Disturbance Observer-Based Anti-Windup Control for Path-Following of Underactuated AUVs via Singular Perturbations: Theory and Experiment

This paper contributes to an anti-windup control scheme for path-following of underactuated autonomous underwater vehicles (AUVs) subject to constrained inputs. First, a disturbance observer is employed to estimate the model uncertainties and external disturbances on the nominal vehicle model, which are referred to as mismatched lumped disturbances. On the basis of that, a three-time scale singular perturbation control law is proposed, taking advantage of the time scale separation caused by the different rates of various variables. Furthermore, a novel disturbance observer-based anti-windup modification is developed to handle possible input saturation. And stability for the singularly perturbed system subject to actuator saturation is also established, using a method called “manifold reconfiguration” in a geometric view. This results in a relatively simple constrained controller and reduces implementation complexity. Finally, simulation and experimental results are presented to substantiate the effectiveness of the proposed method for path-following of underactuated AUVs exposed to unknown disturbances and input constraints.

windup modification, based on the physical perspective and the "geometric view" mentioned above.Simulation results and experiment results suggest that this approach is feasible.Although our motivating application is to the path-following of AUVs in the horizontal plane, the proposed control method is applicable to a variety of problems in control community, in which the time-scale separation widely exists.In future research, we will extend the proposed method to three-dimensional (3D) path-following, as well as apply the singular perturbation technique to reduce the computation complexity of MPC and Neural Network control.Additionally, as the PI/PID controllers usually suffer from the difficulties of selecting proper control gains, it is also expected that the singular perturbation theory can be employed for solving this problem, by taking advantages of the physical perspective.

> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) <
Ming Lei, Ye Li, Tiedong Zhang, and Dapeng Jiang  Abstract-This paper mainly contributes an anti-windup control scheme for path-following of underactuated autonomous underwater vehicles (AUVs) exposed to constrained inputs.First, a disturbance observer (DO) is employed to estimate the model uncertainties and external disturbances on the vehicle which is called mismatched lumped disturbance.Then, taking advantages of the time scale separation caused by different rates of numerous variables, a three-time scale singular perturbation control scheme is proposed for path-following of underactuated AUVs, leading to a control law with relatively simple structure and thus a reduction of the complexity in implementations.Furthermore, a novel disturbance observer-based anti-windup modification is developed to handle the possible input saturation, by means of time scale decomposition and so-called "manifold reconfiguration" in a geometric view.The stability for the overall system is also established.Finally, the results of simulation and experiment are illustrated to substantiate the efficacy of proposed method for path-following of autonomous underwater vehicles subject to disturbances and input constraints in the horizontal plane.
Note to Practitioners-Path-following is a fundamental motion control problem for AUVs.Despite a number of nonlinear control techniques having offered new tools and promising solutions to deal with the path-following problem of AUVs subject to model uncertainties and actuator saturation, they typically nonetheless yield relatively complicated controllers which may be prohibitive in the real world.Motivated by that, this paper aims to develop an alternative anti-windup control scheme, which should be capable of achieving satisfactory control performance, as well as be easy-to-implement in practical cases.To this end, it suggests a new approach using the theory of singular perturbation and time scales.In this paper, we first make good use of the difference between the bandwidths for observer and vehicle dynamics to design and analyze the DO, so

