Predefined-Time Three-Dimensional Trajectory Tracking Control for Underactuated Autonomous Underwater Vehicles

Abstract

This paper addresses the three-dimensional trajectory tracking problem of underactuated autonomous underwater vehicles (AUVs) operating in the presence of external disturbances and unmodeled dynamics by proposing a predefined-time adaptive control scheme. Firstly, the underactuated AUV system was decoupled into drive and non-drive subsystems to facilitate the design of a controller that does not rely on specific model parameters. Radial basis function neural networks (RBFNNs) were employed to estimate the external disturbances. To enhance tracking performance, a predefined-time adaptive control law was designed to ensure that tracking errors converged to a small neighborhood around the origin within the predefined time. The adaptive control law compensated for the unmodeled components. Finally, we used theoretical proofs and simulations to show that our method is effective and superior.

1. Introduction

In recent years, the burgeoning demand for ocean exploration has driven the widespread adoption of AUVs, establishing them as pivotal instruments in marine resource exploration, seabed mapping, and oceanographic research. AUVs have substantially broadened the horizons of oceanic investigations while concurrently mitigating the risks associated with human-operated missions. In practical applications, AUVs are frequently required to accurately and swiftly adhere to designated target trajectories [1,2]. However, within engineering frameworks, AUVs typically operate as underactuated systems, characterized by a limited number of control inputs relative to their degrees of freedom. Moreover, fluctuations in the marine environment introduce significant uncertainties in the parameters of the AUVs’ nonlinear motion models during navigation. Numerous marine operations demand precise maneuvering and positioning of AUVs within three-dimensional space. Consequently, the development of trajectory tracking control techniques for underactuated AUVs in three dimensions has emerged as a central area of research interest among scholars both domestically and internationally in recent years.

To address the performance and stability issues of AUVs in complex marine environments, various control algorithms have been proposed by researchers. In the case of known mathematical models, Paliotta et al. [3] proposed a feedback linearization control method to handle external time-varying ocean current disturbances; Zhou et al. [4] combined backstepping and radial basis function neural networks (RBFNNs) to approximate external disturbances; Li et al. [5] employed backstepping control to solve the 3D trajectory tracking problem; and An et al. [6] introduced a continuous-time disturbance observer to avoid the “complexity explosion” of backstepping. These methods show good performance in compensating for external disturbances, but most focus on planar trajectories and do not consider the effects of unmodeled system components.

In cases where the AUV’s mathematical model and ocean environmental disturbances are unknown, Yan et al. [7] designed an adaptive neural network controller based on sliding mode control, achieving finite-time convergence; Yu et al. [8] proposed a global finite-time control strategy; Chu et al. [9] integrated RBFNNs with sliding mode control to ensure the stability of 3D trajectory tracking; and Chen et al. [10] designed a fixed-time controller for a 2D multi-agent system. Although these methods improve the system’s response speed, the convergence time of finite-time control depends on initial conditions, and the convergence time of fixed-time control is difficult to determine. As a result, predefined-time control methods have gained attention. Dong et al. [11] studied the attitude control problem of rigid spacecraft under external disturbances, and Mei et al. [12] investigated the predefined-time consensus tracking problem for multi-agent systems. However, research on predefined-time trajectory tracking for underactuated AUVs in 3D remains relatively scarce.

In recent years, radial basis function neural networks (RBFNNs) [13,14,15] and predefined-time control [16,17,18] have gained significant attention. In the presence of external disturbances, RBFNNs exhibit superior approximation capabilities compared to traditional adaptive methods. Moreover, in contrast to fixed-time control methods, predefined-time control offers distinct advantages in determining convergence time. Based on these discussions, this paper proposes a predefined-time control strategy for 3D trajectory tracking of underactuated AUVs with unmodeled dynamics and external disturbances. Specifically, the aim of this research is to design an efficient and stable 3D trajectory tracking control method that ensures the AUV can achieve precise trajectory tracking within a predefined time, even in the presence of unmodeled dynamics and external disturbances. The main contributions of this paper are as follows:

1.

A predefined-time control strategy based on RBFNNs is proposed, effectively approximating external disturbances through RBFNNs, thereby enhancing the robustness and approximation performance of the system.

2.

The introduction of predefined-time control technology ensures that tracking errors converge to a neighborhood of zero within a predefined time, offering better convergence speed and performance compared to traditional methods.

2.

By decoupling the control model into driven and non-driven subsystems, the control law derivation is simplified, and the unmodeled parts of the system are effectively compensated, thereby improving the stability and adaptability of the control system.

Compared with existing strategies, the proposed approach guarantees that the tracking error converges to a neighborhood of zero within a predefined time, improving the system’s convergence speed and tracking performance. This approach facilitates the derivation of model-parameter-free control laws, compensates for unmodeled parts using predefined-time adaptive techniques, and estimates external disturbances through RBFNNs, effectively mitigating their adverse impact on the system.

The structure of this paper is organized as follows: the Section 1 presents the mathematical modeling of the underactuated AUV and decouples the model; the Section 2 designs a virtual control law based on backstepping, followed by the design of a predefined-time adaptive control law, with RBFNNs used to compensate for the effects of unknown disturbances; the Section 3 provides stability analysis, proving that the virtual control law, predefined-time control law, adaptive law, and weight update rule all satisfy the Lyapunov stability theorem; the Section 4 presents the experimental results, demonstrating the effectiveness of the proposed method through simulations; and the Section 5 concludes the paper.

2. Problem Formulation and Preliminaries

2.1. Symbolic Representation

The space ${ℝ}^{n\times n}$ is characterized as an $n\times n$ Euclidean space, where $|\cdot |$ denotes the absolute value for scalars. The vector $\mathit{I}={[I_1\hspace{0.33em}{I}_2\hspace{0.33em}\dots \hspace{0.33em}{I}_n]}^T$ and the matrix $\mathbf{A}\in {ℝ}^{n\times n}$ are defined within this space. The norm of the vector $\mathit{I}$ is given by $‖\mathit{I}‖={\displaystyle \sum _j\left|I_j\right|=\sqrt{I_1^2+I_2^2+\dots +I_n^2}}$. Here, ${ \mathrm{I}}_j$ represents the $j-th$ component of the vector $\mathit{I}$ and ${\mathbf{A}}_{ij}$ represents the element located in the $i-th$ row and $j-th$ column of matrix $\mathbf{A}$. Furthermore, the minimum eigenvalue ${\lambda }_{\min }(\mathbf{A})$ is defined as the smallest eigenvalue of the matrix $\mathbf{A}$. The functions $\mathrm{sgn}(\cdot )$ and $\tanh (\cdot )$ represent the sign function and the hyperbolic tangent function, respectively.

