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A homography-based dynamic control approach applied to station keeping of autonomous underwater vehicles without linear velocity measurements

Lam-Hung Nguyen, Minh-Duc Hua, Guillaume Allibert, Tarek Hamel

To cite this version:

Lam-Hung Nguyen, Minh-Duc Hua, Guillaume Allibert, Tarek Hamel. A homography-based dynamic control approach applied to station keeping of autonomous underwater vehicles without linear velocity measurements. IEEE Transactions on Control Systems Technology, 2021, 9 [START_REF] Chaumette | Visual servo control, Part II: Advanced approaches[END_REF]. hal-02069350v2

A homography-based dynamic control approach applied to station keeping of autonomous underwater vehicles without linear velocity measurements Lam-Hung Nguyen, Minh-Duc Hua, Guillaume Allibert and Tarek Hamel

Abstract-A homography-based dynamic control approach applied to station keeping of Autonomous Underwater Vehicles (AUVs) without relying on linear velocity measurements is proposed. The homography estimated from images of a planar target scene captured by a downward-looking camera is directly used as feedback information. The full dynamics of the AUV are exploited in a hierarchical control design with inner-outer loop architecture. Enhanced by integral compensation actions and disturbance torque estimation, the proposed controller is robust with respect to model uncertainties and unknown currents. The performance of the proposed control approach is illustrated via both comparative simulation results conducted on a realistic AUV model and experimental validations on an in-house AUV.

Index Terms-Autonomous Underwater Vehicle, visual servoing, homography-based control, station keeping, nonlinear control

I. INTRODUCTION

Safe and efficient navigation of Autonomous Underwater Vehicles (AUVs) in cluttered environments remains a challenging task. Scientific issues are particularly related to the fact that global acoustic positioning systems become unusable or insufficiently precise in such situations, leading to an obvious interest in developing advanced sensor-based control strategies for AUV applications in close proximity to a complex sea bottom or submarine structures. While acoustic systems have been widely used for sensing underwater environments, cameras offer an appealing alternative due to the rich information captured by images and their high update rate. By using vision sensors as a sensor modality for relative (scaled) position and orientation, the control problems can be cast into Position-Based Visual Servoing (PBVS) or Image-Based Visual Servoing (IBVS) [START_REF] Chaumette | Visual servo control, Part II: Advanced approaches[END_REF]. Classical visual servo control techniques have been initially developed for robotic manipulators and mobile ground vehicles [START_REF] Hutchinson | A tutorial on visual servo control[END_REF], [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] and then for aerial drones [START_REF] De Plinval | Stabilization of a class of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF], [START_REF] Gonc ¸alves | Vision-based autonomous approach and landing for an aircraft using a direct visual tracking method[END_REF], [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF]. In underwater robotics, vision sensors have been used to perform station keeping or positioning [START_REF] Cufi | An approach to vision-based station keeping for an unmanned underwater vehicle[END_REF], [START_REF] Garcia | Positioning an underwater vehicle through image mosaicking[END_REF], [START_REF] Lots | A 2-D visual servoing for underwater vehicle station keeping[END_REF], [START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF], docking [START_REF] Brignone | A fully autonomous docking strategy for intervention AUVs[END_REF], [START_REF] Lwin | Visual docking against bubble noise with 3-D perception using dual-eye cameras[END_REF], [START_REF] Maire | A vision based target detection system for docking of an autonomous underwater vehicle[END_REF], [START_REF] Wirtz | Design and test of a robust docking system for hovering AUVs[END_REF], and pipeline following [START_REF] Allibert | Pipeline following by visual servoing for Autonomous Underwater Vehicles[END_REF], [START_REF] Antich | Development of the control architecture of an underwater cable tracker[END_REF], [START_REF] Matsumoto | Real-time vision-based tracking of submarinecables for AUV/ROV[END_REF], [START_REF] Rives | Underwater pipe inspection task using visual servoing techniques[END_REF], etc.

Both stereo and monocular cameras have been exploited for stabilization and positioning of AUVs. When the vehicle's

The authors are with I3S laboratory (Laboratoire d'Informatique, Signaux et Systèmes de Sophia-Antipolis), CNRS, Université Côte d'Azur, UMR7271, Sophia Antipolis, France. T. Hamel is also with Institut Universitaire de France. Emails: lhnguyen(hua,allibert,thamel)@i3s.unice.fr. pose (i.e. position and orientation) can be estimated, existing PBVS controllers can be directly applied [START_REF] Palomeras | Vision-based localization and mapping system for AUV intervention[END_REF]. In contrast, the case of monocular vision without the assumption of planarity of the visual target and the prior knowledge of its geometry is more involved since full pose reconstruction from visual data is not possible. However, monocular vision can be sufficient to achieve stabilization of an AUV in front of a planar target [START_REF] Brignone | A fully autonomous docking strategy for intervention AUVs[END_REF], [START_REF] Lots | A 2-D visual servoing for underwater vehicle station keeping[END_REF], [START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF]. Recently, an advanced kinematic IBVS control scheme was proposed in [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF] by exploiting the socalled homography that is an invertible mapping relating two camera views of the same planar scene by encoding in a single matrix the camera pose, the distance between the camera and the scene, and the normal direction to the scene [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF], [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. A remarkable feature of that approach is the non-requirement of homography decomposition often computationally expensive (see e.g. [START_REF] Malis | Deeper understanding of the homography decomposition for vision-based control[END_REF]), as opposed to other homography-based visual servo (HBVS) controllers [START_REF] Brignone | A fully autonomous docking strategy for intervention AUVs[END_REF], [START_REF] Malis | 2 1/2D visual servoing[END_REF]. More recently, this kinematic HBVS control approach has been extended in our prior work [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] in order to account for the full dynamics of fully-actuated AUVs and to obtain an enlarged provable domain of stability. The present paper continues to extend our work [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] to the case where linear velocity measurements are unavailable. We restrict, however, our study to the case of a downward-looking camera. One of the main motivations behind this effort is related to the development of a low-cost but efficient solution for station keeping of AUVs without the need of a costly Doppler Velocity Log (DVL) velocity sensor. More precisely, the proposed solution makes use of a minimal and inexpensive sensor suite consisting of an Inertial Measurement Unit (IMU) and an embedded video camera whose total cost can be less than 150C. A similar problem concerning underactuated aerial drones addressed in [START_REF] De Plinval | Stabilization of a class of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF] is in line with our effort in dealing with system's dynamics and in depleting the need of a linear velocity sensor. However, the approach proposed in [START_REF] De Plinval | Stabilization of a class of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF] relies on the assumption that the visual velocity is available for control design, for instance, via the use of a high-gain observer, but a complete stability analysis including such high-gain observer is missing.

The paper is structured into seven sections including the introduction and concluding sections. Notation and problem formulation of HBVS control of fully-actuated AUVs are provided in Section II. Section III presents a recall on system modeling together with a simplified model for control design. In Section IV a novel inertial-aided homography-based dynamic control approach applied to station keeping of fullyactuated AUVs without relying on linear velocity measurements is proposed. Comparative simulation results conducted on a realistic AUV model are reported in Section V, showing the performance and robustness of the proposed approach. Section VI first describes some practical development aspects of the experimental platform then reports extensive experimental validation results in a real environment. Three video links are also provided showing these experimental results.

A primary version of this work has been presented at a conference [START_REF] Nguyen | Inertialaided homography-based visual servo control of autonomous underwater vehicles without linear velocity measurements[END_REF]. Several extensions of [START_REF] Nguyen | Inertialaided homography-based visual servo control of autonomous underwater vehicles without linear velocity measurements[END_REF] are proposed in this paper. For instance, the restrictive assumption of alignment between the AUV's center of buoyancy and center of mass and the camera optical center used in [START_REF] Nguyen | Inertialaided homography-based visual servo control of autonomous underwater vehicles without linear velocity measurements[END_REF] is no longer necessary, allowing one to accommodate any arbitrary camera location on the AUV. The inner-loop controller has been modified by using a bounded nested saturation integrator of the angular velocity error instead of a classical integrator that is prone to the well-known phenomenon of integrator wind-up. Finally, the comparative simulation study and extensive experimental validations on an in-house AUV of this paper have been newly developed.

II. NOTATION AND PROBLEM FORMULATION

A. Notation

P4 P3 P2 P1 d n A B C O B C r C - → e a 3 - → e a 2 - → e a 1 G - → e b 3 r G - → e b 1 - → e b 2 (R, p C ) (R, p)
Fig. 1: An AUV with a downward-looking camera and notation

The AUV is modeled as a rigid body immersed in water. The following notation is used throughout the paper (see Fig. 1).

• Let m and J 0 denote respectively the vehicle's mass and inertia matrix, G and B its center of mass (CoM) and center of buoyancy (CoB). The distance between G and B is denoted as l. The gravity constant is denoted as g.

• The considered inertial frame is denoted as • Let Ω = [ω 1 , ω 2 , ω 3 ] ∈ R 3 denotes the angular velocity of B with respect to A, expressed in B. By denoting V ∈ R 3 and V C ∈ R 3 respectively the vector of coordinates expressed in B of the linear velocity of B and C, one has V = V C -Ω×r C .

A = {O; - → e a 1 , - → e a 2 ,
• The vectors of coordinates of the current velocity expressed in A and B are denoted as v f and V f , respectively. Then the CoB's velocity relative to the current expressed in B is denoted as

V h V -V f . Assume that v f is constant. • Let {e 1 , e 2 ,

B. Problem formulation

Assume that the AUV, equipped with a downward-looking monocular camera, operates sufficiently close to a (near) planar textured seafloor. To perform the station-keeping task, a reference image of the seafloor is first taken at a desired pose. Then, the visual servoing controller must stabilize the AUV about the desired pose by exploiting information encoded in both the current and reference images.

Let us define the inertial frame A attached to the camera's desired pose as depicted in Fig. 1. Assume that the estimation of the so-called homography H ∈ R 3×3 between the current image and the reference image is available for control design. Such a homography matrix contains coupled information about the rotation and translation between the current camera frame C and the inertial A as follows [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF]

H = R - 1 d R p C n (1)
where n ∈ S 2 is the unit normal vector pointing toward the target plane with coordinates expressed in A; and d is the distance between the camera optical center at the desired pose and the target plane (see Fig. 1). The derivative of H satisfies

Ḣ = -Ω × H - 1 d V C n (2)
In addition to the estimate of H, (approximate) measurements of the angular velocity Ω and of the gravity direction R e 3 provided by an IMU are also available for control design.