I. INTRODUCTION
UTONOMOUS underwater vehicle plays an important role in marine activities, since it provides a safe, efficient, and economical way without placing human lives at risk.The motion control scenarios for autonomous underwater vehicles (AUVs) mainly include pathfollowing [1]- [8], trajectory tracking [9]- [10] and formation tracking [11]- [12].In particular, the path-following issue is related to the applications such as oceanographic survey, target carpet searching and pipeline inspection, and so on, thus has drawn compelling interest.During the past few years, numerous path-following controllers have been developed for underactuated AUVs with different focuses.In [1], a variational principle of analytical mechanics and Lagrange multiplier is employed to derive a path-following controller for AUVs.In [2], the neurodynamic optimization technique is used to solve the path-following control problem with velocity and input constraints.In [3], a model-free robust fuzzy adaptive control scheme is proposed for bottom following of a flight-style AUV with input constraints.In [4], an outputfeedback controller is developed for path-following of underactuated AUVs based on extended state observer (ESO) and projection neural networks.In [5], a multi-objective model predictive control (MOMPC) framework is developed > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < for path-following of AUVs.In [6], the disturbance observer (DO) and linear parameter varying (LPV) technique are introduced to enhance robust 3D path-following control of underactuated AUVs with multiple uncertainties.In [7], a deep deterministic policy gradient algorithm based on optimized sample pools and average motion critic network is proposed for path-following of AUVs.In [8], a heuristic fuzzy control scheme is developed for path-following of underactuated AUVs subject to model uncertainties and external disturbances.They have offered new tools and promising solutions to deal with AUV path-following control.However, they usually yield relatively complicated controllers which may be prohibitive in the real world.As a result, from a practical perspective, an easy-to-implement nonlinear pathfollowing control law is of great importance.
Besides, all these methods mentioned above cannot handle the time scale separation caused by different rates of numerous variables.It is worth noting that time scale separation is ubiquitous and naturally arises in control systems, and a direct application of standard control methods, without accounting for the presence of time scale multiplicity, may lead to controller ill-conditioning and/or closed-loop instability [13].For the design and analysis of this kind of multi-time scale dynamic systems, the singular perturbation theory [14] is a more suitable approach.In [15], a robust nonlinear feedback control law is developed for remotely operated vehicles (ROVs) equipped with a robot manipulator, by utilizing the difference between the two-time operation scale between the vehicle and the manipulator.In [16], the singular perturbation theory is introduced to wing-level flight control of underwater glider.In [17], a two-time scale singular perturbation control laws are proposed for path-following of marine surface vessels.In [18], a singular perturbation analysis for the diving control of underactuated AUVs is provided.In [19], a multi-time scale control scheme is proposed for 3D coordinated formation control for underactuated AUVs with uncertainties.As illustration, the singular perturbation theory allows a time scale decomposition of a dynamic system into lower order subsystems, in which the control laws can be designed independently, leading to a reduction of design complexity and a control law with relatively simple structure.With such attractive feature, the design and analysis for path-following of AUVs via singular perturbation technique is a very active topic of research.
Another issue with regard to the AUV motion control is the actuator saturation which is a common phenomenon in practical cases and may deteriorate the performance or even the stability of the closed-loop system if not properly accounted for in control design.Such that, the methods given in [1], [4]- [8] may not be applicable.Generally speaking, there are mainly two categories to cope with actuator saturation: direct design and anti-windup design.The former tries to handle the saturation constraints, meanwhile the nominal performance specifications can also be met.The latter is composed of two stages to handle the performance and the windup, respectively.Singularly perturbed systems with actuator saturation have been studied by many researchers [20]- [22].However, the design and analysis for singularly perturbed systems subject to unknown disturbance and actuator saturation usually are very complex, which holds back its application gravely.Therefore, it is of great significance to surmount this drawback.
In this paper, a method is presented for the path-following problem of underactuated AUVs moving in the horizontal plane.The vehicles are subject to model uncertainties, environmental disturbances and input constraints.Motivated by the aforementioned investigations, a DO is first employed to estimate the mismatched lumped disturbances.Then, the stabilizing controller is developed based on the singular perturbation theory, by taking advantages of the time scale separation caused by different rates of numerous variables.And the stability analysis for a three-time scale singularly perturbed system is performed by constructing a composite Lyapunov function, meanwhile all of the error signals in the overall closed-loop system are proved to be ultimately bounded.Moreover, an anti-windup modification is proposed to stabilize the AUV path-following control system in case of input constraints occur.Finally, the simulation and experiment studies substantiate the efficacy of proposed control scheme for path-following of underactuated AUVs subject to unknown internal and external disturbances, as well as actuator saturation.
Compared with some existing results, the distinct features of the proposed method are as follows. In contrast to the control method as in [1]- [8], the proposed method yields a control law with simple structure, and thus being easy to implement in reality. The difference between the bandwidths for observer and vehicle dynamics is exploited via a singular perturbation model formulation.And analyzing the DO differentially provides a new perspective. A novel anti-windup modification is proposed to handle the possible input saturation, by simply modifying the DO.And a so-called "manifold reconfiguration" method for stability analysis of singularly perturbed systems subject to actuator saturation is developed.They lead to a reduction of the complexity of design and analysis.The remainder of the paper is organized as follows.Section Ⅱ states the problem formulation.Section Ⅲ covers the pathfollowing controller design for the underactuated AUV, which includes disturbance estimation, three-time scale singular perturbation control law design, and stability analysis.Section IV presents a novel anti-windup modification.The results of simulation and experiment are provided to illustrate the efficacy of the proposed method in Section Ⅴ and Section Ⅵ, respectively.Section Ⅶ concludes this paper.

