * -s) . (26)
Thus, since Λ is a positive definite diagonal matrix, s = s * is a stable equilibrium for the closed-loop system. Also, note that minimization of h has no effect on the convergence rate of the task.

For constant gain matrix Λ, convergence of the task will be exponential according to [START_REF] Prats | Vision-tactile-force integration and robot physical interaction[END_REF]. Thus, since ( 25) is a proportional feedback controller, the joint velocities will also follow an exponential trend, an unwanted behaviour which may lead to abrupt velocity changes at task transitions (i.e., when the error suddenly increases). A simple solution to this, is the use, for each task, of an adaptive gain matrix, function of the task error s * -s, inspired by [START_REF] Saab | Dynamic Whole-Body Motion Generation under Rigid Contacts and other Unilateral Constraints[END_REF]:

Λ (s) = Λ * e -α||s * -s|| + β 1 -e -α||s * -s|| . (27)
In [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF], Λ * is the diagonal gain matrix applied when s is close to s * , and α ≥ 0 and β ∈ ]0, 1] are two scalar parameters such that, as the task error norm ||s * -s|| increases, Λ exponentially decreases (with slope dependent on α) to βΛ * , for very large task error. This exponential trend compensates that of the error signal, thus generating a less variable control input q, as will be shown by the experiments. The values of α, β, and Λ * are tuned empirically, so that the robot joint velocities stay roughly constant during operation.

In the next Section, we we first define the reference frames and the main variables of the framework and then, for each of three sensor-based tasks (position, vision and force), we give the expression of s and that of the corresponding Jacobian J.

Sensor-based tasks

Definitions

The reference frames used in our work are (see Fig. B is defined as:

B p A = B t A , B θu A ∈ SE (3)
, with B θu A the angle/axis vector [START_REF] Waldron | Kinematics[END_REF].

For the camera, we use the normalized perspective model. A 3D point with coordinates C X, C Y, C Z in the camera frame, projects in the image as a 2D point with coordinates:

x = C X C Z , y = C Y C Z . (28)
We assume that the pose of the camera in the end effector, E p C , is constant and known through a preliminary calibration step.

For human-robot collaboration, we use n = 3 tasks: positioning, visual, and force task. Each one has dimension k = 6, in order to control all 6 degrees of freedom of the end effector. We will hereby detail each task.

Positioning task

The objective of positioning is to control the end effector pose in the base frame. Hence, the positioning task is:

s p = B p E . (29)
This can be estimated at each iteration, by applying the robot forward kinematics to the measured articular variables, q.

For this task, the Jacobian in ( 21) is simply:

J p = ∂ B p E ∂q . (30)
This Jacobian can be computed, at run time, by applying the technique presented in [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF].

Visual task

The objective of the visual task, is to drive the end effector to a desired pose with respect to a visible target. To this end, we apply the two and onehalf-dimensional (2 1/2 D) visual servo paradigm originally introduced in [START_REF] Malis | 2-1/2D visual servoing[END_REF]. This method combines the advantages of image-based and position-based visual servoing schemes, while trying to avoid their shortcomings [START_REF] Chaumette | Visual servo control[END_REF]. In fact, the task is defined by a combination of image features and 3D characteristics:

s v = x y log C Z C * θu C . (31)
In this equation, x and y are the image coordinates of the target characterized by ( 28), C Z is the target depth in the camera frame, and C * θu C gives the relative rotation between the current and desired poses of the camera.

The Jacobian corresponding to the 2 1/2 D task is [START_REF] Malis | 2-1/2D visual servoing[END_REF]:

J v = L s C V B ∂ B p C ∂q . (32)
In this expression, L s is the interaction matrix relating the task evolution to the camera velocity in frame C:

L s = L 11 x, y, C Z L 12 (x, y) 0 L 22 C * θu C , (33)
while C V B is the spatial motion transform matrix from frame B to frame C:

C V B = C R B C t B × C R B 0 C R B . (34)
The complete expressions of L 11 , L 12 , and L 22 are given in [START_REF] Chaumette | Visual servo control[END_REF], and [t] × is the skew-symmetric matrix associated with vector t. Jacobian J v can be calculated at each iteration, since L s depends on s, C V B on the pose of B in C (determined via forward kinematics B p E plus constant known E T C ), and ∂ B p C /∂q can be calculated again using the technique presented in [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF].

Force task

The objective of force control is to regulate the external wrench h (force and torque vectors f and m), at the contact point between robot and human, to a desired value. This is essential to guarantee safe interaction with the environment and with the human operator. Without loss of generality, in this work, such external wrench is expressed in the end effector frame E.

To realize the force task, we apply an admittance controller [START_REF] Hogan | Impedance control: an approach to manipulation: parts I-III[END_REF], where the deviation of the end effector motion due to the interaction with the environment is related to the contact wrench, through an equivalent mass-spring-damper system with adjustable parameters.

Here, we consider a simple spring system, with null mass and damping, and positive definite diagonal square stiffness matrix K, such that:

E h E -E h * E = -K E p E -E p * E = K E p * E . (35)
Then, the force task is defined as: s f = E h E . Deriving the above equation yields the Jacobian corresponding to this task:

J f = -K ∂ E p E ∂q . (36)
Having defined K, J f can again be calculated with the technique from [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF].

A case study: collaborative screwing

Experimental setup and assumptions

To validate our controller, we focus on a case study, where a robot aids a human operator in a screwing operation. In Fig. 2, we show the setup, along with the frames defined in Sect. 3.1; the screw is denoted S.

Human and robot operate on the opposite sides of a flank, where a series of screws must be inserted. The required operations are respectively:

• for the human: to insert the screws in the holes,

• for the robot: to tighten a bolt on each of the inserted screws, while the human maintains it on the flank.

