Adaptive Variable Threshold Event-Triggered Control for Trajectory Tracking of Autonomous Underwater Vehicles with Actuator Saturation

Jian Xu aCollege of Intelligent Systems Science and Engineering, Harbin Engineering University, Heilongjiang, China
bQingdao Innovation and Development Center of Harbin Engineering University, Qingdao, Shandong, China

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Xing Wang aCollege of Intelligent Systems Science and Engineering, Harbin Engineering University, Heilongjiang, China
bQingdao Innovation and Development Center of Harbin Engineering University, Qingdao, Shandong, China

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Ping Liu cChina Ship Development and Design Center, Wuhan, Hubei, China

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Qiaoyu Duan aCollege of Intelligent Systems Science and Engineering, Harbin Engineering University, Heilongjiang, China
bQingdao Innovation and Development Center of Harbin Engineering University, Qingdao, Shandong, China

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Abstract

This article develops a novel event-triggered sliding mode control (ETSMC) approach with variable threshold to deal with trajectory tracking matters of autonomous underwater vehicles (AUVs) accompanied by actuator saturation and external disturbances, which can effectively reduce the communication burden between the controller and actuator. The proposed scheme will be very practical when some extreme situations occur. First, the closed-loop system is split into two parts: fixed terms determined by the system itself and nonlinear terms caused by uncertain factors. The nonlinear terms are estimated through adaptive technique. Then the apposite event-triggered mechanism, adaptive laws, and modeled actuator saturation characteristics are designed. The correctness of the presented scheme is illustrated via the stability analysis in the sequel, and the Zeno phenomenon is certificated to be excluded simultaneously. Finally, two different reference motion trajectories are adopted in the simulation experiments, which can indicate that the proposed ETSMC possesses performance superiority and only requires to consume a small amount of communication resources in trajectory tracking control of AUVs.

Significance Statement

Through the research of this article, we propose a novel event-triggered sliding mode control method with variable threshold applied to autonomous underwater vehicles (AUVs). When conducting ocean exploration work, we usually need the AUVs to follow particular trajectories. By using the proposed method, it can greatly reduce the loss of communication resources inside the system.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xing Wang, g-wangxing@hrbeu.edu.cn

Abstract

This article develops a novel event-triggered sliding mode control (ETSMC) approach with variable threshold to deal with trajectory tracking matters of autonomous underwater vehicles (AUVs) accompanied by actuator saturation and external disturbances, which can effectively reduce the communication burden between the controller and actuator. The proposed scheme will be very practical when some extreme situations occur. First, the closed-loop system is split into two parts: fixed terms determined by the system itself and nonlinear terms caused by uncertain factors. The nonlinear terms are estimated through adaptive technique. Then the apposite event-triggered mechanism, adaptive laws, and modeled actuator saturation characteristics are designed. The correctness of the presented scheme is illustrated via the stability analysis in the sequel, and the Zeno phenomenon is certificated to be excluded simultaneously. Finally, two different reference motion trajectories are adopted in the simulation experiments, which can indicate that the proposed ETSMC possesses performance superiority and only requires to consume a small amount of communication resources in trajectory tracking control of AUVs.

Significance Statement

Through the research of this article, we propose a novel event-triggered sliding mode control method with variable threshold applied to autonomous underwater vehicles (AUVs). When conducting ocean exploration work, we usually need the AUVs to follow particular trajectories. By using the proposed method, it can greatly reduce the loss of communication resources inside the system.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xing Wang, g-wangxing@hrbeu.edu.cn

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 xRn and any matrix MRn×n, xi (i = 1, 2, …, n) represents the ith element of x. The superscript T on xT and MT denotes the transpose of vector x and matrix M, respectively. The Euclidean norm of x is denoted as x=xTx and the induced 2 norm of M is described as ‖M‖.

2. Problem formulation

a. AUV model

This research adopts the AUV model established in Fig. 1, which can be expressed as follows:
{η˙=J(η)vMv˙+C(v)v+D(v)v+g(η)=τu+τd,
where η=[ξηζθψ]T denotes the position and attitude states relative to the inertial frame, and v=[uυwqr]T describes the linear and angular velocities in body-fixed frame.
Fig. 1.
Fig. 1.

AUV model diagram.

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

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.