A. Model of an AUV in the Horizontal Plane
This subsection introduces a mathematical model for horizontal path-following control of underactuated AUVs.Neglecting the dynamics of heave, roll and pitch, the motion of an underactuated AUV in the horizontal plane can be expressed by kinematic equations (1) and kinetic equations (2) where and are the positions in the inertial frame {I} as shown in Fig. 1; and are the surge and sway velocities in the body-fixed frame {B}; and are the yaw angle and its angular rate, respectively; , , denote mass including hydrodynamic added mass; is the hydrodynamic damping parameter of AUV model in sway direction.The hydrodynamic damping force and torque in surge and yaw directions are totally unknown.
, and represent the mismatched lumped disturbances on the vehicle.It is assumed that , , and their time derivatives are bounded in the present study.and are control inputs.The following constraints are imposed on the control inputs: min max , min max (3) where min , max , min and max are variable bounds on the control inputs.

B. Kinematic Path-Following Error Dynamics
In this subsection, the kinematic error dynamics for pathfollowing is derived.To formulate the path-following problem, we first define a C 1 desired path , where denotes the path variable, as illustrated in Fig. 1.Let be the crosstrack error.It can be taken as (4) Here, and are the positions of the closest point along the desired path from the vehicle's position ; denotes the path-tangential angle for point and is given by , where and .Moreover, let be the yaw angle tracking error, which is expressed as (5) where is commonly known as the sideslip angle.By taking the time derivatives of and , the obtained error dynamics is as follows: (6) Here, the amplitude is recognized as the moving speed.
is referred as the yawing rate for the point along the desired path and is taken as (7) where and .The path variable propagates with (8) In this case, the speed of tracking point is set to synchronize with the vehicle's velocity component along axis.Such that, once the cross-track error converges to zero, the path-following performance is achieved.
Therefore, the control objective herein is to design a control law for and in (2) to stabilize the cross-track error to zero, so as to force an AUV to follow a prescribed path with desired moving velocity.

A. DO Design
In this subsection, a DO for the mismatched lumped disturbance is given.
To facilitate the observer design, first define , , , . Then, the AUV path-following control issue described by ( 2) and ( 6) can be rewritten in a compact form (9) Let be the estimate of .As a result, the disturbance rejection control can be defined as , where is called stabilizing controller that will be designed later.To determine , we resort to the high gain integral control technique, as shown in Fig. 2. Accordingly, it is given by (10)

> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) <
Here is a control gain that satisfies .Using (10), the mathematical model for path following control is put into an extended-state model as follow.(11) where . Hence, the extended dynamics ( 11) can be naturally treated by standard singular perturbation theory.In this case, is slow variable, is fast variable.Prior to the singular perturbation analysis, two operators and are defined, which will be used throughout this paper.For example, and yield the quasi-steady state (QSS) and the boundary layer correction (BLC) of state variable , respectively.And they should satisfy condition: . For more detail, please refer to [14].Accordingly, the extended dynamics (11) can be decomposed into two lower order subsystems, where the fast model is given by (12) and the slow model is given by (13) Here, is a stretched time scale.is an associated root of degenerate equation .As a result, .Remark 1.In a geometric view, is an invariant manifold for system (11).It reveals that as long as the closedloop system (11) is stable, the state variable will finally converge to the vicinity of .Therefore, the DO can be treated as a fast dynamical control law designed to shape the desired space configuration of fast variable, so as to obtain the nominal reference slow model (13), in which the control law for can be designed independently.This is a significantly important point for the anti-windup modification in Section Ⅳ.