The considered control objective consists in asymptotically stabilize H about I 3 , which is equivalent to the stabilization of (R, p C ) about (I 3 , 0). Main difficulties for control design are related to the unknown quantities d and n involved in the expression (1) of the homography H, the coupled rotation and translation transformations encoded in H, and last but not least the unavailability of linear velocity measurements (i.e. DVL is not used).

C. Discussions on an existing HBVS control approach

An advanced HBVS approach without relying on homography decomposition has been proposed in [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF]. This kinematic control approach consists in using (V C , Ω) as control inputs to stabilize the visual errors e p , e Θ ∈ R 3 to zero, where

e p (I 3 -H)m , e Θ vex(H -H) (3)
with m ∈ S 2 an arbitrary unit vector satisfying n m > 0. In the case of station keeping using a downward-looking camera, the angle between the target plane's normal vector and the camera axis at the desired pose should be smaller than 90 degree, which implies that n e 3 > 0. It follows that the simple choice m = e 3 ensures the validity of the condition n m > 0 without any prior knowledge of n . This choice (i.e. m = e 3 ) will be considered throughout the present paper.

The following lemma states the local exponential stability of the kinematic controller proposed in [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF].

Lemma 1 (See [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF]) The following kinematic controller

V C = -k p e p , Ω = -k Θ e Θ ( 4
)
with positive gains k p and k Θ ensures the local exponential stability of the equilibrium (R, p C ) = (I 3 , 0), i.e. H = I 3 .

For AUV control design, forces and torques must be used as control inputs instead of the linear and angular velocities. However, the passage from kinematic control to dynamic control requires these velocity measurements as discussed in our prior work [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]. It is thus inapplicable to the problem here considered due to the unavailability of linear velocity measurements for control design. In the sequel, a novel dynamic HBVS control approach will be developed to address the lack of linear velocity measurements.

III. SYSTEM MODELING A. Recalls on system modeling

The kinematic equations of the vehicle are given by

ṗ = RV (5a) Ṙ = RΩ × (5b)
The vehicle's dynamic equations are derived based on the translational and rotational momentums of the body-fluid system. The total kinetic energy of the body-fluid system is E T = E B + E F where E B and E F are the kinetic energy of the vehicle and of the surrounding fluid, respectively:

E B = 1 2 W h M B W h , with M B mI3 -mr G× mr G× J0 E F = 1 2 W h M A W h , with M A M 11 A M 12 A M 21 A M 22 A with W h [V h , Ω ] ∈ R 6
, and M A ∈ R 6×6 the hydrodynamic added-mass matrix. One thus deduces

E T = 1 2 W h M T W h , with M T = M D D J and M mI 3 + M 11 A , J J 0 + M 22 A , D mr G× + M 21 A .
From there, the translational and rotational momentums are computed as

P h = ∂E T ∂V h = MV h + D Ω Π h = ∂E T ∂Ω = JΩ + DV h (6)
Then, according to the formulation of Leonard [START_REF] Leonard | Stability of a bottom-heavy underwater vehicle[END_REF], the vehicle's dynamics are given by

Ṗh = P h × Ω + F c + F gb + F d (7a) Πh = Π h ×Ω + P h ×V h + Γ c + Γ g + Γ d (7b)
where F c ∈ R 3 and Γ c ∈ R 3 are the control force and torque vertors, F gb (mg -F b )R e 3 the sum of the gravity and buoyancy forces, Γ g mgle 3 ×R e 3 the gravity torque, and F d and Γ d the damping force and torque modeled as the sum of linear and quadratic terms as follows

F d (V h ) = -(D Vl + |V h |D Vq )V h Γ d (Ω) = -(D Ωl + |Ω|D Ωq )Ω (8)
with positive damping matrices

D Vl , D Vq , D Ωl , D Ωq ∈ R 3×3 .
Note that the model ( 8) of F d and Γ d is not used for control design but is only useful for simulation purposes.

B. Simplified model for control design

The translational and rotational dynamics (7a)-(7b) are tightly coupled due to the coupling matrix D involved in the definition (6) of the momentum terms. These complex dynamic couplings are often neglected in the literature by neglecting all terms involving D using the fact that the considered AUV is compact and the distance between the CoB and CoM is relatively small. Moreover, since the linear velocity is not measured, the "Munk moment" (MV h ) × V h is here considered as a disturbance. Finally, all terms involving unknown current velocity V f , together with the damping force and torque, are also considered as disturbances. These considerations result in the following simpler dynamic equations that decouple the translational and rotational dynamics:

M V = (MV)×Ω + F c + F gb + ∆ F (9a) J Ω = (JΩ)×Ω + Γ c + Γ g + ∆ Γ (9b)
with the "disturbance" terms

∆ F -(MV f ) × Ω -MΩ × V f +(D Ω) × Ω -D Ω + F d ∆ Γ (DV h )×Ω + P h ×V h -D Vh + Γ d
In the case of station keeping, ∆ F and ∆ Γ would eventually converge to constant vectors. In the sequel, these terms will be first neglected in the derivation of a basic controller, which later on will be robustified via both integral correction actions and a high-gain observer of ∆ Γ .

IV. CONTROL DESIGN

Homography estimation

Outer-loop By analogy to the cascade inner-outer loop control architecture proposed in [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF], the following modified version (illustrated by Fig. 2) handling the unavailability of linear velocity measurements is adopted:

Inner-loop IMU AUV F c Γ c ω 3r ω3r H Ω, R e 3 Ω, F gb
• An inner-loop controller, governing the rotation dynamics (5b) and (9b), defines the torque control vector Γ c to ensure the asymptotic stabilization of (Ω, R e 3 ) about (Ω r , e 3 ), where the reference angular velocity Ω r is defined by Ω r k g e 3 × R e 3 + ω 3r e 3 (10) with k g > 0 positive gain and ω 3r ∈ R the reference yaw angular velocity to be specified by the outer-loop controller. Note that ω3r must be computable so that the feedforward term Ωr is also computable by the torque controller.

• An outer-loop controller defines the force control vector F c together with the reference yaw angular velocity ω 3r (for the inner-loop controller) to fulfill the main objective of stabilizing H about I 3 , provided that the inner-loop controller ensures the almost global asymptotic stability (almost-GAS) and local exponential stability (LES) of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ). The inner-loop control design is less involved and is postponed after the outer-loop control design, which is the main contribution of this paper.

A. Outer-loop control design 1) Force control design: Analogously to our prior work [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF], the control design for stabilizing the visual error e p defined in (3) about zero is first carried out, but here without linear velocity measurements.

In view of ( 2) and (3) and using the choice m = e 3 , the dynamics of e p satisfy ėp = -Ω × (e p -e 3 ) + a V C [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF] with a n e3 d an unknown number. However, we know that it is positive using the condition n e 3 > 0.

In our prior work [START_REF] Nguyen | Inertialaided homography-based visual servo control of autonomous underwater vehicles without linear velocity measurements[END_REF] the outer-loop controller has been designed based on the assumption that the vehicle's CoB B and CoM G and the camera optical center C are aligned. This restrictive assumption is not used in the present paper. To this purpose let us introduce a modified velocity variable

V V + ω 3r e 3 × r C (12)
As a result of the inner-loop controller (to be designed thereafter) that ensures the exponential convergence of (Ω, R e 3 ) to (Ω r , e 3 ), one ensures that Ω exponentially converges to ω 3r e 3 , which in turn implies that V C exponentially converges to V and Ω × e 3 exponentially converges to zero. Therefore, [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF] can be rewritten as

ėp = -Ω × e p + a V + ε(t) (13)
with exponentially vanishing term

ε(t) Ω × e 3 + a (Ω -Ω r ) × r C (14)
On the other hand, using (9a) the dynamics of V verify

M V = (M V) × Ω + F c + F gb + ∆ F + ω3r M(e 3 × r C ) -ω 3r (M(e 3 × r C )) × Ω = (M V) × Ω + Fc + F gb + ∆ F + ε 1 (t) (15)
with new control force variable

Fc F c + ω3r M(e 3 × r C ) -ω 2 3r (M(e 3 × r C )) × e 3 = F c + ( ω3r I 3 + ω 2 3r [e 3 ] × )M(e 3 × r C ) (16)
and exponentially vanishing term

ε 1 (t) -ω 3r (M(e 3 × r C )) × (Ω -Ω r ) (17)
To expose the main ideas of control design, the outer-loop control design will be first carried out for the case where the disturbance term ∆ F involved in [START_REF] Hutchinson | A tutorial on visual servo control[END_REF] is considered null, i.e. ∆ F ≡ 0. Then, we will show later on how to cope with external disturbances and model uncertainties via integral correction actions. Proposition 1 Consider system including the dynamics (13) of e p and translation dynamics given by [START_REF] Hutchinson | A tutorial on visual servo control[END_REF] with ∆ F ≡ 0. Assume that Ω remains bounded for all time. Assume that the disturbance terms ε(t) and ε 1 (t) remain also bounded for all time and converge asymptotically to zero and with exponential rate of convergence after some time instant T ε ≥ 0. Introduce the augmented system

ėp = -Ω × êp -K 1 êp + K 1 e p , êp (0) ∈ R 3 (18)
with K 1 ∈ R 3×3 positive definite gain matrix. Apply the control force

Fc = mM -1 k 2 êp -(k 2 + k 3 )e p -F gb (19)
with m, k 2 , k 3 > 0. Then, the equilibrium (ê p , e p , V) = (0, 0, 0) is globally asymptotically stable (GAS).

Proof: Define ẽp êpe p . From ( 13) and ( 18) one obtains

ėp = -Ω × ẽp -K 1 ẽp -a V -ε(t) (20) Define x 1 [ẽ p , e p , M V ] , x 2 [ε , ε 1 ]
. One verifies from ( 14) and ( 17) that x 2 is independent with x 1 . Thus, the closed-loop system of system ( 13)+( 15)+( 20) can be rewritten in the following cascaded time-varying system studied by Panteley and Loria [START_REF] Panteley | On global uniform asymptotic stability of non linear time-varying non autonomous systems in cascade[END_REF] 1 :

ẋ1 = A(t)x 1 + Bx 2 (21)
with

A(t)   -Ω × -K 1 0 -a M -1 0 -Ω × a M -1 k 2 mM -1 -k 3 mM -1 -Ω ×  
and B a constant matrix composed of 0 and 1. System (21) can be seen as a nominal system

ẋ1 = A(t)x 1 (22)
perturbed by the output x 2 of a stable system that remains bounded for all time and converges to zero with exponential rate when t ≥ T ε .

