## 1. Introduction

Autonomous underwater vehicles (AUVs) are a kind of handy and practical tool for assisting humans with underwater research. They can be used to explore the ocean floor, observe the marine environment, track shoals of fish, and so on in ecological and environmental studies. In military affairs, the AUVs also play an important role, which can be used to detect the situation of the enemies and equip weapons to defend against the enemies (Bovio et al. 2006). All of these high-end applications need the AUVs to possess better maneuverability and controllability. Nevertheless, a lot of uncertainties caused by ocean current interference are unavoidable in the ocean environment, and the communication in the seawater is very inefficient at the same time. Not only that, the AUV system is nonlinear and strongly coupled, which makes it impossible to get accurate values for many important parameters (Yan et al. 2015). These situations have brought huge challenges to relevant research, but they must be conquered by all the researchers.

Trajectory tracking control is a usual mission for AUV to fulfill a particular requirement. This control method requires AUV to reach a certain place at a certain time for the reference trajectory. With the advancement of technology, researchers have attempted all kinds of methods to deal with the trajectory tracking problem, among which sliding mode control (SMC) is most widely used (Xu et al. 2015). In subsequent research and developments, modified SMC methods such as back-stepping sliding mode control (BSSMC) (Yan et al. 2019a; Li et al. 2019), integral sliding mode control (ISMC) (Li et al. 2019; Qiao and Zhang 2019a, 2020), and terminal sliding mode control (TSMC) (Qiao and Zhang 2020; Elmokadem et al. 2017) put up a good performance of tracking the reference trajectory, and the convergence time of the system is greatly reduced in the meantime. To solve the influences of external disturbance, parameter perturbation, and model uncertainty, researchers have adopted the adaptive technique (Qiao and Zhang 2019b), fuzzy algorithm (Liu et al. 2019; X. Wang et al. 2019; Liang et al. 2020), and neural network (Chu et al. 2017; Yan et al. 2019b; J. L. Zhang et al. 2020) to deal with the trajectory tracking problem of AUV. There are also methods by predicting the future states and the output of the system, then using correct the motion states and system output online in the trajectory tracking process of AUV, such as model predictive control (MPC) (Yan et al. 2020; Gong et al. 2020).

Event-triggered control (ETC) is a type of control method which is different from traditional continuous time-triggered control. Event-triggered control does not calculate and change the control input in each cycle, but in some specific moments determined by the stable relationship of the system. With these designs, it is possible to economize a lot of communication resources. There have also been more and more event-triggered methods applying to nonlinear systems (Ning et al. 2018; A. Wang et al. 2020; L. Wang et al. 2020; Y.-W. Wang et al. 2020). Two types of dynamic output feedback event-trigger controllers containing continuous and sampled measurement outputs in the system are given in J. H. Zhang et al. (2020). It is also a good attempt to apply the event-trigger mechanism to the neural network (Zhang and Bao 2020). Meanwhile, a mass of mature precedents of event-triggered control have been successfully applied to the field of aerospace (Wei et al. 2018; F. Wang et al. 2019; Liu et al. 2020). However, AUV is a typical nonlinear system. Relevant studies about using event-triggered control on AUV are few and far between (Xu et al. 2020; Su et al. 2021). It is the first and bold attempt by employing event-triggered control to solve the trajectory tracking problem of AUVs in this paper.

The rest of the article is constituted as follows: the mathematical model of AUV and the modeled saturation characteristic of actuators are given in section 2. Section 3 presents the designed event-triggered mechanism, adaptive laws and the output of actuators suffering from saturation characteristic in detail. In section 4, rigorous mathematical proofs are proceeded to certify that the closed-loop system is asymptotically stable and the Zeno behavior can be exclude by the presented adaptive ETSMC scheme. Simulation results and conclusions are given in sections 5 and 6, respectively.