B. Stabilizing Control Law Design without Input Constraints
In the previous subsection, a DO was developed for the total uncertainty.This subsection will present the stabilizing control law design in the slow model (13), by using the singular perturbation theory.
Referring to [14], singular perturbation and hence the timescale character is often associated with a small parameter multiplying the highest derivative of the differential equation or multiplying some of the state variables of the state equation describing a physical system.However, modeling a physical system in the singularly perturbed form may not be easy, due to the fact that it is not always clear how to identify the small singular perturbation parameters.To this end, a singularly perturbed form is obtained by using the method as in [23].In this case, we select as fast variable; as slow variable; as ultra-slow variable.Then, system (13) can be rewritten as (14) Here, .and are artificial small parameters in order to provide a singular perturbation analysis.
Firstly, considering the singular perturbation parameter yields a simplified two-time scale problem formed by the following fast and slow subsystems (15) (16) Here, is a stretched time scale.is the QSS of , given by degenerate equation .Note that, and are treated as fixed parameters in the fast subsystem (15).
Similarly, considering the singular perturbation parameter , subsystem (16) also can be decomposed into another two subsystems of lower order.They are respectively defined by ( 17) (18) Here, is another stretched time scale.is the QSS of , given by degenerate equation . Also note, is treated as a fixed parameter in the fast subsystem (18).In a geometric view, and are invariant manifolds for system (14).This is a significantly important point for the anti-windup control design and analysis in Section Ⅳ.
As long as the full dynamics ( 14) is decomposed into three subsystems with different time scales, the control laws can be designed in each subsystem separately, as shown in Fig. 3. To Here, , , . is a positive constant, which denotes the desired surge velocity.In a geometric view, control law (19) aims to drive the state variables to manifold .Then, assuming the fast variable has reached its equilibrium, subsystem ( 17) is asymptotically stable by selecting a virtual feedback control law (20) where . In a geometric view, control law (20) aims to drive the state variables to manifold .Finally, assuming and have reached their equilibrium and evolved on their own manifolds, subsystem (18) will be asymptotically stable by selecting (21) where , .As illustration, the stabilizing control law is given by combining ( 20)- (22).One might find that the design procedure is very concise, and the proposed singular perturbation control law is with a relatively simple structure, and thus is easy-to-implement in reality.

C. Stability Analysis
In this subsection, the stability analysis for proposed unsaturated path-following control law is given.
At first, we prove the input-to-state stability of system (14).According to the time scale decomposition, it can be divided into two steps: the stability analysis for subsystem (16) and the stability analysis for full dynamics (14).
Step 1: stability analysis for lower order subsystem (16) As the result of decomposition, the construction of Lyapunov functions for subsystems is comparatively easy.Following the method as in [14] and [24], we consider to use their weighted sum to construct a Lyapunov function candidate for system (16).It is taken as , where .and are Lyapunov functions for subsystems (18) and (17), respectively.Then the derivative of along ( 16) is computed as Suppose that the following conditions hold: Using the conditions (a)-(d) above and Young's equation, is put into (22) Here, .It can be easily seen that the minimum value of occurs at and is given by (23) Therefore, the origin of system ( 16) is a globally asymptotically stable equilibrium point while the following condition holds: Step 2: stability analysis for full dynamics (14) On the basis of that, the following Lyapunov function candidate for system (14) is considered: , where .and are Lyapunov functions for subsystems (16) and (15), respectively.Straightforward calculation shows that Similarly, suppose that the following conditions hold: (e) ; (f) ; Using the conditions (e)-(h) above and Young's equation yields where .It can be seen that the minimum value of occurs at and is given by (26) In practical cases, and are usually unachievable owing to economic and other practical limitations.But both of them are bounded while the AUV moves with a certain velocity, i.e., there exist positive constants and such that and .As , , , Therefore, the full dynamics ( 14) with the input vector being , and is input-to-state stable (ISS), while the following condition holds: Namely, ( 24) and (28) provide a mathematical bound on the control gains.Next, we prove the input-to-state stability of overall closedlooped system (11), by the following lemma.
Lemma 1.The overall closed-loop system (11) with the input vector being , and is ISS.Proof.The overall closed-loop system (11) can be rewritten as following form Referring to the Step 2: stability analysis for full dynamics (14), one might find that the system (29a) with the input vector being , , and is ISS.The proofs are omitted for the sake of brevity.
Following that, we prove the input-to-state stability of estimate error dynamics (29b).Constructing a Lyapunov function as and differentiating (29b), we have .It can be concluded that the estimate error dynamics (11b) is ISS, as makes .Finally, by [25,Lemma C.4], the cascade system formed by the path-following control system (29a) and the estimate error dynamics (29b) is ISS.Therefore, all error signals in the closed-loop system are ultimately bounded while , , are bounded.