Since the focus here is mainly on our multimodal control framework, we do not implement the physical screwing action; instead, we consider a screw to be tightened, when the end effector touches it with proper alignment. To realize the collaborative screwing operation, we utilize a Kinect, that outputs an RGB-D image of the work scene from a fixed pose, and a black and white camera mounted on the robot. These sensors are respectively dedicated to detecting and tracking the human hand motion, and to tracking newly inserted screws on the flank. Finally, to properly align end effector and screw, an estimation of the external forces is necessary. This force estimation will be explained in Sect. 5.

Our work assumptions are that the flank is perpendicular to the Y axis of the base frame, at known distance from B, and that the Kinect pose in the base frame has been coarsely calibrated.

Hand approach

Pose-based screw approach

One untightened screw is detected The hand is not detected

One untightened screw is detected

The hand is detected

No untightened screws are visible

Halt Screw tightening

The untightened screw is near

The screw is tightened

Force-based screw approach

The untightened screw is not visible anymore To avoid luminosity variations in the image, we maintain the camera orientation with respect to the flank constant throughout operation. Since E p C is constant, we have decided to do this by keeping the end effector perpendicular to the flank, with the axes of frame E placed as in Fig. 2. Hence, we impose the desired rotation matrix from end effector to base to be:

B R * E =   -1 0 0 0 0 -1 0 -1 0   . (37)
In the rest of this Section, we will detail the strategy that has been used to realize collaborative screwing, with controller (25).

Multimodal control strategy

To realize the collaborative screwing task, we utilize four modes, and halting, which simply consists in setting q = 0. The Jacobian used in ( 25) is:

J =   J p J v J f   , (38)
with J p , J v , and J f defined respectively in ( 30), [START_REF] Prats | Reliable non-prehensile door opening through the combination of vision, tactile and force feedback[END_REF], and [START_REF] Saab | Dynamic Whole-Body Motion Generation under Rigid Contacts and other Unilateral Constraints[END_REF]. The modes are operated by the state machine in Fig. 3. As the figure shows, the transitions can be activated either by detection (red) or loss (black) of sensed information, or by success of the mode (blue). The detection/loss of information is determined by sensors processing. Instead, a mode is successful when: with w = [w 1 . . . w 6 ] ∈ R 6 a vector of positive weights, and σ a scalar threshold.

6 i=1 w i ||s * i -s i || < σ, (39)
X + -- + + X × X × X × X (SJ) + Λ X I -(SJ) + SJ X J= J p J v J f ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ S= S p S v S f ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∇h s * = s p * s v * s f * ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ s = s p s v s f ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ s * -s ! q S J s robot + sensors
Our complete framework is summarized in Fig. 4. In the rest of this section, we will focus on each of the four modes, by specifying the selection matrices S p , S v and S f , the desired task vector s * , and the activation condition (39).

Hand approaching mode

If the human operating hand is detected by the Kinect, its position is fed to a controller that moves the robot so that the camera has a good view of the area where the human is operating. Since only the positioning task ( 29) is necessary, the selection matrices are:

S p = I S v = S f = 0. ( 40
)
From ( 3) and ( 29), we can infer the desired task vector:

s * = S p s * p = B p * E ∈ SE(3). (41)
To derive s * p , we introduce the Kinect (K), and operating hand (H) frames (see Fig. 2). The origin of H is the orthogonal projection of the operator hand on the flank, while its orientation is that of B. The hand position in the kinect frame is estimated using OpenNI3 , and then orthogonally projected on the flank4 to obtain K X H , which can then be transformed to B X H in the base frame. Then, our aim is to place H at a fixed desired position in the camera frame:

C X * H = C X * H C Y * H C Z * H 1 , (42)
to increase the chances of visualizing in the image the future inserted screw. Also, since we set H R B = I, B R * E according to [START_REF] Waldron | Kinematics[END_REF], and since C R E is constant and known, the desired orientation of the camera with respect to the hand, H R * C can be derived. Combining C X * H and H R * C , we can obtain the desired camera to hand transformation, H T * C . This can now be used to determine B T * E , and therefore s * p , to be used in [START_REF] Morel | Impedance based combination of visual and force control[END_REF], along with:

S = I 0 0 . ( 43
)
Since the hand approaching mode should be activated and deactivated only by perceived data, the task convergence need not be monitored, and we can set σ = 0 and w = 0, so that ( 39) is never true in this mode.

Pose-based screw approaching mode

If a screw is detected in the image (see Fig. 2, bottom right), its position determines s v according to [START_REF] Ceriani | A constraint-based strategy for task-consistent safe human-robot interaction[END_REF], so that the end effector is driven in front of it. To this end, we exploit the screw position as viewed from the on-board camera (to infer x and y), along with the measures of the robot articular positions for forward kinematics (to infer C Z and C * θu C ). The details on the image processing algorithms used to detect and track the screws are given in [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF].

The shift from hand to screw approaching can lead to abrupt joint accelerations. This problem, which is common when switching between manipulation primitives, has been recently tackled using on-line trajectory generation [START_REF] Kröger | Robot motion control during abrupt switchings between manipulation primitives[END_REF]. Here, we exploit homotopy to better manage the transition. We define t > 0 the screw age (i.e., the time since it has been detected). The visual task selection matrix is then designed to smoothly vary from 0 to I, as t tends to a tuned scalar T :

S v = 1-cos(πt/T ) 2 I if t < T, I otherwise. ( 44
)
The other task selection matrices are set to:

S p = I -S v , S f = 0, (45)
so that, in controller (25):

S = I -S v S v 0 . (46)
In practice, the visual task is gradually activated by S v , while concurrently the hand position task is deactivated by S p .

