Divine Maalouf

Ahmed Chemori
email: ahmed.chemori@lirmm.fr

Vincent Creuze

L 1 Adaptive Depth and Pitch Control of an Underwater Vehicle with Real-Time Experiments

Keywords: Adaptive Control, Autonomous Systems, Underwater vehicles, Robustness, Experimentations

Introduction

Underwater vehicles have gained an increased interest in the last decades given the multiple tasks they can accomplish in various fields ranging from scientific to industrial and military applications. For this reason, it is less risky and more advantageous to send robots in hazardous environments such as deep seas and oceans. In our study, we are particularly interested in the category of tethered underwater vehicles also called Remotely Operated Vehicles (ROV). Different challenges in autonomous control of such systems arise from the inherent high nonlinearities and time varying behavior of the vehicle's dynamics subject to hydrodynamic effects and external disturbances. To deal with these problems, various advanced control techniques have been proposed in litterature such as robust H ∞ approaches [START_REF] Roche | A Hierarchical Varying Sampling H ∞ Control of an AUV[END_REF], backstepping control [START_REF] Lapierre | Nonlinear path-following control of an AUV[END_REF], predictive control [START_REF] Steenson | Experimental Verification of a Depth Controller using Model Predictive Control with Constraints onboard a Thruster Actuated AUV[END_REF] and sliding mode control [START_REF] Pisano | Output-feedback control of an underwater vehicle prototype by higher-order sliding modes[END_REF]. Intelligent control methods using reinforcement learning or artifical intelligence have also been proposed [START_REF] Kim | A novel neuro-fuzzy controller for autonomous underwater vehicles[END_REF] (Chang et al., 2003) (Carreras et al., 2002). A more detailed overview of the proposed controllers for underwater vehicles can be found in [START_REF] Yuh | Design and Control of Autonomous Underwater Robots : A Survey[END_REF] and [START_REF] Yildiz | State-of-the-Art System Solutions for Unmanned Underwater Vehicles[END_REF]. In all of these cited articles, the need for robustness and adaptation is highlighted since the robot's parameters are hard to identify and the environment where it operates is likely to change. For example, the addition of sensors or payloads modifies the weight and the position of the center of mass of the vehicle, furthermore the experimental conditions vary whether the vehicle operates in a sea or in a river resulting in a different buoyancy for each case. In order to avoid a degradation in the performance of the controlled system during a specific mission, the vehicle is expected to possess a self tuning ability and compensate for different kind of disturbances. That is why adaptive controllers are very popular for such systems (Fjellstad & Fossen, 1994a) [START_REF] Antonelli | Adaptive control of an autonomous underwater vehicle: experimental results on ODIN[END_REF] [START_REF] Antonelli | A New Adaptive Control Law for the Phantom ROV[END_REF] [START_REF] Antonelli | An Adaptive Law for Guidance and Control of Remotely Operated Vehicles[END_REF]. Since adaptation was found to be a necessity, it was combined with other techniques such as sliding mode control [START_REF] Fossen | Adaptive control of nonlinear underwater robotic systems[END_REF], fuzzy logic control [START_REF] Marzbanrad | A Robust Adaptive FUzzy Sliding Mode Controller for Trajectory Tracking of ROVs[END_REF] or backstepping control [START_REF] Lapierre | Robust diving control of an AUV[END_REF]. However, various problems are related to the implementation of an adaptive controller on an underwater vehicle. For instance, we notice that most of these adaptive schemes require regressor matrices based on a knowledge of the dynamic model and a large set of parameters to be estimated. In order to remediate to this problem, some non-regressor based controllers emerged such as the one proposed in [START_REF] Yuh | Experimental Study on Adaptive Control of Underwater Robots[END_REF] also detailed in [START_REF] Yuh | Application of non-regressor-based adaptive control to underwater robots: experiment[END_REF] where a combination of unknown bounded constants are estimated. In [START_REF] Zhao | Experimental Sutdy on Advanced Underwater Robot Control[END_REF], a disturbance observer has been added to the initial controller of [START_REF] Yuh | Application of non-regressor-based adaptive control to underwater robots: experiment[END_REF] and interesting experimental results have been obtained in nominal conditions as well as in case of external disturbances rejection and robustness towards parameters' uncertainties. The advantage of this method is that it does not require any a priori knowledge about the system; furthermore the update of the parameters is based on the performances of the closed-loop system. However, the drawbacks of such a method lies in the neglect of the coupling effects between the degrees of freedom due to the spherical shape of the used robot. The model parameters of the dynamic model can be initialized randomly, but the control parameters governing this method are very critical to be tuned and highly dependent on the initial configuration of the robot. In [START_REF] Sun | Adaptive Setpoint for Autonomous Underwater Vehicles[END_REF], an adaptive control scheme based on a saturated proportional derivative feedback law was proposed for a setpoint control case. The contribution of this last paper lies in the use of only one regressor being for gravity instead of a full regressor for the whole dynamic model. In addition to that, an approximation was made when the transpose of the transformation matrix between the body frame and the earth frame was used instead of its inverse. Six adaptive controllers, including the three previously cited (Fjellstad & Fossen, 1994a) [START_REF] Yuh | Application of non-regressor-based adaptive control to underwater robots: experiment[END_REF] [START_REF] Sun | Adaptive Setpoint for Autonomous Underwater Vehicles[END_REF], were compared in [START_REF] Antonelli | On the use of Adaptive/Integral actions for Six-Degreesof-Freedom control of Autonomous Underwater Vehicles[END_REF] in simulation within a study that focuses on the ability of the controllers to compensate for the persistent effects (restoring forces and ocean currents). The non-regressor-based methods [START_REF] Yuh | Application of non-regressor-based adaptive control to underwater robots: experiment[END_REF] [START_REF] Sun | Adaptive Setpoint for Autonomous Underwater Vehicles[END_REF] were unable to compensate for the restoring forces and the model-based methods (Fjellstad & Fossen, 1994a) [START_REF] Fossen | The NEROV autonomous underwater vehicle[END_REF])(Fjellstad & Fossen, 1994b) needed adequate persistent excitation. This will generate a problem at steady state when a static error occurs in presence of waves or current. Indeed, in this scenario, the parameter excitation will be reduced since the error on the velocity is zero while the position error is not, and therefore a corrective adaptive con-trol action cannot be triggered. The adaptive control law introduced in [START_REF] Antonelli | A novel adaptive control law for Autonomous Underwater Vehicle[END_REF] was the one defended in the comparative study of [START_REF] Antonelli | On the use of Adaptive/Integral actions for Six-Degreesof-Freedom control of Autonomous Underwater Vehicles[END_REF] because it accomplishes the desired full compensation, however it still requires the adaptation of nine parameters with a suitable initialization of the restoring parameter vector and a reasonable choice for the adaptation gain. The simulations were performed on an ellipsoidal autonomous vehicle weighing 225kg. Another comparative study between adaptive controllers was made in (Fossen & Fjellstad, 1996) where the two adaptive controllers previously presented in [START_REF] Slotine | Hamiltonian adaptive control of spacecraft[END_REF] and [START_REF] Sadegh | Robustness analysis of a class of adaptive controllers for robotic manipulators[END_REF] were compared in simulations in presence of unknown model parameters and noisy measurements. The conclusion was that by replacing the regressor with the desired state trajectories in [START_REF] Sadegh | Robustness analysis of a class of adaptive controllers for robotic manipulators[END_REF]) a better robustness towards measurement noise was noticed. All of the previously cited papers on adaptive schemes reveal that despite their numerous advantages, these controllers hold some drawbacks that have been highlighted above for our specific application. Without a loss of generality, we notice that the same problems are present in other applications and therefore based on the above discussion we can list some of the main commun drawbacks of adaptive control: i) A wide range of such controllers exhibit undesirable frequency characteristics and are often used with restrictive assumptions [START_REF] Rohrs | Stability problems of adaptive control algorithms in the presence of unmodeled dynamics[END_REF]. The authors showed that sinusoidal reference inputs at certain frequencies and/or sinusoidal output disturbances at any frequency will cause the adaptation gain to significantly increase which will destabilize the control system.

ii) The need for the persistency in excitation can lead to a bad transient behavior [START_REF] Zang | Transient bounds for adaptive control systems[END_REF].

iii) An increase in the adaptation gain drives the closed-loop system closer to instability while a small gain would slow down the convergence rate [START_REF] Narendra | A new adaptive law for robust adaptation without persistent excitation[END_REF]. iv) Any parameter vector to be adapted must be adequately initialized and this choice would depend on the specific configuration of the system. This would be even more critical for the non-model-based controllers [START_REF] Antonelli | On the use of Adaptive/Integral actions for Six-Degreesof-Freedom control of Autonomous Underwater Vehicles[END_REF].