2.2. Lemmas and Assumptions

Consider the Following Dynamical System

$$\dot{x}(t)=\mathit{f}(x,t), \mathit{f}\left(0,t\right)=0,x\in {ℝ}^n$$

(1)

where $\mathit{f}:{ℝ}^n\times {ℝ}^{+}\to {ℝ}^n$ is a continuous nonlinear function, and the initial condition is denoted as $x\left(0\right)=x_0.$

If there exists a predetermined constant $T_{\max }>0$ and $\delta >0,$ such that any solution $x\left(t,x_0\right)$ of the system satisfies $‖x(t;x_0)‖\le \delta$ for all $x_0\in {ℝ}^n$ when $t\ge T_{\max },$ then the origin of the system is said to be uniformly globally stable.

Theorem 1.

Assume there exists a Lyapunov function$V(x,t)$ defined on${ℝ}^n\times {ℝ}^{+}$ that satisfies the following conditions

$$\dot{V}(x)\le -{\displaystyle \frac{\pi }{\beta T_c}}{V}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}^{1-{\displaystyle \frac{\beta }{2}}}+\mathrm{D}$$

(2)

In this case, $0<\beta <1,\mathrm{D}>0$ are constants. The formula $T_c={\left(\sqrt{2}\right)}^{-1}{T}_{\max }$ indicates that the solution of the system remains stable within a predefined-time interval [11]. The system state converges to the following set of differences:

$$\mathsf{\Omega }=\left\{\underset{t\to T_{\max }}{\lim }x\left|V⩽\min \left\{{\left({\displaystyle \frac{2\beta T_c\mathrm{D}}{\pi }}\right)}^{{\displaystyle \frac{2}{2-\beta }}},{\left({\displaystyle \frac{2\beta T_c\mathrm{D}}{\pi }}\right)}^{{\displaystyle \frac{2}{2+\beta }}}\right\}\right.\right\}$$

(3)

Theorem 2

([19]). Define the variables$y_1,y_2,\dots ,y_n$ as follows

$$\sum _{i=1}^n{\left|y_i\right|}^a\ge n^{1-a}{\left(\sum _{i=1}^n\left|y_i\right|\right)}^a,a>1$$

(4)

$$\sum _{i=1}^n{\left|y_i\right|}^a\ge {\left(\sum _{i=1}^n\left|y_i\right|\right)}^a,0<a<1$$

(5)

Theorem 3

([20]). When $\chi >0,$ the following equations are established

$$|x|\le x\tanh (\chi x)+{\displaystyle \frac{\epsilon }{\chi }},\epsilon =0.2785$$

(6)

Theorem 4.

For any defined $x>0,h>0,$ and $y>0,$ the following inequality holds

$$x^{2h}-y^{2h}\le -{(x-y)}^{2h}+2x^{2h}$$

(7)

Theorem 5

([21]). Young’s Inequality

For positive values $a,b\ge 0$ and real numbers $p,q>1$, the following inequality holds

$$ab\le {\displaystyle \frac{a^p}{p}}+{\displaystyle \frac{b^q}{q}}$$

(8)

Theorem 6

([22]). Radial basis function networks (RBFNs) can be used to approximate continuous functions ${\mathit{f}}_{NN}$, expressed as follows

$${\widehat{\mathit{f}}}_{NN}\left(\mathit{x}\right)={\widehat{\mathbf{W}}}^T\mathit{h}\left(\mathit{x}\right)+\mathit{B}$$

(9)

where $x={\left[x_1,x_2,\dots ,x_n,\right]}^T$ is the input vector of the RBFNNs, and  ${\mathit{h}}_j\left(\mathit{x}\right)=\exp \left(-{\displaystyle \frac{{‖\mathit{x}-\mathit{\mu }‖}^2}{2b^2}}\right)$ is the radial basis function. For RBFNs, the following applies: let $\mathit{B}={\left[b_1,b_2,\dots b_m\right]}^T$ be the output bias, and $\widehat{\mathbf{W}}$ is used for estimating optimal weights. In the weight estimation, there exists an optimal weight vector ${\mathbf{W}}^{\ast }$ such that

$${\mathit{f}}_{NN}={\mathbf{W}}^{\ast T}\mathit{h}\left(\mathit{x}\right)-\mathit{\epsilon }$$

(10)

where $\mathit{\epsilon }$ represents the approximation error of the neural network, satisfying $\left|\mathit{\epsilon }\right|\le \overline{\mathit{\epsilon }}.$

2.3. Kinematic and Dynamic Modeling

The kinematic and dynamic model of the underactuated AUV with five degrees of freedom is defined as follows:

$$\dot{\mathit{\eta }}=\mathit{J}(\mathit{\eta })\mathit{\nu }$$

(11)

$$\mathit{M}\dot{\mathit{v}}+\mathit{C}(\mathit{\nu })\mathit{\nu }+\mathrm{D}(\mathit{\nu })\mathit{\nu }+\mathit{G}(\mathit{\eta })+\mathit{f}=\mathit{\tau }$$

(12)

In Figure 1, Here, $\mathit{\eta }={\left[{\mathit{\eta }}_1;{\mathit{\eta }}_2\right]}^T$ represents the position and orientation vector in the earth-fixed frame, where ${\mathit{\eta }}_1={\left[\begin{array}{c}x,y,z\end{array}\right]}^T$ denotes the position and ${\mathit{\eta }}_2={\left[\begin{array}{c}\theta ,\psi \end{array}\right]}^T$ represents the orientation. The velocity vector $\mathit{v}={\left[{\mathit{v}}_1;{\mathit{v}}_2\right]}^T$ includes ${\mathit{v}}_1={\left[\begin{array}{c}u,v,w\end{array}\right]}^T$, the linear velocities, and ${\mathit{v}}_2={\left[\begin{array}{c}q,r\end{array}\right]}^T$, the angular velocities, both expressed in the body-fixed frame.