As proved in Sect. 2, the advantage of our framework is that such a smooth transition can be easily implemented without compromising the controller stability, since, in contrast with hybrid sensors control, the selection matrices do not have to be binary. This is also a fundamental advantage with respect to the method proposed in [START_REF] Kröger | Robot motion control during abrupt switchings between manipulation primitives[END_REF].

From (3), we can infer the desired task vector:

s * = (I -S v ) s * p + S v s * v if t < T, s * v otherwise. ( 47
)
As mentioned, s * varies from a task dependent on both hand and screw, to purely vision-based screw approaching task s * v , that we define as:

s * v = x * S y * S log C Z * S 0 . (48)
This s * v corresponds to driving the screw to image position (x * S , y * S ) at desired depth C Z * S , while zeroing the orientation error between C and C * . Let us now explain how s * v is derived. The image position (x * S , y * S ) (circle in Fig. 2, bottom right) is set so that end effector and screw are aligned at the end of this mode. We set the end effector Cartesian position to have a desired translation with respect to the screw:

E t * S = 0 0 E Z * S , (49)
so that E Z * S > 0 is as small as possible, without end effector occlusion. Then, from the known C T E , and from E X * S , we can derive C X * S , and, from that:

C Z * S , x * S = C X * S / C Z * S , and
y * S = C Y * S / C Z * S .
For rotations, as usual we servo B R * E according to [START_REF] Waldron | Kinematics[END_REF]. Then, C * θu C can be calculated from known C R E and E R * B . The following mode (force-based screw approaching), is triggered when the visual error with respect to the screw is small enough. Hence, we set:

w = 0 ∀t > 0, σ = 0 if t < T, σ > 0 otherwise, (50)
so that (39) is verified only after time T , when the hand task is deactivated.

Force-based screw approaching mode

Once the screw is near enough, force control is activated, to make the end effector compliant in case of contact, while advancing. We activate this mode just before contact, because, in the absence of external contacts, the force signal to noise ratio can lead to inaccurate positioning.

The desired wrench on the end effector, in the end effector frame, is:

E h * E = 0 0 E f * E,Z 0 0 0 . (51)
Through force control ( 35), E f * E,Z < 0 makes the end effector progress forward. All other components are zeroed to make the end effector compliant.

As long as the screw is visible, the end effector can be driven towards it by using visual control. Then, the task selection matrices are:

S p = 0 S v = I 2 0 2×4 0 4×2 0 4 S f = 0 2 0 2×4 0 4×2 I 4 . (52)
Therefore, in controller (25):

S = 0 2×6 I 2 0 2×6 0 2×4 0 4×6 0 4×2 0 4×6 I 4 , (53)
and the desired task is:

s * = S v s * v + S f s * f = x * s y * s E f * Z 0 0 0 . ( 54
)
The transition from this mode to the following is triggered by the loss of the screw, when it is too near to be visible in the image. Then, we set σ = 0 and w = 0, so that (39) is never true.

Screw tightening mode

When the screw is so near that it is not visible any more, the last mode is activated. This relies solely on force control:

S p = 0 S v = 0 S f = I. (55)
Therefore, in controller ( 25):

S = 0 0 I , (56)
and the desired force task is:

s * = s * f = 0 0 E f * E,Z 0 0 0 . (57)
Clearly, if tightening was also to be realized (although this is not the case here, as mentioned in Sect. 4.1), the desired moment around Z, E m * E,Z should also be non-null. To verify that the screw is tightened, we check the force error according to [START_REF] Kröger | Robot motion control during abrupt switchings between manipulation primitives[END_REF], with tuned weights w = 0 and threshold σ > 0.

Experiments

To validate our framework, we have run a series of experiments with a lightweight KUKA LWR IV robot in the scenario illustrated in Fig. 2. Since a tightening tool is not mounted on the end effector, we have used a cylindrical tool of external diameter 14 mm to verify the precision of our method. The LWR is redundant with respect to the end effector operational space dimension (it has j = 7 degrees of freedom, whereas k = 6). Thus, we use the extra degree of freedom to guarantee joint limit avoidance. To this end, in [START_REF] Morel | Impedance based combination of visual and force control[END_REF], we use a scalar, configuration dependent, cost function [START_REF] Liegeois | Automatic supervisory control of configurations and behavior of multibody mechanisms[END_REF]:

h (q) = 1 2 7 i=1 q i -q i,mid q i,M -q i,m 2 , (58)
with [q i,m , q i,M ] the available range for joint i and q i,mid = (q i,M + q i,m ) /2 its midpoint. The values of q computed via (25) are fed to the Reflexxes online trajectory generation library5 for smoothing. To get the interaction wrench E h E , instead of mounting a force sensor on the end effector, we have decided to use the estimated external wrench signal provided by the robot controller through the FRI Interface6 . The camera mounted on the end effector is a Stingray F201B from Allied Vision Technologies, with resolution 1024 × 768 pixels, and we used the ViSP library [START_REF] Marchand | ViSP for visual servoing: a generic software platform with a wide class of robot control skills[END_REF] for visualization purposes. The image processing pipeline takes approximately 60 ms. Thus, although the skeleton processing on the Kinect is slightly faster, we fix the control loop rate at 15 Hz. To highlight the novel contributions of our framework, i.e., force control, and the use of homotopy and adaptive gains, various experiments were run. These are shown in the video attached to this paper.

In a preliminary experiment, (see Fig. 5), three screws are touched with the tip of the tool, using only the hand approaching (HA) and pose-based screw approaching (SA) modes. In this experiment, the homotopy between these two modes is deactivated, and the gain matrix Λ is independent from s. In Fig. 6, we have plotted the components of the error e = s * -s (top) and of the joint velocities q (bottom). The numbers correspond to the inserted screws (1 to 3). It is clear from the curves that the transitions between modes are abrupt in terms of q. This is because homotopy and adaptive gains are not used.