That is why a control approach that can ensure a robustness decoupled from adaptation would be highly desirable. Such a scheme would drive the closed-loop system to its desired trajectory while overcoming the drawbacks listed above. This decoupling has recently been proven to be achievable by the L 1 adaptive control scheme presented in [START_REF] Hovakimyan | L 1 Adaptive Control Theory[END_REF]. It can be compared to a Model Reference Adaptive Controller (MRAC) modified in two ways: a state predictor is used instead of a reference model and a low-pass filter is inserted in the feedback loop to cancel out undesirable high frequencies in the control input. This controller can ensure a good performance with zero parameter initialization and without any necessity for a specific excitation. It is worth to note that, to the best knowledge of the authors, this control scheme was mainly applied to aerial vehicles [START_REF] Kaminer | Path following for unmanned aerial vehicles using L 1 adaptive augmentation of commercial autopilots[END_REF] [START_REF] Kaminer | Path following for unmanned aerial vehicles using L 1 adaptive augmentation of commercial autopilots[END_REF] and mechatronic systems [START_REF] Techy | Nonlinear Control of a Novel Two-Link Pendulum[END_REF])(Fan & Smith, 2008) and has never been applied yet to underwater vehicles. We propose in this paper the theoretical aspects of the de-sign of an L 1 adaptive controller to be applied for the first time on an underwater vehicle. Moreover, we experimentally compare this new controller, in underwater robotics, with the well proven nonlinear state feedback controller proposed in [START_REF] Fossen | Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF]. Various scenarios allow us to compare each controller's performances in nominal conditions and in presence of external disturbances, as well as their robustness towards parameter uncertainties. Furthermore, the vehicles used in the previously cited papers are significantly bigger and heavier than the one used for our study (it weighs only 3kg). Indeed, a small vehicle is more sensitive to any kind of disturbance compared to a larger one thus placing us in difficult experimental conditions. The tether alone, given its weight and diameter relative to the dimensions of the robot, constitutes in itself a persistant disturbance to be compensated. This paper is an extension of the work presented in [START_REF] Maalouf | A novel application of multivariable l1 adaptive control: From design to real-time implementation on an underwater vehicle[END_REF] where only depth and pitch were controlled with an L 1 adaptive controller. In the actual work, not only new experimental results are introduced to highlight the capacity of the L 1 adaptive controller to reject external disturbances, but a thorough quantitative comparative study with the nonlinear adaptive state feedback controller is also provided for different experimental scenarios. This study emphasizes the superiority of the L 1 adaptive controller in terms of fast adaptation and robustness. This paper is organized as follows: in the second section we present the dynamic modeling of the system, the third section describes the two proposed control schemes, the forth section presents the prototype and its experimental setup, in the fifth section the obtained experimental results will be presented and discussed, and in the sixth section the performance of the two controllers will be analyzed and compared. Finally the paper ends with some concluding remarks.

Dynamic modeling of the underwater vehicle

Modeling Background

The dynamics of an underwater vehicle involves two frames of reference: the body-fixed frame and the earth-fixed frame (cf. Figure 1). The vectors with the variables put at stake are ν = [u, v, w, p, q, r] T and η = [x, y, z, φ, ϑ, ψ] T being the vectors of velocities (in the body-fixed frame) and position/Euler angles (in the earth-fixed frame) respectively. By considering the inertial generalized forces, the hydrodynamic effects, the gravity, and buoyancy contributions as well as the effects of the actuators (i.e thrusters), the dynamic model of an underwater vehicle in matrix form, using the SNAME notation and the representation described in [START_REF] Fossen | Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF], can be written as follows: η = J(η)ν

(1)

M ν + C(ν)ν + D(ν)ν + g(η) = τ + w d (2) J(η) ∈ R 6×6
is the transformation matrix mapping the body-fixed frame to the earth-fixed one. The model matrices M, C, and D denote inertia (including added mass), Coriolis-centripetal (including added mass), and damping respectively, while g is a vector of gravitational/buoyancy forces. τ is the vector of control inputs and w d the vector of external disturbances. In the case of our study, the vehicle used has a slow dynamics, and hence it will be moving at velocities low enough to make the Coriolis terms negligible (C(ν) ≈ 0). Considering this assumption, the dynamics (2) can be rewritten as:

M ν + n(ν, η) = τ + w d (3) with n(ν, η) = D(ν)ν + g(η).
Equation ( 2) describes the dynamics of the system in 6 degrees of freedom taking into account the 3 translations and 3 rotations. The input vector τ ∈ R 6 considers 6 actions on the system to fully control it. The presented formulation of the robot's dynamics is expressed in the body-fixed frame and can be transformed to the earthfixed frame by using the kinematic transformations of the state variables and the model parameters as follows:

η = J(η)ν η = J(η)ν + J(η)ν M * (η) = J -T (η) M J -1 (η) D * (ν, η) = J -T (η) D(ν) J -1 (η) g * (η) = J -T (η) g(η) τ * = J -T (η) τ w * d = J -T (η) w d (4)
Equation ( 2) can therefore be expressed in the earth fixed frame as:

M * (η)η + D * (ν, η)η + g * (η) = τ * + w * d (5)

Modeling of the Studied Dynamics

The previous section describes the dynamics of an underwater vehicle in its full six degrees of freedom. In this paper, the proposed controllers will target two degrees of freedom due to the limitations we have in terms of sensors and actuators.

Therefore, we have chosen to study a trajectory of motion in depth which is the vertical translational motion (along the z axis) and in pitch ϑ which is the rotation around the y axis. In fact, the inertial measurement unit (IMU) is the only sensor giving information concerning surge and sway (translational motion along x and y respectively). This information is the acceleration which is not enough to give an adequate estimate of the needed position values: integrations need to be done leading to a drift that cannot be corrected since no other sensor is available on board for this purpose. On the other hand, yaw measurement cannot be used due to the very short distance between the magnetic sensor and motors inducing strong magnetic disturbances. Due to the small size of the vehicle, this distance cannot be increased. The roll is not actuated in this robot and therefore we were left with the depth and pitch to control. These two degrees of freedom are coupled and hold high nonlinearities. For this reason, in the case where the adequate instrumentation would be available (i.e. x, y, and yaw measurements), the controller could easily be extended to other degrees of freedom. We can easily extract our studied dynamics from (2) to get the reduced model of interest in the body-fixed frame as:

           M z 0 0 M ϑ                       ẇ q            +            D z 0 0 D ϑ                       w q            +            -cos(ϕ)cos(ϑ)(W -B) Wr g z sin(ϑ)            =            τ z + w d z τ ϑ + w d ϑ            (6)
The matrices of mass and damping constituted of the terms M z , M ϑ and D z , D ϑ respectively are considered as diagonal given the symmetrical cubic shape of our robot. W is the weight of the robot, B its buoyancy and r g z is the z coordinate of the robot's center of mass.

τ r =            τ z τ ϑ           
is expressed in Newton and is given by:

τ r = T Ku (7)
where u ∈ R 2 is the vector of control inputs in volts generated by the two vertical thrusters acting simultaneously on the depth and pitch, K is the force coefficient in Newton/Volt that has been identified after several tests performed on the motors.

T ∈ R 2×2 is the actuators configuration matrix taking into account the position and orientation of the motors, thus allowing to determine the associated forces in the body frame.

Similarly, we extract from (5) the studied dynamics in the earth frame to be:

           M * z 0 0 M * ϑ                       z θ            +            D * z 0 0 D * ϑ                       ż θ            +            -(W -B) Wr g z cos(ϕ)sin(ϑ)            =            τ * z + w * d z τ * ϑ + w * d ϑ            (8)
All the starred terms can be computed in a straightforward manner using transformations in (4).

Proposed adaptive control schemes

Before introducing the design of the proposed L 1 adaptive controller, we briefly remind the design of the well-known Adaptive Nonlinear State Feedback Controller (ANSF) and its formulation to our application. In fact, this well proven controller will be used in our experiments as a basis of comparison in order to highlight the performances of the L 1 adaptive controller. In the rest of the paper, the subscript r in the model matrices refers to the reduced model ( 8) pertaining to the studied dynamics.

Adaptive Nonlinear State Feedback Controller

Background

The adaptive state feedback controller provides an online estimation of the unknown model parameters in order to ensure a good trajectory following by the closed-loop system [START_REF] Fossen | Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF]. The control law obtained based on the dynamics of the robot presented in equation ( 3) and given by:

τ = Ma b + n(ν, η) (9)
where the hat symbol denotes the parameters' estimates, a b the commanded acceleration described in the body-fixed frame, and n(ν, η) the estimate of n(ν, η) in

(3). Given that the dynamic model is linear in its parameters, the adaptive control law (9) can then be rewritten as:

τ = Φ(a b , ν, η) θ ( 10
)
where Φ is the regressor matrix and θ is the vector of the estimated parameters.