To apply [START_REF] Panteley | On global uniform asymptotic stability of non linear time-varying non autonomous systems in cascade[END_REF]Theorem 2] one needs to verify first that the state x 1 of the nominal system [START_REF] Maire | A vision based target detection system for docking of an autonomous underwater vehicle[END_REF] remains bounded in the time interval [0, T ε ]. Indeed, since Ω, ε and ε 1 are bounded for all time (and subsequently in [0, T ε ]) by assumption there exist two positive numbers

β 1 , β 2 > 0 such that | ẋ1 (t)| ≤ β 1 |x 1 (t)| + β 2 , ∀t ∈ [0, T ε ]
This inequality implies that the solutions x 1 cannot grow faster than exponentially and, thus, do not blow up in finite time. From there, one can conveniently consider System [START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] with initial time t 0 = T ε and apply [START_REF] Panteley | On global uniform asymptotic stability of non linear time-varying non autonomous systems in cascade[END_REF]Theorem 2] to this system. The next step consists in verifying the validation of all assumptions of that theorem.

One of the main requirements of [29, Theorem 2] is that the origin of the nominal system ( 22) is globally uniformly asymptotically stable. Consider the following Lyapunov function candidate

L k 2 2 |ẽ p | 2 + k 3 2 |e p | 2 + a 2 m |M V| 2 (23)
One verifies that the time-derivative of L along any solution to the controlled unperturbed system ( 22) is

L = -k 2 ẽ p K 1 ẽp -a k 2 ẽ p V + a k 3 e p V + a m-1 M V) ( Fc + F gb = -k 2 ẽ p K 1 ẽp (24)
From ( 24) one deduces that L is semi-negative definite and, thus, ẽp , e p , and V remain bounded by initial conditions. One then easily verifies that ėp and L also remain bounded, implying the uniform continuity of L. Then, application of Barbalat's lemma [START_REF] Khalil | Nonlinear systems (3rd Edition)[END_REF] ensures the convergence of L and, thus, of ẽp to zero. Then, from (20) (with ε(t) ≡ 0) and by application of the extended Barbalat's lemma [START_REF] Micaelli | Trajectory tracking for unicycle-type and two-steering-wheels mobile robots[END_REF] one also ensures the convergence of ėp to zero, which in turn implies that V converges to zero. Similar arguments can be used to show the convergence of V to zero. In view of (15) (with ∆ F ≡ 0), ε 1 (t) ≡ 0 and Fc given by [START_REF] Lots | A 2-D visual servoing for underwater vehicle station keeping[END_REF], one ensures that e p converges to zero. The stability of the equilibrium (e p , êp , V) = (0, 0, 0) (or x 1 = 0) is a direct consequence of ( 23) and [START_REF] Malis | Deeper understanding of the homography decomposition for vision-based control[END_REF]. We have proved that the origin of the nominal system ( 22) is GAS. The next step consists in showing [29, inequality [START_REF] De Plinval | Stabilization of a class of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF]] is satisfied with the Lyapunov function [START_REF] Malis | 2 1/2D visual servoing[END_REF] and for the nominal system [START_REF] Maire | A vision based target detection system for docking of an autonomous underwater vehicle[END_REF]. This condition consists in proving the existence of

c 1 > 0, δ 1 ≥ 0 such that ∂L ∂x 1 |x 1 | ≤ c 1 L, ∀|x 1 | ≥ δ 1 (25)
From the definition (24) of L one deduces

α 1 |x 1 | 2 ≤ L ≤ α 2 |x 1 | 2 with α 1 min( k2 2 , k3 2 , a 2 m ), α 2 max( k2 2 , k3 2 , a 2
m ). One also verifies that ∂L ∂x1 ≤ 2

√ 3α 2 |x 1 |. Thus, inequality (25)
is satisfied with c 1 = 2 √ 3α 2 /α 1 and δ 1 = 0. It remains to verify Assumptions A2 and A3 of [START_REF] Panteley | On global uniform asymptotic stability of non linear time-varying non autonomous systems in cascade[END_REF]Theorem 2]. Indeed, Assumption A2 on the interconnection term B is clearly satisfied since B is constant (i.e. ∃c B > 0 : |B| ≤ c B ) which implies that [29, inequality (9)] is satisfied, i.e.

|B(t,

x 1 , x 2 )| ≤ θ 1 (t, |x 2 |) + θ 2 (t, |x 2 |)|x 1 | with θ 1 (t, |x 2 |) = c B and θ 2 (t, |x 2 |) = 0.
One can verify that Assumption A3 is also satisfied using the fact that |x 2 (t ≥ t 0 )| exponentially converges to zero (with initial time t 0 = T ε as explained previously). Indeed, there exist γ 1 , γ 2 > 0 such that

|x 2 (t, t 0 , x 2 (t 0 ))| ≤ γ 1 |x 2 (t 0 )|e -γ2(t-t0) , ∀t ≥ t 0
and, subsequently, [29, inequality [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF]] holds, that is

∞ t0 |x 2 (t, t 0 , x 2 (t 0 ))|dt ≤ φ(|x 2 (t 0 )|) γ 1 γ 2 |x 2 (t 0 )|
with φ(•) a class K function (i.e. strictly increasing and vanishing at zero). From there, the application of [29, Theorem 2] ensures that the equilibrium (x 1 , x 2 ) = (0, 0) is GAS. This concludes the proof.

Note that Proposition 1 applies to the case where the perturbation term ∆ F is negligible. While this basic controller would be able to handle small currents, in practice it is often desirable to enhance control robustness by incorporating integral correction actions. However, in our case the system considered in Proposition 1 (i.e. ( 18)+( 13)+( 15)) is already a third-order time-varying system. Thus, adding an integrator would lead to a fourth-order time-varying system. Too high order system, together with the presence of an unknown multiplicative factor a in [START_REF] Hua | Balanced-force-control of underactuated thrust-propelled vehicles[END_REF] and the unavailability of linear velocity measurements, excludes the possibility of establishing global (or semi-global) stability results similar to Proposition 1. However, it is still possible to state local exponential stability. For simplicity, let us consider the case where M can be roughly approximated by a positive diagonal matrix, i.e. M ≈ diag(m 1,1 , m 2,2 , m 3,3 ). Proposition 2 Consider system including dynamics of e p given by [START_REF] Hua | Balanced-force-control of underactuated thrust-propelled vehicles[END_REF] and translation dynamics given by (15) with constant disturbance ∆ F and diagonal total mass matrix M = diag(m 1,1 , m 2,2 , m 3,3 ). Assume that the disturbance terms ε(t) and ε 1 (t) remaining bounded for all time and converge asymptotically to zero and with exponential rate of convergence after some time instant T ε . Introduce the following integrator

ż = -Ω × z + e p , z(0) ∈ R 3 ( 26
)
and the following augmented system

ėp = -Ω × êp -K 1 êp + K 1 ēp , êp (0) ∈ R 3 (27)
with positive diagonal matrix K 1 ∈ R 3×3 and ēp e p + k I z with positive integral gain k I . Apply the control force

Fc = mM -1 sat η1 (k 2 ẽp ) -sat η2 (k 3 ēp ) -F gb (28)
with positive numbers m, k 2 , k 3 , η 1 , η 2 , and ẽp êpēp . Choose η 2 high enough such that

η 2 > m-1 |M∆ F | (29)
and choose k I satisfying

k I < k 2 λ K1 k 2 + k 3 ( 30
)
with λ K1 the smallest diagonal component of K 1 . Assume that the outer-loop control ω 3r together with the innerloop control Γ c ensures that Ω can be considered as a first order term in first order approximations. Then, the equilibrium (ê p , e p , V, z) = (k I z , 0, 0, z ), with z ( mk 3 k I ) -1 M∆ F , of the controlled system is locally exponentially stable (LES). Moreover, Fc remains bounded by

| Fc | ≤ mλ -1 M (η 1 + η 2 ) + |mg -F b | (31)
with λ M the smallest eigenvalue of M.

Proof: One verifies that the linearized system of Eqs. ( 13), ( 15), ( 27) augmented with integrator (26) around the equilibrium (ê p , e p , V, z) = (k I z , 0, 0, z ), with z ( mk 3 k I ) -1 M∆ F , is given by

ẋ1 = Ax 1 + Bx 2 (32)
with

x 1 [(ê p -k I z ) , e p , (M V) , (z -z ) ] ∈ R 12 x 2 [ε , ε 1 ] ∈ R 6 A     -K1 K1 0 k I K1 0 0 a M -1 0 k2 mM -1 -(k2 +k3) mM -1 0 -k I (k2 +k3) mM -1 0 I3 0 0    
and B ∈ R 12×6 a constant matrix composed of 0 and 1. The 12th-order characteristic polynomial of the nominal

autonomous system ẋ1 = Ax 1 is Q(λ) = Q 1 (λ)Q 2 (λ)Q 3 (λ), with Q i (λ) = λ 4 + k 1i λ 3 + (k 2 + k 3 )a m m 2 i,i λ 2 + k 1i k 3 + (k 2 + k 3 )k I a m m 2 i,i λ + k 1i k 3 k I a m m 2 i,i
with k I satisfying [START_REF] Rives | Underwater pipe inspection task using visual servoing techniques[END_REF]. Application of Routh-Hurwitz criterion ensures the exponential stability of the nominal autonomous system. Since x 2 is exponentially stable by assumption, one deduces that the cascade (32) is also exponentially stable. One can verify that condition ( 29) is necessary so that the integral action can compensate for the disturbance ∆ F . Finally, the bound of the force control vector Fc given by ( 31) is directly deduced from [START_REF] Palomeras | Vision-based localization and mapping system for AUV intervention[END_REF].

