UDE-based nonlinear path-following control of autonomous underwater vehicles with multiple uncertainties and input saturation

A novel nonlinear control scheme is suggested in this research for the path-following (PF) control of an underactuated autonomous underwater vehicle (AUV) in the horizontal plane. Firstly, the proper nonlinear models of the underactuated AUV with multiple uncertainties and input saturation are established. The unknown time-varying sideslip angular speed in the PF error dynamic model is treated as the kinematic uncertainty. Also, the linear superposition of parameters uncertainties, environmental perturbations, and other unmodeled dynamics is considered the lumped uncertainty. The above kinematic and lumped dynamic uncertainties are predicted by the uncertainty and disturbance estimator (UDE) approach. Such treatment is to solve the dependence on a precise mathematical model and attain a simpler controller. Secondly, an UDE-based PF controller is proposed via the backstepping frame augmented by the nonlinear tracking differentiator (NTD) and auxiliary dynamical system. The NTD is adopted to prevent the “explosion of complexity” pertinent to the standard backstepping method, and the auxiliary dynamical system is utilized to solve the issue of input saturation. Lastly, the steadiness of the entire enclosed-loop framework is presented. Conducting comprehensive simulations are presented to assess the efficacy and robustness of the devised controlling scheme.


Introduction
Autonomous underwater vehicles (AUVs) have played a crucial role in various fields of the marine economy, such as offshore oil and gas operations, oceanographic mapping, and surveillance in the last two decades [1][2][3].Path-following control (PFC) of AUVs has a great deal of growing attention since its implementations can be realized in marine surveys, underwater rescue, and pipeline inspection [4].To effectively perform the mentioned operations, constructing an efficient and reliable system to control them is crucial [5].However, the AUVs are usually subject to system uncertainties.
Neglecting the system uncertainties when designing a control system might lead to weak convergence ratios or/and important steady-state errors may cause unsteadiness of the enclosed-loop framework [4].The uncertainties pertinent to the system are generically produced by the change of the hydrodynamic derivatives caused by the alterations in navigation conditions [6].Additionally, waves, currents, and winds considered external perturbations significantly have an impact on the PFC of the vehicle.Therefore, to resolve the PF problem of AUVs when uncertainties and external perturbations, many advanced control methods have been proposed in recent years, including proportional-integral-derivative (PID) control [7], backstepping [8], sliding-mode [9], and neural network [10].Among the mentioned methods, the backstepping method has been widely employed to design controllers for nonlinear systems with strict-feedback forms [11,12].In [13], Lyapunov's theory and the backstepping method have been embraced to establish a PFC approach that forces an underactuated AUV to track a given trajectory.However, the standard backstepping technique requires the calculation of analytical expressions for virtual control variables.Therefore, the traditional backstepping approach often involves the resembling "explosion of complexity" problem.To address the problem presented above, multiple approaches pertinent to control are suggested in the literature, namely, the dynamic surface control (DSC) in [14] employs a filter with first-order to find the derivatives of the desired so-called control commands.For example, Zhao et al. [15] utilized the control method dealing with the linearization of the trajectory and a nonlinear tracking differentiator (NTD) to suggest a robust PF controlling scheme to resolve the PFC of the vehicle.The NTD was employed to obtain the so-called controlling command and its derivative so that the "explosion of complexity" issue in the conventional backstepping method is addressed.Other approaches that have been successfully utilized to handle the system uncertainties or/and environmental perturbations are disturbance observerbased control (DOBC) method [16], extended state observer-based control (ESOC) method [17], and uncertainty and disturbance estimator-based control (UDEC) method [18].In [19], the backstepping method and a nonlinear disturbance observer (NDO) were embraced to propose a novel PF controlling scheme for the PFC of a underwater vehicle.The nonlinear disturbance observer was employed to predict the system uncertainties and external perturbations.Peng et al. [20] implemented the extended state observer (ESO) and the optimization approach based on a neurodynamic framework to suggest a disturbance rejection control method for the PFC of the vehicle, where the ESO was conducted to model uncertainty and outside perturbations.In [21], a novel trajectory tracking controller utilizing linear parameter varying (LPV) was built for an underactuated AUV, and the ESO was utilized to estimate and compensate for the unknown attack and side-slip angular velocity.Despite the fact that the merit of ESO is that it requires the minimum information of observations of the plant [22], it is supposed that the derivatives of perturbations are bounded.The UDE technique is an effectual and robust method for linear and nonlinear systems with unknown dynamics.In [23], the sliding mode (SMC) approach was combined with the UDE for motion control of an AUV, where the UDE technology was utilized to estimate and indemnify the uncertainties existed in the parameters related to hydrodynamic and volatile perturbations.