The reference frames are denoted as $\left\{E\right\}=(x_E,y_E,z_E)$ for the earth-fixed frame and $\left\{B\right\}=(x_B,y_B,z_B)$ for the body-fixed frame. $\mathit{J}(\mathit{\eta })\in {ℝ}^{5\times 5}$ signifies the transformation matrix, $\mathit{M}\in {ℝ}^{5\times 5}$ represents the inertia matrix, $\mathit{C}(\mathit{\nu })\in {ℝ}^{5\times 5}$ denotes the Coriolis and centripetal forces, and $\mathit{D}(v)\in {ℝ}^{5\times 5}$ indicates the hydrodynamic drag, while $\mathit{G}(\mathit{\eta })\in {ℝ}^5$ accounts for gravitational and buoyancy forces. The term $\mathit{f}\in {ℝ}^5$ denotes external disturbances, and $\mathit{\tau }\in {\Re }^5$ represents the control inputs. The detailed expressions of $\mathit{J}(\mathit{\eta }),\mathit{M},\mathit{C}(\mathit{v}),\mathrm{D}(\mathit{v}),\mathit{G},\mathit{\tau }$ are as follows.

$$\mathit{J}=\left[\begin{array}{cc}{\mathit{J}}_1& {\mathit{O}}_{3\times 2}\\ {\mathit{O}}_{2\times 3}& {\mathit{J}}_2\end{array}\right],{\mathit{J}}_1=\left[\begin{array}{ccc}\cos \psi \cos \theta & -\sin \psi & \cos \psi \sin \theta \\ \sin \psi \cos \theta & \cos \psi & \sin \psi \sin \theta \\ -\sin \theta & 0& \cos \theta \end{array}\right],{\mathit{J}}_2=\left[\begin{array}{cc}1& 0\\ 0& 1/\cos \theta \end{array}\right]$$

(13)

$$\mathit{M}=diag\left(\left[M-X_{\dot{u}},M-Y_{\dot{v}},M-Z_{\dot{w}}{I}_{yy}-M_{\dot{q}},I_{zz}-N_{\dot{r}}\right]\right)$$

(14)

$$\mathit{C}(\mathit{v})=\left[\begin{array}{ccccc}0& 0& 0& \left(M-Z_{\dot{w}}\right)w& -\left(M-Y_{\dot{v}}\right)v\\ 0& 0& 0& 0& \left(M-X_{\dot{u}}\right)u\\ 0& 0& 0& -\left(M-X_{\dot{u}}\right)u& 0\\ -\left(M-Z_{\dot{w}}\right)w& 0& \left(M-X_{\dot{u}}\right)u& 0& 0\\ \left(M-Y_{\dot{v}}\right)v& -\left(M-X_{\dot{u}}\right)u& 0& 0& 0\end{array}\right]$$

(15)

$$\mathit{D}(\mathit{v})=diag\left(\left[X_{u\left|u\right|}\left|u\right|+X_u,Y_{v\left|v\right|}\left|v\right|+Y_v,Z_{w\left|w\right|}\left|w\right|+Z_w,M_{q\left|q\right|}\left|q\right|+M_q,N_{r\left|r\right|}\left|r\right|+N_r\right]\right)$$

(16)

$$\mathit{G}={\left[\begin{array}{c}0,0,0,Z_{B}G\sin \theta ,0\end{array}\right]}^T$$

(17)

$$\mathit{f}={[f_1,0,0,f_2,f_3]}^T$$

(18)

$$\mathit{\tau }={\left[\begin{array}{c}{\tau }_u,0,0,{\tau }_q,{\tau }_r\end{array}\right]}^T$$

(19)

This paper simplifies the controller design by decoupling the underactuated AUV system into the actuating subsystem and the non-actuating subsystem. The model of the actuating subsystem is established as follows. [23]

$${\dot{\mathit{\eta }}}_1={\mathit{J}}_1({\mathit{\eta }}_1)\mathbf{\upsilon }$$

(20)

$${\mathit{M}}_a\dot{\mathbf{\upsilon }}+{\mathit{C}}_a(\mathbf{\upsilon })\mathbf{\upsilon }+{\mathit{D}}_a(\mathbf{\upsilon })\mathbf{\upsilon }+{\mathit{G}}_a+{\mathit{f}}_a={\mathit{\tau }}_a$$

(21)

$\mathbf{\upsilon }={\left[\begin{array}{c}u,q,r\end{array}\right]}^T,$${\mathit{M}}_a,{\mathit{C}}_a,{\mathit{D}}_a,{\mathit{G}}_a,{\mathit{f}}_a,{\mathit{\tau }}_a$ as follows:

$${\mathit{M}}_a=diag\left(\left[M-X_{\dot{u}},I_{yy}-M_{\dot{q}},I_{zz}-N_{\dot{r}}\right]\right)$$

(22)

$${\mathit{C}}_a=\left[\begin{array}{ccc}0& \left(M-Z_{\dot{w}}\right)w& -\left(M-Y_{\dot{v}}\right)v\\ \left(Z_{\dot{w}}-X_{\dot{u}}\right)w& 0& 0\\ \left(X_{\dot{u}}-Y_{\dot{v}}\right)v& 0& 0\end{array}\right]$$

(23)

$${\mathit{D}}_a(\mathbf{\upsilon })=diag\left(\left[X_{u\left|u\right|}\left|u\right|+X_u,M_{q\left|q\right|} \left|q\right|+M_q,N_{r\left|r\right|} \left|r\right|+N_r\right]\right)$$

(24)

$${\mathit{G}}_a={\left[\begin{array}{c}0,Z_{B}G\sin \theta ,0\end{array}\right]}^T$$

(25)

$${\mathit{\tau }}_a={\left[\begin{array}{c}{\tau }_u,{\tau }_q,{\tau }_r\end{array}\right]}^T$$

(26)

$${\mathit{f}}_a={[f_1,f_2,f_3]}^T$$

(27)

Due to the limited maximum moment that the actuator can handle, the input moment ${\mathit{\tau }}_a$ is considered to be subject to the following asymmetric constraints.

$${\tau }_{Lu}\le {\tau }_u\le {\tau }_{Hu},{\tau }_{Lq}\le {\tau }_q\le {\tau }_{Hq},{\tau }_{Lr}\le {\tau }_r\le {\tau }_{Hr}$$

(28)

Before presenting the research objectives of this paper, the following assumptions were made:

Assumption 1.