Remark 1 In first order approximations, the force control Fc given by (28) is given by

Fc = k 2 mM -1 êp -(k 2 + k 3 ) mM -1 ēp -F gb ( 33
)
The proof of global asymptotic stability of the equilibrium (ê p , e p , V, z) = (k I z , 0, 0, z ), with Fc given by (33), proceeds analogously to the proof of Proposition 2. The linear approximation (33) of Fc is useful for gain tuning using, for instance, pole placement technique, while its nonlinear expression (28) that involves saturation functions allows us to define explicitly the bound of the force control input Fc as given by [START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF]. The latter property is of particular importance in practice since it is often desirable to take explicitly actuation limitations into account. For instance, [START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF] implies that the desired bound of | Fc | can be set to any value µ

(> |mg -F b |) if η 1 + η 2 ≤ m-1 λ M (µ -|mg -F b |).
2) Control design of the reference yaw angular velocity ω 3r : The previous part of outer-loop control design ensures the convergence of e p to zero, which in turn ensures the convergence of p C to zero. Then, using the fact that the innerloop torque controller ensures the convergence of R e 3 to e 3 , one easily verifies that the element h 1,2 at the first row and second column of H converges to sinψ, with ψ to yaw Euler angle. From there, the design of ω 3r can proceed identically to our prior work [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] and is, thus, recalled here for the sake of completeness.

Proposition 3 (see [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] for proof) Assume that the inner-loop torque controller Γ c ensures the almost-GAS and LES of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ), with Ω r defined by [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF] combined with ω 3r (involved in [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF]) solution to the following system

ω3r = -k Θ2 ω 3r -k Θ1 sat ∆Θ (h 1,2 ), ω 3r (0) ∈ R (34)
with k Θ1 , k Θ2 , ∆ Θ > 0 and h 1,2 the element at the first row and second column of H. Apply the outer-loop force controller F c given either by Proposition 1 (when ∆ F ≡ 0) or Proposition 2 (when ∆ F is constant). Then, the equilibrium H = I 3 is LES. Moreover, this equilibrium is almost-GAS in the case where Fc is given by Proposition 1 and ∆ F ≡ 0.

B. Inner-loop control design

The more involved part concerning the outer-loop control design has been presented. It remains to design an effective inner-loop torque controller that ensures the stability of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ), with Ω r defined by [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF] combined with (34).

In view of the rotation dynamics (i.e. (5b) and (9b)), it is not too difficult to carry out the above-mentioned objective since the sub-system under consideration is fully-actuated and the measurements of both Ω and R e 3 are at our disposal. However, the troublesome term ∆ Γ involved in (9b) should be carefully addressed, especially when the vehicle is subjected to strong currents that excite the Munk moment effects. Since the angular velocity can be measured at high frequency and with good precision, we propose to estimate ∆ Γ using a high-gain observer similarly to the idea proposed in [START_REF] Hua | Balanced-force-control of underactuated thrust-propelled vehicles[END_REF].

Lemma 2 Consider the following observer of ∆ Γ :

J Ω = (JΩ)× Ω + Γ c +Γ g + ∆Γ + k 0 J(Ω -Ω) ∆Γ = a 2 0 k 2 0 J(Ω -Ω) (35)
with Ω and ∆Γ the estimates of Ω and ∆ Γ , respectively; Ω(0) ∈ R 3 , ∆Γ (0) ∈ R 3 ; a 0 , k 0 some positive gains. Assume that ∆Γ is uniformly ultimately bounded (u.u.b.). Then for any

a 0 ∈ (1 - √ 2/2, 1 + √ 2/2),
1) The errors Ω -Ω and ∆Γ -∆ Γ are u.u.b. by a positive constant ε(k 0 ) that tends to zero when k 0 tends to +∞. Moreover, these terms converge exponentially to zero for any

k 0 > 0 if ∆ Γ is constant. 2) ∆Γ is u.u.b. by a constant independent of k 0 .
The proof proceeds identically to the proof of [START_REF] Hua | Contributions to the automatic control of aerial vehicles[END_REF]Proposition 8]. Now, we can use the estimate ∆Γ as a feedforward term for the inner-loop torque control design.

Define the angular velocity error variable Ω Ω -Ω r . From (9b), one obtains the following error equation

J Ω = (JΩ) × Ω + Γ c + Γ g + Γ + ∆Γ + ∆Γ (36)
with Γ (JΩ) × Ω r -J Ωr and ∆Γ ∆ Γ -∆Γ .

Proposition 4 Consider error equation given by (36). Assume that the unknown perturbation term ∆Γ is constant and bounded by a known value ¯ . Define an anti-windup integrator I Ω solution to the following differential equation

İΩ = -kI Ω + sat ∆1 kI Ω + sat ∆2 (Q Ω) , I Ω (0) ∈ R 3 (37)
with k a positive gain, ∆ 1 , ∆ 2 some positive constants, and

Q ∈ R 3×3 satisfying Q Q = J.
Apply the control torque

Γ c = -sat η3 (K Ω Ω) -k iΩ Q I Ω + max 0, |Q Ω| ∆ 2 -1 Γ g -Γ -∆Γ (38)
with K Ω ∈ R 3×3 positive diagonal gain matrix, k iΩ positive gain, η 3 a positive number, and Ω r defined by (10) combined with (34

). If k k iΩ ||Q -||¯ + ∆ 2 ≤ ∆ 1 (39)
then, the following properties hold.

1) The error state ( Ω, I Ω , R e 3 ) converges either to (0, I Ω , e 3 ) or (0, I Ω , -e 3 ) for all initial conditions, with

I Ω k -1 iΩ Q - ∆Γ .
2) The "desired" equilibrium ( Ω, I Ω , R e 3 ) = (0, I Ω , e 3 )

is almost-GAS and LES. The "undesired" equilibrium ( Ω, I Ω , R e 3 ) = (0, I Ω , -e 3 ) is unstable.

The proof is given in Appendix A.

V. COMPARATIVE SIMULATION RESULTS

The proposed control approach applied to station keeping of fully-actuated AUVs without the need of linear velocity measurements is, in fact, inspired by the one proposed in [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]Remark 7] which corresponds to the particular case of using a downward-looking camera and which makes use of linear velocity measurements. Although the novel approach already has a practical advantage by depleting the need of a costly DVL, it is even more desirable if its performance is also comparable to the previous approach. Therefore, comparative simulation results of the two approaches using a realistic model of a fully-actuated AUV and in presence of a constant horizontal current

v f = [ 1 2 √ 2 , 1 2 √ 2 ,
0] (m/s) will be reported thereafter. For convenience, let us call the proposed controller (c.f. Propositions 2, 3, 4) and the one proposed in [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] as Controller 1 and Controller 2, respectively.

The simulated vehicle is the BlueROV 2 . Its physical parameters are provided in Tab. I, where the added-mass, addedinertia and damping coefficients are roughly identified from

k 2 = 8 3 s 2 a , k 3 = 1 3 s 2 a , s = √ 2 k I = 0.7, η 1 = 1.8, η 2 = 2.3 kg = 1, k Θ1 = 1, k Θ2 = √ 2, ∆ Θ = 1 Inner-loop K Ω = diag(3, 3, 3), k iΩ = 2 k = 10, ∆ 1 = 6.25, ∆ 2 = 2 a 0 = 0.5, k 0 = 20, η 3 = 8
k p2 = √ 2 ku = 1, k Θ1 = 1, k Θ2 = √ 2, ∆ Θ = 1 Inner-loop K V = diag(5, 5, 5), K iV = 0.4K V K Ω = diag(3, 3, 3), k iΩ = 2 k = 10, ∆ 1 = 6.25, ∆ 2 = 2 a 0 = 0.5, k 0 = 20, η 3 = 8
(0) = [-2, -1.5, -1] (m), R(0) = R { π 18 ,-π 18 ,π} , V(0) = Ω(0) = 0.
A very large initial yaw error is chosen (i.e. ψ = π) to verify the large stability domain of the control algorithms.

• Simulation with Controller 1 (i.e proposed approach):

The proposed control approach including the outer-loop controller given in Propositions 2-3 and the inner-loop controller given in Proposition 4 is simulated, with gains and parameters given in Tab. II. The control gains have been obtained based on the classical pole placement technique using a coarse estimation of a equal to 1. Due to the current velocity of significant magnitude (i.e. 0.5[m/s]), the drag force and Munk moment are not negligible. This shows the need of control robustification using, for instance, the integrator and high-gain observer techniques proposed in this paper. As observed from the simulation results reported in Figs. 3a-6a, the estimate ∆Γ and the integrators in both inner-and outer-loop controllers successfully counteract for these abovementioned disturbances as well as the imprecise estimated system parameters. Fig. 3a shows the convergence of the AUV's position and orientation to zero without large overshoots. The sea current on horizontal plane has a clear impact on the integral variable z, the estimate terms êp and ∆Γ , as well as the control force and torque. More precisely, z 1 , z 2 , êp1 , êp2 , ∆Γ3 , F c1 , F c2 and Γ c3 converge to non-null values as shown in Figs. 4a-6a. On the other hand, the vertical component of F c3 converges to a positive value since the vehicle is positively buoyant.

• Simulation with Controller 2 (i.e. approach proposed in [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]): To make a fair comparison, the inner-loop controller of [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] has been revised by also incorporating a high-gain observer of the perturbation torque induced by sea current, similarly to the one proposed in Lemma 2. Control parameters and gains of the controller are given in Tab. III. They have been chosen so that the time evolution of the AUV's position has almost the same settling time as in the previous simulation. Simulation results are reported in Figs 3b-6b. In overall, the time evolutions of the vehicle's pose and of the visual error e p are quite reminiscent of the corresponding ones resulted by Controller 1. This implies that the performance of the proposed controller is comparable to our prior control approach [START_REF] Krupínski | An inertialaided homography-based visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] that in contrast relies on a costly DVL for linear velocity measurements.

VI. EXPERIMENTAL VALIDATIONS A. Experimental setup

The implementation of the proposed algorithm with realtime homography estimation has been performed on an inhouse AUV experimental platform (see Fig. 7). This platform has been developed based on a purchased BlueROV. The on-board electronic components (see Fig. 8) include a widely-used 3DR Pixhawk flight controller and a compact and low-cost Odroid XU-4 octa-core single board computer. A laptop is used as a ground station, that is connected with Odroid XU-4 over the high speed Ethernet connection. In the AUV architecture depicted in Fig. 8, ROS plays the role of a middleware, providing communication (i.e. sending and receiving messages between the distributed components). The open-source PX4 autopilot widely-used for unmanned aerial vehicles is exploited and adapted for our research purpose.

