Jesus Guerrero
email: jguerrero@abasolo.tecnm.mx

Ahmed Chemori
email: ahmed.chemori@lirmm.fr

Jorge Torres
email: jtorres@ctrl.cinvestav.mx

Vincent Creuze
email: vincent.creuze@lirmm.fr

Time-delay high-order sliding mode control for trajectory tracking of autonomous underwater vehicles under disturbances

Keywords:

This paper presents a new generalized super-twisting algorithm (GSTA) controller with a timedelay estimator (TDE) for the tracking control of an autonomous underwater vehicle (AUV) under disturbances. For exploration tasks in confined areas, AUVs are typically equipped with sensors such as Doppler velocity logs (DVL) or acoustic modems to measure their position and speed. However, these sensors have a disadvantage in that their acquisition rates are low. This disadvantage, combined with the influence of external disturbances, directly influences position control because it requires partial knowledge of the system for proper operation. To solve this problem, the introduction of a TDE that estimates the vehicle hydrodynamics with delayed sensor information, and consequently, improves the controller performance is proposed. For controller validation, a stability analysis is presented, and two results are provided in the simulation. In the first test, the AUV is controlled to track a parameterized spiral-shaped path to validate the vehicle movement in the three directions of the plane ( , , ). In the second test, the AUV performs path tracking in yaw and depth motions. In all numerical simulations, three types of disturbances/uncertainties are considered: (i) system parameters, (ii) sensor readings, and (iii) control.

Introduction and related work

Autonomous underwater vehicles (AUVs) have become popular in recent years owing to their wide range of application in military, commercial, and scientific fields. Indeed, numerous scanning and monitoring tasks are today performed in deep water, where intrinsic human limitations do not allow access. Therefore, AUVs have positioned themselves as an acceptable choice for this type of task.

To perform tasks efficiently, AUVs require a degree of autonomy; for example, to position the vehicle at a specific pose or to track a path for mapping an area of interest. Accordingly, the control of AUVs is one of the most challenging requirements in performing any task. Therefore, in recent years, different control techniques have been proposed.

Linear controllers for AUVs are based on PD/PIDs Jalving (1994); [START_REF] Busquets | Low-cost auv based on arduino open source microcontroller board for oceanographic research applications in a collaborative long term deployment missions and suitable for combining with an usv as autonomous automatic recharging platform[END_REF]; [START_REF] Fossen | Guidance and control of ocean vehicles[END_REF] and their variants [START_REF] Smallwood | Model-based dynamic positioning of underwater robotic vehicles: Theory and experiment[END_REF]; Rathore and Kumar (2015a). These are simple to implement yet lack robustness against external disturbances [START_REF] Tabar | Sliding mode control of autonomous underwater vehicles[END_REF], even though nonlinear variants have been proposed Guerrero et al. (2020bGuerrero et al. ( , 2019b)). Robust controllers have also been proposed, including sliding mode controllers (SMC) [START_REF] Salgado-Jimenez | Using a high order sliding modes for diving control a torpedo autonomous underwater vehicle[END_REF]; [START_REF] Elmokadem | Trajectory tracking sliding mode control of underactuated auvs[END_REF], fuzzy techniques [START_REF] Kanakakis | Fuzzy-logic based navigation of underwater vehicles[END_REF], neuronal approaches [START_REF] Amin | Neural networks control of autonomous underwater vehicle[END_REF], adaptive techniques [START_REF] Huang | Adaptive stabilization and tracking control of a nonholonomic mobile robot with input saturation and disturbance[END_REF][START_REF] Huang | Adaptive output feedback tracking control of a nonholonomic mobile robot[END_REF]; [START_REF] Tijjani | Robust adaptive tracking control of underwater vehicles: Design, stability analysis and experiments[END_REF]; Guerrero et al. (2020a); [START_REF] Maalouf | A novel application of multivariable l1 adaptive control: From design to real-time implementation on an underwater vehicle[END_REF]; Guerrero et al. (2019a), global stable tracking control with input saturation [START_REF] Huang | Global stable tracking control of underactuated ships with input saturation[END_REF]; [START_REF] Campos | Saturation based nonlinear depth and yaw control of underwater vehicles with stability analysis and real-time experiments[END_REF]; Guerrero et al. (2019b); [START_REF] Campos | A nonlinear controller based on saturation functions with variable parameters to stabilize an AUV[END_REF], backstepping control [START_REF] Guerrero | Autonomous underwater vehicle robust path tracking: Generalized super-twisting algorithm and block backstepping controllers[END_REF], predictive control [START_REF] Shen | Trajectory tracking control of an autonomous underwater vehicle using lyapunov-based model predictive control[END_REF], and adaptive variations [START_REF] Li | A neural network adaptive controller design for free-pitch-angle diving behavior of an autonomous underwater vehicle[END_REF]; [START_REF] Chen | Adaptive terminal sliding mode ndo-based control of underactuated auv in vertical plane[END_REF]; [START_REF] Rezazadegan | A novel approach to 6-dof adaptive trajectory tracking control of an auv in the presence of parameter uncertainties[END_REF]. For a detailed review of the literature on AUV control, please refer to [START_REF] Tijjani | A survey on tracking control of unmanned underwater vehicles: Experiments-based approach[END_REF]. Sliding mode control (SMC) is a popular and robust technique used in several fields including marine robotics. This technique provides finite-time convergence and robustness against parametric uncertainties and matched external disturbances. In its basic implementation (firstorder SMC), this controller can exhibit aggressive control input behavior (chattering phenomenon) owing to the signum function, which can damage the system's actuators. This negative effect appears during the reaching condition and tends to be sensitive to inaccurate mathematical models. The classical approaches to chattering attenuation in SMC systems are as follows.