The transformation matrix${\mathit{J}}_1\left(\mathit{\eta }\right)$ is invertible, and there is a known constant $\widehat{\mathit{J}}$, such that ${\sup }_{\eta }‖{\mathit{J}}_1\left(\mathit{\eta }\right)‖\le \widehat{\mathit{J}}$.

Assumption 2.

The derivative of the transformation matrix ${\mathit{J}}_1\left(\mathit{\eta }\right)$ with respect to time has the following form:

$${\dot{\mathit{J}}}_1={\mathit{J}}_1\mathit{\omega }$$

(29)

where $\mathit{\omega }=\left[\begin{array}{ccc}0& -r& q\\ r& 0& 0\\ -q& 0& 0\end{array}\right]$ is an antisymmetric matrix for vector  $\mathit{I}$. The following theorem follow.

$${\mathit{I}}^T\mathit{\omega }\mathit{I}=0$$

(30)

Assumption 3.

For the parameter${\mathit{M}}_a,{\mathit{C}}_a,{\mathit{D}}_a,{\mathit{G}}_a,{\mathit{f}}_a$, the following constraints apply:

$$m_1\le ‖{\mathit{M}}_a‖\le m_2,‖{\mathit{C}}_a‖\le C\parallel \mathbf{\upsilon }\parallel ,‖{\mathit{D}}_a‖\le d_1+d_2\parallel \mathbf{\upsilon }\parallel ,‖{\mathit{G}}_a‖\le g,‖{\mathit{f}}_a‖\le f$$

(31)

where $m_1,m_2,c,d_1,d_2,g,f$, is an unknown constant.

Remark 1.

Some of the system parameters or external influences were not adequately considered due to environmental uncertainties and technical measurement limitations. In addition, the complex dynamic behavior of real systems is difficult to capture entirely through mathematical models. Therefore, the model parameters ${\mathit{M}}_a,{\mathit{C}}_a,{\mathit{D}}_a,{\mathit{G}}_a$ are not modeled accurately enough. In this paper, we explore the effect of the unmodeled part on the system and derive controllers that do not depend on the model parameters through the described decoupling algorithm.

This study aimed to investigate the problem of three-dimensional trajectory tracking for underactuated autonomous underwater vehicles (AUVs) in the presence of unmodeled dynamics and external disturbances. The primary objective was to ensure that the tracking error converges to a small bounded region within a predefined time frame. Under the assumption that the desired trajectory is smooth and continuously differentiable, the tracking error can be mathematically expressed as

$${\mathit{\eta }}_e={\mathit{\eta }}_1-{\mathit{\eta }}_{1d},$$

(32)

A predefined-time adaptive controller without model parameters is derived such that the trajectory error satisfies the following conditions:

$$\underset{t\to T}{\lim }\left|{\eta }_{ei}\right|\le \zeta ,(i=1,2,3)$$

(33)

$T$ is the settling time constant, and the competition is $\zeta$ positive parameter.

3. Predefined-Time Backstepping Control

Define the virtual desired trajectory ${\mathit{\eta }}_g$ as

$${\mathit{\eta }}_g={\mathit{\eta }}_{1d}+{\mathit{J}}_1\mathit{M}$$

(34)

where $\mathit{M}={\left[{\rho }_1,{\rho }_2,{\rho }_3\right]}^T, {\rho }_1,{\rho }_2,{\rho }_3$ is the positive parameter to be designed d. The virtual desired trajectory error ${\mathit{\eta }}_{ge}$ is

$${\mathit{\eta }}_{ge}={\mathit{\eta }}_1-{\mathit{\eta }}_g$$

(35)

Convert the virtual desired trajectory error ${\mathit{\eta }}_{ge}$ into the Earth coordinate system:

$${\mathit{e}}_{ge}={\mathit{J}}_1^T{\mathit{\eta }}_{ge}$$

(36)

Define the virtual trajectory tracking error ${\mathit{z}}_1$ as

$${\mathit{z}}_1={\mathit{e}}_{ge}+\mathit{M}$$

(37)

Substituting (34)–(36) into (37) gives

$${\mathit{z}}_1={\mathit{J}}_1^T{\mathit{\eta }}_{ge}+\mathit{M}={\mathit{J}}_1^T({\mathit{\eta }}_1-{\mathit{\eta }}_{1d}-{\mathit{J}}_1\mathit{M})+\mathit{M}={\mathit{J}}_1^T{\mathit{\eta }}_e$$

(38)

Since ${\mathit{J}}_1^T$ is an orthogonal matrix and the determinant is not zero, it follows from (38) that ${\mathit{\eta }}_e=0$ when ${\mathit{z}}_1=0$. Therefore, the control objective of this paper converts into how to make ${\mathit{z}}_1$ stable in a predefined time.

According to (30), (34)–(38), the derivative of ${\mathit{z}}_1$ with respect to time is solved to be equal to

$$\begin{array}{ll}{\dot{\mathit{z}}}_1& =-\mathit{\omega }{\mathit{J}}_1^T{\mathit{\eta }}_e+{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_e+\dot{\mathit{M}}\\ & =-\mathit{\omega }{\mathit{z}}_1-\mathit{\omega }\mathit{M}+{\mathit{v}}_1-{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g+\dot{\mathit{M}}\\ & =-\mathit{\omega }{\mathit{z}}_1+\Lambda \mathbf{\upsilon }+{\mathit{v}}_1^{\ast }-{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g+\dot{\mathit{M}}\end{array}$$

(39)

where $\Lambda =\left[\begin{array}{ccc}1& {\rho }_3& -{\rho }_2\\ 0& 0& -{\rho }_1\\ 0& {\rho }_1& 0\end{array}\right]$ is the system connection matrix, which simplifies the controller design; $\mathbf{\upsilon }={\left[\begin{array}{c}u,q,r\end{array}\right]}^T$ is the velocity vector in the direction of the driving degrees of freedom; and ${\mathit{v}}_1={\left[\begin{array}{c}u,v,w\end{array}\right]}^T,{\mathit{v}}_1^{\ast }={\left[0,v,w\right]}^T$ is the intermediate matrix such that the virtual control error is ${\mathbf{\upsilon }}_e=\mathbf{\upsilon }-\mathit{\alpha },$ $\mathit{\alpha }$, which is the virtual control law to be designed.