The J(η) mentioned above is Jacobian transformation matrix with the following expression:
J(η)=[J1(η)03×202×3J2(η)],
J1(η)=[cosθcosψsinψsinθcosψcosθsinψcosψsinθsinψsinθ0cosθ],
J2(η)=[1001/cosθ],
where M denotes the inertia matrix including additional quality items, C(v) indicates the matrix of the Coriolis and centripetal forces, D(v) represents the matrix of the hydrodynamic damping terms, g(η) depicts restoring forces including gravity and buoyancy, τd elucidates the vector of the time-varying external disturbances, and τu shows the control forces and torques provided by the actuators. These matrixes M, C(v), D(v) and vector g(η) can be broken down into fixed terms determined by the system and nonlinear terms caused by uncertain factors, which can be written as
{M=Mf+MnC(v)=Cf(v)+Cn(v)D(v)=Df(v)+Dn(v)g(η)=gf(η)+gn(η),
where Mf, Cf(v), Df(v), and gf(η) denote the fixed terms, and Mn, Cn(v), Dn(v), and gn(η) represent the nonlinear terms:
Mf=[m1100000m2200000m3300000m5500000m66],
Cf(v)=[000m33wm22υ0000m11u000m11u0m33w0m11u00m22υm11u000],
Df(v)=[d1100000d2200000d3300000d5500000d66],
gf(η)=[000ρgGMLsinθ0]T,
where ρ is the density of water, g denotes acceleration of gravity, and GML indicates vertical metacentric height, which means the vertical distance between the center of gravity and buoyancy.
Then Eq. (1) can be rewritten as follows:
Mfv˙+Cf(v)v+Df(v)v+gf(η)=τu+τn,
where τn indicates the lump of all nonlinear terms, which is shown as
τn=τd[Mnv˙+Cn(v)v+Dn(v)v+gn(η)].

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 dm so as to satisfy ǁτdǁ < dm.

Assumption 3 (Qiao and Zhang 2019a). The uncertain terms of lumped system τn can be constrained by the following form:
τn<ϑ0+ϑ1v+ϑ2v2,
where ϑ0, ϑ1, and ϑ2 are unknown positive constants. In ǁτnǁ, the constant term is strictly smaller than ϑ0, the coefficient of ǁvǁ is strictly smaller than ϑ1, and the coefficient of ǁvǁ2 is strictly smaller than ϑ2.

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.

Therefore, the saturation characteristic of thruster is expressed as follows:
τi=sat(τi,force)={τi,fmax,τi,force>τi,fmaxτi,force,τi,fminτi,forceτi,fmaxτi,fmin,τi,force<τi,fmin,
where τi (i = 1, 2, 3) denote the output of actuator, in other words, the amount of thrust force provided by each thruster. τi,force (i = 1, 2, 3) stand for the output passed by the controller. τi,fmax and τi,fmin represent the critical value of positive and negative thrust that can be provided, respectively.
Analogously, the steering engine subjects to saturation characteristic with the following form:
τj=sat(τj,force)={τj,fmax,τj,force>τj,fmaxτj,force,τj,fminτj,forceτj,fmaxτj,fmin,τj,force<τj,fmin,
where τj (j = 4, 5) demonstrate the steering torque produced by steering engine. τj,torque (j = 4, 5) depict the steering command transmitted from controller to actuator. τj,tmax and τj,tmin 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 ηd=[ξdηdζdθdψd]T, among which θd=arctan(ζ˙d/ξ˙d2+η˙d2) and ψd=arctan(η˙d/ξ˙d). According to the method adopted by most researchers today, a virtual object with velocity status vr is structured to track the reference trajectory ηd. Then, the task requirement of trajectory tracking can be accomplished by designing appropriate velocity v of AUV to track the virtual velocity vr. The system structure of adaptive ETSMC is as shown in Fig. 2.

Fig. 2.
Fig. 2.

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

The following two error variables are defined:
e1=ηηd,
e2=vvr,
where e1 denotes the tracking error of position and attitude between the virtual object and the reference trajectory, and e2 shows the velocity error of AUV relative to the virtual speed.
Subsequently, the first sliding mode surface is designed as
S1=e1.
The second sliding mode surface is introduced by
S2=e2.
Taking the time derivative of (17) along (18), it yields that
S˙1=η˙η˙d=J(η)vrη˙d,
S˙2=v˙v˙r.

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.

For the second sliding mode surface described in (18), it can bring the following error variable in the triggering interval t ∈ [tk, tk+1):
εS2(t)=S2(t)S2(tk),
where tk (kN) 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 εS2(t) returns to zero at the trigger instant as long as an event occurs. The value of error variable is a cyclic process changing from the beginning of accumulating to reset to zero.