IV. ANTI-WINDUP MODIFICATION
In this section, an anti-windup modification is presented, based on the following assumptions: (1) The variable bounds min , , max known.
(2) The control inputs could completely reject the estimates of unknown disturbances, that is, min max and max Considering the following input constraints: where the input bias caused by the actuator saturation, given by .The control law has been given in Section Ⅲ-B.According to the proposed method for DO design as in Section Ⅲ-A, the DO is modified as It is clear that, as long as the closed-loop system (30) is stable, the estimate will finally converge to the vicinity of even there exists input constraints. . is a solution of univariate cubic equation .In the light of the characteristics of univariate cubic function which satisfies conditions and , one might find that there must be solution while the aforementioned assumption (2) holds.
Remark 2: We care less the exact solutions and , but their existence.Note that, and are bounded in case of is continuously differentiable and is bounded.It will prove convenient.This is one of the main contributions of the paper.
The stability analysis for system (30) with input constraints can be performed by replacing the control gains , , , by their reconstructed values , , , Specifically, referring to the stability analysis in Section Ⅲ-C, one find that the conditions (24) (28) hold for and Such that, recalling Lemma 1, it is easy to demonstrate that the system (30a) with the input vector being , , , , and is ISS.The proofs are omitted for the sake of brevity.Besides, we observe the input-to-state stability of estimate error dynamics (30b).Finally, the cascade system (30) formed by the path-following control system (30a) and the estimate error dynamics (30b) is therefore ISS, following the [25,Lemma C.4].As a result, all error signals in the closed-loop system are ultimately bounded while , , , and are bounded.Remark 3: The modified controller can stabilize the AUV path-following control system in case of there is no enough control ability to track the nominal reference.And once the saturation ends, the modified controller degenerates to the nominal reference which can drive an underactuated AUV to the prescribed path with good performance.

V. SIMULATION RESULTS
In this section, the simulation results are provided to illustrate the effectiveness of the proposed controller, using an AUV model used in earlier study [26].
Simulations are conducted using a fourth-order Runge-Kutta fixed-step integration algorithm with an integration step of 0.01 s.Two representative cases are considered, i.e., a pathfollowing mission with unconstraint inputs (Case A) and a path-following mission with bounded inputs (Case B).The desired path is given by The initial states of the AUV are set to , , , , , .The desired surge velocity is set to .The bound for is taken as .To guarantee the stability of the closed-loop system, the design parameters for proposed controller are selected as and , , , and .

A. Simulation for the AUV Without Input Constraints
In order to demonstrate the superiority of the proposed unsaturated controller, the backstepping unsaturated control and PID unsaturated control are also tested, respectively.Based on [26], the backstepping unsaturated control law is given by Here, and .
is the path curvature.denotes the LOS guidance law, and given .For more detail, please refer to [26].The PID unsaturated control law is given by Here, and .The control gains used in the simulation are listed as follows: , . Relevant simulation results are given in Figs.5-10.The motion trajectories and path-following error norms of different controllers are shown in Figs. 5 and 6, respectively.The perfect path-following performance of the proposed controller in spite of severe parametric uncertainties and external disturbances is observed, which has verified the analysis in Lemma 1.And from the enlargements in Figs. 5 and 6, it is as expected that proposed unsaturated control and backstepping unsaturated control achieve better performance during the steady phase, since the PID unsaturated control suffers from a    The control inputs under different control schemes are shown in Fig. 7.It is clear that the peak values occur at the beginning and in the sudden change of desired path, because the pat-following errors between the AUV and the path is relatively large and the corresponding control laws are such updated that the AUV can quickly converge to the desired path.Note that all of them might be outside of the permitted range in practical cases.And from the enlargements in Fig. 7, we can see that the backstepping unsaturated controller generates a large and shaking yaw torque curve in the first 1 s and the sudden change of desired path, which likely results from the first-order and second-order derivative calculation.Although the control commands of PID does not shake, it leads to larger oscillation behaviors, compared with the proposed unsaturated control.The velocities profiles of different controllers are given in Fig. 8.
As shown in Fig. 9, by using the proposed saturated control method, the cross-track error , boundary correction and asymptotically decrease to the vicinity of zero.It is also clear to see that the dynamic response speed of is the fastest while the dynamic response speed of is faster than that of , which proves reasonable to apply the theory of singular perturbation and time scale to path-following control problem of underactuated AUVs.The estimation performance of proposed DO is shown in Figs. 10.It can be seen that the designed DO can rapidly estimate the unknown disturbances.