Except for uncertainties and outside perturbations pertinent to systems, the actuator saturation issue has been also crucial once the PF controller is designed [24].In practical engineering applications, performance degradation and instability may occur when constructing a PF controller without considering the issue called the saturation of the input [25].In [26], an angle-based PF controller was combined with fuzzy logic in an adaptive manner and adaptive compensation control law for the PF of an AUV when the actuator saturation is not known.In [27], a finite-time PF controller was proposed for a USV under error constraints and input saturations.A hyperbolic tangent function was proposed to approximate the saturation term, and the tan-type barrier Lyapunov function approach was utilized to deal with the restrictions of the error.In [28], a sliding-model-based adaptive controller was combined with an NDO for attitude control of an AUV with input nonlinearities.An auxiliary dynamic system was conducted to compensate for the unknown impacts of the saturation of the rudder.The input saturation issue has been effectively resolved by using an anti-windup compensator similar to [6] and [29].Hence, this paper suggests an original horizontal PF controller for the underactuated AUV when multiple uncertainties and the saturation of the input are under consideration simultaneously.The fundamental novelties of the paper are summarized below.
(i) The unknown time-varying sideslip angular speed in the PF error dynamic model is treated as the kinematic uncertainty.Also, the linear superposition of model parameter uncertainties, environmental perturbations, and other unmodeled dynamics in the dynamic model is taken as the lumped dynamic uncertainty.The UDE technology is then employed to estimate and compensate for the above kinematic and lumped dynamic uncertainties.Such treatment is to solve the dependence on an accurate mathematical model and attain a simple controller.
(ii) A novel augmented backstepping controller is proposed by simultaneously taking into account multiple uncertainties and the input saturation.Compared with the DOBC method proposed in [11], because the proposed controller does not contain the inverse of the nominal plant model, it can be easily conducted in practical applications.Also, compared with the ESOC method presented in [21], because the proposed controller relaxes the constraint that the derivatives of the perturbations to be observed are bounded, it is more suitable for challenging ocean settings.
The sections of the paper are briefly articulated as follows.Section 2 exhibits the horizontal PF problem of the underwater vehicles.Section 3 introduces the nonlinear PFC method.The steadiness of the closed-loop system is examined in Section 4. Section 5 deals with simulations and presents discussions.Section 6 exhibits the key remarks related to the conducted research and prospective research.

The statement of the problem
The three-DOF mathematical models of the vehicle containing multiple uncertainties and input saturation are defined as the horizontal PF problem.Fig. 1 illustrates the horizontal PF problem of the underactuated AUV, and the representations of the related frames adopted in the paper are depicted in Table 1.
where the place and yaw angle of the vehicle are denoted by and , respectively when the inertial frame, is used; the surge, sway, and yaw speeds of the vehicle are denoted by u, v, and r, respectively when the body-fixed frame is used.
According to the previous study [30], the dynamic model of the vehicle when multiple uncertainties are under consideration is given by with , , and , where denote the mass terms combining with the added mass, and denotes the bulk and rotational inertia of the vehicle, respectively, , Since the actuators of the vehicle cannot supply infinite force of control and torque, the impact of input saturation should be regarded.Therefore, the actual force of the force and the moment are expressed [24] by (5) where and represent maximum and minimum limits of the input of the control, respectively, and denote the control signals when the saturation issue of the input is neglected.

{I} {B} {SF}
cos sin sin cos The illustrated frames in the PF issue of the vehicle.