(i) Partial knowledge of the system's parameters. Under the SMC technique, the controller is typically divided into two terms: the nominal part, which compensates and cancels the nonlinearities of the model through a partial knowledge of the system, and the weighted switching function, which is responsible for canceling the effect of the external disturbances. As expected, if the system's model parameters are known, the gain of the switch function can be maintained at a small value, which reduces the chattering amplitude. If only minimum information regarding the parameters is available, the gain must be increased to compensate for the modeling errors and disturbance effects. However, the amplitude of the chattering is also increased.

(ii) Replacing the discontinuous control function by saturation or sigmoid functions. This action smooths the control signal. However, this is at the cost of loss of robustness because it constrains the sliding system's trajectories to the sliding surface vicinity [START_REF] Shtessel | Sliding mode control and observation[END_REF].

(iii) Adopting a high-order sliding mode control (HOSMC) approach. This methodology exploits quasi-continuous control, which allows driving to the origin the sliding surface and its derivative in the presence of matched external disturbances.

Furthermore, it is well known that the sliding mode technique is sensitive to large time delays in a controlled system, which can degrade controller performance [START_REF] Utkin | Sliding mode control in electromechanical systems[END_REF]; [START_REF] Fridman | Steady modes and sliding modes in the relay control systems with time delay[END_REF]; [START_REF] Gouaisbaut | Robust control of delay systems: a sliding mode control design via lmi[END_REF].

In underwater vehicles, sensors can induce large delays. Consider, for example, the case of the Doppler velocity log (DVL), which is used to measure the ground speed of a vehicle based on acoustic measurements. This time delay must be considered in the control design to minimize its negative effect on the closed-loop performance.

In this study, we propose the design of a robust controller that considers these issues. On one hand, the large time delays induced by the sensors or the communication system are considered. However, we also assume that some of the hydrodynamic parameters of the AUV are unknown. The proposed controller is based primarily on two techniques. The first is the generalized super twisting algorithm (GSTA) [START_REF] Moreno | A linear framework for the robust stability analysis of a generalized super-twisting algorithm[END_REF], which is the generalization of the classical super-twisting (STW) controller introduced by [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]. This controller includes a linear version of the algorithm, standard STW, and STW with extra linear correction terms that provide more robustness and convergence velocity. It is noteworthy that this method continues to require partial information regarding the parameters of the system. The second is the time-delay estimator (TDE) that estimates the hydrodynamic parameters of underwater vehicles. This methodology was introduced initially in Hsia (1989) to estimate certain parameters and disturbances in the model using the time-delayed information of the state derivatives and input control. Furthermore, this method considers the time delay induced by sensors when estimating the parameters. TDE has been employed in the field of mobile robotics for the control of robotic manipulator arms, where certain parameters are difficult to estimate Hsia et al. (1991a); [START_REF] Jin | Robust compliant motion control of robot with nonlinear friction using time-delay estimation[END_REF]. Later, this method was introduced in the field of underwater vehicles as an option for estimating the hydrodynamic parameters [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF].

This study introduces an approach that is different from that proposed in [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF]. The main contributions of this study can be summarized as follows:

1. The robust trajectory tracking problem is considered.

The assumption of having decoupled yaw motion

from surge and sway motions, which could be restrictive when considering trajectory-tracking problems is avoided.

3. TDE estimates the AUV parameters through the timedelayed information of the available states and control input only. Then, these parameters are inserted into the GSTA control law to maintain the gains at a low value to reduce the chattering amplitude. Note that information a priori regarding the parameters of the system is not required.

4. The closed-loop stability of the GSTA controller with TDE estimator is analyzed by Lyapunov arguments, such that practical stability can be concluded.

The remainder of this paper is organized as follows. In Section 2, the dynamic model of the AUV is introduced. In Section 3, the proposed controller based on GSTA and TDE is detailed. Section 4 is devoted to closed-loop stability analysis based on Lyapunov arguments while considering disturbances and non-modeled hydrodynamics. The numerical simulation results for tracking the three-dimensional (3D) trajectories are provided in Section 5. Finally, Section 6 presents the conclusions and future work.

Dynamic model of the vehicle

Dynamic models of underwater vehicles have been described in several studies [START_REF] Fossen | Guidance and control of ocean vehicles[END_REF]; Fossen;[START_REF] Prestero | Development of a six-degree of freedom simulation model for the remus autonomous underwater vehicle[END_REF]; [START_REF] Kinsey | A survey of underwater vehicle navigation: Recent advances and new challenges[END_REF]. First, let us consider the mathematical model of a marine vehicle given by

̇ + ( ) + ( ) + ( ) = ,
(1)

̇ = ( ) , ( 2
)
where = [ , , , , , ] is the state vector of the velocity relative to the body-fixed frame and = [ , , , , , ] represents the vector of the position and orientation relative to the inertial frame (see Figure 1). ∈ ℝ 6×6 is the matrix of inertia, where the effects of added mass are included;

( ) ∈ ℝ 6×6 is the Coriolis-centripetal matrix; ( ) ∈ ℝ 6×6 represents the hydrodynamic damping matrix; ( ) ∈ ℝ 6 is the vector of gravitational and buoyancy forces and moments. Finally, ∈ ℝ 6 is the control vector acting on the underwater vehicle and ( ) ∈ ℝ 6×6 is the transformation matrix mapping from the body-fixed frame to the earth-fixed frame.