B. Simulation for the AUV With Input Constraints
To show the anti-windup ability of the proposed controller, simulation on the vehicle model subject to bounded inputs is also presented, in which the following input constraints are considered: and .Besides, the backstepping saturated control and PID saturated control are also employed here for comparison.The backstepping saturated control scheme is obtained by using the proposed anti-windup modification.And the PID saturated control scheme is obtained by isolating the integrator when input saturation occurs.
Simulation results under the above three saturated control schemes are illustrated in Figs.11-16.Besides the similar behaviors as those in Case A, some interesting phenomena can be found due to the possible input saturation.A simple observation of Figs.11 and 12 shows that the AUV is capable of tracking the given path under the proposed saturated control scheme, in the presence of internal and external disturbances, as well as input constraints.It also can be seen that the backstepping saturated control under the proposed anti-windup modification works well, which has verified the effectiveness of the proposed anti-windup modification in some extent.And from the enlargements in Figs.11 and 12, it is as expected that both of the proposed    Besides some kind of nature and characteristic of their own, one of the main reasons is that the PID saturated control scheme cannot actively approximate and eliminate the timevarying disturbance anymore due to the isolation of integrator and compensation for the anti-windup purpose.However, the proposed anti-windup modification can utilize the DO to estimate and reject the unknown disturbances, even the saturation occurs.This can be seen in Fig. 16.
Relevant control inputs are illustrated in Fig. 13.It is interesting to find that the backstepping saturated control eliminates the large and shaking yaw torque curve (see Fig. 7).One of the main reasons might be that the saturation keeps a lid on the sharp dynamic response, leading to a reduction of the first-order and second-order derivative of virtual control signals.And as expected, the control inputs under the PID saturated control scheme have undesirable oscillations due to the sudden change of the control structure.This inevitably degrades the tracking performance the transient phase, which can be seen from the comparisons of the motion trajectories in Fig. 11.On the other side, both of the proposed saturated control and the backstepping saturated control generate sufficiently smooth signals, and this benefits from such a novel anti-windup mechanism that can modify the tracking references with the assistance of the proposed DO.Obviously, the velocities also have the similar behavior in Fig. 14.
As shown in Fig. 15, by using the proposed saturated control method, the cross-track error , boundary correction and converges to the vicinity of zero, even there is possible input saturation.It proves the effectiveness of the proposed anti-windup modification as in Section Ⅳ.The estimation performance of proposed DO is shown in Figs.16.The comparison between given mismatched lumped disturbances and estimated values is given.It reveals that the unknown disturbances, adding on the closed-loop control system, can be accurately estimated by using the proposed DO, even though there are input constraints.