2.2
The error dynamics of the PF Fig. 1 depicts the point F, an arbitrary, on the desired trajectory is the starting point of the Serret-Frenet (i.e., the socalled moving target), and the underactuated AUV is driven to follow the virtual moving target F in the inertial frame.
Considering that the wanted path to be traced is represented by parameters utilizing a scalar attribute denoted by , the path-tangential angle at the so-called moving target F is expressed [30] by (6) where and .
Then, the error equation between the underactuated AUV and the virtual moving target F is represented as [30]: where , , and are the along-track, cross-track, and yaw angle errors, respectively, in the course angle of the vehicle, which can be defined as , and is the vehicle's sideslip angle.
Differentiating Eq. ( 6) and defining , the differentiation of Eq. ( 6) is written as follows [13]: (8) where s denotes the curvilinear abscissa of F along with the trajectory, is the curve of the planned path at the virtual moving target F computed by , and is the total speed defined as .

Control objective
Taking into account the physical constraints on the underactuated AUV during the horizontal PF, the practical assumptions are summarized as the following: Assumption 1.The unknown time-varying sideslip angular velocity is bounded, i.e., , where represents a constant, which is positive and unknown [26].
Assumption 2. The unknown time-varying dynamic uncertainties in Eq. ( 2) is considered to be, i.e., , and satisfy , where are fixed, positive and unknown [26].
Assumption 3. A finite desired path is considered, such that , , , and are bounded [21].
According to the above assumptions and analyses, the PFC objective can be expressed as follows: Considering the underactuated AUV model described by Eqs. ( 2) and (7), design an appropriate controller to generate control force and moment .Thus, the vehicle could track a preset geometric trajectory at a given fixed surge velocity when the constraints are denoted by Assumptions 1-3, such that where denotes a small positive constant around zero.It means that the controller should be properly constructed to retain the tracking errors within an arbitrarily narrow range around zero in a relatively short time.
Fig. 2 The structure of the designed controller

Designing the PFC algorithm
A horizontal PF controller for the vehicle with multiple uncertainties and input saturation is constructed.Fig. 2 depicts the edifice of the constructed controlling scheme.The proposed controlling algorithm consists of three components: lineof-sight (LOS) guidance law, kinematic controller, and dynamic controller.The LOS guidance law is designed to attain the wanted LOS guidance angle.The attitude controller is devised utilizing the theory of Lyapunov to extract the wanted yaw angular speed in the kinematic controller.In the dynamic controller, the control signal related to angle and surge speed is constructed to force the underactuated AUV to trace the planned yaw velocity and the surge velocity, respectively.

Designing the guidance law of the LOS
The LOS guidance law is built to attain the planned angle of the LOS.The LOS guidance problem and the associated attributes are depicted in Fig. 1.The vehicle's maneuvering performance depends notably on the distance ahead .
Precisely, a lower value causes more aggressive steering.Therefore, procedures with a time-varying are conducted to assure versatile conduct.To mitigate the issue of complexity, a constant is picked, and the LOS guidance law is where represent the LOS guidance angle.

3.2
The design of the controller of the kinematics

Designing the control law of the attitude
The attitude control law is constructed to attain the wanted speed of yaw .Consider the candidate for a Lyapunov function presented in Eq. ( 9): (10) Taking derivatives of Eq. ( 9) w.r.t time and plugging Eq. ( 7) into it leads to: The desired yaw speed is designed as: (12) where denotes a positive constant to be designed.Plugging Eq. ( 11) into Eq.( 10) leads to: (13)

Designing the movement control law of the virtual target
Consider the candidate of Lyapunov function presented in Eq. ( 13): Differentiating Eq. ( 13) w.r.t time and plugging into Eq.( 7) into it leads to: The so-called control law is built by (16) where denotes a positive constant to be designed.

Designing the control law of the angular velocity
The control law of the angular speed is constructed to construct the control moment to tend the yaw speed to the desired one acquired from the attitude control law.Suppose that the candidate of Lyapunov function is presented in Eq. ( 16): where represent the state of the auxiliary dynamical system.
The following compensator dealing with anti-windup is constructed to handle the impact of the input saturation [14]: where , and are positive constants to be designed, and is a small positive parameter.
Design the control moment as: (20) where and denote the positive constants to be designed.