The event-triggered condition is devised as follows:
tk+1=inf{t>tk|εS2(t)max{βS2(tk),m}},
where 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, ǁS(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

On account of general uncertainties and time-varying external disturbances, from (12), it can be discovered that the lumps of all nonlinear terms can be bounded by the function of actual velocity. On the basis of assumption 3 with the same principle, when the time t ∈ [tk, tk+1), it must be possible to take suitable values of ϑ¯i(i=0,1,2) to make the following formula to be true:
τn<ϑ¯0+ϑ¯1v(tk)+ϑ¯2v(tk)2.

The parameters ϑ¯i(i=0,1,2) are difficult to get specific values by means of measurement, so the adaptive technology is employed to estimate these parameters.

The adaptive laws are designed as forms below:
ϑ^˙0=δ0(1+β)S2(tk),
ϑ^˙1=δ1(1+β)S2(tk)v(tk),
ϑ^˙2=δ2(1+β)S2(tk)v(tk)2,
where the adaptive gains δi (i = 0, 1, 2) are positive constants.
Define a new error variable:
ϑ˜i=ϑ¯iϑ^i(i=0,1,2),
where ϑ^i(i=0,1,2) denote the estimated values through the adaptive laws.

d. Controller design and output of actuators

The control command provided by the controller is designed as follows:
τi(t)=τ(t)S2i(t)S2(t),
where τi(t) (i = 1, 2, …, 5) denote the control command at each DOF,
τ(t)=1+β1β{F(t,η,v,vr)+[ϑ^0+ϑ^1v(t)+ϑ^2v(t)2+b]},
where b > 0 is a constant control parameter, and
F(t,η,v,vr)=Mfv˙r(t){Cf[v(t)]+Df[v(t)]}v(t)gf[η(t)].

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.

Therefore, during the triggering interval t ∈ [tk, tk+1), the output of actuators can be drawn that as follows according to assumption 5:
τui=sat[τi(tk)]={τmax,τi(tk)>τmaxτ(tk)S2i(tk)S2(tk),τminτi(tk)τmaxτmin,τi(tk)<τmin,
where τui(i=1,2,,5) describe the output of actuator at each DOF, and τmax and τmin 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 ∈ [tk, tk+1), then the whole closed-loop system is also asymptotically stable.

Consider the first sliding mode surface S1, the virtual velocity vr is designed as follows:
vr=J1(η)(η˙dk1e1),
where k1 is a positive constant.
Construct the following Lyapunov control function:
V1=12S1TS1.
Take the time derivative of (33) and substitute (19) and (32) into which, it can obtain that
V˙1=S1T[J(η)vrη˙d]=k1e1Te1=k1e120,
where V˙1 is negative semidefinite; therefore, the tracking error of position and attitude e1 converges to a compact set near zero.
Theorem 1. Consider the ETSMC of the closed-loop system, if the event-triggered mechanism is implemented as (22), and the adaptive laws are updated by (24)(26), and the outputs of actuators are described as (31). In each trigger interval t ∈ [tk, tk+1), as long as the appropriate control parameter b and the event-triggered parameter β are chosen which satisfy
F(t,η,v,vr)F(tk,η,v,vr)<b,
0<β<PΩP+Ω,
where
P=5τ(tk)min{τmax2,τmin2},
Ω=F(t,η,v,vr)+[ϑ^0+ϑ^1v(tk)+ϑ^2v(tk)2].

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

Proof 1. The following candidate Lyapunov function is constructed during each trigger interval t ∈ [tk, tk+1):
V2(t)=12S2T(t)MfS2(t)+12i=02δi1ϑ˜i2.
The derivative of (39) is obtained as
V˙2(t)=S2T(t)MfS˙2(t)+i=02δi1ϑ˜iϑ˜˙i.
Substitute (20) and (27) into the above equation yields
V˙2(t)=S2T(t)Mf{Mf1(τu+τnCf[ν(t)]ν(t)Df[v(t)]v(t)gf[η(t)])v˙r(t)}i=02δi1(ϑ¯iϑ^i)ϑ^˙i=S2T(t)[τu+τn+F(t,η,v,vr)]i=02δi1(ϑ¯iϑ^i)ϑ^˙i.
Substitute (21) into (41), it yields that
V˙2(t)=[S2(tk)+εS2(t)]T[τn+F(t,η,v,vr)]+[S2(tk)+εS2(t)]Tτui=02δi1(ϑ¯iϑ^i)ϑ^˙i.
Situation 1. Think about the event-triggered condition ǁS2(tk)ǁ > m, the event-triggered scheme is denoted as follows:
tk+1=inf{t>tk|εS2(t)βS2(tk)}.
In fact, the following inequality holds until the next event occurs:
εS2(t)βS2(tk).
So, it yields the following inequation:
[S2(tk)+εS2(t)]TS2(tk)+εS2(t)(1+β)S2(tk),
and the εS2(t) can be rewritten as the following form:
εS2(t)=kS2(tk),
where k ∈ [−β, β].
The second term of (42) is derived:
[S2(tk)+εS2(t)]Tτu=(1+k)S2T(tk)τu.