The computed input is calculated in the body-fixed frame but the trajectory following is performed in the earth-fixed frame and therefore a b is calculated from a simple transformation between the body and the earth fixed frames and is given by:

a b = J -1 (a n -Jν) (11)
where a n is the commanded acceleration in the earth-fixed frame and J the transformation matrix from the body-fixed frame to the earth-fixed frame with J its first derivative. To guarantee that the error converges to zero, a n is chosen as follows:

a n = ηdes -K P η -K I t 0 η dt -K D η (12)
with η = ηη des and η is its first derivative, η des is the desired trajectory and ηdes is its corresponding acceleration.

The vector of the estimated parameters is updated according to the following update law:

θ = -ΓΦ T (a b , ν, η)J -1 y A ( 13
)
where Γ is a diagonal positive definite matrix representing the adaptation gain, and y A the combined error defined as follows:

y A = c 0 η + c 1 η (14)
c 0 and c 1 are constant positive gains. The choice of their values is governed by the algorithm presented in [START_REF] Fossen | Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF] guaranteeing the convergence of the error to zero. The proof of stability is made by applying Barbalat's lemma.

Application to the studied dynamics

The vector of parameters to be estimated includes the elements of the matrices M * r and D * r and the parameters in the vector g * r . From equations ( 9) to ( 14), we extract the explicit formulation of our controller as:

τ r = Φ r θr (15)
with the vector of the estimated parameters being:

θr = Mz Dz W -B Mϑ Dϑ x G W -x B B z G W -z B B T (16) (W -B)
is the parameter representing the difference between the weight and the buoyancy. x G and z G are the coordinates of the center of gravity in the body-fixed frame and x B and z B are the ones of the center of buoyancy. Given that the origin of our body-fixed frame is in fact the center of buoyancy and that we consider both centers of gravity and buoyancy aligned due to the symmetry of the robot, we have x G = x B = z B = 0 and we can therefore simplify our parameters' vector to the following:

θr = Mz Dz W -B Mϑ Dϑ z G W T ( 17
)
with this simplification taken into account, the regressor matrix takes the form:

Φ r =            a b z w -cos(ϑ)cos(ϕ) 0 0 0 0 0 0 a b ϑ q sin(ϑ)            (18)
the commanded acceleration in the earth frame is given by:

a n r =            zdes θdes            -K P            z -z des ϑ -ϑ des            -K I t 0            z -z des ϑ -ϑ des            dt -K D            ż -żdes θ -θdes            (19)
K P , K I and K D ∈ R 2×2 are diagonal positive constant matrices representing respectively the proportional, the integral and the derivative gains. The commanded acceleration in the body-fixed frame is:

a b r =            cos(ϕ)cos(ϑ) 0 0 cos(ϕ)            . a n r -            -qsin(ϑ) -pcos(ϑ)sin(ϕ) 0 0 -φsin(ϕ)                       w q            (20)
the parameters adaptation law writes:

θr = -Γ r Φ T r            cos(ϕ)cos(ϑ) 0 0 cos(ϕ)            y A r (21)
the combined error is expressed by:

y A r = c 0            z -z des ϑ -ϑ des            + c 1            ż -żdes θ -θdes            (22)
Finally, given the configuration matrix T and the force coefficient K explained in equation ( 7) the control input will take the form:

u = T -1 τ r K (23)
It is important to emphasize that a good parameter convergence is more guaranteed to occur when the reference trajectory is rich enough to excite the parameters to be estimated [START_REF] Slotine | Applied Nonlinear Control[END_REF]. These parameters will converge to a set of values that allows trajectory following. Moreover, the vector of parameters is seen to be bounded according to Barbalat's lemma as shown in the proof of stability given by [START_REF] Fossen | Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF].

L 1 adaptive controller

Background

To control the system described in (5), an L 1 adaptive controller is proposed.

The choice of this controller is motivated by its architecture characterized by the decoupling between adaptation and robustness. High adaptation gains can then be chosen securing a fast convergence with a smooth transient response. This architecture described in [START_REF] Hovakimyan | L 1 Adaptive Control Theory[END_REF] is shown in the following block-diagram (cf. Figure 2). The closed-loop system holds a prediction phase and an adaptation phase. In the feebdack, a low pass filter is added to cancel out the high frequencies that might occur in the control input. This last component ensures fast adaptation without harming the robustness. The details of the different blocks of Figure 2 are the following:

• Controlled System: We will start by considering the following class of nonlinear systems which including our underwater vehicle:

ẋ1 (t) = x 2 (t) x 1 (0) = x 1 0 ẋ2 (t) = f (t, x(t)) + B 2 ωu x 2 (0) = x 2 0 y(t) = Cx(t) (24)
where x 1 ∈ R n and x 2 ∈ R n are the states of the system forming the complete state vector:

x(t) = [x T 1 (t), x T 2 (t)] T . u(t) ∈ R m is the control input vector (m ≤ n) and ω ∈ R m×m is the uncertainty on the input gain. B 2 ∈ R n×m
is a constant full rank matrix. C ∈ R m×n is a known full rank constant ma-trix, y ∈ R m is the vector of measured outputs and f (t, x(t)) is an unknown nonlinear function representing the nonlinear dynamics. The partial derivatives of this function are assumed to be semiglobaly uniformly bounded and f (t, 0) is assumed to be bounded. The previous system of equations can be transformed into a semi linear one with some assumptions on the boundedness of the parameters and function f (t, x(t)) as described in [START_REF] Cao | Robust adaptive control of underwater vehicles: A comparative study[END_REF]. It is concluded that this function can be rewritten

as f (t, x(t) = A 2 x 2 + θ(t)||x(t)|| L ∞ + σ(t)
with θ(t) and σ(t) unknown varying parameters. Therefore the system can be rewritten as follows:

ẋ(t) =            0 n×n I n 0 n×n A 2                       x 1 x 2            +            0 n×1 θ            ||x|| L ∞ +            0 n×1 σ            +            0 n×m B 2            ωu (25) y(t) = Cx(t) (26) Let A =            0 n×n I n 0 n×n A 2           
be the state matrix describing the actual open-loop system dynamics. It should be modified into a Hurwitz matrix describing the desired closed-loop dynamics using a static feedback gain k m . We would

therefore get A m = A -B m k m with B m =            0 n×m B 2            .
The system can then be finally rewritten in compact form as:

ẋ(t) = A m x(t) + B m ωu a + θ(t)||x(t)|| L ∞ + σ(t) ; x(0) = x 0 y(t) = Cx(t) (27)
Given their structure, the vectors θ and σ can be summed to the control input as shown above. In case the vector B 2 is not an identity matrix, these two uncertain varying parameters would be scaled by the constants contained in B 2 . Since they are unknown, the notation for these two variables will not be changed. u a is the control input used for adaptation after the transformation of the matrix A into A m . The final control input to be applied to the system

is u = u m + u a with u m = -k m x.
• State Predictor: The states of the system are calculated at each iteration using the estimated parameters obtained from the adaptation phase along with the control input. The prediction equation is then written as:

ẋ(t) = A m x(t) + B m ω(t)u a (t) + θ(t)||x(t)|| L ∞ + σ(t) (28)
• Adaptation Phase: This stage uses the error between the measured states and the estimated ones to adapt the parameters while considering a projection method in order to ensure their boundedness. In fact, a projection operator avoids the parameter drift using the gradient of a convex function and a maximal bound on the parameters to be estimated. The adapation law for each estimated parameter is given by:

θ(t) = ΓPro j( θ(t), -( xT (t)PB m ) T ||x(t)|| L ∞ ) σ(t) = ΓPro j( σ(t),-( xT (t)PB m ) T ) ω(t) = ΓPro j( ω(t), -( xT (t)PB m ) T u T a (t)) (29)
The parameter P is the solution to the algebraic Lyapunov equation:

A T m P + PA T m = -Q for any arbitrary symmetric matrix Q = Q T > 0.
Γ is the adaptation gain and x(t) the prediction error.

• Control law formulation: The last stage pertains to the formulation of the control input characterized by the addition of a low pass filter. It is written as: This control architecture with the equations included can be summarized in Figure 3.

u a (s) = -kD(s)(η l (s) -k g r) ( 30

Application to the studied dynamics

This control architecture is applied on two degrees of freedom: the depth z and the pitch ϑ of our underwater vehicle. Combining ( 8), ( 25) and ( 26) we get the full dynamics in the earth frame to be:

           η1 η2            =            0 2×2 I 2 0 2×2 -D * r M * r                       η 1 η 2            -            0 2×1 g * r M * r - w * dr M * r            +            0 2×2 1 M * r            ωτ * r ( 31
)
with η 1 = [z ϑ] T and η 2 = [ż, θ] T . In this case ω ∈ R 2×2 is considered to be an identity matrix. Rewriting (31) in the form of ( 28) in terms of the state matrix A m and the parameters ω, θ and σ, we get:

           η1 η2            = A m            η 1 η 2            +            0 2×2 1 M * r            ωu a + θ(t)||η(t)|| L ∞ + σ(t) (32)
y =            1 0 0 0 0 1 0 0                       η 1 η 2            =            z ϑ            (33)
where A m is obtained from a choice of k m rendering the state matrix Hurwitz,

with A m ∈ R 4×4 and B m = [0 2×2 , 1 M * r ] T ∈ R 4×2 .
The parameters' vector θ ∈ R 2 includes the uncertainties on the damping coefficients and is given by: θ

= ∆(-D * z ) , ∆(-D * ϑ )
T . The parameter σ ∈ R 2 is a lumped parameter regrouping the gravitational and buoyancy forces as well as the external disturbances

σ = -g * z + w * d z , -g * ϑ + w * d ϑ T
. The parameter ω ∈ R 2×2 is considered constant and will not be adapted for this study as we have a precise knowledge of the motors' features. The expression ||η(t)|| L ∞ refers to the infinity norm of the state vector η at instant t. As shown in equation ( 33), the outputs of the system are z and ϑ. The control input is computed in the earth fixed frame and consequently should be transformed into the body fixed frame. The system's control input is computed as follows:

u = K -1 T -1 J T (u a + u m ) ∈ R 2
, with u a and u m as explained above.