Designing the control law of the surge velocity
Controlling the surge velocity is established to construct the force of the control force .Thus, the surge velocity tends to be the planned one .Suppose that the candidate of the Lyapunov function is presented in Eq. ( 22): (23) where represents the status of the auxiliary dynamic system.
Similar to the process that designs the control law of the angular speed, a compensator is utilized for anti-windup to defeat the issue of the input saturation [14]. ( ) .
and are positive fixed values to be designed, and is a small positive parameter.

The UDE-based controller design
It is noted that the controllers devised above include the unknown uncertainties , , and .Therefore, the above controllers cannot be directly utilized.Moreover, the above-designed controllers suffer from the inherent calculational complexness in the standard backstepping approach.In this subsection, the UDE technology predicts the uncertainties about kinematic and dynamic aspects, and the NTD technology is utilized to avoid the "explosion of complexity" associated with the conventional backstepping method.
While the kinematic controller is designed, according to the definition of the sild-slip angle and the entire speed , we can deduce that .Thus, the dynamic model (Eq.( 2)) is utilized to compute , resulting in a highly complex expression.Deal with the underlined problem is considered a kinematic uncertainty, and the UDE technology is used for its estimation.Therefore, the desired yaw speed is simplified as follows: (30) where represents the predicted uncertainty pertinent to the kinematic aspect.
The control law of the angular velocity defined by Eq. ( 19) implies the computation of differentiation of the desired yaw speed .Nevertheless, contains a complex parameter , leading to the differential extension issue led by the conventional backstepping computational complexness.Hence, an NTD is adopted to derive the differentiation of the planned yaw velocity.The NTD is defined by [31]: where , , and are the acceleration parameter, the filter factor, and the sampling period, respectively.An exemplification of and a comprehensive description of the NTD is provided in [31].
Based on the theoretical outcomes of the NTD approximation in [32], the subsequent outcomes can be inferred: Corollary 1. Suppose that , an input signal is differentiable and bounded and , arbitrarily picked small numbers, exits that the inequality is met by (32) Thus, the kinematic and dynamic controllers are simplified as follows: According to the dynamic model (Eq.( 2)) and PF dynamic model (Eq.( 7)), the kinematic and dynamic uncertainties are reformulated as the following： (34) Therefore, though the kinematic and dynamic uncertainties , , and can be acquired from the dynamic model and the control inputs, they cannot be directly utilized to construct the control laws.In [33], the UDE technique is utilized to obtain the estimation of a signal.It is assumed that describes the impulse response of a strictly appropriate filter , and the pass-band of the selected low-pass filter includes the whole frequency content of the estimated uncertainties.Thus, the steady-state error is minimized [33].Then, , , and can be precisely estimated utilizing the UDEs as： where , , and are the estimations of , , and , respectively, and ' ' is the convolution operator.
Therefore, the above-designed controllers can be rewritten as： where ' ' represents the Laplace inverse transform operator.
A low-pass filter is considered as one with the impulse response of , where , and the unit step of the signal is denoted by .Then, the controller is further simplified as follows: (37)

The analysis of the stability
The steadiness of the whole enclosed-loop framework is verified.The errors of the tracking are defined by , , Theorem 1.Consider the vehicle with multiple uncertainties and the saturation of the input (Eqs.( 1), (2), and (4)), and Assumptions 1-3 are fulfilled is considered.If the planned angle of the LOS is derived from Eq. ( 8), the wanted parameter of the path is recomputed by Eq. ( 15), and the input saturation is compensated by Eqs. ( 17) and ( 23), the derivative of the wanted yaw velocity is attained via Eq.( 30), and the control signals are computed by Eq. ( 36), then the following propositions hold: (1) The errors of tracking position ( and ), the tracking errors of the attitude , and the tracking error of the speed ( and ) finally tend to be the compact sets close to the origin point.
(2) The sway speed not regulated straightly is uniformly eventually bounded.
Proof: (1) The proof includes the subsequent two states.