Next, Eq. (47) is analyzed separately.

Case 1. When the control command at each DOF τi(tk) > τmax, which can derive that from Eq. (28):
S2i(tk)<τmaxS2(tk)τ(tk).
Under the circumstances τui=τmax (i = 1, 2, …, 5), so (47) can evolve into
(1+k)S2T(tk)τu=(1+k)i=15S2iT(tk)τui<(1+k)5τmax2S2(tk)τ(tk)<(1β)5τmax2S2(tk)τ(tk).
Take (45) and (49), and the adaptive laws (24)(26) in Eq. (42), it yields that
V˙2(t)(1+β)S2(tk)[τn+F(t,η,v,vr)](1β)5τmax2S2(tk)τ(tk)(1+β)S2(tk)[(ϑ¯0ϑ^0)+(ϑ¯1ϑ^1)v(tk)+(ϑ¯2ϑ^2)v(tk)2].
Then it yields that
V˙2(t)(1+β)S2(tk)(F(t,η,v,vr)+(τnϑ¯0ϑ¯1v(tk)ϑ¯2v(tk)2)+(ϑ^0+ϑ^1v(tk)+ϑ^2v(tk)2))(1β)5τmax2S2(tk)τ(tk).
Substitute (23) into (51), it yields that
V˙2(t)(1+β)S2(tk)(F(t,η,v,vr)+(ϑ¯0+ϑ¯1v(tk)+ϑ¯2v(tk)2)(1β)5τmax2S2(tk)τ(tk).
From Eq. (38), it yields that
V˙2(t)<(1+β)S2(tk)Ω(1β)5τmax2S2(tk)τ(tk).
Consider P1=5τmax2/τ(tk) and P2=5τmin2/τ(tk), it yields that
V˙2(t)<(1+β)S2(tk)Ω(1β)S2(tk)P1.
Consider Eq. (36), it yields that
V˙2(t)<(1+PΩP+Ω)S2(tk)Ω(1PΩP+Ω)S2(tk)P1=2PΩP+ΩS2(tk)2P1ΩP+ΩS2(tk)=2ΩP+ΩS2(tk)(PP1).

Remark 5: If τmax2>τmin2, that is to say P1 > P2, then it can get that P1 > P2 = P; substitute this conclusion into Eq. (55), V˙2(t)<0 can be obtained. On the other hand, if τmax2<τmin2, that is to say P1 < P2, then it can get that P1 = P; substitute this conclusion into Eq. (55), V˙2(t)=0 can be obtained.

From the above analysis, it can be known that V˙2(t)<0 can always be established. Therefore, the asymptotical stability of S2(t) can be verified in this case.

Case 2. When the control command at each DOF τi(tk) < –τmin, similarly it yields that
S2i(tk)>τminS2(tk)τ(tk).
At this point, τui=τmin (i = 1, 2, …, 5), then (47) can transform into
(1+k)S2T(tk)τu=(1+k)i=15S2iT(tk)τui<(1+k)5τmin2S2(tk)τ(tk)<(1β)5τmin2S2(tk)τ(tk).

Use the same derivation method in case 1, it can also be concluded that V˙2(t)<0, the asymptotical stability of S2(t) can be guaranteed in case 2.

Case 3. When the control command at each DOF −τminτi(tk) ≤ τmax, it can be observed that τui=τ(tk)[S2i(tk)/S2(tk)], (47) can be rewritten as
(1+k)S2T(tk)τu=(1+k)i=15S2iT(tk)τui=(1+k)τ(tk)S2(tk)2S2(tk)=(1+k)τ(tk)S2(tk).
Substitute (58) into (42), it yields that
V˙2(t)(1+β)S2(tk)[τn+F(t,η,v,vr)](1+k)τ(tk)S2(tk)(1+β)S2(tk)Ψ,
where Ψ=(ϑ¯0ϑ^0)+(ϑ¯1ϑ^1)v(tk)+(ϑ¯2ϑ^2)v(tk)2.
Equation (59) can be further developed as follows:
V˙2(t)(1+β)S2(tk)[τn+F(t,η,v,vr)Ψ](1β)τ(tk)S2(tk).
Substitute (29) into (60), it yields that
V˙2(t)(1+β)S2(tk){τn[ϑ¯0+ϑ¯1v(tk)+ϑ¯2v(tk)2]+[F(t,η,v,vr)F(tk,η,v,vr)b]}.