Remark 2.

By (34)–(39), the underactuated AUV system is linked to the drive subsystem, so the control law of the underactuated AUV can be solved by solving the control law of the drive subsystem and substituting it into the main system to solve the control law of the underactuated AUV, which facilitates the derivation of the control law without the model parameters.

3.1. Control Law Design

The design of the controller is mainly divided into two parts:

• Constructing the Lyapunov function to design the virtual control law;
• Adopting the RBF neural network to estimate the external disturbances of the system and identifying the unmodeled part of the system by the adaptive law.

The predefined-time adaptive control law is designed in detail as follows:

Step 1:

Designing the virtual control law $\mathit{\alpha }$, substituting ${\mathbf{\upsilon }}_e$ into (39)

$${\dot{\mathit{z}}}_1=-\mathit{\omega }{\mathit{z}}_1+\Lambda {\mathbf{\upsilon }}_e+\Lambda \mathit{\alpha }+{\mathit{v}}_1^{\ast }-{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g+\dot{\mathit{M}}$$

(40)

Constructing alternative Lyapunov functions

$$V_1={\displaystyle \frac{1}{2}}{\mathit{z}}_1^T{\mathit{z}}_1$$

(41)

Compute the derivative with respect to time for $V_1$

$${\dot{V}}_1={\mathit{z}}_1^T\left(-\mathit{\omega }{\mathit{z}}_1+\Lambda {\mathbf{\upsilon }}_e+\Lambda \mathit{\alpha }+{\mathit{v}}_1^{\ast }-{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g+\dot{\mathit{M}}\right)$$

(42)

Design the virtual control law $\mathit{\alpha }$ as

$$\mathit{\alpha }={\Lambda }^{-1}(-C_1{\mathit{K}}_{1}diag\left({\left|{\mathit{z}}_1\right|}^{1+\beta }\right)\mathrm{sgn}\left({\mathit{z}}_1\right)-C_2{\mathit{K}}_{2}diag\left({\left|{\mathit{z}}_1\right|}^{1-\beta }\right)\mathrm{sgn}\left({\mathit{z}}_1\right)-{\mathit{v}}_1^{\ast }+{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g-\dot{\mathit{M}})$$

(43)

where $\mathit{\alpha }={\left[{\alpha }_u,{\alpha }_q,{\alpha }_r\right]}^T,{\alpha }_u,{\alpha }_q,{\alpha }_r$ is the virtual control law in the direction of the three driving degrees of freedom. ${\mathit{K}}_1,{\mathit{K}}_2$ is a positive definite symmetric matrix defined as ${\mathit{K}}_1=\mathrm{diag}\left(\right[k_{11},k_{12},k_{13}]),{\mathit{K}}_2=\mathrm{diag}\left(\right[k_{21},k_{22},k_{23}]),k_{11},k_{12},k_{13},$ $\beta \in (0,1)$ is a predefined time control parameter, $C_1,C_2$ is a predefined time control parameter satisfying the following definitions $C_1={\displaystyle \frac{\pi }{3^{\beta }\beta T_c{k}_{1\min }{2}^{1+\beta }}},C_2={\displaystyle \frac{\pi }{\beta T_c{k}_{2\min }{2}^{1-\beta }}}$, and $T_c$ is a predefined time.

Step 2:

Design of drive subsystem controllers ${\mathit{\tau }}_a$

Extended alternative Lyapunov functions:

$$V_3=V_1+V_2={\displaystyle \frac{1}{2}}{\mathit{z}}_1^T{\mathit{z}}_1+{\displaystyle \frac{1}{2m}}{\mathit{z}}_2^T{\mathit{M}}_a{\mathit{z}}_2$$

(44)

Here, ${\mathit{z}}_2={\mathbf{\upsilon }}_e,m_1\le ‖{\mathit{M}}_a‖\le m_2,m>m_2$, which is derived by differentiating with respect to time relative to $V_3$:

$$\begin{array}{ll}{\dot{V}}_3& ={\mathit{z}}_1^T\left(-\mathit{\omega }{\mathit{z}}_1+\Lambda {\mathbf{\upsilon }}_e+\Lambda \mathit{\alpha }+{\mathit{v}}_1^{\ast }-{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g+\dot{\mathit{M}}\right)+\frac{1}{2m}{\mathit{z}}_1^T\Lambda {\mathit{z}}_2+\frac{1}{2m}{\mathit{z}}_2^T{\mathit{M}}_a{\dot{\mathit{z}}}_2\\ & ={\mathit{z}}_1^T\left(-\mathit{\omega }{\mathit{z}}_1+\Lambda {\mathbf{\upsilon }}_e+\Lambda \mathit{\alpha }+{\mathit{v}}_1^{\ast }-{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g+\dot{\mathit{M}}\right)+\frac{1}{2m}{\mathit{z}}_1^T\Lambda {\mathit{z}}_2+\frac{1}{2m}{\mathit{z}}_2^T\left(m^{-1}\left({\mathit{\tau }}_a-{\mathit{f}}_a+\Theta \right)\right)\end{array}$$

(45)

where $m>m_2,\Theta =m^{-1}\left(-{\mathit{C}}_a\mathit{v}-{\mathit{D}}_a\mathit{v}-{\mathit{G}}_a-{\mathit{M}}_a{\dot{\mathit{v}}}_d\right)$. By Assumption 3, $\Theta$ satisfies the following properties:

$$\parallel \Theta \parallel \le m^{-1}\left(c\parallel \mathit{v}{\parallel }^2+{\mathit{d}}_1\parallel \mathit{v}\parallel +{\mathit{d}}_2\parallel \mathit{v}{\parallel }^2+g+m_2{\dot{\mathit{v}}}_{\mathit{d}}\right)\le l_0+l_1\parallel \mathit{v}\parallel +l_2\parallel \mathit{v}{\parallel }^2$$