Let us now focus on the second, complete experiment (see Fig. 7), where all modes, as well as homotopy and adaptive gains, were applied. This time, we verify that the cylindrical tool successfully encircles new, non-tightened screws (see photo on the left of Fig. 7). A high accuracy is required, since the screw external diameter and tool internal diameters are respectively 5 and 9 mm. Although, the specifications are more strict than in the previous experiment, force control facilitates the insertion, by correcting slight orientation errors. Hence, we are confident that, with a tightening tool, the approach should also work. Convergence of the hand approaching and screw approaching modes has been discussed just above. Let us now focus on the final mode, when force control intervenes. In Fig. 8, we have E f E,Z and B Y E during this final mode.

The plots start at time t = 60 seconds, when the tool comes into contact with the flank. Correspondingly, E f E,Z decreases from the null value, until the desired value is reached (in (51), we set E f * E,Z = -25 N). Then, at t ≈ 65 s, the end effector stops (see bottom graph). After a few seconds, a user (see snapshots in Fig. 7 and video) moves by hand the robot last joint. The forces are detected (see E f E,Z in Fig. 8), and admittance control induces the small variations of B Y E . This experiment shows that the framework is capable of force stabilization, for safe human-robot interaction.

Finally, we ran experiments with and without the adaptive gains for Λ (experiments noted AG and FG), and with and without the smooth homotopy from hand to screw approaching (noted H and NH). To compare the experiments, in the beginning of screw approaching, we have only plotted the curves for these phases. The mode change is visible in the plots, and occurs after approximately 5 seconds. The joint velocity norm has been chosen, since it is a good indicator of the movement smoothness. In the first experiment (NH+FG, green curve), which reproduces the approach used in [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF], strong variations appear during the HA mode; these come from the variability of the positioning error s * p -s p , due to hand motion. Replacing the fixed gain with an adaptive one yields the cyan curve. As the curves show, the use of an adaptive gain, reduces the shaky motion. Another improvement is obtained by adding an homotopy of duration T = 1 second when the screw is seen. In this case (red curve), the transition from hand to screw tracking is much smoother. To further demonstrate the smooth transitions obtained thanks to the adaptive gains and homotopies, in Fig. 10, we have compared the joint velocity components obtained with NH+FG (left), and with H+AG (right). As the figure shows, the curves are much smoother, and less control effort is required, with the new approach. We have also compared the average values of | q| and |q| over each experiment. These, shown in Table 1, confirm the cited properties. For both metrics, AG outscores FG, by realizing the same operation in the same time with less velocity and acceleration. Also, as expected, homotopy reduces |q| (from 0.145 to 0.068 rad s -1 ) since it realizes a smoothing effect on the joint velocities, but has no influence on | q|. Reducing |q| is crucial for safe human-robot interaction, since most robot safety metrics (see [START_REF] Versace | A review of severity index[END_REF]), depend on accelerations measured at impacts.

In summary, the approach with both homotopy and adaptive gains (H+AG) should be selected. The advantages are numerous: less energy is required, the motion is smoother (facilitating image processing), and faster operation can be obtained. In fact, although, for the purpose of these comparisons, the gains were all tuned so that the duration of the experiments be the same, in other experiments we have fine tuned the gains of H+AG, to achieve screw approaching in 40 seconds, i.e., approximately 80% faster than in [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF].

Conclusions

In this paper, we have generalized the multimodal framework for humanrobot interaction originally introduced in [START_REF] Cherubini | Multimodal control for human-robot cooperation[END_REF]. The generalized framework can operate by activating or deactivating various tasks, according to the sensed data and to the needs of the application. This is of particular interest when numerous sensing devices are to be used for control, as is often the case in HRI. Typically, in this work, we have applied the framework to a collaborative screw tightening experiment, where vision, kinect, position and force data must be alternatively controlled. To avoid abrupt accelerations, important features such as adaptive gains and homotopy are included in the framework.

This preliminary work opens numerous avenues for future research. In the future, we plan to use our framework for full human-robot cooperation, with direct physical interaction. It would then be possible to verify its robustness to unknown dynamic parameters, resulting from the interaction with the human body (e.g., arm and hand).

Figure 1 :

 1 Figure 1: Reference frames used in our multimodal framework for human-robot interaction.

Figure 2 :

 2 Figure 2: Collaborative screwing case study. Top: experimental setup. Bottom left: view of the camera and end effector. Bottom center: Kinect image. Bottom right: camera image.

Figure 3 :

 3 Figure 3: Collaborative screwing state machine, selecting the appropriate control mode, according to the sensed data.

Figure 4 :

 4 Figure 4: Multimodal framework block diagram.

Figure 5 :

 5 Figure 5: Six consecutive snapshots of the first experiment of collaborative screwing.

Figure 6 :

 6 Figure 6: The six components of the error e = s * -s (top) and the seven components of the joint velocities q (bottom) during the first experiment.

Figure 7 :

 7 Figure 7: Left: the tightening task. Right: consecutive snapshots of the second experiment.

Fig. 9 ,Figure 8 :

 98 Figure 8: Evolution of E f E,Z (top) and B Y E (bottom) over time, during force-based screw approach and tightening.

Figure 9 :

 9 Figure 9: Evolution of | q| over time, during hand and screw approaching, using NH+FG (green), NH+AG (cyan), and H+AG (red).

Figure 10 :

 10 Figure 10: Evolution of q components over time, during hand and screw approaching, using NH+FG (left), and H+AG (right).