The notations used in this paper are as follows. Scalars are denoted by normal math fonts; vectors or matrices are denoted by bold math fonts. For any vector **x** ∈ *R ^{n}* and any matrix

*R*

^{}

^{n}^{×}

^{n}^{},

*x*(

_{i}*i*= 1, 2, …,

*n*) represents the

*i*th element of

**x**. The superscript T on

**x**

^{T}and

^{T}denotes the transpose of vector

**x**and matrix

**x**is denoted as

## 2. Problem formulation

### a. AUV model

Remark 1: In theory, an AUV has 6 degrees of freedom (DOFs) to be controlled. But the rolling stability is verified through stability calculations during the overall design of AUV. Therefore, the rolling motion does not require additional control in practical application. That is, the rolling angle *φ* and rolling angular velocity *p* are defaulted to zero.

**) mentioned above is Jacobian transformation matrix with the following expression:**

*η***v**) indicates the matrix of the Coriolis and centripetal forces,

**v**) represents the matrix of the hydrodynamic damping terms,

**g**(

**) depicts restoring forces including gravity and buoyancy,**

*η*

*τ**elucidates the vector of the time-varying external disturbances, and*

_{d}*τ*shows the control forces and torques provided by the actuators. These matrixes

_{u}**v**),

**v**) and vector

**g**(

**) can be broken down into fixed terms determined by the system and nonlinear terms caused by uncertain factors, which can be written as**

*η**,*

_{f}*(*

_{f}**v**),

*(*

_{f}**v**), and

**g**

*(*

_{f}**) denote the fixed terms, and**

*η**,*

_{n}*(*

_{n}**v**),

*(*

_{n}**v**), and

**g**

*(*

_{n}**) represent the nonlinear terms:**

*η**ρ*is the density of water,

*g*denotes acceleration of gravity, and

**∇**

*GM*indicates vertical metacentric height, which means the vertical distance between the center of gravity and buoyancy.

_{L}

*τ**indicates the lump of all nonlinear terms, which is shown as*

_{n}Assumption 1. The motion of AUV is approximated to the general motion of rigid body in fluid.

Assumption 2. Assume that external disturbances that change over time are bounded and have a positive constant *d _{m}* so as to satisfy ǁ

*τ**ǁ <*

_{d}*d*.

_{m}

*τ**can be constrained by the following form:*

_{n}*ϑ*,

_{0}*ϑ*

_{1}, and

*ϑ*

_{2}are unknown positive constants. In ǁ

*τ**ǁ, the constant term is strictly smaller than*

_{n}*ϑ*

_{0}, the coefficient of ǁ

**v**ǁ is strictly smaller than

*ϑ*

_{1}, and the coefficient of ǁ

**v**ǁ

^{2}is strictly smaller than

*ϑ*

_{2}.

### b. Actuator saturation characteristic

The actuators of AUVs suffer from different saturation characteristics, including thrusters and steering engines.

Assumption 4. Each actuator can provide output in both positive and negative directions.

*τ*(

_{i}*i*= 1, 2, 3) denote the output of actuator, in other words, the amount of thrust force provided by each thruster.

*τ*

_{i}_{,force}(

*i*= 1, 2, 3) stand for the output passed by the controller.

*τ*

_{i}_{,}

_{f}_{max}and

*τ*

_{i}_{,}

_{f}_{min}represent the critical value of positive and negative thrust that can be provided, respectively.

*τ*(

_{j}*j*= 4, 5) demonstrate the steering torque produced by steering engine.

*τ*

_{j}_{,torque}(

*j*= 4, 5) depict the steering command transmitted from controller to actuator.

*τ*

_{j}_{,}

_{t}_{max}and

*τ*

_{j}_{,}

_{t}_{min}also represent the maximum and minimum steering torque.

Assumption 5. To facilitate analysis and calculation, it can be assumed that the thruster and steering engine have the same saturation characteristic parameters.

## 3. Adaptive ETSMC design

When dealing with the trajectory tracking problem of AUV, it usually gives a reference trajectory to be tracked. The reference trajectory is defined as **v*** _{r}* is structured to track the reference trajectory

*η**. Then, the task requirement of trajectory tracking can be accomplished by designing appropriate velocity*

_{d}**v**of AUV to track the virtual velocity

**v**

*. The system structure of adaptive ETSMC is as shown in Fig. 2.*

_{r}Block diagram of the adaptive ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Block diagram of the adaptive ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Block diagram of the adaptive ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

### a. Design of sliding mode surface

**e**

_{1}denotes the tracking error of position and attitude between the virtual object and the reference trajectory, and

**e**

_{2}shows the velocity error of AUV relative to the virtual speed.