Remark 1: In the matrices A 2 and B 2 , the elements M * z , M * ϑ and D * z , D * ϑ are likely to vary since they depend on the orientation of the vehicle given that they are computed in the earth frame. We have mentioned before that A m and B m should be constant and for the sake of consistency, we replace these starred model elements with M z , M ϑ , D z and D ϑ . This will guarantee for A m a constant desired dynamics.

All the uncertainties will be compensated in the vectors of the controlled parameters θ and σ that are to be adapted. It has to be noticed that unlike for the ANSF, no a priori knowledge of these parameters is required and they have been initialized to zero.

Experimental testbed

The AC-ROV submarine (cf. Figure 4) is an underactuated vehicle. Its propulsion system consists of six thrusters driven by DC motors and controlling five degrees of freedom. The motors 1, 2, 3 and 4 control simultaneously translations along x and y axes and rotation around the z axis (yaw angle). Motors 5 and 6 control depth and pitch. Roll is left uncontrolled but remains naturally stable due to the

Real-time experimental results

In this section, the obtained experimental results are presented and discussed.

They result from the application of the controllers detailed in section III to the 1 and2 summarizes the values of all the parameters and gains used in these experiments.

The reader can refer to the following internet link for a video with experimental results: www.lirmm.fr/∼creuze/ocean/

Proposed Experimental Scenarios

Three experimental scenarios were performed, namely:

i) Scenario 1: Control in nominal conditions

The objective of this scenario is to control the depth of the AC-ROV without considering any external disturbances. The gains for each controller have been tuned (cf. Tables 1 and2 to accomodate this case and were kept unchanged for the rest of the experiments.

ii) Scenario 2: Punctual external disturbance rejection

In this scenario, when the robot reached its steady state position, an external punctual disturbance was applied by giving the robot a quick kick pushing it downwards. The objective of this experiment is to see whether the controllers are able to drive the system back to its regulated position. Such a situation corresponds to the case where the vehicle hits a rock or an underwater structure, or collides with another vehicle or floating obstacle.

iii) Scenario 3: Robustness towards parameter uncertainty

The model of the vehicle was changed right before starting this scenario by the addition of a rectangular piece of polyester introducing then a change of buoyancy of +0.2 N and bringing a variation of approximately 15% with respect to the minimal value of the parameter (W -B). Such a variation corresponds for instance to the situation where the vehicle encounters a sudden change in the water's salinity. with 75 s while the pitch angle was able to follow the desired trajectory. Both degrees of freedom do not exhibit any overshoot. The similarity in the responses of these two controllers is seen through the root mean square errors (cf. Table 3), but we can still deduce that the L 1 adaptive controller was faster and this is due to the better trajectory following achieved. We observe that for both controllers we have a smooth response of the thrusters that exert a total force of 1.25 N (cf. ). On the latter two figures we observe that the parameters of the ANSF were initialized with a rough knowledge of the system. Moreover, small adaptation gains have been used in order to converge easily to a suitable set of parameters. The controlled parameters of the L 1 adaptive controller were all initialized to zero and although the adaptation gains were very high a smooth convergence is observed. We conclude from this scenario that the L 1 adaptive controller is able to ensure a faster output convergence without the necessity of having any a priori knowledge of the model parameters. It has to be noticed that regardless of the considered controller, the pitch angle shows some oscillations during the first 40 s. This behavior is due to the fact that the vehicle starts its trajectory from the surface, thus leading to disturbances that cannot be compensated since the propellers are not fully immersed during this initial period of time.

Control in nominal conditions

Punctual external disturbance rejection

As specified earlier, an external punctual disturbance has been applied on the vehicle after it reached the steady state position (cf. Figures 7,8. Due to the experimental setup of this scenario, the desired pitch angle was set to 0. The disturbance was applied at time t = 245 s to cause a positive depth error of 15 cm. The recovery time was of 15 s for the ANSF controller against 10 s for the L 1 adaptive controller. A small overshoot can be noticed with the L 1 adaptive controller for the depth whereas the ANSF converged with no overshoot but a static error of about 5 cm is conserved for 30 s; the pitch angle for this controller was more severely affected by this disturbance but it was also able to stabilize again in 15 s.

The root mean square errors in this scenario favours the ANSF controller for the pitch since it maintained its stable position more accurately whereas the depth was better compensated with the L 1 adaptive controller. These differences in the system's response of each controller are also reflected in Figures 7-(b) and 8-(b)

where the control input of the L 1 adaptive controller is seen to react smoothly with a more significant change in its estimated parameters as seen in Figure 8-(c).

Robustness towards parameter uncertainty

The additional buoyancy added to the system disturbs in a persistant way the motion of the vehicle that would tend to float more. The depth response of the ANSF is seen to have a delay of 10 s compared to the nominal case (cf. Figure 9) furthermore the slope of the response of this degree of freedom was attenuated at the middle of the trajectory due to the additional time needed for a complete parameter convergence of the parameter (W -B) as seen in Figure 9-(c). This parameter influencing greatly the dynamics in depth converged to -1.1 N in this scenario compared to -0.95 N in the nominal case. This parameter is the most dominant one for the dynamics studied. The pitch angle's convergence time was delayed by 20 s and the parameter z G W showed a change from 0.035 mN to 0.015 mN, since the position of the center of buoyancy is the center of our frame of reference and it gets modified with this added persistent disturbance. On the contrary, when the L 1 adaptive controller was applied, the system reponse times for both degrees of freedom were kept unchanged. However, we observed a stronger level of noise in the response of the pitch angle. The differences in the root mean square errors between the controllers are significant on the depth compared to the nominal case.

Given that no static erros are observed at steady state, this shows how the system was slowed down with the ANSF. As for the depth, the residual oscillations made the RMSE for this degree of freedom not very relevant after the average was made for both controllers. The fact that no additional delays were observed for the L 1 adaptive controller despite the added buoyancy, can be explained by

Comparison study of the proposed adaptive control schemes

Table 3 below summarizes in a quantitative manner the comparisons performed above between the two proposed controllers for the different proposed experimen-tal scenarios. Some relevant criteria have been chosen to perform this comparison.

Indeed, throughout the performed experiments, it was noticed that the L 1 adaptive controller drived the system faster to the desired state due to the decoupling aspect between robustness and adaptation. On the contrary, the adaptation gains of the ANSF controller had to be chosen relatively small in order to avoid oscillating responses and even instability of the system. For this reason, we found important discrepancies among both controllers' behaviors when a parameter of the model was modified. Both showed to be robust to this persistant disturbance but the ANSF controller needs more time to overcome this variation while no significant difference compared to the nominal case was observed with the L 1 adaptive controller. For this reason, the latter controller reacted also faster in presence of an external disturbance and therefore recovered faster. An important point to be reminded concerns the initialization of the parameters' vector to be estimated for each controller. The ANSF controller's parameters need to be initialzed with some suitable values requiring a sufficient knowledge of the system while the parameters of the L 1 adaptive controller can be initialized to 0.

Conclusion and Future Work

This paper deals with control of an underwater vehicle considering the challenges arising from the high nonlinearities of the system's dynamics and the variations of its parameters. The proposed solution includes the design and real-time implementation of an L 1 adaptive controller known by its particular architecture where robustness and adaptation are decoupled. To the best knowledge of the authors,

  ) D(s) is an m × m strictly proper transfer function leading to the stable closed-loop filter: C(s) = ωkD(s)I m+ωkD(s) . k is a positive feedback gain, k g = -(CA -1 m B m ) -1 is a feedforward prefilter applied to the reference signal r(t)and ηl = ω(t)u a (t) + θ||x(t)|| L ∞ . To ensure the stability of the closed-loop system, the feedback gain k and the filter D(s) must be chosen in order to fulfill the L 1 norm condition. The reader can refer to[START_REF] Hovakimyan | L 1 Adaptive Control Theory[END_REF] for a detailed proof of stability.