(46)

$l_0, l_1, l_2$ is the unknown positive parameter; ${\mathit{\tau }}_a$ is the controller to be designed; and the design of the actual control law is ${\mathit{\tau }}_a$.

$${\mathit{\tau }}_a=m\left(-{\mathbf{\Lambda }}^T{\mathit{z}}_1-\sigma \tanh \left({\mathit{z}}_2\right)-C_3{\mathit{K}}_3{\left|{\mathit{z}}_2\right|}^{1-\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-C_4{\mathit{K}}_4{\left|{\mathit{z}}_2\right|}^{1+\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-{\widehat{\mathit{f}}}_{NN}\right)$$

(47)

where ${\mathit{z}}_2={\mathbf{\upsilon }}_e,$${\mathit{K}}_3,{\mathit{K}}_4, C_3, C_4$ are the control gains of the actual control law; ${\mathit{K}}_3=\mathrm{diag}\left(\right[k_{31},k_{32},k_{33}]),{\mathit{K}}_4=\mathrm{diag}\left(\right[k_{41},k_{42},k_{43}])$ is a positive definite symmetric matrix; $k_{31},k_{32},k_{33},k_{41},k_{42},k_{43},$ is a given parameter; $C_3,C_4$ is a predefined time control parameter defined as follows: $C_3={\displaystyle \frac{\pi }{\beta T_c{k}_{3\min }{2}^{1-\beta }}},C_4=\frac{\pi }{3^{\beta }\beta T_c{k}_{4\min }{2}^{1+\beta }}$; $\sigma ={\widehat{l}}_0+{\widehat{l}}_1\parallel \mathbf{\upsilon }\parallel +{\widehat{l}}_2\parallel \mathbf{\upsilon }{\parallel }^2$ is an estimate of the unmodeled part of the system; and ${\widehat{l}}_0, {\widehat{l}}_1, {\widehat{l}}_2$ is a predefined time-adaptive law that is an estimate of $l_0, l_1, l_2$ that is updated for and follows the following law:

$${\dot{\widehat{l}}}_i={\gamma }_i‖{\mathit{z}}_2‖{‖\mathbf{\upsilon }‖}^i-C_{\min }2\beta {\gamma }_i^{2-\beta }{\widehat{l}}_i^{1-\beta }-C_{\min }2\beta {\gamma }_i^{2+\beta }{\widehat{l}}_i^{1+\beta },i=1,2,3$$

(48)

where $C_{\min }=\min \left\{C_5,C_6\right\},C_5=-\frac{\pi }{T_c\beta }{\left(\frac{1}{2}\right)}^{1-\frac{\beta }{2}},C_6=-\frac{\pi }{T_c\beta }{\left(\frac{1}{2}\right)}^{1+\frac{\beta }{2}}{2}^{\frac{\beta }{2}}$ is the predefined-time adaptive law parameter; ${\gamma }_0,{\gamma }_1,{\gamma }_2$ is the adaptive law gain; and ${\widehat{\mathit{f}}}_{NN}$ is the estimated value of ${\mathit{f}}_{NN}$. The neural network weight update law $\dot{\widehat{\mathbf{W}}}$ is designed as follows, as defined in (9).

Define the cost function as [24]:

$$I(t)={\displaystyle {\int }_t^{\infty }{e}^{-\frac{l-t}{\psi }}\phi (l)}dl$$

(49)

$\psi$ is the positive constant, and $\phi (l)$ is the instantaneous cost function defined as

$$\phi (t)={\left({\mathit{\eta }}_1-{\mathit{\eta }}_{1d}\right)}^T\mathit{D}\left({\mathit{\eta }}_1-{\mathit{\eta }}_{1d}\right)+{\mathit{\tau }}_a^T\mathit{R}{\mathit{\tau }}_a$$

(50)

where $\mathit{D},\mathit{R}$ is a positive definite diagonal matrix used to regulate the effect of trajectory tracking errors and input moments on weight updating, and the loss function $\vartheta (t)$ is defined according to (50) as

$$\vartheta (t)=\phi (t)-{\displaystyle \frac{1}{\psi }}{\widehat{\mathit{f}}}_{NN}+{\dot{\widehat{\mathit{f}}}}_{NN}$$

(51)

According to the first-order gradient descent rule, the update law of the weight $\widehat{\mathbf{W}}$ has the following form:

$$\dot{\widehat{\mathbf{W}}}=-\delta {\displaystyle \frac{\partial E}{\partial \widehat{\mathbf{W}}}}$$

(52)

where $E_c=(1/2){\vartheta }^2$, $\delta >0$ is the learning rate, obtained by substituting (51) into (52):

$$\begin{array}{l}\dot{\widehat{\mathbf{W}}}=-\delta \vartheta \frac{\partial \vartheta }{\partial \widehat{\mathbf{W}}}\\ =-\delta \vartheta {\displaystyle \frac{\partial [\phi (t)-(1/\psi ){\widehat{\mathit{f}}}_{NN}+{\dot{\widehat{\mathit{f}}}}_{NN}]}{\partial \widehat{\mathbf{W}}}}\\ =-\delta \vartheta \left[-\frac{1}{\psi }\frac{\partial {\widehat{\mathit{f}}}_{NN}}{\partial \widehat{\mathbf{W}}}+\frac{\partial }{\partial \widehat{\mathbf{W}}}\left(\frac{\partial {\widehat{\mathit{f}}}_{NN}}{\partial {\mathit{z}}_1}\right){\dot{\mathit{z}}}_1+\frac{\partial }{\partial \widehat{\mathbf{W}}}\left({\displaystyle \frac{\partial {\widehat{\mathit{f}}}_{NN}}{\partial {\mathit{z}}_2}}\right){\dot{\mathit{z}}}_2\right]\\ =-\delta \left(\phi (t)+\widehat{\mathbf{W}}\Xi \right)\Xi =-\delta \mathit{\rho }\left(\widehat{\mathbf{W}},{\mathit{z}}_1,{\dot{\mathit{z}}}_1,{\mathit{z}}_2,{\dot{\mathit{z}}}_2\right)\end{array}$$