The overall dimensions of the vehicle are 0.39[m] × 0.33[m] × 0.65[m] in height, width and length. The original dry weight of BlueROV is only 7.6[kg]. However, for our research a longer tube has been used to accommodate the main electronic components and two smaller tubes have been added for camera and battery housing. After incorporating all these components, the total volume of the AUV is almost double to its original design. Consequently, the buoyancy force is significantly increased. To overcome that issue, additional weight blocks have been attached to the vehicle to achieve near neutral buoyancy. The weight blocks together with additional buoyant blocks give two degrees of freedom to adjust the center of mass G and the center of buoyancy B such that the line connecting them (i.e BG) is parallel with the direction of the downward-looking camera. The resulting total weight of the AUV is approximately 16[kg] in air. The vehicle is equipped with three vertical thrusters for heave, pitch and roll actuations, two horizontal thrusters for yaw and surge actuations, and one lateral thruster for sway actuation.

For performing the real-time homography estimation, the AUV is equipped with a myAHRS+ IMU sensor providing measurement output at 100[Hz] and an oCam downward-looking monocular camera providing color images of 640[px] × 480[px] at 20 [Hz]. For homography estimation, we have used the HomographyLab library3 which has been developed by our team based on the homography observer proposed in [START_REF] Hua | Featurebased recursive observer design for homography estimation and its application to image stabilization[END_REF]. This library has been implemented in C++ combined with OpenCV for image processing (for instance, the FAST Feature Detector and ORB Descriptor Extractor algorithms available in the OpenCV library are employed for carrying out feature detection and descriptor extraction in images). HomographyLab has been evaluated at the TRL 7 (Technology Readiness Level) and is protected by the French Agency for the Protection of Programs. Real-time and robustness (with respect to fast camera motions, occlusions, image blurs, sudden changes in light intensity, poor image quality, etc.) are the two principle distinguished features of HomographyLab and the implemented algorithm with respect to the state-of-the-art codes and algorithms (see [START_REF] Hua | Featurebased recursive observer design for homography estimation and its application to image stabilization[END_REF] for Fig. 7: AUV with downward-looking camera (inside yellow circle) Fig. 8: AUV's architecture more detail). In the reported experiments, HomographyLab has been implemented on the ground station laptop with an Intel Core i7-7700HQ octa-core CPU running at 2.8 GHz. The transmission of the data from the camera and the IMU to the ground station is carried out through the high speed Ethernet cable. The laptop has a Linux-based operating system and is responsible for the following tasks: 1) interfacing with the camera and IMU hardwares and acquisition of images and IMU data 2) real-time estimation of the homography at 10[Hz] 3) perform outer-loop control at 10[Hz], and 4) interfacing with the joystick and Pixhawk via Odroid to remotely control the vehicle.

To perform the station-keeping task, the UAV is initially in the teleoperation mode and is manually positioned at a Fig. 9: Experimental validations in lake Saint-Cassien certain depth above the lakebed so that the latter is visible by the camera ensuring a sufficient number of detected features for good homography estimation. Such unknown depth thus varies according to lighting conditions and water turbidity. In the reported experiment, the vehicle was positioned at about 0.5[m] above the lakebed.

The parameters and control gains involved in the computation of the control inputs are given in Tab. IV and Tab. V.

B. Experimental results

Experimental results carried out in lake Saint-Cassien (France) are reported next (see Fig. 9). Due to space limitation, only brief but most representative parts of total results are presented in the video https://youtu.be/p oiISPOtgw. However, the reader can also view two other videos showing the capability of long-term station keeping (more than 30 minutes) as well as other trials carried out during our research process with different water turbidity conditions and target images:

• https://youtu.be/mkAAjX3mgVk • https://youtu.be/KjAfYu1jG18 Regarding the reported video, experimental results including the time evolution of the control force F c , the visual error e p and its estimate êp , the homography component h 12 , and

Controller

Gains and other parameters Proposition 2 K 1 = diag(3s, 3s, 3s)

k 2 = 8 3 s 2 a , k 3 = 1 3
s 2 a , s = 0.9 In Fig. 11 one observes that right after finishing the interaction, the AUV went back to its stabilized pose with relatively small overshoot. The control force generated for pushing the AUV back to the reference position is shown in Fig. 10. The settling time is about 30 seconds. The transient response also shows the efficiency of the integrator in eliminating the static error caused by the current.

k I = 0.1, η 1 = 1.8, η 2 = 2.3 Proposition 3 kg = 1, k Θ1 = 1/2, k Θ2 = √ 2, ∆ Θ = 2 √ 2 Proposition 4 K Ω = diag
In general, one can clearly observe the practical convergence of the AUV's pose to the desired one as illustrated by the practical convergence of the Frobenius norm ||H -I|| to zero (see Fig. 13). In particular, the convergence of the AUV's position is attested by the convergence of visual error e p to zero (Fig. 11) whereas the component h 12 converges near to zero (Fig. 12) showing the practical convergence of the AUV's yaw angle to the desired one.

The effectiveness of the integrator correction in dealing with unknown currents and model uncertainties can be appreciated from Figs. 10 and 11. In steady state, the horizontal components of the control force F c1 and F c2 (Fig. 10) converge to non-zero values required to compensate for the horizontal current. The vertical component F c3 ultimately remains far from zero since the AUV is positively buoyant. In Fig. 11 it can be seen that e p converges near to zero and that the offsets between the corresponding components of e p and êp As can be observed from the reported video, the experiment has been carried out on a sunny day and the waves generate moving bright spots in the lakebed that periodically alter local brightness of the images captured. However, very good and robust quality of homography estimation can be appreciated despite that change in light illumination as well as large occlusions due to a panel fixed to the stick in between 310[s] and 360 [s]. It is also worth noting that, theoretically, the homography estimation uses a planar target. In the experiment, this assumption does not hold true since the lakebed is covered by 3D reliefs such as sand rocks, and additionally the camera is relatively close to the lakebed. However, the proposed controller along with real-time homography estimation keeps working efficiently, showing the robustness of the control approach.

One can observe some small oscillations during the convergence. This is essentially due to the fact that the outer-loop control runs at a low frequency (10[Hz]) and also due to the imperfect thrust generated by the thrusters inside its deadband. This phenomenon becomes more prominent when the outer-loop poles are large. Hence, smaller pole values have been chosen while tuning the gains in the actual experiment as compared to the ones used in the reported simulation.

In conclusion, the experimental results for the overall control approach (i.e inner and outer-loop controls) are quite convincing even though some vehicle's physical parameters (e.g. added mass and added moment of inertia) are only roughly estimated and the AUV was subjected to various challenging conditions (e.g. significant unknown currents, water turbidity, lightning variation, target occlusion, etc.).

VII. CONCLUSION

In this paper a homography-based dynamic control approach of fully-actuated underwater vehicles is proposed. An advanced feature with respect to existing works on the topic is the non-requirement of a costly DVL for linear velocity measurements. A potential application to station keeping has been demonstrated in a real environment with very encouraging results despite challenging conditions such as important disturbances induced by currents, water turbidity, lightning variation, target occlusion, etc. The proposed approach will undoubtedly enlarge the working envelop of low-cost remotely operated underwater vehicles (ROVs) as a costly DVL is not required. Even for industrial or professional ROVs and AUVs equipped with DVLs, the approach also allows for overcoming the common problem of measurement imprecision of DVLs due to threshold violation when operating close to underwater structures or a sea bottom. A potential extension of this work concerns the station-keeping task of an AUV equipped with a forward-looking camera observing a (near) vertical target. Acknowledgement: This work was supported by the French ANR Astrid CONGRE (ANR-18-ASTR-0006) and FUI GreenExplorer projects.

APPENDIX PROOF OF PROPOSITION 4

Proof: Consider the positive storage function (43) where the last inequality is obtained using condition (39) and the fact that ∀a, b ∈ R 3 , ∆ ∈ R + one has | -a + sat ∆ (b + a)| ≤ |b| if |a| ≤ ∆ (see [START_REF] Hua | Contributions to the automatic control of aerial vehicles[END_REF] for the proof). Clearly L1 is negative semi-definite. Remark that system (41) and (37) is not autonomous due to the time-varying term Ω and consequently La Salle's principle does not apply. However, Ω and I Ω are bounded with respect to initial conditions. Since Ω r and its derivative are bounded thanks to the expressions [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF] and (34), one deduces from (41) that Ω is also bounded. Then it is straightforward to verify that L is also bounded, implying the uniform continuity of L. Then, application of Barbalat's lemma ensures that L and, thus, Ω and e 3 × R e 3 converge to zero. Next, using Barbalat-like arguments it can be shown that Ω also converges to zero, implying the convergence of I Ω to I Ω . The convergence of e 3 × R e 3 to zero implies that Re 3 converges to either e 3 or -e 3 . So far we have proved that ( Ω, I Ω , Re 3 ) converges either to (0, I Ω , e 3 ) or (0, I Ω , -e 3 ).

V      1 2 |Q Ω| 2 if |Q Ω| ≤ ∆ 2
It remains to show that the "desired" equilibrium ( Ω, I Ω , Re 3 ) = (0, I Ω , e 3 ) is LES and the "undesired" equilibrium ( Ω, I Ω , Re 3 ) = (0, I Ω , -e 3 ) is unstable. Note that the almost-GAS of the "desired" equilibrium then directly follows. In the first-order approximations, one has R ≈ I + Θ × with Θ = [φ, θ, ψ] and, subsequently, e 3× R e 3 ≈ [-φ, -θ, 0] . Denoting [ω 1 , ω2 , ω3 ] Ω and using the approximation Θ ≈ Ω, one obtains the following linearized system of (36) and (37)

         φ ≈ ω1 -k g φ θ ≈ ω2 -k g θ Ω ≈ -J -1 K Ω Ω-J -1 k iΩ Q ĨΩ +mglJ -1 [-φ,-θ, 0] İΩ = Q Ω ( 44
) Consider the following Lyapunov function candidate

L Ω = 1 2 Ω J Ω + 1 2 k iΩ | ĨΩ | 2 + 1 2 mgl φ 2 + θ 2 (45)
One verifies that LΩ = -Ω K Ω Ω -k g mgl(φ 2 + θ 2 ) ≤ 0 (46)

From here, LaSalle's principle ensures that Ω, φ and θ and, thus, Ω converge to zero, which implies the convergence of z Ω to zero. The convergence of φ and θ to zero is equivalent to the convergence of R e 3 to e 3 . Since the equilibrium ( Ω, I Ω , R e 3 ) = (0, 0, e 3 ) of the linearized system (44) is asymptotically stable, it is also exponentially stable. Now, the Chetaev's theorem is used to prove the instability of the equilibrium ( Ω, I Ω , R e 3 ) = (0, 0, -e 3 ). Define y = e 3 + R e 3 . Consider the positive function S 1 (y) y e 3 = 1 + e 3 R e 3 , satisfying S 1 (0) = 0. Define U r {y|S 1 (y) > 0, |y| < r} for some number 0 < r < 1. Note that U r is nonempty. By neglecting all high-order terms, one verifies that

Ṡ1 ≈ e 3 RΩ r× e 3 = k g |e 3× R e 3 | 2 = k g |e 3× y| 2
For all y ∈ U r , the fact that y e 3 > 0 is equivalent to |e 3× y| 2 > 0, which implies that Ṡ1 > 0. Since all conditions of Chetaev's theorem are now united [START_REF] Khalil | Nonlinear systems (3rd Edition)[END_REF], one concludes that the origin of the linearized system about the undesired equilibrium (so that y = 0) is unstable.