Table 1 :

 1 Comparison between | q| and |q| in the four configurations. configuration FG AG NH H | q| (rad s -1 ) 0.091 0.067 0.079 0.079 |q| (rad s -2 ) 0.132 0.082 0.145 0.068

Otherwise, specific strategies for avoiding singularities, which are out of the scope of this paper, are to be devised.

Throughout the paper, we refer to controllers as optimal when they provide the least squares solution to the task, i.e., they minimize the control effort.

http://www.openni.org

It is trivial to derive the flank plane equation in K from B T K and B Y H .

www.reflexxes.com

http://cs.stanford.edu/people/tkr/fri/html/

Acknowledgements

Work supported by the ANR (French National Agency) ICARO project.


Alice Combernoux

Frédéric Pascal

Guillaume Ginolhac

Marc Lesturgie

ASYMPTOTIC PERFORMANCE OF THE LOW RANK ADAPTIVE NORMALIZED MATCHED FILTER IN A LARGE DIMENSIONAL REGIME

Keywords: Low rank detection, Random matrix theory, Asymptotic distribution, Adaptive Normalized Matched Filter

The paper addresses the problem of approximating the detector distribution used in target detection embedded in a disturbance composed of a low rank Gaussian noise and a white Gaussian noise. In this context, it is interesting to use an adaptive version of the Low Rank Normalized Matched Filter (LR-ANMF) detector, which is a function of the estimated projector onto the low rank noise subspace. We will show that the traditional approximation of the LR-ANMF detector distribution is not always the better one. In this paper, we propose to perform its limits when the number of secondary data K and the data dimension m both tend to infinity at the same rate m/K → c ∈ (0, ∞). Then, we give the theoretical distributions of these limits in the large dimensional regime and approximate the LR-ANMF detector distribution by them. The comparison of empirical and theoretical distributions on a jamming application shows the interest of our approach.

INTRODUCTION

In the context of target detection in a noise composed of a low rank (LR) Gaussian noise and an AWGN (Additive White Gaussian Noise), one could use the Low Rank Normalized Matched Filter (LR-NMF) detector [START_REF] Rangaswamy | Robust adaptive signal processing methods for heterogeneous radar clutter scenarios[END_REF] in order to exploit this low rank structure. Indeed, although its full rank version (NMF detector [START_REF] Scharf | Matched subspace detectors[END_REF]) depends on the covariance matrix, the LR-NMF detector only requires the projector onto the LR noise subspace.

However, in practice, this projector and the covariance matrix are unknown and it is necessary to estimate them using K secondary data which share the same properties as the tested data. Traditionally, the projector is estimated using the estimated covariance matrix: the Sample Covariance Matrix (SCM). Then, the adaptive detector is obtained substituting the true projector by the estimated one in the LR-NMF detector. The obtained detector is then named LR-ANMF (LR Adaptive NMF) detector. One also could note that it needs much less secondary data as its classical version (ANMF detector [START_REF] Scharf | Adaptive matched subspace detector and adaptive coherence[END_REF]) for equivalent performances in terms of filtering [START_REF] Kirstein | Adaptive detection using a low rank approximation to a data matrix[END_REF][START_REF] Haimovich | Asymptotic distribution of the conditional signal-to-noise ratio in an eigenanalysis-based adaptive array[END_REF], which is useful in practice.

Hence, engineers should want to calibrate their detector with an automatic tuning of the threshold in order to obtain a certain false alarm probability. In order to do this, they need the knowledge of the distribution of the employed detector, here the LR-ANMF detector. Unfortunately, its distribution is not known. That is why, we classically approximate it by the well-known distribution of the LR-NMF detector [START_REF] Rangaswamy | Robust adaptive signal processing methods for heterogeneous radar clutter scenarios[END_REF].

Nevertheless, depending on the data dimension m and the number of secondary data K, this approximation could not be always the most appropriated. However, using the Random Matrix Theory (RMT), one could better approximate the distribution of the LR-ANMF detector. Indeed, it is composed of 3 quadratic forms and, under the large dimensional regime (i.e. m, K → ∞ at the same rate m/K → c ∈ (0, +∞)), their limits differ and depend on c and on the SCM model. Thus, we propose to study the LR-ANMF detector convergences when m, K → ∞ at the same ratio under the two main data/SCM models considered in RMT. The first model is a basic model proposed by [START_REF] Mestre | Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates[END_REF][START_REF] Mestre | Modified subspace algorithms for DoA estimation with large arrays[END_REF] where the multiplicity of the covariance matrix's eigenvalues grows with the dimension m at a fixed ratio. The second data model, the spiked model introduced by [START_REF] Johnstone | On the distribution of the largest principal component[END_REF] considers that the eigenvalues multiplicity corresponding to the low rank signal is fixed at the same value even if m increases and leads to a deterministic limit of the quadratic forms [START_REF] Couillet | Fluctuations of spiked random matrix models and failure diagnosis in sensor networks[END_REF]. Then, the distribution of the limits in the large dimensional regime will be established in an intermediate form which will be finalized in a forthcoming paper. They will allow to more precisely approximate the LR-ANMF detector distribution.

As an illustration of the interest of the proposed approximations, the jamming application is studied. The purpose of this application is to detect a target despite jammers which emit energy in order to disrupt the target signal. In this case, the LR Gaussian noise is composed of the different jammers. The simulations will compare the empirical distributions of the LR-NMF, the LR-ANMF and their RMT limits as well as their theoretical distributions.

The paper is organized as follows: Section 2 presents the problem statement and the definition of the LR-NMF and the LR-ANMF detectors. Section 3 contains the main theoretical contributions of this work, i.e. the limits of the LR-ANMF detector in the large dimensional regime for two models of covariance matrix and their theoretical distributions. Finally, Section 4 shows the jamming application which illustrates the obtained results.