Ⅵ. EXPERIMENT RESULTS
In this section, the experiment results are provided to illustrate the control performance of proposed method, using an AUV, as shown in Fig. 17.The main model parameters are listed in Table Ⅰ.The experiment studies were conducted in an indoor pond.The main hardware architecture of AUV control system for experiment studies can be seen in Fig. 18.The control inputs are generated by a propeller in surge direction and a rudder in yaw direction.The acceleration and yaw rate of the AUV are measured by the magnetic compass integrated in the JY901 inertial navigation module of WITMOTION SHENZHEN Co., LTD.It is equipped on the AUV.The position of AUV is given by using the two receiving terminals of ultra-wide band positioning system (UWBPS), which are highlighted by blue circles in Fig. 17.And the yaw angle is derived by using the relative position of these two receiving terminals.As the UWBPS suffers from the indoor environment, especially the steel facilities, the measurement errors of position and yaw angle are unavoidable.In the   present study, the effect of measurement errors is simply treated as mismatched disturbance.In this case, the experiment results will prove the robustness of proposed controller.Also note, the initial position of AUV is near the X-Y carriage.As a result, the position and yaw angle might extremely vibrate at the beginning (see Figs. 19,20,22,23).
The experiment studies consist of two different cases: -       controller has satisfactory performance in driving the vehicle to the prescribed fold line and sinusoidal curve, in the presence of model uncertainties and actuator saturation.And as expected, the proposed saturated control method realizes better control performance compared with the classical PID saturated control.Figs.20 and 23 show the system speed evolutions for cases 1 and 2, respectively.It can be seen that the surge speed converges to within a short time.The results also indicate that both of the sway speed and yaw rate are bounded.The yaw angle and control commands in surge and yaw directions are plotted in Figs.21 and 24.Comparing the stern control surface deflection commands for proposed controller and PID controller, one might observe that the proposed method have smoother controller input, whereas the PID controller suffers an obvious overshoot.Such that, the yaw angle and its rate, by using the proposed controller, exhibits smaller oscillation behavior.In all, the relative tendency between the proposed controller and the PID controller is same as the results in simulation, but with more shaking in reality, caused by the sensor noises.Accordingly, effective filter might be an important part in improving the control performance.
As stated above, the proposed control method has been thoroughly realized in an AUV without other expensive equipment, which verifies the its effectiveness and being easyto-implement in practical cases.

Ⅶ. CONCLUSION
This paper considered the path-following control problem of underactuated AUVs subject to unknown internal and external disturbances, as well as actuator saturation.The proposed controller has the ability to estimate the lumped disturbances and we need not make complicated analysis on a variety of uncertainties and disturbances in the design process.A threetime scale singular perturbation control method is proposed to obtain a nonlinear path-following controller with simple structure and lower the complexity in implementation.Considering the possible actuator saturation, an anti-windup modification is developed to stabilize the AUV to desired path when input constraints occur, meanwhile the stability can be guaranteed.
To verify the designed controller, two experiment studies are conducted in an indoor pond.The experiment results show that the proposed controller can achieve satisfactory pathfollowing performance with smooth control inputs in spite of the possible input saturation, which demonstrates the effectiveness and superiority of our proposed control method.
Although our motivating application is to the path-following of AUVs in the horizontal plane, the singular perturbation control method is applicable to a variety of problems in AUV control community, in which the time-scale separation widely exists.For example, in the diving motion system of AUV, the pitch motion is much fast than the error dynamics.Therefore, the proposed method can be extended to 3D path-following.Nonetheless, many important issues still remain untouched.Future work will focus on MPC and Neural Networks (NNs) control based on singular perturbation technique, so as to reduce the computation complexity.Additionally, as the PI/PID controllers usually suffer from the difficulties of selecting proper control gains, it is also expected that the singular perturbation theory can be employed for solving this problem, by taking advantages of the physical perspective.
Therefore, the conditions (g) and (h) hold in case of , and .
positivedefinite functions.The derivation of the coefficients and functions in these assumptions is outlined in Appendix.
positivedefinite functions.The derivation of the coefficients and functions in these assumptions is outlined in Appendix.

Fig. 5 .
Fig. 5. Motion trajectories different in Case A. Fig. 6.Path-following error norms in Case A, where .

Fig. 7 .
Fig. 7. Control commands of different controllers in Case A.

Fig. 8 .
Fig. 8. Velocities profiles of different controllers in Case A.

Fig. 9 .
Fig. 9. Path-following errors of proposed control method in Case A.

Fig. 10 .
Fig. 10.Estimation performance by using the DO in Case A.

Fig. 11 .
Fig. 11.Motion trajectories of different controllers in Case B.

Fig. 13 .
Fig. 13.Control commands of different controllers in Case B.

Fig. 14 .
Fig. 14.Velocities profiles of different controllers in Case B.

Fig. 15 .
Fig. 15.Path-following errors of proposed control method in Case B.

Fig. 16 .
Fig. 16.Estimation performance by using the proposed DO in Case B.
Case 1: fold line -Case 2: sinusoidal curve I THE MAIN MODEL PARAMETERS OF YX-1 PID control is also employed here for comparison.But the backstepping control is not involved, owing to economic and other practical limitations.Experiment results under the two different controllers are provided in Figs.19-24.A simple observation of Figs.19 and 22 shows that the proposed
conditions (c) and (d) hold, with and Referring to(23), one has (