  θ ϑ initial Initial value for the parameter θ of ϑ 0 Γ ϑ Adaptation gain of ϑ 1300 relative position of buoyancy and gravity centers. The robot weighs 3 kg and has a rectangular shape with height 203 mm, length 152 mm and width 146 mm. For measurement purposes, the prototype is equipped with different sensors. A 6-DOF IMU (Inertial Measurement Unit) measures roll, pitch, and yaw along with their respective velocities. A pressure sensor is used for depth measurement. Once the control law has been computed by the control PC, the control inputs are transmitted to the power stage. Then, six PWM modulated signals are sent to the motors of the AC-ROV through a 40-meter long tether. Figure 4-(b) shows a schematic view summarizing the various components of the system's hardware and their interactions. The real-time experiments have been performed in a 5 m 3 pool. The tether has been sufficiently deployed to avoid inducing additional drag into the dynamics of the vehicle. The feedback gains have been tuned for the nominal conditions and kept unchanged for the rest of the experimental scenarios despite the eventual changes in the model or its environment in order to evaluate the robustness of the proposed controller. The information concerning the velocity in the z direction is estimated by an alpha-beta observer.

  underwater vehicle testbed described in section IV. The different performed scenarios are explained and then the obtained results are presented through Figures 5 to 10 and analyzed. The considered initial position of the vehicle is the surface (horizontal static position) then it is controlled to reach a depth of 0.8 m and a pitch angle of 10 deg. In fact, the tested trajectories (in pitch and depth) have been chosen so as to avoid thrusters' saturation (i.e a maximum force of 1 N) and that is why the desired depth and pitch are reached within 40 seconds and the pitch is limited to 10 deg. It is worth to note that these results can easily be extended to larger scale and faster trajectories in case of more powerful actuators. For each experimental scenario, the evolution of the control inputs, generated by thrusters 5 and 6 (cf. Figure4-(a)) controlling these two degrees of freedom as well as the evolution of the estimated parameters are plotted. Only parameters pertaining to gravity and buoyancy are retained for the ANSF controller since they are the ones with a great impact on the dynamical model. The other parameters did not vary enough and therefore are not displayed. Tables

Figures 5

 5 Figures 5 and 6 display the evolution of the vehicle's position for each of the proposed controllers. The robot is expected to follow a trajectory in depth going from the surface and reaching 0.8 m in 40 s and another one in pitch going from 0 deg at t=100 s and reaching ϑ = 10 deg in 40 s. The ANSF controller in Figures 5-(a) needs around 85 s to reach the steady state depth (5% of the final value) with no significant overshoot. The desired pitch of the same controller was reached much faster (⋍ 65 s) with no significant overshoot either. The L 1 adaptive controller reveals to have a similar convergence slope for the depth but remains slightly faster

Figures 5 -

 5 Figures 5-(b), 6-(b)) as well as a parameter convergence to steady state values

  its fast adaptation that guarantees the convergence of the parameters to their new values. (In the nominal case we had θ = [7, -26] T and σ = [10, -100] T becoming θ = [6, -22] T and σ = [12, -135] T with added buoyancy). As stated in[START_REF] Slotine | Applied Nonlinear Control[END_REF], It is worth to note that adaptative controllers do not necessarly ensure the convergence of the updated parameters to their desired values to obtain the convergence of the system to its desired position. The control law itself (13) in the case of the ANSF controller, and the parameter projection (29) in the case of the L 1 adaptive controller ensure the boundedness of the parameters but not necessarily their convergence to the real values. The control inputs generated by both controllers, are depicted in Figures9-(b) and 10-(b). Compared to the nominal case, we can observe that the robot's actuators are exerting more effort in order to immerse the vehicle while keeping the desired pitch angle; we have a combined force of 1.45 N compared to 1.25 N in the nominal case.

  this is the first study evaluating the performance of such a controller on an underwater vehicle. Indeed, to highlight the advantages of this controller it was experimentally compared with the well proven ANSF controller. Various scenarios are proposed to evaluate the closed-loop system behavior in nominal conditions as well as in presence of parameters' changes and external disturbances. This study showed that both controllers are capable of compensating the introduced uncertainties and external disturbances. However, the L 1 adaptive controller was faster in adaptation despite the zero initialization of its parameters. Future work could involve the modeling of the thrusters's dynamics and their incorporation in the controlled system. systems in the presence of unmodelled dynamics: Part II. American Control Conference, Seattle, WA. Carreras, M., Yuh, J., & JBatlle, J. 2002. High-Level control of autonomous robots using a behavior-based scheme and reinforcement learning. In: 15th Triennial World Congress of the International Federation of Automatic Control. Chang, M., Chang, W., & Liu, H.H. 2003. Model-Based fuzzy modeling and control for autonomous underwater vehicles in the horizontal plane. Journal of Marine Sciences and Technology, 11, 155-163. Dobrokhodov, V.N., Xargay, E, Hovakimyan, N, Kaminer, I.I., Kitsios, I, Cao, C, Gregory, I, & Valavani, L. 2010. Experimental Validation of L 1 Adaptive Control: Rohrs' Counterexample in Flight. Journal of Guidance, Control, and Dynamics,. Fan, X, & Smith, R.C. 2008. L 1 adaptive control of hysteresis in smart materials. Proceedings of the SPIE Smart Structures/NDE, 15th Annual International Symposium, San Diego, CA. Fjellstad, O.-E., & Fossen, T.I. 1994a. Singularity-free tracking of unmanned underwater vehicles in 6 DOF. Pages 1128-1133 of: Proceedings of the 33rd IEEE Conference on Decision and Control. Florida, USA: IEEE. Fjellstad, O.E, & Fossen, T.I. 1994b. Position and attitude tracking of AUV's: A quaternion approach. IEEE Journal of Ocean Engineering, 19(4), 512-518.

Figure 1 :

 1 Figure 1: View of the AC-ROV and its reference frames (x i y i z i : earth-fixed frame, x b y b z b : body-fixed frame).Figure 2: Block diagram of the closed-loop L 1 adaptive controller.Figure 3: Detailed block-diagram of the L 1 adaptive control architecture.Figure 4: (a): Description of the AC-ROV and its components, (b): Schematic view of its hardware architecture.Figure 5: ANSF controller (control in nominal case): (a) the system outputs' responses (z and ϑ), (b) the control inputs, and (c) the estimated parameters M r and ĝr . Figure 5-(a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 5-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 2 :

 2 Figure 1: View of the AC-ROV and its reference frames (x i y i z i : earth-fixed frame, x b y b z b : body-fixed frame).Figure 2: Block diagram of the closed-loop L 1 adaptive controller.Figure 3: Detailed block-diagram of the L 1 adaptive control architecture.Figure 4: (a): Description of the AC-ROV and its components, (b): Schematic view of its hardware architecture.Figure 5: ANSF controller (control in nominal case): (a) the system outputs' responses (z and ϑ), (b) the control inputs, and (c) the estimated parameters M r and ĝr . Figure 5-(a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 5-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 3 :

 3 Figure 1: View of the AC-ROV and its reference frames (x i y i z i : earth-fixed frame, x b y b z b : body-fixed frame).Figure 2: Block diagram of the closed-loop L 1 adaptive controller.Figure 3: Detailed block-diagram of the L 1 adaptive control architecture.Figure 4: (a): Description of the AC-ROV and its components, (b): Schematic view of its hardware architecture.Figure 5: ANSF controller (control in nominal case): (a) the system outputs' responses (z and ϑ), (b) the control inputs, and (c) the estimated parameters M r and ĝr . Figure 5-(a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 5-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 4 :

 4 Figure 1: View of the AC-ROV and its reference frames (x i y i z i : earth-fixed frame, x b y b z b : body-fixed frame).Figure 2: Block diagram of the closed-loop L 1 adaptive controller.Figure 3: Detailed block-diagram of the L 1 adaptive control architecture.Figure 4: (a): Description of the AC-ROV and its components, (b): Schematic view of its hardware architecture.Figure 5: ANSF controller (control in nominal case): (a) the system outputs' responses (z and ϑ), (b) the control inputs, and (c) the estimated parameters M r and ĝr . Figure 5-(a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 5-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 5 :

 5 Figure 1: View of the AC-ROV and its reference frames (x i y i z i : earth-fixed frame, x b y b z b : body-fixed frame).Figure 2: Block diagram of the closed-loop L 1 adaptive controller.Figure 3: Detailed block-diagram of the L 1 adaptive control architecture.Figure 4: (a): Description of the AC-ROV and its components, (b): Schematic view of its hardware architecture.Figure 5: ANSF controller (control in nominal case): (a) the system outputs' responses (z and ϑ), (b) the control inputs, and (c) the estimated parameters M r and ĝr . Figure 5-(a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 5-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 5 -

 5 Figure 5-(c): Time history of the evolution of the parameters M r = [ M z M ϑ ] T

Figure 6 -

 6 (a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 6-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 6 -

 6 Figure 6-(c): Time history of the evolution of the parameters θ = [ θz θϑ ] T and the nonlinear terms σ = [ σz σϑ ] T . Figure 7: ANSF Controller (punctual external disturbance rejection): (a) the system outputs' responses (z and ϑ), (b) the control inputs, and (c) the estimated parameters M r and ĝr . Figure 7-(a): Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure 7-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 7 -

 7 Figure 7-(c): Time history of the evolution of the parameters M r = [ M z M ϑ ] T

Figure 8 -Figure

 8 Figure 8-(c): Time history of the evolution of the parameters θ = [ θz θϑ ] T and the nonlinear terms σ = [ σz σϑ ] T . Figure 9: ANSF Controller (robustness towards parameter uncertainty): (a)

Figure 9 -

 9 Figure 9-(c): Time history of the evolution of the parameters M r = [ M z M ϑ ] T

  : Time history of the measured depth position z and pitch angle ϑ as well as their respective desired trajectories. Figure10-(b): Time history of the force generated by the two thrusters controlling z and ϑ.