- → e a 3 }.

 3 The body frame attached to the AUV with the origin at the CoB is denoted as B = {B; -→ e b 1 , -→ e b 2 , -→ e b 3 }. The camera frame with base vectors parallel to those of B is denoted as C = {C; -→ e c 1 , -→ e c 2 , -→ e c 3 }. The vectors of coordinates expressed in B of --→ BC and --→ BG are respectively denoted as r C ∈ R 3 and r G = le 3 ∈ R 3 , with e 3 = [0, 0, 1] . • The orientation (i.e. attitude) of the body frame B relative to the inertial frame A is represented by a rotation matrix R ∈ SO(3). By denoting p and p C the vector of coordinates expressed in A of B and C respectively, one verifies that p = p C -Rr C .

e 3 }

 3 denote the canonical basis of R 3 and I 3 denote the identity of R 3×3 . For all u ∈ R 3 , u × denotes the skew-symmetric matrix associated with the cross product byu, i.e., u × v = u × v, ∀v ∈ R3 . Let vex(•) denote an operator such that vex(u × ) = u. The transpose operator is denoted as (•) . The Euclidean norm in R n and the Frobenius norm in R n×n are denoted as | • | and || • ||, respectively. The notation sat δ (•) ∈ R n , with δ > 0, denotes the classical saturation function, i.e., sat δ (x) x min (1, δ/|x|) , ∀x ∈ R n .

Fig. 2 :

 2 Fig. 2: Control architecture of the proposed HBVS

2 Fig. 3 : 2 Fig. 4 :

 2324 Fig. 3: (Simulation) AUV's position and attitude (Euler angles) vs. time

2 Fig. 5 : 2 Fig. 6 :

 2526 Fig. 5: (Simulation) Control force F c and torque Γ c vs. time

  3 + M11 A [kg] diag(17.868, 23.868, 21.024) Ĵ = J 0 + M22 A [kg.m 2 ] diag(0.3105, 0.8486, 1.0) D = mle 3× [kg.m] 0.4e 3×

  (1.863, 5.0916, 5.0), K iΩ = 0.1K Ω a 0 = 0.5, k 0 = 20, η 3 = 6

Fig. 10 :

 10 Fig. 10: Control force F c vs. time

Fig. 11 :Fig. 12 : 13 :

 111213 Fig. 11: e p and êp vs. time

1 2 ( 2 -

 122 2|Q Ω| -∆ 2 )∆ 2 otherwiseUsing the fact that J = Q Q one verifies thatV = min 1, ∆ 2 |Q Ω| Ω J Ω(40)From (36), (38) and the definition of Ω one obtainsJ Ω = (JΩ) × Ω -sat η3 (K Ω Ω) + max 1, |Q Ω| ∆ 2 Γ g -k iΩ Q I Ω + ∆Γ(41)Consider the Lyapunov function candidateL 1 V + mgl(1 -e 3 R e 3 ) + 1 2 k iΩ | ĨΩ | 2 (42)with ĨΩ I Ω -I Ω . Calculating the time-derivative of L 1 using the expression (40) of V, equation (36), the torque control expression (38), the definition[START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF] of Ω r , the expression (37)of İΩ and the fact that sat∆2 (Q Ω) = min(η3 (K Ω Ω) -k iΩ Ĩ Ω sat ∆2 (Q Ω) -mglΩ r e 3 × R e 3 + k iΩ Ĩ Ω İΩ = -min(1, ∆ 2 |Q Ω| ) Ω sat η3 (K Ω Ω) -mglk g |e 3 × R e 3 | kk iΩ | ĨΩ | 2 + k iΩ Ĩ Ω -kI iΩ -sat ∆2 (Q Ω) + sat ∆1 k ĨΩ + kI Ω + sat ∆2 (Q Ω) ≤ -min(1, ∆ 2 |Q Ω| ) Ω sat η3 (K Ω Ω) -mglk g |e 3 × R e 3 | 2

TABLE I :

 I Specifications of the simulated AUV

	Controller	Gains and other parameters
	Outer-loop	K 1 = diag(3s, 3s, 3s)

TABLE II :

 II Control gains and parameters of Controller 1

	Controller	Gains and other parameters
	Outer-loop	k p1 = 0.4145,

TABLE III :

 III Control gains and parameters of Controller 2

	the given shape. To test the robustness of the controllers
	with respect to model uncertainties, the following "erroneous"
	estimates of M and J are used:
	M = mI 3 + M11 A = diag(8.712, 12.712, 10.816) [kg] Ĵ = Ĵ0 + M22 A = diag(0.1642, 0.5643, 0.5116) [kg.m 2 ]
	One notes that these estimated parameters are quite different
	from the corresponding "real" ones.
	The homography H is directly computed using
	(1) with d	= 1(m) and n	= R { π 18 , π 6 ,0} e 3 =
	[0.4924, -0.1736, 0.8529] . The initial conditions are
	p C

TABLE IV :

 IV Specifications of the experimental AUV

TABLE V :

 V Control gains and parameters in experiment

The class of cascade time-varying systems studied by Panteley and Loria is of the form: ẋ1 = f (t, x 1 )+g(t, x 1 , x

)x 2 . Thus, in our case f (t, x 1 ) = A(t)x 1 and g(t, x 1 , x 2 ) = B.

http://sdb3.i3s.unice.fr/homographyLab/


Md Sahidullah

Achintya Kumar Sarkar

Ville Vestman

Xuechen Liu

Romain Serizel

Tomi Kinnunen

Zheng-Hua Tan

Emmanuel Vincent

UIAI System for Short-Duration Speaker Verification Challenge 2020

Keywords: Text-dependent speaker verification, Utterance verification, Fusion, Bottleneck feature, SdSV challenge 2020

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Automatic speaker verification (ASV) is the task of verifying whether a speech utterance has been spoken by a claimed speaker [START_REF] Kinnunen | An overview of textindependent speaker recognition: From features to supervectors[END_REF][START_REF] Hansen | Speaker recognition by machines and humans: A tutorial review[END_REF]. State-of-the-art ASV systems show promising performance when several minutes of audio data have been collected in controlled conditions for both enrollment and verification. The amount of enrollment and verification speech is an important factor [START_REF] Poddar | Speaker verification with short utterances: a review of challenges, trends and opportunities[END_REF]. While having more speech typically improves recognition performance, short-duration utterances are often preferable for practical deployment. The short-duration speaker verification (SdSV) challenge 2020 primarily focuses on the duration factor where speakers are enrolled and verified with a few seconds of audio data [START_REF] Zeinali | Shortduration speaker verification (SdSV) challenge 2020: the challenge evaluation plan[END_REF]. Besides, the speech corpus for the challenge was recorded in realistic environments, and the collection protocol was designed to incorporate various kinds of noises in the speech corpus which introduced mismatches between the enrollment and the verification phases. The challenge has two independent tasks. Our entry focuses on Task 1, which concerns textdependent ASV.

In text-dependent ASV, the spoken phonetic contents for enrollment and verification are assumed to be identical. Typically, a short sentence or phrase is shared by all users. However, considering the practical fact that the test speaker can also utter a wrong phrase [START_REF] Larcher | Textdependent speaker verification: Classifiers, databases and RSR2015[END_REF][START_REF] Lee | The RedDots data collection for speaker recognition[END_REF], there may be four types of trials, respectively defined as target correct (TC) where the target speaker utters the correct phrase, target wrong (TW) where the target speaker utters a different phrase, impostor correct (IC) where an impostor utters the same phrase as in speaker enrollment, and impostor wrong (IW) where an impostor utters a different phrase compared to speaker enrollment. For evaluation purposes, TC trials are considered as genuine trials to be accepted, and the remaining three as impostor trials to be rejected.

Although methods developed for text-independent ASV are applicable to text-dependent ASV without any modification, they do not generalize well [START_REF] Larcher | Textdependent speaker verification: Classifiers, databases and RSR2015[END_REF]. In particular, the performance severely degrades in the TW condition due to the similarities in speaker information. The solutions proposed for text-dependent ASV verify the speaker identity and the spoken phrase in an integrated manner. Here, the phrase information is incorporated by capturing contextual information with a hidden Markov model (HMM) [START_REF] Larcher | Textdependent speaker verification: Classifiers, databases and RSR2015[END_REF][START_REF] Sarkar | Text dependent speaker verification using un-supervised HMM-UBM and temporal GMM-UBM[END_REF][START_REF] Zeinali | HMMbased phrase-independent i-vector extractor for textdependent speaker verification[END_REF][START_REF] Zeinali | Deep neural networks and Hidden markov models in i-vector-based text-dependent speaker verification[END_REF][START_REF] Dey | Template-matching for text-dependent speaker verification[END_REF], dynamic time warping (DTW) [START_REF] Dey | Template-matching for text-dependent speaker verification[END_REF][START_REF] He | A low-power textdependent speaker verification system with narrow-band feature pre-selection and weighted dynamic time warping[END_REF] and pass-phrase identification [START_REF] Sarkar | Incorporating pass-phrase dependent background models for text-dependent speaker verification[END_REF]. Alternatively, standalone utterance verification (UV) and ASV modules can be developed, and fused together at the score or decision level [START_REF] Kinnunen | Utterance verification for text-dependent speaker recognition: A comparative assessment using the RedDots corpus[END_REF].