Combine (23) and (35) with (61), it can be obtained that V˙2(t)<0; therefore, the asymptotical stability of S2(t) can be guaranteed by the output of actuator which is shown in (31).

Situation 2. Consider the event-triggered condition ǁS2(tk)ǁ ≤ m, the event-triggered scheme is denoted as follows:
tk+1=inf{t>tk|εS2(t)m}.
Before the next event is activated, the following equations are existent:
εS2(t)m,
S2(tk)mβ.
Combine (63) and (64) with (21), and it is derived that
S2(t)S2(tk)+m(1+β)mβ.

Therefore, S2(t) is bounded by (1+β)m/β.

Remark 6: After an event is triggered, the error variable εS2(t) accumulates gradually. Once the value of εS2(t) exceeds m, the next event is triggered and it must be satisfied that S2(tk+1)[m/β,(1+β)m/β]. So, at the next trigger moment ǁS2(tk+1)ǁ ≥ m, the event-triggered mechanism changes from situation 2 to situation 1. However, S2(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 e1 and e2 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.

Proof 2. Define et(t)=εS2(t) during each trigger interval:
ddtet(t)=ddtεS2(t)=ddtεS2T(t)εS2(t)ddtεS2(t).
By virtue of (20) and (21), it is derived that
ddtet(t)ddt[S2(t)S2(tk)]=ddtS2(t)=v˙(t)v˙r(t)=Mf1[τu+τn+F(t,η,v,vr)].
Profiting from the previous analysis, it is known that the closed-loop system is bounded. In fact, it can be assumed τu, τn, and F(t,η,v,v˙r) are norm bounded B1, B2, and B3, respectively. Denoting κ = B1 + B2 + B3. From (67), one has
ddtet(t)Mf1κ.
By recalling the comparing lemma (Khalil 2007) with initial status et(tk) = 0, it is derived that
et(t)κMf1(ttk).
At the next trigger moment, one has
et(tk+1)=εS2(tk+1)max{βS2(tk),m}>0.
Selecting Tk = tk+1tk be the lower bound of trigger time interval. From (69) and (70), it obtains that
Tk=max{βS2(tk),m}κMf1>0.

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 external disturbances acting on AUV are chosen as follows:
τd=d(t)={2sin(0.5πt)+rand(·)3sin(0.3πt)+rand(·)3sin(0.2πt)+rand(·)cos(0.2πt)+rand(·)2cos(0.3πt)+rand(·),
where rand(⋅) is Gaussian random noise with value between 0 and 1.

The event-triggered parameters are given by β = 0.2 and m = 0.05. The parameter k1 of the virtual velocity vr is set as k1 = 1.

The adaptive gains are chosen as δ0 = δ1 = δ2 = 0.05, and the initial values estimated by adaptive laws are selected as ϑ^0=ϑ^1=ϑ^2=5.

The control parameter is set as b = 3, and the maximum values of positive and negative output of actuator are chosen as τmax = τmin = 300.

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

Table 1

Parameter values of the AUV model.

Table 1

a. Trajectory tracking of spiral helical curve

The performances of ETSMC scheme are demonstrated by tracking the spatial helical reference curve described as ξd(t) = 10 sin(0.1t) m, ηd(t) = 10 cos(0.1t) m, ζd(t) = 0.1t 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.

Fig. 3.
Fig. 3.

Trajectory tracking response under ETSMC scheme.

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

Fig. 4.
Fig. 4.

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.

Fig. 5.
Fig. 5.

(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

Fig. 6.
Fig. 6.

(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

Fig. 7.
Fig. 7.

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 ξd(t) = t m, ηd(t) = 10 sin (0.1t) m, ζd(t) = 10 + 10sin (0.1t) 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.

Fig. 8.
Fig. 8.

Trajectory tracking response under ETSMC scheme.

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

Fig. 9.
Fig. 9.

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.

Fig. 10.
Fig. 10.

(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

Fig. 11.
Fig. 11.

(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

Fig. 12.
Fig. 12.

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.

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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  • Wang, A., L. Liu, J. Qiu, and G. Feng, 2020: Finite-time adaptive fuzzy control for nonstrict-feedback nonlinear systems via an event-triggered strategy. IEEE Trans. Fuzzy Syst., 28, 21642174, https://doi.org/10.1109/TFUZZ.2019.2931228.

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  • Wang, F., M. Hou, X. Cao, and G. Duan, 2019: Event-triggered backstepping control for attitude stabilization of spacecraft. J. Franklin Inst., 356, 94749501, https://doi.org/10.1016/j.jfranklin.2019.09.010.