### b. Design of event-triggered mechanism

From the system structure diagram in Fig. 2, it can be observed that the position, attitude, and velocity of AUV measured by sensor are calculated and then transmitted to the controller. The event-triggered mechanism is applied to the transmission channel between controller and actuator, which decides whether the control command is transmitted from controller to actuator.

*t*∈ [

*t*,

_{k}*t*

_{k}_{+1}):

*t*(

_{k}*k*∈

*N*) represents the trigger instant.

Remark 2: At the same time, the control command is transferred to actuator. Afterward, the output of the actuator is maintaining unaltered until next event is triggered. The error variable

*β*∈ (0, 1) and

*m*> 0 are the event-triggered parameters.

Remark 3: Due to the existence of the initial error of the system, it operates normally as designed at the beginning. As the program runs, when the system gradually tends to be stable, the error variables approach to zero; that is to say, ǁ*t*)ǁ gets close to zero. If just a fixed threshold is adopted when designing the event-triggering mechanism, it may cause events to be triggered frequently, which is contrary to what we hope to achieve. Hence, an event-triggered scheme with variable threshold is created. When the system tends to stabilize, the error variables are accumulated gradually. The event is triggered only when the accumulated error values exceed the set threshold *m*, which has better manifestations of control performance. At the same time, it can effectively avoid Zeno phenomenon.

### c. Adaptive laws

*t*∈ [

*t*,

_{k}*t*

_{k}_{+1}), it must be possible to take suitable values of

The parameters

*δ*(

_{i}*i*= 0, 1, 2) are positive constants.

### d. Controller design and output of actuators

*τ*(

_{i}*t*) (

*i*= 1, 2, …, 5) denote the control command at each DOF,

*b*> 0 is a constant control parameter, and

Remark 4: In fact, the event-triggered mechanism plays a role in the channel between controller and actuator. When an event is activated, the control command is transmitted to the actuator. Then the output of actuator will remain the same until the next event is triggered.

*t*∈ [

*t*,

_{k}*t*

_{k}_{+1}), the output of actuators can be drawn that as follows according to assumption 5:

*τ*

_{max}and

*τ*

_{min}represent the maximum values of positive and negative output, respectively.

## 4. Stability analysis

From the previous designs, a conclusion can be drawn that as long as the system is asymptotically stable over the time interval of triggering *t* ∈ [*t _{k}*,

*t*

_{k}_{+1}), then the whole closed-loop system is also asymptotically stable.

_{1}, the virtual velocity

**v**

*is designed as follows:*

_{r}*k*

_{1}is a positive constant.

**e**

_{1}converges to a compact set near zero.

*t*∈ [

*t*,

_{k}*t*

_{k}_{+1}), as long as the appropriate control parameter

*b*and the event-triggered parameter

*β*are chosen which satisfy

Then the proposed ETSMC scheme can guarantee the closed-loop system converge to a small neighborhood of the origin.

*t*∈ [

*t*,

_{k}*t*

_{k}_{+1}):

_{2}(

*t*)ǁ >

_{k}*m*, the event-triggered scheme is denoted as follows:

*k*∈ [−

*β*,

*β*].

Next, Eq. (47) is analyzed separately.

*τ*(

_{i}*t*) >

_{k}*τ*

_{max}, which can derive that from Eq. (28):

*i*= 1, 2, …, 5), so (47) can evolve into

Remark 5: If *P*_{1} > *P*_{2}, then it can get that *P*_{1} > *P*_{2} = *P*; substitute this conclusion into Eq. (55), *P*_{1} < *P*_{2}, then it can get that *P*_{1} = *P*; substitute this conclusion into Eq. (55),

From the above analysis, it can be known that _{2}(*t*) can be verified in this case.