Notations: An italic letter stands for a scalar quantity, boldface lowercase (uppercase) characters stand for vectors (matrices) and (.) H stands for the conjugate transpose. IN is the N × N identity matrix. diag(.) denotes the diagonalization operator such as (A)i,i = (diag(a))i,i = (a)i and equal to zero otherwise. [[a, b]] is the set defined by x ∈ Z : a x b, ∀(a, b) ∈ Z 2 . X ⊥ Y for the random variables X and Y stands for independence between them. The abbreviation iid stems for independent and identically distributed.

LOW RANK DETECTION

Problem formulation

In an observation vector x ∈ C m×1 , the problem consists in detecting a complex signal d corrupted by an additive noise c + n. This detection problem can be formulated as the following binary hypothesis test:

H0 : x = c + n x k = c k + n k , k ∈ [[1, K]] H1 : x = d + c + n x k = c k + n k , k ∈ [[1, K]] (1)
where the target response d = αa(Θ d ), α is the unknown amplitude and a(Θ d ) is the target localization steering vector. The x k 's∈ C m×1 correspond to the K iid signal-free data (secondary data) used for the noise parameters estimation. This noise is composed of an AWGN,

n k ∈ C m×1 (resp. n) ∼ CN (0, σ 2
to R = m i=1 λiuiu H i with λ1 = γ1+σ 2 > • • • > λr = γr+σ 2 > λr+1 = • • • = λm = σ 2 .
Then, we define the projector onto the LR Gaussian noise subspace Πc and the projector onto the subspace orthogonal to the LR Gaussian noise subspace Π ⊥ c :

Πc = r i=1 uiu H i Π ⊥ c = Im -Πc = m i=r+1 uiu H i (2)

LR-NMF and LR-ANMF detectors

A filtering preprocessing is applied on the observation vector x in order to remove the LR Gaussian noise. This leads to the following standard signal detection problem:

H0 : r = U H 0 x = n0 H1 : r = U H 0 x = d0 + n0 (3)
where

U0 = [ur+1, • • • , um].
Solving this detection problem and considering the white noise power of n0 as unknown leads to the LR-NMF [START_REF] Rangaswamy | Robust adaptive signal processing methods for heterogeneous radar clutter scenarios[END_REF] detector:

Λ(Θ) = |a(Θ) H Π ⊥ c x| 2 (a(Θ) H Π ⊥ c a(Θ))(x H Π ⊥ c x) H 1 ≷ H 0 δ (4)
where δ stands for the detection threshold. However, in practice, the covariance matrix R and the projector Π ⊥ c are unknown and have to be estimated. Classically, the Sample Covariance Matrix (SCM) is computed from the K secondary data and can be written as

RSCM = 1 K K k=1 x k x H k = m i=1 λi ûi ûH i
where λi and ûi are the SCM eigenvalues and eigenvectors respectively. Finally, the estimated projectors are:

Πc,SCM = r i=1 ûi ûH i Π⊥ c,SCM = Im -Πc,SCM = m i=r+1 ûi ûH i , (5)
leading to the LR-ANMF detector, written as:

Λ(Θ) = |a(Θ) H Π⊥ c,SCM x| 2 (a(Θ) H Π⊥ c,SCM a(Θ))(x H Π⊥ c,SCM x) H 1 ≷ H 0 ξ (6)
where ξ stands for the detection threshold.

STATISTICAL ANALYSIS OF THE LR-ANMF DETECTOR

This section contains the main theoretical contributions of this work. First, the limits of the LR-ANMF detector are established for different covariance matrix models in the large dimensional regime. Then, the corresponding asymptotic distributions are derived.

Convergences

It is well known that the LR-ANMF is consistent for a fixed m when K → ∞, i.e. it converges to the LR-NMF. However, it has been proved in [START_REF] Combernoux | Random matrix theory applied to low rank STAP detection[END_REF], using RMT techniques, that the LR-ANMF converges to another undetailed quantity when both m, K → ∞ with m/K → c. This quantity depends on the covariance matrix model (and of course on c). This paper focuses on the two most prevalent models: the model proposed by [START_REF] Mestre | Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates[END_REF][START_REF] Mestre | Modified subspace algorithms for DoA estimation with large arrays[END_REF] (named GMUSIC model in the remainder of the paper) and the spiked model [START_REF] Johnstone | On the distribution of the largest principal component[END_REF][START_REF] Couillet | Fluctuations of spiked random matrix models and failure diagnosis in sensor networks[END_REF][START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF].

The GMUSIC model considers that the multiplicity of the eigenvalues of R increases with m at a fixed ratio. On the other hand, since our problem consists in a LR Gaussian noise and a white Gaussian noise, the spiked model better fits with our data as it considers the signal of interest as a sum of r fixed rank perturbations, e.g. there are r eigenvalues of fixed multiplicity as m increases. To prove the convergences, we assume the following standard hypotheses.

(As1) R has uniformly bounded spectral norm ∀m ∈ N * , i.e. ∀i ∈ [[1, m]], λi < ∞.
(As2) The vectors s1, s2 ∈ C m×1 used in the quadratic forms (here a(Θ) and x) have uniformly bounded Euclidean norm ∀m ∈ N * . In practice, they will be fixed to 1.

(As3) Let Y ∈ C m×K having iid entries yij ∼ CN (0, 1), absolutely continuous and with E[|yij| 8 ] < ∞.