In our study, we assume that the AUV is designed to be intrinsically stable in roll and pitch [START_REF] Salgado-Jiménez | Control of rovs using a model-free 2nd-order sliding mode approach[END_REF]; Rathore and Kumar (2015b); [START_REF] Caccia | Guidance and control of a reconfigurable unmanned underwater vehicle[END_REF]. This assumption is exploited to simplify the model of the vehicle and reduce its dynamics to only four degrees of freedom, given by

( ) ̇ ( ) = ( ) ( ) ( ) - ( ) - | | ( )| ( )| + ( ) + ( ), (3)
( ) ̇ ( ) = -( ) ( ) ( ) - ( ) - | | ( )| ( )| + ( ) + ( ), (4)
( ) ̇ ( ) = - ( ) - | | ( )| ( )| + ( ) + ( ) + ( ), (5)
( ) ̇ ( ) = -( ( ) -( )) ( ) ( ) - ( ) - | | ( )| ( )| + ( ) + ( ), (6)
where the surge ( ), sway ( ),and heave ( ) are the translational speeds along , , , respectively. The yaw ( ) is the rotational velocity along the z-axis, ( ), ( ), ( ), and are the vehicle mass and moment of inertia (including added mass and inertia). The terms ( ), ( ), and ( ) represent the forces applied by the thrusters along the surge, sway, and heave directions, respectively. is the torque on the yaw motion and ( ) is the buoyancy force on the heave. Note that hydrodynamic damping is modeled as a linear and quadratic term, and and∖ | | are the linear and quadratic damping coefficients in surge, sway, heave, and yaw, respectively. Finally, ( ), ( ), ( ), and ( ) are bounded disturbances along each direction.

∖ | | , ∖ | | , ∖ | | ,
Under the assumption that the vehicle is intrinsically stable in pitch and roll, the transformation matrix ( ) in ( 2) is simplified, and the vehicle kinematics modeled as follows:

̇ ( ) = ( ) ( ( )) -( ) ( ( )), (7)
̇ ( ) = ( ) ( ( )) + ( ) ( ( )), (8)
̇ ( ) = ( ), (9)
̇ ( ) = ( ). (10)
In [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF], yaw dynamics were not considered in the controller design procedure. In this approach, yaw dynamics are considered for the trajectory tracking controller design. Now, let us consider 1 ( ) = [ ( ), ( ), ( ), ( )] and 2 ( ) = [ ( ), ( ), ( ), ( )] . Then, the set of nonlinear differential equations ( 3)-( 10) can be rewritten as follows:

̇ 1 ( ) = ( ) 2 ( ), (11) ̇ 2 ( ) = ( ) + ( ) + Ω( ), (12)
where the mass matrix is defined as = { ( ), ( ), ( ), ( )} and the control inputs are synthesized in vector form as ( ) = [ ( ), ( ), ( ), ( )] . The vector of the external disturbances is defined as Ω( ) = [ ( ), ( ), ( ), ( )] . Finally, the matrices ( ) and ( ) are given by

( ) = ⎡ ⎢ ⎢ ⎢ ⎣ ( ) -( ) 0 0 ( ) ( ) 0 0 0 0 1 0 0 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎦ , ( 13
) ( ) = ⎡ ⎢ ⎢ ⎢ ⎣ ( ) ( ) ( ) ( ) ⎤ ⎥ ⎥ ⎥ ⎦ , ( 14
)
with

( ) = ( ) ( ) ( ) - ( ) - | | ( )| ( ), | (15) ( ) = -( ) ( ) ( ) - ( ) - | | ( )| ( ), | (16)
( ) = - ( ) - | | ( )| ( ), | (17)
( ) = -( ( ) -( )) ( ) ( ) - ( )- (18)
- | | ( )| ( ).| (19)

Proposed control design

In this section, the design of two robust controllers for trajectory tracking of an underwater vehicle is proposed. First, based on the assumption that the hydrodynamic parameters of the vehicle are unknown, the construction of the GSTA nominal design is presented. In the second step, the TDE approach is adapted to the underwater vehicle system to estimate the unknown dynamics using information stored in the central computer of the vehicle. Then, the estimation made by the TDE method is introduced into the previous controller nominal design to enhance it.

GSTA nominal design

First, model (11)âĂŞ-( 12) must be expressed in a proper form. Therefore, the following state variables are chosen:

Finally, using the relations stated above, system (11)âĂŞ-( 12) can be rewritten as follows:

̇ 1 ( ) = 2 ( ), (23)
̇ 2 ( ) = 1 ( ) + 2 ( ) + 1 ( ) ( ) + Ω( ), (24)
where the vectors 1 ( ) and 2 ( ), and matrix 1 ( ) are given by

1 ( ) = ̇ ( ) ( ) -1 2 ( ), ( 25
) 2 ( ) = ( ) -1 ( 1 , 2 ), ( 26
) 1 ( ) = ( ) -1 , ( 27
) Ω( ) = ( ) -1 Ω( ). (28)
The next step is to design the controller to enforce the sliding mode on the manifold.