*τ*(

_{i}*t*) < –

_{k}*τ*

_{min}, similarly it yields that

*i*= 1, 2, …, 5), then (47) can transform into

Use the same derivation method in case 1, it can also be concluded that _{2}(*t*) can be guaranteed in case 2.

*τ*

_{min}≤

*τ*(

_{i}*t*) ≤

_{k}*τ*

_{max}, it can be observed that

Combine (23) and (35) with (61), it can be obtained that _{2}(*t*) can be guaranteed by the output of actuator which is shown in (31).

_{2}(

*t*)ǁ ≤

_{k}*m*, the event-triggered scheme is denoted as follows:

Therefore, _{2}(*t*) is bounded by

Remark 6: After an event is triggered, the error variable *m*, the next event is triggered and it must be satisfied that _{2}(*t _{k}*

_{+1})ǁ ≥

*m*, the event-triggered mechanism changes from situation 2 to situation 1. However,

_{2}(

*t*) is asymptotically stable, which leads to the event-triggered scheme switching to situation 2 again. So the event-triggered mechanism is used interchangeably between situation 1 and situation 2.

According to the description in Qiao and Zhang (2019a), the error variable **e**_{1} and **e**_{2} constitute a cascaded system, the full closed-loop cascaded system is uniformly finite-time stable.

Theorem 2. The designed event-triggered mechanism shown in (22) can effectively exclude the Zeno behavior.

*τ**,*

_{u}

*τ**, and*

_{n}_{1}, B

_{2}, and B

_{3}, respectively. Denoting

*κ*= B

_{1}+ B

_{2}+ B

_{3}. From (67), one has

**e**(

_{t}*t*) = 0, it is derived that

_{k}*T*=

_{k}*t*

_{k}_{+1}−

*t*be the lower bound of trigger time interval. From (69) and (70), it obtains that

_{k}Therefore, the designed event-triggered mechanism can never be triggered countless times in finite time. The Zeno behavior can be ruled out effectively.

## 5. Simulation results

In this section, two numerical simulation experiments are conducted to demonstrate the feasibility and the advantages in saving communication resources of the proposed ETSMC scheme which suffers from external disturbances and actuator saturation characteristics.

The event-triggered parameters are given by *β* = 0.2 and *m* = 0.05. The parameter *k*_{1} of the virtual velocity **v*** _{r}* is set as

*k*

_{1}= 1.

The adaptive gains are chosen as *δ*_{0} = *δ*_{1} = *δ*_{2} = 0.05, and the initial values estimated by adaptive laws are selected as

The control parameter is set as *b* = 3, and the maximum values of positive and negative output of actuator are chosen as *τ*_{max} = *τ*_{min} = 300.

The parameters related to AUV model are as follows in Table 1.

Parameter values of the AUV model.

### a. Trajectory tracking of spiral helical curve

The performances of ETSMC scheme are demonstrated by tracking the spatial helical reference curve described as *ξ _{d}*(

*t*) = 10 sin(0.1

*t*) m,

*η*(

_{d}*t*) = 10 cos(0.1

*t*) m,

*ζ*(

_{d}*t*) = 0.1

*t*m. The initial positions of AUV are chosen as

*ξ*(0) = −5 m,

*η*(0) = 10 m,

*ζ*(0) = 0 m,

*θ*(0) =

*ψ*(0) = 0 rad. The 3D figure of trajectory tracking under the ETSMC scheme is shown in Fig. 3, which illustrates the AUV can track the desired trajectory very well. The interval time of event-triggered mechanism is presented in Fig. 4, which shows that the control commands are transmitted to the actuator 234 times in 100 s. Nevertheless, the conventional time-triggered control method is transmitted 10 000 times, which illustrates that the proposed ETSMC scheme can save a lot of communication resources.