(As4) Let Y ∈ C m×K defined as in (As3). Then, its distribution is invariant by left multiplication by a deterministic unitary matrix. Moreover, the eigenvalues empirical distribution function of 1 K YY H almost surely (a.s.) converges to the Marcenko-Pastur distribution [START_REF] Marcenko | Distributions of eigenvalues for some set of random matrices[END_REF] with support

[(1 - √ c) 2 , (1 + √ c) 2 ].
(As5) The maximum (resp. minimum) eigenvalue of 1 K YY H tends a.s. to

(1 + √ c) 2 (resp. to (1 - √ c) 2 ).
As a comment, Y can be related to RSCM with the relation

RSCM = R 1/2 1 K YY H R 1/2
. Finally, the last assumption, namely the separation condition depends on the model of the SCM: (As6-GMUSIC) 1/c is lower bounded (see [START_REF] Mestre | Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates[END_REF]).

(As6-SPIKED) The eigenvalues of R satisfy |λi -1| > √ c, ∀i ∈ [[1, r]].
Under (As1-As5, As6-GMUSIC), [START_REF] Mestre | Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates[END_REF] proved that:

s H 1 Π⊥ c,SCM s2 a.s. -→ m,K→∞ m/K→c<∞ s H 1 Π⊥ c,G s2 (7)
with Π⊥ c,G = m i=1 w G i uiu H i and

w G i =    1 - 1 m-r-1 r n=1 σ 2 λn-σ 2 -µm λn-µm if i > r σ 2 λ i -σ 2 -µm λ i -µm else (8) where µ1 • • • µm are the eigenvalues of diag(λ)-c m √ λ √ λ T , λ = [λ1, • • • , λm]
T and σ 2 is the white noise power.

Under (As1-As5, As6-SPIKED), [START_REF] Couillet | Fluctuations of spiked random matrix models and failure diagnosis in sensor networks[END_REF] proved that

s H 1 Π⊥ c,SCM s2 a.s. -→ m,K→∞ m/K→c<∞ s H 1 Π⊥ c,S s2 (9)
with Π⊥ c,S = m i=1 w S i uiu H i and

w S i =    1 if i > r 1 - 1 -c(λi -1) -2 1 + c(λi -1) -1 else (10)
Thus, using the Slutsky's theorem and Eqs.( 7) and ( 9), one obtains the limits of the LR-ANMF when m, K → ∞ with m/K → c: GMUSIC model

Λ(Θ) P -→ m,K→∞ m/K→c<∞ ΛG (Θ) = |a(Θ) H Π⊥ c,G x| 2 (a(Θ) H Π⊥ c,G a(Θ))(x H Π⊥ c,G x) (11)
Spiked model

Λ(Θ) P -→ m,K→∞ m/K→c<∞ ΛS (Θ) = |a(Θ) H Π⊥ c,S x| 2 (a(Θ) H Π⊥ c,S a(Θ))(x H Π⊥ c,S x) (12)
Notice that, since the convergence occurs for a given model and when m → ∞, these limits are expected to be better approximations of the LR-ANMF than the LR-NMF. This will be highlighted in the simulations section.

Asymptotic distributions

This section is devoted to the derivation of the theoretical distributions of the LR-ANMF limits, under H0, permitting to obtain the threshold for a fixed false alarm probability. These distributions will allow to have a better approximation of the LR-ANMF distribution than the classical one given by the LR-NMF distribution [START_REF] Rangaswamy | Robust adaptive signal processing methods for heterogeneous radar clutter scenarios[END_REF]. This is of utmost interest for instance, to obtain a better false alarm regulation. This is the purpose of the following theorem.

Theorem 1. Let w (resp. Λ(Θ)) stands for w G or w S (resp. ΛG (Θ) or ΛS (Θ)) according to the underlying model (GMUSIC or spiked). Then, under H0, (As1-As5) and the separation condition (As6), Λ(Θ) shares the distribution of the following quantity:

m i=1 αiXi + 2 m i,j=1 i<j βij ( (zij)Yij + (zij)Zij) C m i=1 γiXi (13)
where

     Xi iid ∼ χ 2 i (2) Yij ∼ Laplace Y ij (0, 1/2) Zij ∼ Laplace Z ij (0, 1/2) (14)
and, if we define

U = [u1, • • • , um],              αi = 1 2 w 2 i λi|a(Θ) H ui| 2 βij = wiwj λiλj zij = u H i a(Θ)a(Θ) H uj γi = wiλi C = 1 2 ||diag([w1, • • • , wm])U H a(Θ)|| 2 2 (15)
Moreover, Yij (resp. Zij) is independent of Y i j (resp. Z i j ) for {i, j} = {i , j } and Yij and Xi, Zij and Xi, and Yij and Zij are uncorrelated (but not independent) for all i, j.

Proof. Due to the lack of space, the detailed proof is omitted and will appear in a forthcoming paper. However, the proof outline is as follows. First, some rotations are applied on the different quadratic forms appearing in Λ(Θ) in order to obtain functions of standard complex Gaussian random variables. Then, after some algebraic manipulations, Λ(Θ) can be reduced to the form given by Eq.( 13).

Observe that the distribution of Eq.( 13) provides a distribution completely independent of the observation vectors x and x k 's. Although this relationship is exploited in the paper in order to better approximate the distribution of the LR-ANMF, this result can be interpreted as follows. An immediate consequence of Theorem 1 is to automatically and theoretically regulate the false alarm probability or equivalently the threshold ξ (Eq.( 6)) as a function of Θ and the ratio c (abusively equal to m/K) which is not the case with the LR-NMF distribution.

SIMULATIONS

In this section, the limits of the LR-ANMF and the corresponding distributions under H0 are analysed through Monte-Carlo simulations.

Parameters

To illustrate the interest of these convergences, the jamming application is chosen. The main purpose of this application is to detect a target thanks to a uniform linear antenna composed of m sensors despite the presence of jamming. The response of the jamming, c is composed of signals similar to the target response. In this section, except for the convergences when m, K → ∞ at the same rate c, we choose m = 100 in order to have a large number for the data dimension. Even if, in practice, this number could not be always realistic, it allows to highlight the interest of RMT where matrices are built from high dimensional observations. Moreover, such a value can be realistic in some applications as STAP [START_REF] Ward | Space-time adaptive processing for airborne radar[END_REF].