We investigate both strategies for this challenge. Our single system applies joint spoken phrase and speaker identity verification where phrase information is incorporated with a pass phrase-dependent background model (PBM) [START_REF] Sarkar | Incorporating pass-phrase dependent background models for text-dependent speaker verification[END_REF]. The primary system integrates modules developed for separate tasks. Here, we adopt a cascade fusion strategy where UV is performed before ASV. We first compute the decision threshold associated with the equal error rate (EER) for UV on the development set. A verification trial is assigned with an arbitrarily low ASV score if its UV score is lower than the threshold. On the other hand, a trial is passed to the ASV module for scoring if its UV score is greater than or equal to the threshold. The threshold computed on the development set is adopted for scoring on the evaluation set. Our UIAI team is a multi-site collaboration involving four research labs. Given the emphasis on text-dependent ASV, we investigate different UV and text-dependent approaches. We develop an i-vector-based UV method that includes channel variability compensation. We introduce a text-dependent ASV method employing phone-based bottleneck features with PBM. We also explore how a standard ASV system can be improved in short-duration conditions with different frame-level acoustic features and utterance-level speaker embeddings. Finally, we study different system combination strategies suitable for combining UV and ASV systems. Our primary system, which is a multi-level fusion of seven different subsystems, has achieved the fifth rank out of 19 submissions in the challenge whereas the single system has shown substantial improvement over the two challenge baselines.

The rest of the paper is organized as follows. Section 2 describes different subsystems developed for the challenge. Section 3 describes the experimental setup. Section 4 presents the experimental results. Finally, we conclude in Section 5.

SYSTEM DESCRIPTION

In this section, we summarize the subsystems used for our primary submission to the SdSV challenge.

Utterance verification

Our UV system relies on speaker embeddings. Although speaker embeddings mainly encode speaker information, they contain a considerable amount of information about the spoken content [START_REF] Raj | Probing the information encoded in x-vectors[END_REF][START_REF] Wang | What does the speaker embedding encode?[END_REF]. This makes them potentially useful for UV tasks (besides ASV). The UV task in the SdSV challenge is a closed-set task as there are no out-of-set phrases. We use an i-vector representation [START_REF] Dehak | Front-end factor analysis for speaker verification[END_REF] and a probabilistic linear discriminant analysis (PLDA) back-end for this task. The setup is similar to the one commonly used in ASV scenarios, except that we treat utterance identifiers (rather than speakers) as the class labels. The i-vectors are projected onto a 9-dimensional space using linear discriminant analysis (LDA). Whitening and length normalization are applied to the projected i-vectors. We then use Gaussian PLDA with a full-rank subspace to compute the pairwise UV score between the average i-vector of the claimed phrase and the i-vector of the test phrase. Finally, we apply score normalization suitable for this closed-set scenario where the number of possible hypotheses is fixed. We use Max norm which subtracts the maximum score of the other (competing) phrases from the hypothesized phrase score [START_REF] Kinnunen | Utterance verification for text-dependent speaker recognition: A comparative assessment using the RedDots corpus[END_REF]. We also tried Mean norm but it performed worse.

Speaker verification

We develop four standalone ASV systems based on x-vector-PLDA [START_REF] Snyder | X-vectors: Robust DNN embeddings for speaker recognition[END_REF] and Gaussian mixture model-universal background model (GMM-UBM) [START_REF] Reynolds | Speaker verification using adapted Gaussian mixture models[END_REF] approaches.

X-vector-PLDA system: The network architecture for extracting x-vector speaker embeddings is given in Table 1. It differs from the x-vector architecture originally reported in [START_REF] Snyder | X-vectors: Robust DNN embeddings for speaker recognition[END_REF] by adding squeeze-and-excitation (SE) [START_REF] Hu | Squeeze-and-excitation networks[END_REF] modules to each frame-level time-delay neural network (TDNN) layer. In addition, three of the frame-level layers are replaced by residual (RES) blocks [START_REF] He | Deep residual learning for image recognition[END_REF]. The global mean and standard deviation pooling layer is replaced with a learnable dictionary encoder (LDE) [START_REF] Zhang | Deep TEN: Texture encoding network[END_REF][START_REF] Cai | A novel learnable dictionary encoding layer for end-to-end language identification[END_REF]. The LDE layer is similar to a GMM: it assigns features into components and computes statistics locally. We used a variant of LDE with a diagonal covariance matrix shared across all components. For more details of the speaker embedding extractor, see [START_REF] Vestman | Neural i-vectors[END_REF]. The trials are scored with a PLDA module [START_REF] Garcia-Romero | Analysis of i-vector length normalization in speaker recognition systems[END_REF] trained on the training embeddings. Finally, adaptive symmetric score normalization (AS-norm) [START_REF] Cumani | Comparison of speaker recognition approaches for real applications[END_REF] is applied.

GMM-UBM systems:

The success of GMM-UBM systems in speaker verification with short utterances [START_REF] Poddar | Speaker verification with short utterances: a review of challenges, trends and opportunities[END_REF] as well as text-dependent ASV [START_REF] Larcher | Textdependent speaker verification: Classifiers, databases and RSR2015[END_REF][START_REF] Kinnunen | Utterance verification for text-dependent speaker recognition: A comparative assessment using the RedDots corpus[END_REF][START_REF] Sahidullah | Local spectral variability features for speaker verification[END_REF] motivates us to explore this approach for the SdSV challenge. Our GMM-UBM systems are similar to standard GMM-UBM systems except that the UBMs are trained in a more efficient way. Instead of training a UBM on the full dataset, we train 10 GMMs on audio data for each of the 10 phrases independently and create the UBM by combining the components of the 10 GMMs and normalizing the mixture weights to unit sum. In the following, we consider three GMM-UBM systems relying on different acoustic features.

Joint utterance and speaker verification

We perform joint verification of spoken content and speaker identity based on PBMs [START_REF] Sarkar | Incorporating pass-phrase dependent background models for text-dependent speaker verification[END_REF]. In this approach, PBMs are first derived from the GMM-UBM using maximum a posteriori (MAP) adaptation with pass phrase-specific audio data. During the enrollment phase, target speaker models are created by MAP adaptation from the best-matched PBM instead of the single UBM in a GMM-UBM system. The best-matched PBM is found in the maximum likelihood (ML) sense. In the test phase, we first determine the best-matched PBM for the test utterance. Finally, we compute the log-likelihood ratio score between the target model and the best-matched PBM. Our primary submission includes two PBM systems based on two different sets of bottleneck features (BN) extracted with deep neural networks (DNNs).

Phone-discriminative bottleneck (Phone-BN) features are extracted by a DNN trained on phone labels obtained using an HMM. We use the in-domain audio data of the SdSV challenge along with phone sequence information. We chose the flat-start method [START_REF] Young | The HTK Book (for version 3.4)[END_REF] for HMM training as phone-level alignments are not available. First, we train HMM-based monophone models by pooling all the utterances and their corresponding phone sequences. Parameters of the phonebased HMMs are then re-estimated in an iterative manner using the Baum-Welch algorithm. We consider 42 HMMs that correspond to the number of unique phones in the training data. We consider a 3-state (excluding start and end-state) left-to-right topology without skipping state transitions. We also model silences and pauses between words. We generate phone-level alignments and discard silences and pauses. Then we train a DNN to classify phones. The frame-level output of one hidden layer [START_REF] Sarkar | Time-contrastive learning based deep bottleneck features for text-dependent speaker verification[END_REF] is then projected using principal component analysis (PCA) to obtain lower-dimensional Phone-BN features. Figure 1 illustrates the Phone-BN feature extraction system.

Stream-wise time-contrastive learning based BN (sTCL-BN) features are extracted in a similar way, except that the phone classes used as DNN training targets are found in an unsupervised way [START_REF] Sarkar | Time-contrastive learning based deep bottleneck features for text-dependent speaker verification[END_REF][START_REF] Sarkar | Time-contrastive learning based DNN bottleneck features for text-dependent speaker verification[END_REF]. First, all the speech utterances are randomly concatenated into a single stream. This stream is segmented into fixed-duration chunks that capture shortterm context. Given the desired number N of classes, N chunks are taken at a time, and data points in the n-th chunk are assigned to class label n where n ∈ {1, 2, ..., N }. The process is repeated until all data points have been assigned. A segment-based clustering algorithm [START_REF] Sarkar | Time-contrastive learning based deep bottleneck features for text-dependent speaker verification[END_REF][START_REF] Sarkar | Time-contrastive learning based DNN bottleneck features for text-dependent speaker verification[END_REF] is applied to group similar chunks together and update the class labels until convergence. We then train a DNN to discriminate the obtained labels. The frame-level output of a hidden layer is projected into a low dimensional space using PCA to obtain sTCL-BN features.

EXPERIMENTAL SETUP

Dataset description

The audio data for the SdSV challenge was created from the DeepMine dataset collected by crowdsourcing [START_REF] Zeinali | DeepMine speech processing database: Text-dependent and independent speaker verification and speech recognition in Persian and English[END_REF][START_REF] Zeinali | A multi purpose and large scale speech corpus in Persian and English for speaker and speech recognition: the DeepMine database[END_REF]. Task 1 involves speech files from 963 speakers as in-domain audio data. There is no development set which could be used for parameter tuning and optimization. We have created a development set by randomly choosing a subset of 63 speakers from this in-domain data. Similarly to the evaluation set, we enroll each speaker with three utterances for phrase-specific targets.

If adequate data for a speaker-phrase combination is not available, we disregard that target model. The remaining sentences are used for test. We obtain a total of 519 target models and 4,810 speech utterances for test in the development set. The number of trials is summarized in Table 2. Similarly to the evaluation set, there is no cross-lingual trial. We have also discarded same-gender trials by clustering i-vectors for the 63 speakers (averaged over all utterances) into two pseudogender classes and keeping the trials for which the speakers in the enrollment model and the test utterance fall into the same class. The remaining in-domain data from 900 speakers consisting of 94,661 utterances are considered for system development in addition to the other permitted audio data, such as VoxCeleb and LibriSpeech. The evaluation set consists of 12,404 enrollment models, 69,542 utterances, and 8,306,700 trials.