Trajectory tracking response under ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Trajectory tracking response under ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Trajectory tracking response under ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The interval time of event-triggered mechanism.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The interval time of event-triggered mechanism.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The interval time of event-triggered mechanism.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The desired velocities and the actual velocities are exhibited in Fig. 5, where the actual linear velocity *u* converges to a small neighborhood around 1 m s^{−1} and the actual angular velocity *r* converges to a small neighborhood around −0.1 rad s^{−1}. The position and attitude tracking errors and the linear velocity and angular velocity tracking errors are displayed in Fig. 6, which can observe that all the tracking errors converge to small bounded fields near zero. The thrusts and torques provided by actuator are shown in Fig. 7.

(a) The desired linear velocities and angular velocities of AUV; (b) the actual linear velocities and angular velocities of AUV.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

(a) The desired linear velocities and angular velocities of AUV; (b) the actual linear velocities and angular velocities of AUV.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

(a) The desired linear velocities and angular velocities of AUV; (b) the actual linear velocities and angular velocities of AUV.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

(a) The position and attitude tracking errors; (b) the linear velocity and angular velocity tracking errors.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

(a) The position and attitude tracking errors; (b) the linear velocity and angular velocity tracking errors.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

(a) The position and attitude tracking errors; (b) the linear velocity and angular velocity tracking errors.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

### b. Trajectory tracking of spatial sine curve

To illustrate the applicability of the designed ETSMC scheme, a more complicated spatial sine trajectory described as *ξ _{d}*(

*t*) =

*t*m,

*η*(

_{d}*t*) = 10 sin (0.1

*t*) m,

*ζ*(

_{d}*t*) = 10 + 10sin (0.1

*t*) m. The initial positions of AUV are chosen as

*ξ*(0) = −5 m,

*η*(0) = 0 m,

*ζ*(0) = 10 m,

*θ*(0) =

*ψ*(0) = 0 rad.

Figure 8 gives the 3D view of tracking spatial sine trajectory under the ETSMC scheme. Figure 9 shows the interval time of event-triggered mechanism, where the control commands are transmitted to the actuator 280 times in 100 s. Combining Figs. 8 and 9, the events are triggered less frequently in the time of 10–20, 45–55, and 75–85 s at the corner of the spatial sine curve, which is in line with the design idea.

Trajectory tracking response under ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Trajectory tracking response under ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Trajectory tracking response under ETSMC scheme.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The interval time of event-triggered mechanism.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The interval time of event-triggered mechanism.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The interval time of event-triggered mechanism.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The desired and actual velocities are shown in Fig. 10. The actual velocities tend to be roughly the same as the expected velocities. The position and attitude tracking errors and velocity tracking errors are displayed in Fig. 11. Due to the presence of the limit of radius of steering curvature and actuator saturation of AUV, all the errors cannot strictly converge to zero, but they converge to compact sets more or less around the zero. It is acceptable with the maximum position error = 0.5 m and the maximum attitude error = 0.3 rad, which demonstrates the feasibility of the designed ETSMC scheme. Figure 12 gives the thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

The thrusts and torques provided by actuator.

Citation: Journal of Atmospheric and Oceanic Technology 39, 12; 10.1175/JTECH-D-22-0056.1

## 6. Conclusions

In this article, an adaptive ETSMC scheme with variable threshold is proposed for the trajectory tracking control of AUVs subject to external disturbances and actuator saturation. The designed event-triggered mechanism is acted on the transmission channel between controller and actuator, which has been demonstrated to economize abundant communication resources compared to traditional time-triggering schemes. The adaptive technique is adopted to estimate the nonlinear terms of the whole closed-loop system caused by the model uncertainties and unknown external disturbances. Rigorous mathematical proofs are proceeded to illustrate that all the tracking errors are uniform ultimate boundedness and the Zeno behavior can be excluded by the presented adaptive ETSMC scheme. By using two of reference trajectories with different form, the simulation results testify the feasibility and practicability of the given adaptive ETSMC scheme.

## Acknowledgments.

This work was supported by the National Natural Science Foundation of China (Grant 5217110503) and the Natural Science Foundation of Shandong Provincial (Grant ZR202103070036).

## Data availability statement.

Datasets of the parameters related to AUV model are included in Yan et al. (2019b) at https://doi.org/10.1016/j.oceaneng.2019.01.008.

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