    • Search Google Scholar
    • Export Citation
  • Wang, L., C. L. P. Chen, and H. Li, 2020: Event-triggered adaptive control of saturated nonlinear systems with time-varying partial state constraints. IEEE Trans. Cybern., 50, 14851497, https://doi.org/10.1109/TCYB.2018.2865499.

    • Search Google Scholar
    • Export Citation
  • Wang, X., Y. Zhang, and Z. Xue, 2019: Fuzzy sliding mode control based on RBF neural network for AUV path tracking. 12th Int. Conf. on Intelligent Robotics and Applications, Shenyang, China, ICIRA, 637–648, https://doi.org/10.1007/978-3-030-27532-7_56.

  • Wang, Y.-W., Y. Lei, T. Bian, and Z.-H. Guan, 2020: Distributed control of nonlinear multiagent systems with unknown and nonidentical control directions via event-triggered communication. IEEE Trans. Cybern., 50, 18201832, https://doi.org/10.1109/TCYB.2019.2908874.

    • Search Google Scholar
    • Export Citation
  • Wei, C., J. Luo, C. Ma, and H. Dai, 2018: Event-triggered neuroadaptive control for postcapture spacecraft with ultralow-frequency actuator updates. Neurocomputing, 315, 310321, https://doi.org/10.1016/j.neucom.2018.07.025.

    • Search Google Scholar
    • Export Citation
  • Xu, J., M. Wang, and L. Qiao, 2015: Dynamical sliding mode control for the trajectory tracking of underactuated unmanned underwater vehicles. Ocean Eng., 105, 5463, https://doi.org/10.1016/j.oceaneng.2015.06.022.

    • Search Google Scholar
    • Export Citation
  • Xu, Y., T. Li, and S. Tong, 2020: Event-triggered adaptive fuzzy bipartite consensus control of multiple autonomous underwater vehicles. IET Control Theory Appl., 14, 36323642, https://doi.org/10.1049/iet-cta.2020.0706.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., H. Yu, W. Zhang, B. Li, and J. Zhou, 2015: Globally finite-time stable tracking control of underactuated UUVs. Ocean Eng., 107, 132146, https://doi.org/10.1016/j.oceaneng.2015.07.039.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., Z. Yang, J. Zhang, J. Zhou, A. Jiang, and X. Du, 2019a: Trajectory tracking control of UUV based on backstepping sliding mode with fuzzy switching gain in diving plane. IEEE Access, 7, 166 788166 795, https://doi.org/10.1109/ACCESS.2019.2953530.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., M. Wang, and J. Xu, 2019b: Robust adaptive sliding mode control of underactuated autonomous underwater vehicles with uncertain dynamics. Ocean Eng., 173, 802809, https://doi.org/10.1016/j.oceaneng.2019.01.008.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., P. Gong, W. Zhang, and W. Wu, 2020: Model predictive control of autonomous underwater vehicles for trajectory tracking with external disturbances. Ocean Eng., 217, 107884, https://doi.org/10.1016/j.oceaneng.2020.107884.

    • Search Google Scholar
    • Export Citation
  • Zhang, J. H., H. Xu, L. Dai, and Y. Xia, 2020: Dynamic output feedback control of systems with event-driven control inputs. Sci. China Inf. Sci., 63, 150202, https://doi.org/10.1007/s11432-019-2659-3.

    • Search Google Scholar
    • Export Citation
  • Zhang, J. L., X. Xiang, Q. Zhang, and W. Li, 2020: Neural network-based adaptive trajectory tracking control of underactuated AUVs with unknown asymmetrical actuator saturation and unknown dynamics. Ocean Eng., 218, 108193, https://doi.org/10.1016/j.oceaneng.2020.108193.

    • Search Google Scholar
    • Export Citation
  • Zhang, Y. J., and Y. G. Bao, 2020: Event-triggered hybrid impulsive control for synchronization of memristive neural networks. Sci. China Inf. Sci., 63, 150206, https://doi.org/10.1007/s11432-019-2694-y.

    • Search Google Scholar
    • Export Citation
Save
  • Bovio, E., D. Cecchi, and F. Baralli, 2006: Autonomous underwater vehicles for scientific and naval operations. Annu. Rev. Control, 30, 117130, https://doi.org/10.1016/j.arcontrol.2006.08.003.

    • Search Google Scholar
    • Export Citation
  • Chu, Z., D. Zhu, and C. Luo, 2017: Adaptive neural sliding mode trajectory tracking control for autonomous underwater vehicle without thrust model. 13th IEEE Int. Conf. on Automation Science and Engineering, Xi’an, China, IEEE, 1639–1644, https://doi.org/10.1109/COASE.2017.8256339.