State 1. Convergence proof of position is presented as the following
Reconsider the candidate of Lyapunov function: (38) Considering Eqs. ( 6) and ( 8), the derivative of Eq. ( 37) regarding time is attained by Define , , and .Eq. ( 38) can be further derived as: (40) where the designed control parameter is chosen , so that represents a positive-definite matrix.Moreover, that decrease will tend to be the errors in the tracking of the boundary .

State 2. Convergence proof of attitude tracking and velocity tracking.
The estimated error for the uncertainties is defined as follows:

The outcomes of the simulation
To assess both the efficacy and robustness of the suggested controlling scheme, the conduct of simulations is realized.
Table 2 summarizes the parameters pertinent to the horizontal-plane hydrodynamics of the vehicle.
Table 2 The horizontal-plane nominal parameters of the vehicle [36] , , , For a fair comparison, similar to Refs.[13,37], the desired path is shown in Eq. ( 48), and Table 3 depicts the where the variable is attainted by taking the integration .
Table 3 The parameters of the desired path , ,

The simulation of the curvilinear PF
The performance comparison is carried out in the following conditions: (a) the controlling scheme with no saturation of input and no compensation of uncertainties, which is expressed by Without SA+Without UDE in the graphs of the simulations; (b) the controlling scheme without considering input saturation but with uncertainties compensation, which is marked as Without SA+With UDE; (c) the controller with the auxiliary system but without the compensation of the uncertainties, expressed by With SA+Without UDE; (d) the proposed controller with the auxiliary system and perturbations compensation, which is marked as With SA+With UDE.
Table 4 summarizes the primary status of the vehicle, the saturated control force and moment, and the parameters related to the controlling scheme.The below gives the dynamic uncertainties. ( Table 4 The primary status of the vehicle, the saturated control inputs, and the parameters related to devised control parameters The primary status and stance of the vehicle , , , The saturated control force and moment , , , and 6 present the simulation outcomes of position, PF errors, velocities, and control inputs, respectively.
Figs. 3 and 4 summarize the outcomes indicating that the Without SA+With UDE method and the proposed With SA+ With UDE method exhibit the improved performance with higher tracking accuracy.This improved efficacy due to the potential of UDEs in attaining an exact estimation of the uncertainties instantly.The results shown in Fig. 5 demonstrate that the Without SA+With UDE method and the proposed With SA+ With UDE method have more satisfactory steadystate performance than the other two methods.Additionally, it is noted that the surge speed of the Without SA+With UDE method provides a transient response faster than the proposed With SA+With UDE method.However, the control inputs of the Without SA+With UDE method are far beyond the actuators' operation condition, which means that they cannot be employed in practical situations.On the contrary, as presented in Fig. 6, the control signals of the presented With SA+With UDE method all remained in the specified region.
Based on the simulation results and analyses, the presented With SA+With UDE method is more efficient and operable.

Comparative analysis for the horizontal PF problem
To present the efficacy of the devised controller when multiple uncertainties and the saturation of the input is under consideration simultaneously, the suggested controller is compared with the standard backstepping approach suggested in Ref. [13], and the PID approach is defined as follows: (47) where and .
The initial conditions of the vehicle, the saturated control force, and the saturated control moment are presented in Table 5.The traditional backstepping controller parameters can be attained in [13], and Table 6 summarizes the designed control parameters of the devised and PID controllers.
Table 5 The primary status of the vehicle and the saturated control force and moment Initial position and posture of the underactuated AUV , , , The saturated control force and moment , , ,   all controllers force the vehicle to converge to the planned path.Fig. 8 designates that the PF errors of all three controllers tend to a neighborhood around zero.However, the backstepping and proposed controllers show smaller steady-state errors than the PID controller.Fig. 9 shows that the surge speed of all three controllers swiftly converges to the desired speed.
However, the backstepping and proposed controllers have more satisfactory steady-state performance than the PID controller.The control signals of all three controllers are illustrated in Fig. 10.Both the PID and backstepping controllers breach the constraints pertinent to the input, whereas the proposed controlling scheme could assure that the control signals will be kept in the constraint range.
Considering the outcomes of the simulations and investigation presented above, the proposed controlling scheme is efficient for the curved PF.   diminishes the tracking errors, and the presented controller provides the minimum errors.The mentioned efficiency may be due to the potential of UDEs in achieving a precise estimate of the uncertainties in real-time.Figs. 13 and 17 show that both surge and yaw angular velocities of three controllers tend to approximate to the desired values in a limited time, and the uncontrolled sway velocities of all three controllers are still kept bounded.It should be noted that the presented controller provides an improved efficiency with higher tracking precision and smaller chattering.Besides, based on the simulation results given in Figs. 14 and 18, the suggested controlling scheme assures that the control signals will be contained in the constraint boundary, whereas both the PID and backstepping controllers violate the input constraints.
By comparing the presented controller with the PID and backstepping controllers, the efficiency of the suggested controller under multiple uncertainties and input saturation can be validated, and it is observed that the proposed controller results in an improved efficacy under input saturation and multiple uncertainties.