Figure 10 -

 10 Figure 10-(c): Time history of the evolution of the parameters θ = [ θz θϑ ] T and the nonlinear terms σ = [ σz σϑ ] T .

  Figure

Table 1 :

 1 Parameters' values of the ANSF controller used in the experiments.

	Parameter	Description	Value	Parameter	Description	Value
	c o	Constant gain	0.1	K P z	Proportional gain of z	0.42
	c 1	Constant gain	0.05	K I z	Integral gain of z	0.055
	M z initial	Initial value of mass of z	1 kg	K D z	Derivative gain of z	0.05
	D z initial	Initial value of damping of z	0.2 N.s/m	K P ϑ	Proportional gain of ϑ	0.2
	M ϑ initial	Initial value of mass of ϑ	0.117 kg.m	K I ϑ	Integral gain of ϑ	0.055
	D ϑ initial	Initial value of mass of damping of ϑ	0.01 N.s/m	K D ϑ	Derivative gain of ϑ	0.05
	(W -B) initial Initial value of the floatability	0.7 N	Γ z	Adaptation gain of z	0.2
	z G W initial	Initial value of the restoring torque of ϑ 0.055 N.m	Γ ϑ	Adaptation gain of ϑ	1

Table 2 :

 2 Parameters' values of the L 1 adaptive controller used in the experiments.

	Parameter	Description	Value Parameter	Description	Value
	σ z initial	Initial value for the nonlinear parameter of z	0	k z	Feedback gain of z	0.4
	σ ϑ initial	Initial value for the nonlinear parameter of ϑ	0	k ϑ	Feedback gain of ϑ	0.3
	θ z initial	Initial value for the parameter θ of z	0	Γ z	Adaptation gain of z	10000

Table 3 :

 3 Controllers Performance Comparison

	Nominal Conditions	ANSF	L 1 Controller
		z z z	ϑ ϑ ϑ	z z z	ϑ ϑ ϑ
	Settling Time	85 s	65 s	75 s	40 s
	Maximum Overshoot	0%	0%	0%	0%
	Root Mean Square Error 25.3 cm 5.6 deg 24.4 cm 5.1 deg
	Punctual Disturbance
		z z z	ϑ ϑ ϑ	z z z	ϑ ϑ ϑ
	Recovery Time	15 s	15 s	10 s	10 s
	Maximum Overshoot	0%	50%	6%	50%
	Root Mean Square Error 5.8 cm 0.6 deg 2.7 cm	1.5 deg
	Change in Buoyancy
		z z z	ϑ ϑ ϑ	z z z	ϑ ϑ ϑ
	Settling Time	95 s	85 s	75 s	40 s
	Maximum Overshoot	0%	0%	0%	0%
	Root Mean Square Error 30.1 cm 5.3 deg 26.1 cm 5.3 deg


A Chemori
email: ahmed.chemori@lirmm.fr

K Kuusmik

T Salumäe

M Kruusmaa
email: maarja.kruusmaa@ttu.ee

Depth Control of the Biomimetic U-CAT Turtle-Like AUV With Experiments in Real Operating Conditions*

Control of underwater vehicles is a thoroughly investigated subject but still an open problem, because of the environmental disturbances, the highly nonlinear behaviour of vehicles, the complexity of the vehicle hydrodynamics, etc. In this paper, we are interested in depth control of a bioinspired U-CAT underwater AUV in real operating conditions. Two depth control schemes are proposed, including a PID controller and a nonlinear RISE feedback controller. The proposed controllers are implemented on the robot, then tested in an open water environment. The obtained results are presented and discussed through different experimental scenarios to illustrate the efficiency of the proposed controllers, not only to successfully control the depth, but also to be robust towards external disturbances and parameters uncertainties. we conclude that RISE controller is more robust towards environmental disturbances and outperforms the PID controller when the robot is tested in real operating condition.

I. INTRODUCTION

Biomimetic underwater robots propose alternatives for conventional propeller-driven underwater vehicles. Median and paired fin (MPF) locomotion is usually suggested as a viable alternative if high maneuverability and hovering capability is required. In fishes, such a propulsion mechanism usually means lower speeds (as opposed to body and caudal fin propulsion) but is advantageous when low speed and precision maneuverability is desired [START_REF] Palmisano | A comprehensive allometric analysis of bio-mimetic mpf-type uuvs[END_REF]. A particular type of MPF propulsion is sea turtle like 4-fin locomotion. Attempts to copy the locomotion of those agile and versatile reptiles reach back at least a decade with "Turtle 2005" [START_REF] Konno | Development of turtle-like submergence vehicle[END_REF] and Madeline [START_REF] Kemp | Madeleine: an agile auv propelled by flexible fins[END_REF]. Other examples include Finnegan, the RobotTurtle [START_REF] Licht | Biomimetic oscillating foil propulsion to enhance underwater vehicle agility and maneuverability[END_REF] and iRobot Transiphibian, developed by the same author [START_REF] Licht | Biomimetic robots for environmental monitoring in the surf zone & in very shallow water[END_REF]. Another line of development is represented by AQUA [START_REF] Dudek | Aqua: An amphibious autonomous robot[END_REF] and AQUA2 [START_REF] Plamondon | Adaptive controller for a biomimetic underwater vehicle[END_REF] four finned amphibian robots that are unique in the way the propulsors are used both for swimming and crawling in and out of water. [START_REF] Geder | Maneuvering performance of a four-fin bioinspired uuv[END_REF] describes a four-finned robot with a controllable fin surface and the authors in particular focus on optimizing the performance of ribbed fins by actively changing the surface contour [START_REF] Palmisano | Robotic pectoral fin thrust vectoring using weighted gait combinations[END_REF], [START_REF]Design of a biomimetic controlled-curvature robotic pectoral fin[END_REF]. Four finned propulsion is also realized by deploying a scaffold structure actively controlled by shape memory alloy (SME) wires [START_REF] Kim | A turtle-like swimming robot using a smart soft composite (ssc) structure[END_REF]. This paper describes an autonomous underwater robot U-CAT that is developed in European Union 7th Framework project ARROWS (Archeological Robot Systems for the World Seas) [START_REF] Allotta | The arrows project: adapting and developing robotics technologies for underwater archaeology[END_REF]. As opposed to the previous examples, four-finned design of this vehicle is motivated solely by the end-user requirements and environmental constraints of the tasks in this, specifically shipwreck penetration. As part of the team of ARROWS robots, it would closely video-inspect underwater objects. U-CAT can be operated in a tethered mode through an onboard camera feedback or autonomously, in which case it would enter a construction though an opening, inspect the space with a reflexive obstacle avoidance algorithms, return to the opening using a hydrophone array detecting a pinger and bringing back video footage from the interior of the shipwreck or any other construction. Currently, the robot prototype is undergoing field testing and continuous development of the autonomous capabilities (obstacle avoidance, navigation, localisation).

Control of underwater vehicles is a thoroughly investigated subject but still proposes difficulties, among the other things, because of environmental disturbance, highly nonlinear behaviour of vehicles, complexity of the vehicle hydrodynamics etc [START_REF] Yuh | Design and control of autonomous underwater robots: A survey[END_REF]. Biomimetic underwater vehicles usually control motion by changing the locomotion primitives of the fins, usually frequency, amplitude, phase shift, or in more complicated cases also the angle of attack, stiffness or surface area. Previously, MPF actuated 4 finned robots have been controlled using adaptive control [START_REF] Plamondon | Adaptive controller for a biomimetic underwater vehicle[END_REF], PID control [START_REF] Geder | Maneuvering performance of a four-fin bioinspired uuv[END_REF] but also bio-inspired CPG control [START_REF] Yao | Development of a turtle-like underwater vehicle using central pattern generator[END_REF]. U-CAT uses 1 DOF flexible fins but as opposed to the previously cited work, the fins are placed in corners of the vehicle under an angle so that the thrust vectors created by each individual fin are not collinear. All 4 fins are independently driven, making the robot holonomic. As opposed to a propeller driven equivalent, such a finned design proposes additional difficulties due to its delayed response and complex hydrodynamics. For example, if changing the direction of the translational velocity vector (e.g. when stabilizing or hovering) the fins have to turn 180 degrees which take finite time and creates hydrodynamic reaction forces from the environment.