Let us denote Θ = θ where θ is the AoA (Angle of Arrival). As the limits are studied under the H0 hypothesis, d = 0m×1. The jamming is composed of three synthetic targets with AoA -20 • , 0 • and 20 • . Thus, the jamming (low rank Gaussian noise) has a rank equal to r = 3. Then, the signal wavelength is l0 = 0.667m, the AWGN n power is σ 2 = 1 and the Jamming to Noise Ratio is JN R dB = 10dB. Finally, the theoretical covariance matrix of the total noise (jamming plus AWGN) is settled to

R = JN R × Ucdiag([6, 2, 1])U H c + σ 2 Im (16) with Uc = [u1 • • • ur]

Empirical cumulative density functions

We will first observe the empirical cumulative density function (ECDF) of Λ and its different limits: Λ (Eq.( 4)), ΛG (Eq.( 11)) and ΛS (Eq.( 12)) under the hypothesis H0. Then, the distributions of ΛG and ΛS (Eq.( 13)) will be plotted and compared to the ECDFs to validate the results provided in Theorem 1 as well as to illustrate the interest of using the proposed limits instead of the classical LR-NMF. In order to observe the pertinence of the convergences and the distributions, the quantities are plotted for two different values of K (K = 2r and K = 2m) and two different values of θ (θ = 20 • i.e. exactly on the jamming angle and θ = 25 • i.e. far from the jamming). On all the figures, the red color (resp. the blue color) corresponds to the case where K = 2r (resp. K = 2m) for the different studied scenarios while the thick black curves correspond to the LR-NMF case (which does not depend on K).

We are first interested in Fig. 1 where the ECDFs of Λ (triangles), Λ, ΛG (stars) and ΛS (circles) are plotted for θ = 20 • at the top and θ = 25 • at the bottom. Thus we observe that, despite the ECDF of the LR-NMF (Λ), the other ECDFs vary as a function of K, especially for θ = 20 • . We also note that ΛS (circles) is always close to Λ (triangles) contrary to ΛG (stars) for K small and θ = 20 • , and Λ (thick black line) for θ = 20 • . As a consequence, the LR-NMF detector is not a good approximation of the LR-ANMF detector and the spiked model seems to be the right model to use, i.e. ΛS is robust to the number K and the value θ. Finally, one could remark that when we are far from the jamming, all the curves are almost superimposed. The interest of our study is consequently mainly located when we are on or close to the jamming. However, the limits still remain useful far from the jamming. We then observe Fig. 2 where we compare the ECDFs of the LR-ANMF limits ΛG (triangles) and ΛS (circles) to their distributions (resp. stars and squares) from theorem 1 for θ = 20 • and K = 2r. One could note that the ECDFs and their associated distributions are in perfect agreement which validate our theorem. Henceforth, the visualization of the ECDFs, dependent of the observation vector, is useless.

Thus, we finally observe Fig. 3 where the distributions of the RMT limits of Λ, ΛG and ΛS , are plotted at the top and at the bottom respectively. For visual readability reasons, the ΛG distribution for θ = 20 • and K = 2r is not plotted in Fig. 3 but can be observed in Fig. 2. Then, the observations concerning the dependence of ΛG and ΛS on the ratio c (abusively equal to m/K) are the same as those previously exposed with the ECDFs in Fig. 1.

CONCLUSION

In this paper, we proposed to study the asymptotic performances of the LR-ANMF detector. That is why, using RMT tools, we performed its convergences and their theoretical distributions when both the number of secondary data K and the data dimension m tend to infinity at the same rate with 2 models of covariance matrix: the GMUSIC model and the spiked model. The obtained distributions allow to approximate the LR-ANMF distribution. Thus, we observed that the spiked model better describes the performances of the LR-ANMF detector as a function of K than the LR-NMF detector and the GMUSIC model and is robust to the variations of the AoA and K. Finally, this paper allowed to conclude that the spiked model has to be used in a such application and that RMT tools are highly useful to obtain detector performance as a function of K or c.

In the next future, we would like to determine the theoretical distribution of the limit of the LR-ANMF detector without contribution of several random variables when both m, K → ∞ at the same rate c for the different models of covariance matrix exposed in this paper. It would allows us to theoretically determine the distribution of the false alarm probability as a function of the threshold. We will also be interested in the derivation of the detection probability.

  Im) and a LR Gaussian noise c k ∈ C m×1 (resp. c), modeled by a zero-mean complex Gaussian vector with a normalized covariance matrix C (tr(C) = m), i.e. c k ∼ CN (0, C). Consequently, the covariance matrix of the secondary data is R = C + σ 2 Im ∈ C m×m . Moreover, considering a LR Gaussian noise, one has rank (C) = r m and hence, the eigendecomposition of C is C = r i=1 γiuiu H i where γi and ui are the non-zero eigenvalues and the associated eigenvectors of C respectively and unknown in practice. This leads

Fig. 1 .

 1 Fig. 1. Empirical cumulative density function of the limits of the LR-ANMF: (up) θ = 20 • , (down) θ = 25 • with JN R dB = 10dB over 10 4 realisations.

Fig. 2 .

 2 Fig. 2. Comparison between the ECDFs of the LR-ANMF RMT limits and their theoretical distribution for θ = 20 • and K = 2r over 10 4 realisations.

Fig. 3 .

 3 Fig. 3. CDF of the distributions of the limits of the LR-ANMF: ΛS (up) and ΛG (down) with JN R dB = 10dB over 10 4 realisations.