(53)

$\Xi =-\left(\overline{\mathit{\alpha }}/\psi \right)+\nabla \overline{\mathit{\alpha }}{\dot{\mathit{z}}}_1+\nabla \overline{\mathit{\alpha }}{\dot{\mathit{z}}}_2$. Here, the parametric projection method is used to ensure the boundedness of the weight vector, so that the upper bound of the weights $\overline{\mathbf{W}}=[{\overline{\mathrm{W}}}_1,{\overline{\mathrm{W}}}_2,\dots {\overline{\mathrm{W}}}_r]$ satisfies $\left|{\widehat{\mathrm{W}}}_r\right|\le {\overline{\mathrm{W}}}_r(r=1,2,\dots l).$ The update law of the weights is modified to be

$$\dot{\widehat{\mathbf{W}}}=\left\{\begin{array}{c}-\delta \mathit{\rho }\begin{array}{cc}\begin{array}{cc}& \end{array}& \parallel \widehat{\mathbf{W}}\parallel \le \parallel \overline{\mathbf{W}}\parallel \mathrm{or} \parallel \widehat{\mathbf{W}}\parallel = \parallel \overline{\mathbf{W}}\parallel ,{\widehat{\mathbf{W}}}^T\mathit{\rho }>0\end{array}\\ -\delta \mathit{\rho }+\delta \mathit{\xi }\begin{array}{ccc}\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& \end{array}& & \end{array}& \hspace{1em}& ‖\widehat{\mathbf{W}}‖=‖\overline{\mathbf{W}}‖,{\widehat{\mathbf{W}}}^T\mathit{\rho }\le 0\end{array}\end{array}\right.$$

(54)

where $\mathit{\xi }=\left(\frac{\left({\widehat{\mathbf{W}}}^T\mathit{\rho }\right)}{\left({\left({\widehat{\mathbf{W}}}^T\right)}^2\right)}\right),\widehat{\mathbf{W}},\mathit{\rho },\mathit{\xi }$ are upper bounded.

Redesign the predefined-time controller as

$${\mathit{\tau }}_a=m\left(-{\Lambda }^T{\mathit{z}}_1-\sigma \tanh \left({\mathit{z}}_2\right)-C_3{\mathit{K}}_3{\left|{\mathit{z}}_2\right|}^{1-\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-C_4{\mathit{K}}_4{\left|{\mathit{z}}_2\right|}^{1+\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-{\widehat{\mathit{f}}}_{NN}\right)$$

(55)

At this point, the predefined time adaptive control law is designed, and the controller structure is shown in Figure 2.

Remark 3.

Designing predefined-time adaptive control laws without model parameters under systems containing unmodeled parts is a difficult task. Thus, the model is decoupled into actuated and unactuated subsystems, Therefore, the model is decoupled into driven and non-driven subsystems, and a self-adaptation law is constructed to offset the effect of the unmodeled part of the system, while the radial-based neural network is designed to compensate for the effect of external perturbations on the system to avoid the appearance of steady state error.

4. Stability Analysis

Theorem 7.

For an underactuated autonomous underwater vehicle system satisfying Assumptions 1–3, consider (40), if ${\mathbf{\upsilon }}_e=0$ and $\mathit{\alpha }$  are of the form (43), ${\mathit{z}}_1$ will converge to a small neighborhood of zero in a predefined-time.

Proof of Theorem 7.

When ${\mathbf{\upsilon }}_e=0,$ and considering (30), we have ${\mathit{z}}_1^T\mathit{\omega }{z}_1=0,$. Substituting (43) into (42):

$${\dot{V}}_1={\mathit{z}}_1^T\left(-C_1{\mathit{K}}_{1}diag\left({\left|{\mathit{z}}_1\right|}^{1+\beta }\right)\mathrm{sgn}\left(z_1\right)-C_2{\mathit{K}}_{2}diag\left({\left|{\mathit{z}}_1\right|}^{1-\beta }\right)\mathrm{sgn}\left({\mathit{z}}_1\right)\right)$$

(56)

By Lemma 2, it follows that

$$\begin{array}{l}-C_1{\mathit{z}}_1^T{\mathit{K}}_{1}diag\left({\left|{\mathit{z}}_1\right|}^{1+\beta }\right)\mathit{s}\mathit{g}\mathit{n}\left({\mathit{z}}_1\right)\le -C_1{k}_1{\mathit{z}}_1^{T}diag\left({\left|{\mathit{z}}_1\right|}^{1+\beta }\right)\mathit{s}\mathit{g}\mathit{n}\left({\mathit{z}}_1\right)\\ \le -C_1{2}^{1+\beta }{3}^{\beta }{k}_1{\left({\displaystyle \frac{1}{2}}{\mathit{z}}_1^T{\mathit{z}}_1\right)}^{1+{\displaystyle \frac{\beta }{2}}}\le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_1^{1+{\displaystyle \frac{\beta }{2}}}\end{array}$$

$$\begin{array}{l}-C_2{\mathit{z}}_1^T{\mathit{K}}_{2}diag\left({\left|{\mathit{z}}_1\right|}^{1-\beta }\right)\mathit{s}\mathit{g}\mathit{n}\left({\mathit{z}}_1\right)\le -C_2{k}_2{\mathit{z}}_1^{T}diag\left({\left|{\mathit{z}}_1\right|}^{1-\beta }\right)\mathit{s}\mathit{g}\mathit{n}\left({\mathit{z}}_1\right)\\ \le -C_2{2}^{1-\beta }{k}_2{\left({\displaystyle \frac{1}{2}}{\mathit{z}}_1^T{\mathit{z}}_1\right)}^{1-{\displaystyle \frac{\beta }{2}}}\le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_1^{1-{\displaystyle \frac{\beta }{2}}}\end{array}$$

(57)

Combining (57), we obtain

$${\dot{V}}_1\le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_1^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_1^{1-{\displaystyle \frac{\beta }{2}}}$$

(58)

Proof end □

Theorem 8.