( ) = ̇ ( ) + Γ ( ), (29)
where the sliding surface is defined as

( ) = [ 1 ( ), 2 ( ), 3 ( ), 4 ( )]
. The tracking error and its time derivative are defined as = 1 ( ) -1 ( ) and ̇ = ̇ 1 ( )-2 , respectively, where 1 ( ) = [ ( ), ( ), ( ), ( )] is the desired trajectory, and ̇ 1 ( ) represents the time derivative of the desired trajectory. Γ = (Γ 1 , Γ 2 , Γ 3 , Γ 4 ) denotes a definite positive matrix. Before proposing the control law, it is necessary to make the following assumption. Assumption 1. The perturbation Ω( ) is a Lipschitz continuous function and its time derivative is bounded by

| Ω ( )| ≤ | 2 ( )|, = 1, 4, (30)
where ≥ 0 is a finite unknown boundary. In this study, we assume that the waves and currents satisfy Assumption 1. The control law based on the sliding-mode controller is given by

( ) = 1 ( ) -1 ̈ 1 ( ) + Γ ̇ -1 ( ) -2 ( ) -, (31)
where the GSTA is defined as follows:

= -1 Φ 1 ( ) + ̇ = -2 Φ 2 ( ), (32)
where

Φ 1 ( ) = [ 11 , 12 , 13 , 14 ] and Φ 2 ( ) = [ 21 , 22 , 23 , 24 ] , each element of which is given by 1 ( ) = 1 | | 1∕2 ( ) + 2 , 2 ( ) = 1 2 2 1 ( ) + 3 2 1 2 | | 1∕2 ( ) + 2 2 , ( 33
)
where 1 , 2 ≥ 0 with = 1, 4, and (⋆) is the signum function, defined as

(⋆) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 if ⋆ > 0 -1 if ⋆ < 0 , ∈ [-1, 1] if ⋆ = 0, (34)
where is scalar. The gain matrices are diagonal and positive definite, that is, 1 = { 11 , 12 , 13 , 14 } and 2 = { 21 , 22 , 23 , 24 }. Finally, assume that information regarding the underwater vehicle hydrodynamic parameters are not available, meaning ( ) is unknown. Then, in (31), the term 2 ( ) is not considered and the controller can be rewritten as follows:

( ) = 1 ( ) -1 ̈ 1 ( ) + Γ ̇ -1 ( ) -. (35)

GSTA design with TDE

In this section, to overcome the lack of information regarding the hydrodynamic parameters of the AUV, the proposed controller, based on the GSTA, is improved by introducing the TDE into the final control law expression. As mentioned above, the procedure is divided into three main steps, detailed in the following.

Step 1. Before proceeding to express the AUV model in a proper form, which allows us to obtain the SMC, it is necessary to introduce certain modifications. First, following the procedure used by [START_REF] Jin | Robust compliant motion control of robot with nonlinear friction using time-delay estimation[END_REF], we introduce the matrix defined as = ( , , , ), which contains the control parameters , , , and into (12). Then, Equations ( 11) and ( 12) can be rewritten as follows:

̇ 1 ( ) = ( ) 2 ( ), (36)
̇ 2 ( ) + ( ) = ( ), (37)
where the elements of ( ) = [ℎ ( ), ℎ ( ), ℎ ( ), ℎ ( )] are defined as follows:

ℎ ( ) = ( ( ) -) ̇ ( ) + ( ) + | | ( )| ( )|, -( ) ( ) ( ) -( ), .ℎ ( ) = ( ( ) -) ̇ ( ) + ( ) + | | ( )| ( )|, + ( ) ( ) ( ) -( ), .ℎ ( ) = ( ( ) - ) ̇ ( ) + ( ) + | | ( )| ( )|, -( ) -( ). ℎ ( ) = ( ( ) -) ̇ ( ) + ( ) - | | ( )| ( )| + ( ( ) -( )) ( ) ( ) -( ).
By choosing the same state variables given by Equation ( 20) and introducing Equations ( 21) and ( 22) into (37), the dynamic system (36)-( 37) can be represented as follows:

̇ 1 ( ) = 2 ( ), (38) ̇ 2 ( ) = 1 ( ) + 3 ( ) + 2 ( ) ( ), (39)
where the function 1 ( ) is defined by (25), and the vector 3 ( ) and matrix 2 ( ) are given by

3 ( ) = ( ) -1 ( ), ( 40
) 2 ( ) = ( ) -1 . ( 41
)
Step 2. As in the nominal case stated earlier, it is necessary to introduce the following assumption.

Assumption 2. The TDE error is defined as

( ) = ( ) -1 [ ( ) -Δ ̂ ( ), ] (42)
where its derivative is considered to be bounded. The proof is given by [START_REF] Jin | Robust compliant motion control of robot with nonlinear friction using time-delay estimation[END_REF] and Hsia et al. (1991b), and adapted to the AUV paradigm by [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF].

Employing the same sliding surface given by Equation ( 29), the sliding mode controller is constructed as follows:

( ) = 2 ( ) -1 ̈ 1 ( ) + Γ ̇ -1 ( ) -3 ( ) -, ( 43
)
where the GSTA is defined in Equations ( 32) and (33).

Step 3. It is noteworthy that for the control law presented by ( 43), the exact knowledge of 1 ( ) and 3 ( ) is necessary.

The DVL provides information regarding the position of the AUV, which means that the term 1 ( ) is known. However, term 3 ( ), which is directly related to the underwater vehicle model, is unknown. Thus, the controller given by ( 43) is not feasible. To overcome this restriction, we propose using the TDE to estimate ( ) from ( 40).

In this sense, an efficient engineering solution to estimate the function ( ) is brought closer through its delayed function ( -ℎ), where ℎ is the given delay Hsia et al. (1991b). By defining the estimated matrix as ̂ ( ), from (37), we obtain the following:

̂ ( ) = ( -ℎ) = ( -ℎ) - ̇ 2 ( -ℎ). ( 44
)
This method is known as TDE. It is simple to implement because it only requires information stored in the central computer. However, there is a trade-off between the proposed method and error ̃ ( ) = ( ) -̂ ( ), which increases as ℎ increases [START_REF] Jung | Force tracking impedance control of robot manipulators under unknown environment[END_REF]. This is related to the sampling frequency of the sensors; that is, if the sampling frequency is low, the error tends to increase.