In this paper we develop and test two non-model-based one-dimensional (depth) control approaches of U-CAT: PID control and RISE feedback control. The PID control is a feasible approach to start with either because it may be sufficient to solve the problem with a satisfactory precision, or, alternatively, because it can serve as a benchmark for comparison with more advanced control approaches (such as the second controller, i.e. the nonlinear RISE feedback control). For instance, [START_REF] Geder | Maneuvering performance of a four-fin bioinspired uuv[END_REF] reported good results using PID heading and depth control. In [START_REF] Plamondon | Adaptive controller for a biomimetic underwater vehicle[END_REF], experiments where run in open dynamic environment with adaptive control, reporting satisfactory roll performance, lag in pitch tracking and also concluding that the adaptive law converged to a value similar obtained in the linearization.

The rest of the paper is organized as follows. The next section shortly describes the U-CAT underwater robot. Then we describe the proposed depth controllers in section III. The obtained experimental results in open water environment with environmental disturbances are presented and discussed in section IV. The paper ends with conclusions and description of our future work directions.

II. U-CAT BIOMIMETIC ROBOT DESIGN CONCEPT

U-CAT is a biomimetic robot (cf. Fig. 1) designed specifically to meet end user requirements of underwater archaeologists. The design principles of U-CAT are laid down in [START_REF] Salumae | The arrows project: adapting and developing robotics technologies for underwater archaeology[END_REF] which describes the constraints and specific requirements posed by the nature of ship penetration on an archaeological missions. As a consequence, a 4-flipper design emerged, not because there was a specific aim to design a bio-inspired robot but rather because the biomimetic design was most suitable for solving the problem. Particularly, the crucial requirements where the small size and high maneuverability of the vehicles, that required a fully actuated robot with as few as possible actuators. In U-CAT, 6DOF actuation is realized using 4 motors. Another requirement was quiet motion, that was best realized by flippers. As opposed to the propellers, flippers have a large actuation area and therefore the propelled mass around the robot disturbs bottom sediments less. This makes close video inspection of a ship interior more feasible. The current version of U-CAT, pictured in Fig. 1, is a fully equipped field deployable robotic platform, designed to dive up to 100m depth. The 0,6m long cylindrical robot houses batteries, electronics, flipper motors and their drivers as well as on board sensors (IMU, hydrophone array, 8 sonars for obstacle avoidance, camera), lighting and modem for underwater communication in untethered mode.

Fig. 2 shows how the 4 independently driven flippers are used to achieve maneuverability in 6 DOF; whereas, Fig. 3 shows the flippers positioning as well as their rotation axes. In addition to the flipper based control, U-CAT also has a buoyancy control unit that gives an additional means for controlling depth, however, depth control in this paper is only realized by controlling the fins and keeping the buoyancy constant. The robot is also equipped with a depth sensor that in the experiments described in this paper, is used as an output measurement.

III. PROPOSED CONTROL SCHEMES

Before going further by introducing the two proposed control schemes, let's summarize first the 6-DOF dynamics of the robot as well as the one-dimensional dynamics of interest to be controlled (i.e depth dynamics).

A. The dynamics of the robot

By considering the inertial generalized forces, the gravity and buoyancy contributions, the hydrodynamic effects, and the generalized forces generated by the actuators, the hydrodynamic model of an underwater vehicle [START_REF] Fossen | Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF] writes:

9 η9 η9 η " Jpηq Jpηq Jpηqν ν ν M 9 ν M 9 ν M 9 ν `Cpνqν Cpνqν Cpνqν `Dpνqν Dpνqν Dpνqν `gpηq gpηq gpηq " τ τ τ (1)
where ν ν ν " ru, v, w, p, q, rs T , η η η " rx, y, z, ϕ, ϑ, ψs T represent the vectors of the vehicle velocities in the body-fixed frame and positions in the earth-fixed frame respectively. Jpηq Jpηq Jpηq P R 6ˆ6 represents a transformation matrix, mapping from the body-fixed frame to the earth-fixed frame. The matrices M M M , C C C, and D D D denote the inertia including added mass, the Coriolis-centripetal including addeid mass, and the damping respectively. The vector g g g represents the gravitational/buoyancy forces and moments. The term τ τ τ represents the vector of control inputs. Equation ( 1) describes the dynamic of the system in six DOF, including the three translations and the three orientations. The input vector τ τ τ P R 6 considers six actions on the system to fully control it. Given the available sensors and instrumentation of the robot, we propose, in this work, studying the control of the vehicle in its translational motions along the z axis (depth control). With the assumption of slow dynamics, involving low velocities, the Coriolis terms can be neglected. Therefore, the dynamics (1) can be rewritten as:

M 9 ν M 9 ν M 9 ν `npν, ηq npν, ηq npν, ηq " τ τ τ (2)
with npν, ηq npν, ηq npν, ηq " Dpνqν Dpνqν Dpνqν `gpηq gpηq gpηq. The hydrodynamic parameters of our robot have been experimentally idientified on U-CAT prototype. Since we are interested in depth control, the dynamics of the robot along this direction can easily be extracted from the previous equation and written as:

M z 9 w `Dz w ´cospϕqcospϑqpW ´Bq " τ z (3)
The gravitational and buoyancy forces are now brought to a single term which is a combination of W (the weight) and B (the buoyancy). τ z is one dimensional control input representing the generated force along the z-axis and controlling the depth.

In the case of our robot U-CAT, due to the flexibility of the flippers, the control input is a complex nonlinear function depending on three parameters: the orientation, the amplitude and the frequency of oscillations of the flippers, that is:

τ z " HpΦ Φ Φ, A osc A osc A osc , f osc f osc f osc q (4)
where H is a highly nonlinear function, A osc A osc A osc , f osc f osc f osc P R 4 represent the vectors of oscillations' amplitudes (respectively frequencies), and Φ Φ Φ P R 4 is the vector of orientation angles, around which the flippers will oscillate (an illustration of these orientations to control independently the different DOFs of the robot is shown in Fig. 2). In our case (depth control), the orientation of the flippers is as illustrated in Fig. 2 (z case, with an orientation upwards or downwards depending on the sign of the depth tracking error). For the case of depth control, the control input τ z is interpreted as a force (in Newtons). This force produced by the fins has been experimentally measured at a range of different amplitudes given a fixed frequency. Indeed, the frequency of oscillations of the flippers is fixed as well as their orientation, consequently the new control input will be the amplitude of the oscillations. In section IV, the experimental identification of the inverse of the nonlinear function H is introduced. In the sequel, the two proposed depth control schemes (The PID controller and the RISE feedback control) will be introduced.

B. First Controller : PID control

The PID is a well known and well established controller [START_REF] Wang | PID Controller Design Approaches -Theory, Tuning and Application to Frontier Areas[END_REF]. It is the most popular controller in industry, characterized by three feedback gains; namely, a proportional, a derivative, and integral gains.

1) Control Law Formulation: In the case of our robot U-CAT, the depth is measured in real-time thanks to a pressure sensor. The controller has to achieve depth control for the tracking of a desired reference trajectory. The control action is given by: τ z,P ID " K P 1 pz des ´zq `KI 1 ż t 0 pz des ´zq dt `KD 1 p 9 z des ´9 zq [START_REF] Licht | Biomimetic robots for environmental monitoring in the surf zone & in very shallow water[END_REF] where τ z,P ID is the control input, representing the force to be generated by the oscillations of the robot's flippers along the z axis. The parameters K P1 , K I1 and K D1 are positive constants representing respectively the proportional, the integral and the derivative feedback gains. z des is the desired depth (a time varying trajectory in our case) and z is the measured one; 9 z des and 9 z represent their respective time derivatives.

2) PID Controller Design: The expression of the control input ( 5) can be reformulated as follows:

uptq " K P reptq `1 T i ż t 0 eptq dt `Td deptq dt s (6)
with eptq " pz des ´zq being the tracking error, K P the proportional gain, T i the integral time and T d the derivative time.

C. Second controller : Nonlinear RISE feedback control

The RISE (for Robust Integral of the Sign of the Error) [START_REF] Fischer | Nonlinear rise-based control of an autonomous underwater vehicle[END_REF] strategy is a feedback control technique that yields semiglobal asymptotic tracking, even in the case of presence of uncertainties in the system's dynamics and parameters [START_REF] Xian | A continuous asymptotic tracking control strategy for uncertain multi-input nonlinear systems[END_REF]. This control scheme, using an integral signum term, can accommodate for a wide class of uncertainties based on some assumptions on the disturbance and the controlled system [START_REF] Patre | Dynamics and control of redundantly actuated parallel manipulators[END_REF].