Conclusion
The paper proposed a novel nonlinear robust controller for an accurate PFC of the vehicle when multiple uncertainties and the saturation of the input is under consideration simultaneously.The kinematic controller is built employing the backstepping method and the theory of Lyapunov to attain the desired yaw velocity.The dynamic controller is constructed to track the planned yaw angular velocity generated by the kinematic controller and the given surge speed.The UDE technology is employed to predict uncertainties led by both kinematic and dynamic, the auxiliary dynamical system is designed to eliminate the impact on input saturation, and the NTD is conducted to deal with the differential expansion issue led by the standard backstepping method.It is shown that the track errors of the whole closed-loop system are uniformly and ultimately bounded.The extensive simulations are run to make comparisons and investigation for both the efficacy and robustness of the proposed controlling scheme.The suggested controlling scheme will be improved for the collaborative PFC of the vehicle in prospective research.

About Data Availability Statements
represent the hydrodynamic damping coefficients, and denote the force surge of the control and yaw torque, respectively, and denote the dynamic uncertainties which are the linear superposition of the parameter uncertainties of the model, outside perturbations, and other unmodeled dynamics.

( 2 )
Suppose that the candidate of the Lyapunov function is expressed by: (47) Considering Eqs.(2) and (3), together with Assumptions 1-3, the derivative of Eq. (46) regarding time is described by (48) It is noted that , and are bounded, while and , parameters of the hydrodynamic aspect, are positive constants.Utilizing Chapter 4.8 of Ref. [35], the sway speed of the vehicle is uniformly bounded.

Case 1 :Case 2 :Case 3 :
The whole parameters of the vehicle are picked as nominal values to assess the efficacy of the suggested controlling scheme.To assess the performance and robustness of the devised controller against the uncertainties of the model parameter and external perturbations, the dynamic uncertainties are picked as: , , and .To assess both efficacy and robustness of the proposed controller when the model parameters exhibit uncertainties, outside perturbations, and other unmodeled dynamics, the dynamic uncertainties are chosen as: for Case 1 are summarized in Figs. 7, 8, 9, and 10.The results shown in Fig. 7 indicate that

Fig. 8 .Fig. 9 .
Fig. 8.The outcomes of the simulations of the PF errors when Case 1 is used.(a) along-track error.(b) cross-track error.(c) yaw angle error.

Fig. 10 .
Fig. 10.Simulation results of control signals when Case 1 is used.(a) surge force.(b) yaw torque.The outcomes of the simulation regarding Cases 2 and 3 are summarized in Figs.11, 12, 13, 14,15,16,17, and 18.As shown in Figs.11 and 15, all three controllers can drive the vehicle quickly and tend to the desired trajectory under multiple uncertainties and input saturation.Figs. 12 and 16 show that the tracking errors of all three controllers ultimately converge to a small compact set near zero.However, the PID controller gives significant tracking errors, the backstepping controller

Table 1
Notations of the relevant frames

Table 6
Control parameters of the suggested controlling scheme PID controller