For AUV localization in the ( -) plane, a DVL sensor or acoustic modem is used, which has a sampling frequency of approximately 3 to 5

. This sampling time is considerably greater than that of other sensors such as an IMU or pressure sensor. Therefore, the TDE error increases, as stated above. To minimize this error, in [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF], a PD + SMC action with a weighted TDE term Δ ̂ ( ) was considered. In this work, we extend the analysis to include a high-order sliding mode controller, namely GSTA, which decreases the convergence in finite time and softens the control action, as discussed below.

Therefore, the estimated term ̂ ( ) provided by Equation ( 44) is inserted into control law (43), leading to

( ) = ( ) -1 ̈ 1 ( ) + Γ ̇ -1 ( ) --Δ ̂ ( ), (45)
where Δ = {Δ 1 , Δ 2 , Δ 3 , Δ 4 } is an adjustable and positive-gain matrix. Moreover, the control law (45) can be rewritten using relations ( 44) and ( 22) to obtain the following controller in terms of the available variables:

( ) = ( ) -1 ̈ 1 ( ) + Γ ̇ -̇ ( ) ( ) -1 2 ( ) - -Δ ( -ℎ) -[ ( ) -1 ̇ 2 ( -ℎ)- -( ) -1 ̇ ( ) ( ) -1 2 ( -ℎ)] . ( 46
)
Finally, if the gains are selected as 1 , 2 > 0 and are sufficiently large, it can be concluded that ̇ ( ) → 0 in finite time, and based on equation ( 29), lim →∞ = 0 and lim →∞ ̇ = 0. The proof of the finite-time convergence of variables 1 ( ) and 2 ( ) into a small neighborhood of the desired reference signals 1 ( ) and 2 ( ) is given as follows.

Remark 1. When the control objective is achieved, ̇ ( ) → 0 and then ( ) → 0. Using Equation (29), one can note that ( ) → 0 in finite time. Besides, the tracking error is defined as ( ) = 1 ( ) -1 ( ), and when the error approaches zero, it is implied that 1 ( ) → 1 ( ) in finite time.

Remark 2. The controller law developed in Equation (31) coincides with the law developed in the study of [START_REF] Guerrero | Autonomous underwater vehicle robust path tracking: Generalized super-twisting algorithm and block backstepping controllers[END_REF]. Moreover, following the assumption that information regarding the dynamic model of the underwater vehicle is not available, we modified the controller as indicated in Equation (35), where the term 2 , which is related to the hydrodynamic parameters, was suppressed. In our approach, as only information from the sensors is provided, we applied the TDE to estimate the hydrodynamic parameters, as we can see in the proposed control law (46), which depends only on the available information.

Stability analysis

To formalize the stability analysis of robust path tracking based on sliding mode control theory, we introduce the following theorem:

Theorem 1. Consider system (38)-( 39) and GSTA (32) with perturbation terms. Suppose that the perturbation terms of system (56) are bounded. Then, gains 1 and 2 can be selected to be sufficiently high such that the error trajectories satisfy the strong practical stability notion, i.e., the error trajectories ( ) = 1 ( ) -1 ( ) are bounded and converge in finite time to the origin.

Proof. From the derivative of (29) and substituting ( 45) and ( 32) into (39), one obtains the closed-loop dynamics given by

̇ = -1 1 | | 1 2 ( ) + 2 -2 ∫ 0 1 2 2 1 ( ( )) + 3 2 1 2 | ( )| 1 2 ( ( )) + 2 2 ( ) + ( ) -1 ( ) -Δ ̂ ( ) . ( 47
)
Then, considering Assumption 2, Equation ( 47) can be rewritten as follows:

̇ = -1 Φ 1 ( ) -2 ∫ 0 Φ 2 ( ) + ( ). ( 48
)
The equation above can be rewritten in an explicit form as

̇ = -1 1 | | 1 2 ( ) + 2 + ( ) -2 ∫ 0 1 2 2 1 ( ( )) + 3 2 1 2 | ( )| 1 2 ( ( )) + 2 2 ( ) . ( 49
)
Remark 3. Note that (47) and (49) clearly indicate the influence of TDE error on the dynamics, i.e., ( ) causes the resulting dynamics to deviate from the ( ) dynamics. Therefore, in this proof, ( ) is considered a disturbance term. This result corroborates the conclusions of [START_REF] Jin | Robust compliant motion control of robot with nonlinear friction using time-delay estimation[END_REF] and [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF]:

Now, let:

1 = , ( 50
) 2 = -2 ∫ 0 2 ( ( )) + ( ), (51)
̇ ( ) = ( ). (52)
Then, Equation ( 49) can be rewritten in scalar form (for = 1, 4) as follows:

̇ 1 = -1 1 | 1 | 1 2
( 1 ) + 2 1 + 2 , (53)

̇ 2 = -2 1 2 2 1 ( 1 ) + 3 2 1 2 | 1 | 1 2 ( 1 ) + + 2 2 1 + . ( 54
)
Without loss of generality, we can represent the system with simplified notation as follows:

̇ 1 = -1 1 | 1 | 1 2 ( 1 ) + 2 1 + 2 , ( 55
)
̇ 2 = -2 1 2 2 1 ( 1 ) + 3 2 1 2 | 1 | 1 2 ( 1 ) + 2 2 1 + . ( 56
)
Note that 2 ( 1 ) = ′ 1 ( 1 ) 1 ( 1 ) and selecting the vector

= [ 1 , 2 ] = [ 1 ( 1 ), 2 ] and = ( ) ′ 1 ( 1 )
, it is possible to rewrite system (56) as follows:

̇ = ′ 1 ( 1 ) + , ( 57
)
where matrices and are defined as follows:

= -1 1 -2 0 , = 0 1. ( 58
)
Case I -Unperturbed system: Consider the case when the approximation of ( ) through Δ ̂ ( ) is exact, i.e., ( ) = 0. Then, choosing the Lyapunov function candidate (LFC) as follows:

= , ( 59
)
where denotes a positive definite matrix that satisfies the Lyapunov equation

+ = -, (60)
where is any given positive definite matrix, let denote the smallest eigenvalue of .