  • Elmokadem, T., M. Zribi, and K. Youcef-Toumi, 2017: Terminal sliding mode control for the trajectory tracking of underactuated autonomous underwater vehicles. Ocean Eng., 129, 613625, https://doi.org/10.1016/j.oceaneng.2016.10.032.

    • Search Google Scholar
    • Export Citation
  • Gong, P., Z. Yan, W. Zhang, and J. Tang, 2020: Lyapunov-based model predictive control trajectory tracking for an autonomous underwater vehicle with external disturbances. Ocean Eng., 232, 109010, https://doi.org/10.1016/j.oceaneng.2021.109010.

    • Search Google Scholar
    • Export Citation
  • Khalil, H. K., 2007: Nonlinear Systems. 3rd ed. Prentice-Hall, 763 pp.

  • Li, J., H. Guo, H. Zhang, and Z. Yan, 2019: Double-loop structure integral sliding mode control for UUV trajectory tracking. IEEE Access, 7, 101 620101 632, https://doi.org/10.1109/ACCESS.2019.2925570.

    • Search Google Scholar
    • Export Citation
  • Liang, X., X. Qu, N. Wang, R. Zhang, and Y. Li, 2020: Three-dimensional trajectory tracking of an underactuated AUV based on fuzzy dynamic surface control. IET Intell. Transp. Syst., 14, 364370, https://doi.org/10.1049/iet-its.2019.0347.

    • Search Google Scholar
    • Export Citation
  • Liu, Q., M. Liu, Y. Shi, and J. Yu, 2020: Event-triggered adaptive attitude control for flexible spacecraft with actuator nonlinearity. Aerosp. Sci. Technol., 106, 106111, https://doi.org/10.1016/j.ast.2020.106111.

    • Search Google Scholar
    • Export Citation
  • Liu, X., M. Zhang, and E. Rogers, 2019: Trajectory tracking control for autonomous underwater vehicles based on fuzzy re-planning of a local desired trajectory. IEEE Trans. Veh. Technol., 68, 11 65711 667, https://doi.org/10.1109/TVT.2019.2948153.

    • Search Google Scholar
    • Export Citation
  • Ning, Z., J. Yu, Y. Pan, and H. Li, 2018: Adaptive event-triggered fault detection for fuzzy stochastic systems with missing measurements. IEEE Trans. Fuzzy Syst., 26, 22012212, https://doi.org/10.1109/TFUZZ.2017.2780799.

    • Search Google Scholar
    • Export Citation
  • Qiao, L., and W. Zhang, 2019a: Double-loop integral terminal sliding mode tracking control for UUVs with adaptive dynamic compensation of uncertainties and disturbances. IEEE J. Oceanic Eng., 44, 2953, https://doi.org/10.1109/JOE.2017.2777638.

    • Search Google Scholar
    • Export Citation
  • Qiao, L., and W. Zhang, 2019b: Adaptive second-order fast nonsingular terminal sliding mode tracking control for fully actuated autonomous underwater vehicles. IEEE J. Oceanic Eng., 44, 363385, https://doi.org/10.1109/JOE.2018.2809018.

    • Search Google Scholar
    • Export Citation
  • Qiao, L., and W. Zhang, 2020: Trajectory tracking control of AUVs via adaptive fast nonsingular integral terminal sliding mode control. IEEE Trans. Ind. Inf., 16, 12481258, https://doi.org/10.1109/TII.2019.2949007.

    • Search Google Scholar
    • Export Citation
  • Su, B., H. Wang, Y. Wang, and J. Gao, 2021: Fixed-time formation of AUVs with disturbance via event-triggered control. Int. J. Control Autom. Syst., 19, 15051518, https://doi.org/10.1007/s12555-020-0127-0.

    • Search Google Scholar
    • Export Citation
  • Wang, A., L. Liu, J. Qiu, and G. Feng, 2020: Finite-time adaptive fuzzy control for nonstrict-feedback nonlinear systems via an event-triggered strategy. IEEE Trans. Fuzzy Syst., 28, 21642174, https://doi.org/10.1109/TFUZZ.2019.2931228.

    • Search Google Scholar
    • Export Citation
  • Wang, F., M. Hou, X. Cao, and G. Duan, 2019: Event-triggered backstepping control for attitude stabilization of spacecraft. J. Franklin Inst., 356, 94749501, https://doi.org/10.1016/j.jfranklin.2019.09.010.