Dataset and parameters for system training

We refer the seven subsystems in the UIAI entry as S1-S7.

S1 is the UV system and it uses mel-frequency cepstral coefficient (MFCC) features. We extract 20-dimensional static MFCCs, apply a RASTA filter [START_REF] Hermansky | Rasta processing of speech[END_REF], and compute deltas and double-deltas to create 60-dimensional features. Utterancelevel cepstral mean and variance normalization (CMVN) is applied after discarding non-speech frames using energybased activity detection (SAD). A 512-component UBM is trained on in-domain training data. The T-matrix is estimated with 600 factors on the same audio data. The extracted ivectors are projected to 9 dimensions using LDA based on phrase labels.

S2 is the x-vector-PLDA ASV system. The x-vector extractor is trained on YouTube audio data from VoxCeleb-1 [START_REF] Nagrani | VoxCeleb: A large-scale speaker identification dataset[END_REF] and VoxCeleb-2 [START_REF] Chung | Vox-Celeb2: deep speaker recognition[END_REF]. Recordings from the same YouTube video source are concatenated together. The scoring back-end (PLDA, centering, whitening, AS-norm) is trained on the in-domain data. For PLDA, the training labels are pairs of speaker and phrase IDs. Both in-domain and VoxCeleb data are augmented 5-fold using Kaldi's [START_REF] Povey | The Kaldi speech recognition toolkit[END_REF] augmentation recipes which include reverberation and additional noise, babble, or music. The input features for the embedding extractor are 60-dimensional, cepstral mean normalized MFCCs extracted with Kaldi. Kaldi's energy based VAD is applied to remove non-speech frames.

S3-5 are the GMM-UBM based ASV systems. Ten phrase-specific 512-component GMMs are trained on the in-domain data and merged into a 5120-component UBM. The target speaker models are created by MAP adaptation with a relevance factor of 3. We use three different acoustic front-ends. System S3 is based on 60-dimensional MFCCs including deltas and double-deltas. System S4 uses linearfrequency cepstral coefficients (LFCCs) with the same dimension. System S5 uses 66-dimensional overlapped block transform coefficients (OBTCs) computed with two blocks of sizes 9 and 13 [START_REF] Sahidullah | Design, analysis and experimental evaluation of block based transformation in MFCC computation for speaker recognition[END_REF]. The pre-and post-processing stages are identical for the three feature sets. We use 20 mel filters and retain the energy coefficients. The features are processed with RASTA filtering and utterance-level CMVN. We do not apply SAD since this degraded performance.

S6 is the PBM-based joint verification system with Phone-BN features. The DNN feature extractor consists of 7 fullyconnected layers with 1024 neurons and sigmoid activation. Its inputs are 57-dimensional RASTA-filtered MFCCs including deltas and double-deltas with a context of 11 frames. The number of outputs is 42. The DNN is trained on the in-domain data using CNTK [START_REF]An introduction to computational networks and the computational network toolkit[END_REF]. We compute BN features by projecting the output of the second hidden layer on each time frame into a 57-dimensional vector using PCA. The PCA matrix is computed on 10,000 randomly selected utterances from the training part of LibriSpeech. We use the open-source robust voice activity detector (rVAD) [START_REF] Tan | rVAD: An unsupervised segment-based robust voice activity detection method[END_REF] to discard non-speech frames from the enrollment and test utterances before applying utterance-level CMVN. The HMM used for extracting the phone labels does not discard non-speech frames, but it uses utterance-level CMVN. We use HTK [START_REF] Young | The HTK Book (for version 3.4)[END_REF] to build the HMM system. The Phone-BN features are then used with the PBM system where a gender-independent GMM-UBM with 2048 Gaussian components is trained using 60, 000 utterances from LibriSpeech. We create the target models using MAP adapta-tion with 3 iterations. We adapt only the mean vectors with a relevance factor of 10.

S7 is the PBM-based joint verification system with sTCL features. The DNN feature extractor uses the same input features as S6. We use rVAD [START_REF] Tan | rVAD: An unsupervised segment-based robust voice activity detection method[END_REF] to discard non-speech frames, and we consider N = 10 unsupervised classes as in our previous study [START_REF] Sarkar | Time-contrastive learning based deep bottleneck features for text-dependent speaker verification[END_REF]. The DNN and the PBMs are trained on the same audio data using the same hyper-parameters as S6.

Performance evaluation

The primary metric for the challenge is the normalized minimum detection cost function (minDCF) [START_REF] Zeinali | Shortduration speaker verification (SdSV) challenge 2020: the challenge evaluation plan[END_REF]. The EER is also reported. These are computed by treating TC trials as genuine and by pooling the remaining three as impostor. We computed those metrics on the development set using our own scoring script while the challenge organizers computed performance on the evaluation set. We also report the performance for the three sub-conditions and their average values on the development set.

RESULTS

The ASV performance of the individual subsystems on the development set is shown in Table 3. The first row shows the ASV result of the UV system S1 which performs well in wrong pass-phrase conditions (TW and IW). However, the performance for the IC condition is random with about 50% EER as this system does not use speaker information. The next row shows the performance for the x-vector-based ASV system S2. Although it yields promising performance in the IC and IW conditions, it performs poorly in the TW condition. Similarly, the GMM-UBM systems perform relatively well in the IC and IW conditions but they fail in the TW condition. Our results also indicate that the GMM-UBM systems give competitive performance compared to the x-vector system. The short-term OBTC features outperform MFCCs in all cases. Out of the two PBM-based methods, the one based on Phone-BN features performs consistently better and both of these methods outperform other subsystems in terms of average EER and minDCF. S6 achieves the lowest average and pooled EERs as well as minDCFs. For this reason, we select it as the single system for the challenge.

We build the primary system submitted to the challenge by combining the modules for UV and ASV. We fuse S1, S6 and S7 for the UV task and all the subsystems except S1 for the ASV task. The subsystems for each task are combined by linear score weighting where the weights are optimized on the development set by linear search. The UV and ASV system are then combined by cascade fusion as described earlier. The trials with wrong pass-phrases as detected by the fused UV system are assigned an ASV score of -100, while the trials with correctly detected pass-phrases are retained with fused ASV score. Table 4 summarizes the results achieved by the single system and the primary system on the development and evaluation sets. The single system outperforms the two challenge baselines, especially in terms of minDCF. Figure 2 shows the DET plots of the submitted systems along with the baselines.

CONCLUSIONS

We have described the UIAI systems submitted to the SdSV challenge 2020 for text-dependent ASV. The systems developed are a fusion of different subsystems using various frontends and back-ends. To deal with the pass-phrase verification problem, we combined the UV system with ASV in a cascade mode. Our development set created with a subset of limited in-domain data generalized well to the evaluation set by estimating suitable fusion parameters and by demonstrating systematic improvement. However, there was a substantial performance gap between the development and evaluation set possibly due to the large number of speakers and presence of unknown noises in the evaluation set.

Systems studied for the challenge submissions were developed independently by four different sites using various features and classifiers. This work could be extended towards a more systematic comparison of front-end acoustic features and different back-end classifiers. For instance, our phonebottleneck features could be studied with x-vector system adopted for this work.

ACKNOWLEDGMENT

Experiments presented in this paper were partially carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000.fr). This work has been partially sponsored by Academy of Finland (proj. no. 309629).

Fig. 1 .

 1 Fig. 1. Phone-BN feature extraction system.

Fig. 2 .

 2 Fig. 2. DET plots of the UIAI systems along with baselines.

Table 1 .

 1 Architecture of speaker embedding extractor network. The layers from 1 to 8 and layer 10 are followed by leaky ReLU activations and batch normalization. The embeddings are extracted from layer 10 before applying the activation function.

	#	Layer type	CNN kernel size Output dim.
	1	TDNN-SE	5	512
	2,3 TDNN-RES-SE	5	512
	4,5 TDNN-RES-SE	7	512
	6,7 TDNN-RES-SE	1	512
	8	TDNN-SE	1	128
	9	LDE aggregation	-	8,192
	10 FC	-	512
	11 FC-softmax	-	#speakers

Table 2 .

 2 Number of trials per condition in the development set.

	TC	TW	IC	IW	Total
	4,810	19,236	119,737	478,946	622,729

Table 3 .

 3 Results (EER in % / minDCF) in the development set for the individual subsystems included in the UIAI primary submission.

	ID	Method	Short-term features	Task	TW	IC	IW	Pooled	Avg.
	S1	i-vector	MFCC	UV	0.08 / 0.005	51.64 / 1.00	0.08 / 0.005	18.29 / 1.00	17.27 / 0.337
	S2	x-vector	MFCC	ASV	9.11 / 0.575	0.82 / 0.041	0.12 / 0.004	0.93 / 0.085	3.35 / 0.207
	S3	GMM-UBM	MFCC	ASV	8.35 / 0.431	0.91 / 0.040	0.60 / 0.019	1.36 / 0.090	3.29 / 0.164
	S4	GMM-UBM	LFCC	ASV	10.69 / 0.535	1.27 / 0.057	0.77 / 0.028	1.68 / 0.116	4.24 / 0.206
	S5	GMM-UBM	OBTC	ASV	7.82 / 0.405	0.85 / 0.035	0.60 / 0.016	1.22 / 0.085	3.09 / 0.152
	S6	PBM	Phone-BN	Both	0.07 / 0.008	1.31 / 0.051	0.01 / 0.001	0.81 / 0.029	0.46 / 0.020
	S7	PBM	sTCL-BN	Both	0.14 / 0.011	1.74 / 0.062	0.01 / 0.001	1.07 / 0.038	0.63 / 0.025

Table 4 .

 4 Results (EER in % / minDCF) of single and primary systems in the development and evaluation sets. The x-vector based baseline achieves 9.05% EER and 0.529 minDCF while the i-vector/HMM baseline achieves 3.49% EER and 0.146 minDCF.

	System	TW	IC	IW	Pooled	Avg.	Eval set. (Pooled)
	Single (S6)	0.07 / 0.008	1.31 / 0.051	0.01 / 0.001	0.81 / 0.029	0.46 / 0.020	3.83 / 0.118
	Primary (fusion S1-S7)	0.06 / 0.006	0.32 / 0.015	0.00 / 0.000	0.17 / 0.007	0.13 / 0.007	2.14 / 0.072