Note that the proposed LFC is a continuous, positive definite, and differentiable function that satisfies the following form:

( )‖ ‖ 2 2 ≤ ( ) ≤ ( )‖ ‖ 2 2 , ( 61
)
where ( ) and ( ) are the smallest and largest eigenvalues of , respectively, and

‖ ‖ 2 2 = | 1 | + 2| 1 | 3 2 + 2 1 + 2 2 is the Euclidean norm of , noting that | ( 1 )| ≤ ‖ ‖ 2 ≤ 1 2 ( ) 1 2 min ( ) .
The time derivative of along the trajectories of the system leads to

̇ ( ) = 2 ̇ = ′ 1 ( 1 ) + = -′ 1 ( 1 ) ≤ -′ 1 ( 1 ) ‖ ‖ 2 2 ≤ -2 + 1 2| | 1 2 ‖ ‖ 2 ≤ - 2 ‖ ‖ 2 - 1 2 ‖ ‖ 2 | | 1 2 ≤ -1 -2 2 √ where 1 = 2 ( ) ; 2 = 1 1 2 ( ) ( ) .
Note that ̇ is a continuously decreasing function, and, as we can observe in the following, we can conclude that the equilibrium point is reached in finite time.

Because the solution of its analog differential equation

̇ = -1 -2 1 2 , (0) ≥ 0 (62)
is given by

( ) = exp(-1 ) (0) 1 2 + 2 1 1 -exp( 1 2 ) 2 (63)
and using the comparison principle, the solution converges to the origin, as previously stated. Finally, → 0 in finite time, and based on (29), it is implied that lim →∞ = 0 and lim →∞ ̇ = 0.

Remark 4. From the stability proof, one can guarantee finite-time convergence of vector = [ 1 , 2 ] to zero. Note from Equation (50) that 1 is related to , which means that → 0, then by Equation (29), one can deduce that ( ) → 0 and ̇ ( ) → 0 as time increases.

Remark 5. Note that for system (57) in the absence of disturbances, the necessary and sufficient condition for convergence is that matrix is Hurwitz. This is equivalent to conditions 1 > 0 and 2 > 0.

Case II. Perturbed system: Consider the case when the approximation of the ( ) term is overestimated or underestimated, i.e., the TDE error ( ) ≠ 0. Then, the robustness of the GSTA controller minimizes the influence of this disturbance. In this case, we consider the LFC defined in (59). Remark 6. It is assumed that the transformed perturbation ℎ( ) satisfies the sector condition [START_REF] Moreno | A linear framework for the robust stability analysis of a generalized super-twisting algorithm[END_REF], which implies that ( ,

) = | 1 | ( ) satisfies | ( , )| ≤ | 1 |. Thus, ( , ) = -2 ( , )+ 2 2 ≥ 0, Then, this expression can be rewritten as ( , ) = -2 ( , ) + 2 ≥ 0 with = [1, 0].
Remark 7. For design reasons, it is important to note that the gain matrix in (58) can be rewritten as

= 0 -0 0 , ( 64
)
where

0 = 0 1 0 0 , 0 = 1 2 , 0 = 1 0 . ( 65
)
The time derivative of along the trajectories of the system gives

̇ =2 ̇ = ′ 1 ( 1 ) ( + ) + + = ′ 1 ( 1 ) + 0 ≤ ′ 1 ( 1 ) + 0 + ( , ) = ′ 1 ( 1 ) + + 2 -1 = ′ 1 ( 1 ) + + 2 + -1 -′ 1 ( 1 ) = ′ 1 ( 1 ) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎣ -0 0 - 0 0 + + 0 + 0 + 2 -1 ⎤ ⎥ ⎥ ⎦ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ( 0 , | , ) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ -′ 1 ( 1 ) = ′ 1 ( 1 ) ( 0 , | , ) - .
Assume that 0 is selected such that > 0 and > 0 providing ( 0 , | , ) ≤ 0. The time derivative of can then be expressed as follows:

̇ ( ) ≤ -′ 1 ( 1 ) (66) = -1 2| 1 | 1 2 -2 (67) ≤ - 1 1 2 min ( ) 2 1 2 -2 . ( 68
)
The fact that the derivative of is negative definite is reached by selecting positive gains 1 and 2 sufficiently high to satisfy the condition ( 0 , | , ) ≤ 0. Therefore, it can be concluded that the equilibrium point is reached in a finite time from any initial condition.