    • Search Google Scholar
    • Export Citation
  • Wang, L., C. L. P. Chen, and H. Li, 2020: Event-triggered adaptive control of saturated nonlinear systems with time-varying partial state constraints. IEEE Trans. Cybern., 50, 14851497, https://doi.org/10.1109/TCYB.2018.2865499.

    • Search Google Scholar
    • Export Citation
  • Wang, X., Y. Zhang, and Z. Xue, 2019: Fuzzy sliding mode control based on RBF neural network for AUV path tracking. 12th Int. Conf. on Intelligent Robotics and Applications, Shenyang, China, ICIRA, 637–648, https://doi.org/10.1007/978-3-030-27532-7_56.

  • Wang, Y.-W., Y. Lei, T. Bian, and Z.-H. Guan, 2020: Distributed control of nonlinear multiagent systems with unknown and nonidentical control directions via event-triggered communication. IEEE Trans. Cybern., 50, 18201832, https://doi.org/10.1109/TCYB.2019.2908874.

    • Search Google Scholar
    • Export Citation
  • Wei, C., J. Luo, C. Ma, and H. Dai, 2018: Event-triggered neuroadaptive control for postcapture spacecraft with ultralow-frequency actuator updates. Neurocomputing, 315, 310321, https://doi.org/10.1016/j.neucom.2018.07.025.

    • Search Google Scholar
    • Export Citation
  • Xu, J., M. Wang, and L. Qiao, 2015: Dynamical sliding mode control for the trajectory tracking of underactuated unmanned underwater vehicles. Ocean Eng., 105, 5463, https://doi.org/10.1016/j.oceaneng.2015.06.022.

    • Search Google Scholar
    • Export Citation
  • Xu, Y., T. Li, and S. Tong, 2020: Event-triggered adaptive fuzzy bipartite consensus control of multiple autonomous underwater vehicles. IET Control Theory Appl., 14, 36323642, https://doi.org/10.1049/iet-cta.2020.0706.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., H. Yu, W. Zhang, B. Li, and J. Zhou, 2015: Globally finite-time stable tracking control of underactuated UUVs. Ocean Eng., 107, 132146, https://doi.org/10.1016/j.oceaneng.2015.07.039.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., Z. Yang, J. Zhang, J. Zhou, A. Jiang, and X. Du, 2019a: Trajectory tracking control of UUV based on backstepping sliding mode with fuzzy switching gain in diving plane. IEEE Access, 7, 166 788166 795, https://doi.org/10.1109/ACCESS.2019.2953530.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., M. Wang, and J. Xu, 2019b: Robust adaptive sliding mode control of underactuated autonomous underwater vehicles with uncertain dynamics. Ocean Eng., 173, 802809, https://doi.org/10.1016/j.oceaneng.2019.01.008.

    • Search Google Scholar
    • Export Citation
  • Yan, Z., P. Gong, W. Zhang, and W. Wu, 2020: Model predictive control of autonomous underwater vehicles for trajectory tracking with external disturbances. Ocean Eng., 217, 107884, https://doi.org/10.1016/j.oceaneng.2020.107884.

    • Search Google Scholar
    • Export Citation
  • Zhang, J. H., H. Xu, L. Dai, and Y. Xia, 2020: Dynamic output feedback control of systems with event-driven control inputs. Sci. China Inf. Sci., 63, 150202, https://doi.org/10.1007/s11432-019-2659-3.

    • Search Google Scholar
    • Export Citation
  • Zhang, J. L., X. Xiang, Q. Zhang, and W. Li, 2020: Neural network-based adaptive trajectory tracking control of underactuated AUVs with unknown asymmetrical actuator saturation and unknown dynamics. Ocean Eng., 218, 108193, https://doi.org/10.1016/j.oceaneng.2020.108193.

    • Search Google Scholar
    • Export Citation
  • Zhang, Y. J., and Y. G. Bao, 2020: Event-triggered hybrid impulsive control for synchronization of memristive neural networks. Sci. China Inf. Sci., 63, 150206, https://doi.org/10.1007/s11432-019-2694-y.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    AUV model diagram.

  • Fig. 2.

    Block diagram of the adaptive ETSMC scheme.

  • Fig. 3.

    Trajectory tracking response under ETSMC scheme.

  • Fig. 4.

    The interval time of event-triggered mechanism.

  • Fig. 5.

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

  • Fig. 6.

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

  • Fig. 7.

    The thrusts and torques provided by actuator.

  • Fig. 8.

    Trajectory tracking response under ETSMC scheme.

  • Fig. 9.

    The interval time of event-triggered mechanism.

  • Fig. 10.

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

  • Fig. 11.

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

  • Fig. 12.

    The thrusts and torques provided by actuator.

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