To quantify the control objective, let's now consider the tracking error e 1 ptq P R, and the filtered tracking error e 2 ptq P R. These errors are defined by the following:

e 1 " z d ´z (7)
e 2 " 9 e 1 `α1 e 1 , α 1 ą 0

where z d P R represents the desired depth position of the robot. According to [START_REF] Xian | A continuous asymptotic tracking control strategy for uncertain multi-input nonlinear systems[END_REF], the RISE control input can be defined as follows :

τ z,RISE "pk s `1qe 2 ´pk s `1qe 2 pt 0 q `ż t t0 rpk s `1qα 2 e 2 pσq `β sgnpe 2 pσqqs dσ (9)
where k s , α 2 , β P R denote positive constant gains and sgnpξq denotes the sign function of ξ. Although, RISE feedback control has been applied to various systems, including direct-drive motors [START_REF] Patre | Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure[END_REF], hard disc drives [START_REF] Taktak | Rise feedback control for a r/w head track following in hard disk drives[END_REF], propelleractuated AUVs [START_REF] Fischer | Nonlinear control of an autonomous underwater vehicle: A rise-based approach[END_REF], parallel kinematic manipulators (PKMs) [START_REF] Bennehar | A novel rise-based adaptive feedforward controller for redundantly actuated parallel manipulators[END_REF], etc, to the best knowledge of the authors, it has never been applied to a biomimetic underwater vehicle.

IV. EXPERIMENTS IN REAL CONDITIONS

The objective here is to present the experimental results obtained through real-time application of the proposed controllers to U-CAT robot. After some first tests in a swimming pool, the idea was to validate the proposed solutions in real operating conditions in a lake with natural disturbances.

A. Some experimental issues

The following experimental details are important to underline:

' The parameters of both controllers have been tuned first in a swimming pool; then a fine tuning was performed during the tests in the lake. The obtained parameters are summarized in ' For the investigated case of depth control, the frequency of oscillations is fixed to f osc " 1.1 Hz for all the flippers, that is f osc f osc f osc " rf osc f osc f osc f osc s T . The orientation of the flippers is also fixed (upwards or downwards, depending on the sign of the depth tracking error). The new control input is the amplitude of oscillation of the fins. Based on the above assumptions and equation ( 4), the inverse of the nonlinear function H was experimentally identified, leading to the following relation between the required force and the new control input (amplitude of oscillations):

A osc " ´0.01170p τ z 4 q 2 `0.4363p τ z 4 q `0.2480 [START_REF]Design of a biomimetic controlled-curvature robotic pectoral fin[END_REF] This expression gives the oscillation amplitude of one flipper, therefore the vector of oscillation amplitudes will be as follows:

A osc A osc A osc " rA osc A osc A osc A osc s T .
' The experiments were conducted in two stages. In the first stage the following two experimental scenarios have been carried out to compare the linear PID controller with the nonlinear RISE feedback controller: (i) Control in the nominal case, (ii) External disturbance rejection. Then, in the second stage, the best controller (according to the first stage) among the two proposed controllers will be tested in the third experimental scenario (iii) Robustness towards uncertainties in the parameters (buoyancy change). ' All the proposed controllers were implemented on the robot using ROS (Robot Operating System). ' All the relevant real-time data were saved to ROS log files (rosbags). Using rosbags enables inspecting the data exactly in the same form as used by the robot software. The rosbag files were then downloaded from the robot after each experiment. ' It is worth noting that the weather conditions were not favorable (presence of environmental disturbances such as wind, current, surface waves, etc).

B. Experimental results

1) Scenario 1 : Control in nominal case: The obtained results (in terms of trajectory tracking as well as tracking error) are depicted in Figs. 4 and5. According to these two figures, it can be clearly observed that both controllers are able to steer the system output in order to track the desired trajectory. The computation of the Root Mean Square (RMS) error may be a good quantitative criterion to compare both controllers. The evaluation of this criterion gives the values 2.9683 ˆ10 ´2 (for the PID) and 2.8648 ˆ10 ´2 (for RISE controller). The tracking performance of RISE controller is consequently better than the one of the PID controller. It is worth to note, for both controllers, that at the beginning and at the end of each depth experiment, the robot is very close to the surface, and consequently subject to highly disturbing waves and currents. For both controllers, in the beginning and at the end of each depth experiment, the robot is very close to the surface subjected to highly disturbing waves and currents, which is also reflected in higher tracking errors. The evolution of the roll and pitch angles of the robot are depicted in Fig. 6, where one can observe that they remain most of the time around zero, except when the robot arrives to the surface. 2) Scenario 2 : External disturbances rejection: The experimental procedure consists of tracking of the desired reference trajectory moving from the surface to the desired constant depth, followed by a steady state value at this desired depth, then moving back to the surface. The disturbance consists of lifting the robot to the surface during the steady state phase to test weather the controllers are able to reject this external disturbance. The obtained results, with both controllers, for this scenario are depicted in Fig. 7, for the reference tracking; and in Fig. 8, for the tracking errors. In the case of the PID controller, the external disturbance is applied around at 53 sec. Despite of the big value of the disturbance, the controller is able to steer the robot back to the reference trajectory with a recovery time of approximately 10 sec. The stability of the closed-loop system is preserved despite of the big amplitude of the external disturbance. In the case of RISE controller, two external disturbances have been applied to the robot: The first one around approximately 57 sec and the second one around 77 sec. According to Figs. 7 and8, the controller is able to reject these external disturbances and bring the robot back around its steady state value in a very stable way.

To sum up for this scenario, the tracking performance of the RISE controller is better than the one of the PID, even though both controllers guarantee a recovery after the applied external disturbances. Indeed, more oscillations can be noticed around 38 sec for the case of the PID; whereas, no oscillations for RISE controller. Furthermore, when the robot moves back to the surface, it can also be noticed that the tracking error for the case of the PID is bigger than the case of the RISE controller. Consequently, the RISE controller outperforms the PID controller. In the following the robustness of the former is shown through an experimental scenario involving a change in one dynamic parameter of the system, namely the buoyancy.

3) Scenario 3 : Robustness towards buoyancy change: In this experimental scenario, the idea lies in introducing an uncertainty on one of the parameters of the robot (the buoyancy) and to check weather the RISE controller is able to compensate for this uncertainty. Fig. 9 illustrates how the buoyancy of the robot has been modified. All the parameters of the controller have been kept the same as for the previous experimental scenarios. The obtained results for this scenario are shown in Fig. 10, representing the depth Increased buoyancy Fig. 9. Illustration of the case of increased buoyancy: The piece of float fixed to the body of the robot will increase its buoyancy. trajectory tracking as well as the evolution of the tracking error. According to the obtained results one can notice that the RISE controller, despite this uncertainty, is able to track the reference trajectory. The tracking performance is slightly degraded, but still acceptable. Indeed, except of the immersion of the robot, the tracking error is close to zero. One can conclude that the RISE controller is robust towards this uncertainty of buoyancy of the robot.

V. CONCLUSIONS AND FURTHER WORK

In this work, automatic depth control of U-CAT biomimetic underwater AUV was studied. Two depth controllers have been proposed, implemented and tested in real operating conditions. Namely a PID controller and a nonlinear RISE feedback controller. Both controllers have been tested on the robot in real operating conditions (Rummu lake in Estonia). The nonlinear RISE feedback controller outperforms the PID controller in terms of tracking performance. Although the obtained results are satisfactory, several issues remain to be investigated in future work. For instance, one could consider more advanced controllers, such as modelbased or robust and adaptive controllers. Furthermore, the consideration of control of multiple degrees of freedom (yaw and depth for instance) should also be considered.

Fig. 1 .

 1 Fig. 1. View of the biomimetic U-CAT AUV operating in an open water environnement, in RUMMU lake (Tallinn, Estonia).

Fig. 2 .

 2 Fig. 2. Actuation of U-CAT: Illustration of how the 4 independently driven flippers are used to actuate the 6 DOF of the robot.

Fig. 3 .

 3 Fig. 3. Flippers positioning and their rotation axes.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Depth tracking for nominal case. Top: The PID controller, bottom: RISE controller.

Fig. 6 .

 6 Fig. 6. Evolution of pitch and roll angles versus time. Top: For the PID controller, bottom: For RISE controller.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Depth tracking for external disturbance rejection case. Top: The PID controller, bottom: RISE controller.

Fig. 10 .

 10 Fig. 10. Depth tracking for robustness towards uncertainties (RISE controller). Top: Trajectory tracking, bottom: Evolution of the tracking error.

TABLE I .

 I

	PID		RISE
	Parameter	Value	Parameter	Value
	K P 1	10	ks	15
	K d 1	7	α 1	0.75
	K I 1	5	α 2	0.5
	Windup limit	0.1	β	0.5
			windup limit	0.1
		TABLE I
	PARAMETERS OF THE PROPOSED CONTROLLERS AND THEIR VALUES.

*This work was supported within ARROWS European project and PHC-PARROT French-Estonian joint collaboration project.