Following the same procedure as in the previous case, the solution of the analog differential equation of ( 68) is given by

( ) = (0) 1 2 + 1 1 2 min ( ) 2 2 1 -exp 2 2 2 ⋅ ⋅ exp(-2 ), (69)
it follows that the solution converges in finite time to the origin at the latest time , computed as follows:

= 2 2 ln ⎛ ⎜ ⎜ ⎜ ⎝ 2 2 1 1 2 min ( ) (0) 1 2 + 1 ⎞ ⎟ ⎟ ⎟ ⎠ . ( 70
)
Finally, using the comparison principle, it can be stated that 1 ( ) converges to 1 ( ) at latest in time given by ( 70). Remark 8. The gains of the proposed controller were selected by solving as stated in the proof of Theorem 1. Matrix ( 0 , | , ) < 0 is a bilinear matrix inequality (BMI) owing to the product of and 0 . To solve this problem as a linear matrix inequality (LMI), the following matrix is introduced:

= 0 . ( 71
)
Matrix can be rewritten as follows:

= ⎡ ⎢ ⎢ ⎣ -0 - 0 + 0 + 0 + 2 -1 . ⎤ ⎥ ⎥ ⎦ (72)
This representation of can be considered as an LMI on and . Note that it is necessary to know the bound of the disturbance and a fixed positive constant value > 0 to solve LMI (72) and find the the GSTA-ESO gains through the following relationship:

0 = -1 (73)
The TDE gains were selected based on the procedure presented in [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF].

Numerical simulation results

In this section, the performance of the proposed controllers given by Equations ( 35) and ( 46) is tested under external disturbances. The objective is to demonstrate that the estimation of the dynamics of the vehicle using the TDE method improves the performance of the nominal controller, thereby reducing the chattering amplitude. The main parameters used for the simulation were obtained from the Cyclops underwater vehicle reported in [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF]. The vehicle had a length of 1477 mm and a width of 868 mm. The parameters are summarized in Table 1 and the control gains are listed in Table 2.

Four scenarios were developed to test the performance of the proposed controllers. In the first, the vehicle followed a parameterized spiral-shaped trajectory. The sampling period was fixed to = 50 to prove that the uncoupled controllers were independently working correctly. In the second scenario, the previous test was repeated with the sampling period increased to = 100 . Based on the previous test, the influence of varying the delay of the TDE on the estimation of the hydrodynamic parameters of the vehicle was indicated. In certain cases, AUVs are programmed to perform pipeline-monitoring tasks, for example, where cameras are used as sensors to measure the distance between the vehicle and pipeline to be followed. This measurement is used as a variable to maintain a constant distance during tracking. This problem inspired the fourth scenario, wherein it was proposed that the vehicle should follow a depth profile by introducing a delay of ℎ = 0.4 while a desired sinusoidal

Buoyancy force in heave [N]

-5

Table 2 Control gains used for simulation tests.

Gain Value (20,20,20,5) trajectory was programmed in the yaw motion. Moreover, different types of disturbances were considered in all numerical simulations. First, perturbations with sinusoidal profiles were introduced in the sway and surge control, emulating an imbalance caused by ocean currents. For heave control, a constant disturbance was introduced such that a change in buoyancy occurred in the vehicle. Finally, it is important to note that in the entire set of experiments, we compared the proposed approach, namely GSTA with TDE, to the nominal design of the GSTA provided in [START_REF] Guerrero | Autonomous underwater vehicle robust path tracking: Generalized super-twisting algorithm and block backstepping controllers[END_REF] to demonstrate the advantages of the proposed controller.

1 (3, 3, 3, 3) 2 (5, 5, 5, 5) Γ (1, 1, 1, 1) Δ (0.2, 0.1, 0.1, 0.1) 10 *
Remark 9. The trajectory profile for this scenario was selected because it is frequently used to test the performance of 3D position controllers ( , , ).

Scenario 1: Nominal case

In this scenario, the sampling period was set to = 50 . The reference trajectory was a 3D spiral function given by the following expression (74):

⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ (2 ) (2 ) -1 . ⎤ ⎥ ⎥ ⎦ (74)
The performances of the nominal GSTA [START_REF] Guerrero | Autonomous underwater vehicle robust path tracking: Generalized super-twisting algorithm and block backstepping controllers[END_REF]) and proposed GSTA with TDE for trajectory tracking without considering external disturbances are displayed in Figure 2. From this figure, it can be observed that both ) with a frequency = 10 , respectively. The perturbation terms were introduced within 15 s of starting the simulation. The same disturbances were considered for all disturbed simulation scenarios. Despite the considered perturbations, the vehicle converged to the desired trajectory. It is worth noting that the GSTA controller with nominal design did not contain any information regarding the hydrodynamic parameters of the vehicle, meaning that the gain of the controller compensated for the effect of both parametric uncertainties and external disturbances. In addition, the behavior of the GSTA with TDE methodology remained close to the nominal design. Finally, it is important to note that, in general, the controllers based on sliding modes were sensitive to the sampling time; therefore, it would be interesting to consider a change in the sample time. This is discussed in the next section.

Effect of increasing the sampling period

In this test, based on the results of the study by [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF] and for a fair comparison, the sampling period of the simulations was set to = 100 . Time delay ℎ, used to estimate the hydrodynamic parameters of the vehicle, was fixed at ℎ = 400 because the DVL-provided data was at a frequency of 3 .

The obtained results of the trajectory tracking of the vehicle considering the effects of external disturbances are displayed in Figure 4. From this figure, it can be observed that the controller with a nominal design was not capable of following the desired trajectory properly. Indeed, an oscillatory behavior of the vehicle around the reference trajectory can be observed, which can be explained by two arguments. First, because the hydrodynamics of the vehicle were not considered in the controller, the gain of the GSTA was required to counteract the disturbance effect. Secondly, as mentioned above, the SMC is sensitive to changes in the sampling period. To obtain superior tracking, it is mandatory to reduce the sampling time, as indicated in the results of the previous section.

The error in the tracking test of the submarine is displayed in Figure 5. As expected, the GSTA with TDE methodology significantly reduced the tracking error. The evolution of the controller input is indicated in Figure 6. These results confirm that adopting the GSTA with TDE approach can attenuate the chattering effect compared with the GSTA with a nominal design. Again, the chattering present in this simulation was due to the high sampling time.

Remark 10. It is worth noting that the main objective of these scenarios was to demonstrate the effectiveness and robustness of the proposed method with respect to the nominal design. When the sampling period was increased, the time delay induced by the sensors also increased.

Estimation of ( ): Varying the time delay

In the previous section, it was noted that a superior estimate of ( ) could be obtained if the gains of the estimator were properly tuned or the time delay of ̂ ( ) was reduced. To illustrate the influence of delay on the estimation of ( ), in this simulation, the spiral trajectory was traced varying the time delay. and a visible improvement in the trajectory tracking of the controller was observed (see Figure 8).

Depth and yaw trajectory tracking

Finally, the last scenario was inspired by a realistic situation, where the vehicle was programmed to perform pipeline monitoring tasks while the depth was regulated at a constant desired value. In this context, a depth profile is tracked along with a sinusoidal signal for yaw dynamics. The results for this scenario are displayed in Figure 9. By zooming around the inflection point in the depth profile, we can observe an oscillation in the control signal due to the selected sampling period. Moreover, the tracking error increased because the depth reference function did not have a second derivative at that point. In Figure 10, yaw tracking can be observed where the same tracking effect is observed, as in the case of depth dynamics. Oscillatory behavior was clearly observed during the tracking test. However, the amplitude of the oscillations was small and did not influence the tracking task. Finally, the tracking errors of this scenario are displayed in Figure 11. Remark 11. This study was inspired by the interesting results presented in [START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF]. However, to perform a fair comparison, we decided to compare our approach with the GSTA nominal design, instead of comparing it with the scheme described in the cited work, where the dynamic couplings betweenand the yaw are neglected.

Conclusion and future work

In this study, two controllers were proposed to solve the problem of trajectory tracking of an AUV in the ( , , ) plane and yaw dynamics. For the controller description, a nonlinear AUV model was used. The high-order slidingmode nonlinear control technique, called the super-twisting algorithm, was improved with a TDE term. The considered TDE term estimates the vehicle hydrodynamics through time-delayed measurements, and this information is fed back to the controller to improve its performance. A stability analysis of the resulting closed-loop system was presented, and the numerical simulation results demonstrated the effectiveness and robustness of the proposed control technique. Several conditions were considered to study the influence on the system behavior, where both variations in the sampling period and time delay were considered.

A real-time implementation and validation of the proposed controller will be the subject of future research. In addition, it is should be noted that the GSTA was tested on an experimental platform; however, in the damping tests, its performance was limited because we had no method to estimate certain parameters of the vehicle model. The extension of the present results to the case of output-feedback sliding modes appears to be a challenging problem.

Figure 1 :

 1 Figure 1: Illustration of vehicle's reference frames. The inertial fixed frame is denoted ( , , , ); the body-fixed frame is denoted ( , , , ).

Figure 2 :

 2 Figure 2: AUV's reference tracking of spiral-shaped path without external disturbances at sampling time of = 50 . The reference is represented by the black solid line, the nominal design GSTA (Guerrero et al. (2018)) is the solid blue line, and the solid red line is the proposed GSTA with TDE controller.

Figure 3 :

 3 Figure 3: Underwater vehicle trajectory tracking of spiralshaped path under external disturbances at sampling time of = 50 . The reference is represented by the black solid line, the nominal design GSTA (Guerrero et al. (2018)) is the solid blue line, and the solid red line is the proposed GSTA with TDE controller.

Figure 5 :Figure 6 :

 56 Figure5: Robust reference tracking errors in surge ( ), sway ( ), heave ( ), and yaw ( ). The tracking errors were minimized by adopting the GSTA with TDE approach (solid red line) instead the GTSA nominal design represented by the solid blue line[START_REF] Guerrero | Autonomous underwater vehicle robust path tracking: Generalized super-twisting algorithm and block backstepping controllers[END_REF]). The sampling period was set to = 100 .

Figure 4 :Figure 7 :Figure 8 :

 478 Figure 4: Underwater vehicle trajectory tracking test under persistent external disturbances at sampling time of = 100 . The reference is represented by the black solid line, the nominal design GSTA (Guerrero et al. (2018)) is the dashed blue line, and the solid red line is the proposed GSTA with TDE controller.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure9: Robust trajectory tracking of depth profile (dashed black line) when time delay of ̂ ( ) is set at ℎ = 400 and sampling time is fixed at = 100 . The zoomed area displays the oscillatory behavior of the GSTA with TDE approach (solid red line).

Table 1

 1 List of parameters used in simulation (values from[START_REF] Kim | Time-delay controller design for position control of autonomous underwater vehicle under disturbances[END_REF]).

	Parameter [Units]	Symbol Value
	Rigid body mass [Kg]		219.8
	Mass in surge [Kg]		391.5
	Linear drag coefficient in surge		16
	Quadratic drag coefficient in surge		229.4
	Mass in sway [Kg]	| |	693.6
	Linear drag coefficient in sway		131.8
	Quadratic drag coefficient in sway		328.3
	Mass in heave [Kg]	| |	639.6
	Linear drag coefficient in heave		65.6
	Quadratic drag coefficient in heave
		| |

( )

( ) = 1 ( ) ̇ 1 ( ) . (20)Then, from Equation (11), we obtain the following useful algebraic relations:2 ( ) = ( ) -1 ̇ 1 ( ), (21) ̇ 2 ( ) = ( ) -1 ̈ 1 ( ) -( ) -1 ̇ ( ) ( ) -1 ̇ 1 ( ). (22)