Considering underdriven underwater robots with RBF neural network control, the directionality and convergence of the weights $\widehat{\mathbf{W}}$ of the RBF neural network can be ensured by updating the weights $\widehat{\mathbf{W}}$ using the parameter projection method that satisfies (54).

Proof of Theorem 8.

The alternative Lyapunov function is chosen to be

$$V_{NN}={\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}$$

(59)

There are two distinct scenarios to consider:

Case 1.

When $\parallel \widehat{\mathbf{W}}\parallel \le \parallel \overline{\mathbf{W}}\parallel$ or  $\parallel \widehat{\mathbf{W}}\parallel = \parallel \overline{\mathbf{W}}\parallel$ is available:

$$\begin{array}{l}{\dot{V}}_{NN}={\displaystyle \frac{1}{{\delta }_C}}{\tilde{\mathbf{W}}}^T\dot{\widehat{\mathbf{W}}}=-{\tilde{\mathbf{W}}}^T\mathit{\rho }\\ \le \left(1+‖\overline{\mathrm{W}}‖‖\Xi ‖\right)\left(‖\overline{\mathrm{W}}‖+‖{\mathrm{W}}^{\ast }‖\right)‖\Xi ‖\le 0\end{array}$$

(60)

Let ${\mathrm{D}}_1=\left(1+‖\overline{\mathbf{W}}‖‖\Xi ‖\right)\left(‖\overline{\mathbf{W}}‖+‖{\mathbf{W}}^{\ast }‖\right)‖\Xi ‖$, and construct some auxiliary terms to obtain:

$$\begin{array}{l}\begin{array}{ll}{\dot{V}}_{NN}& \le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1+{\displaystyle \frac{\beta }{2}}}\\ & +{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1-{\displaystyle \frac{\beta }{2}}}+{\mathrm{D}}_1\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1-{\displaystyle \frac{\beta }{2}}}+{\mathrm{D}}_2\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_{NN}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_{\mathit{C}}}}{V}_{NN}^{1-{\displaystyle \frac{\beta }{2}}}\end{array}\\ {\mathrm{D}}_2={\mathrm{D}}_1+{\displaystyle \frac{\pi }{\beta T_c}}{\left(‖\overline{\mathbf{W}}‖+‖{\mathbf{W}}^{\ast }‖\right)}^{1+{\displaystyle \frac{\beta }{2}}}+{\displaystyle \frac{\pi }{\beta T_c}}{\left(‖\overline{\mathbf{W}}‖+‖{\mathbf{W}}^{\ast }‖\right)}^{1-{\displaystyle \frac{\beta }{2}}}\end{array}$$

(61)

Case 2.

When $‖\widehat{\mathbf{W}}‖=‖\overline{\mathbf{W}}‖,{\widehat{\mathbf{W}}}^T\mathit{\rho }\le 0$ is available:

$${\dot{V}}_{NN}={\displaystyle \frac{1}{{\delta }_C}}{\tilde{\mathbf{W}}}^T\dot{\widehat{\mathbf{W}}}=-{\tilde{\mathbf{W}}}^T\mathit{\rho }+{\displaystyle \frac{{\tilde{\mathbf{W}}}^T\mathit{\rho }}{{\widehat{\mathbf{W}}}^2}}{\tilde{\mathbf{W}}}^T\widehat{\mathbf{W}}\le 0$$

(62)

The proof is similar to Case 1, constructing some auxiliary terms, and we obtain

$$\begin{array}{ll}{\dot{V}}_{NN}& \le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1+{\displaystyle \frac{\beta }{2}}}\\ & +{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1-{\displaystyle \frac{\beta }{2}}}\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\widehat{\mathbf{W}}\right)}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2\delta }}{\tilde{\mathbf{W}}}^T\tilde{\mathbf{W}}\right)}^{1-{\displaystyle \frac{\beta }{2}}}\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_{NN}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_{NN}^{1-{\displaystyle \frac{\beta }{2}}}\end{array}$$

(63)

Proof end □

Theorem 9.

Combining the mathematical models (11)–(28) for underdriven underwater robots, if the controller is defined in the form of (47) and the conditions of (43) and (48) are satisfied, the error ${\mathit{\eta }}_e,{\mathit{v}}_e$ will converge to a small neighborhood of zero in predefined time.

Proof of Theorem 9.

Substituting (47), (48), and (58) into (45) gives

$$V_5=V_1+V_2+V_4={\displaystyle \frac{1}{2}}{\mathit{z}}_1^T{\mathit{z}}_1+{\displaystyle \frac{1}{2m}}{\mathit{z}}_2^T{\mathit{M}}_a{\mathit{z}}_2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}{\tilde{l}}_i^2$$

(64)

$\tilde{l}=\widehat{l}-l,$ is the adaptive law estimation error, and the derivative of $V_2={\displaystyle \frac{1}{2m}}{\mathit{z}}_2^T{\mathit{M}}_a{\mathit{z}}_2, V_4={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}{\tilde{l}}_i^2$, $V_1$ has been discussed in the previous section, so the derivative with respect to time is taken for $V_2,V_4$ here:

$$\begin{array}{l}{\dot{V}}_2+{\dot{V}}_4={\mathit{z}}_2^T\left(\mathit{F}-\sigma \tanh \left({\mathit{z}}_2\right)-C_3{\mathit{K}}_3{\left|{\mathit{z}}_2\right|}^{1-\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-C_4{\mathit{K}}_4{\left|{\mathit{z}}_2\right|}^{1+\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)\right)+{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}({\widehat{l}}_i-l_i){\dot{\widehat{l}}}_i\\ ={\mathit{z}}_2^T\left(-C_3{\mathit{K}}_3{\left|{\mathit{z}}_2\right|}^{1-\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-C_4{\mathit{K}}_4{\left|{\mathit{z}}_2\right|}^{1+\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)\right)-‖{\mathit{z}}_2‖{\displaystyle \sum _{i=0}^2}{\widehat{l}}_i\parallel \mathit{v}{\parallel }^i+{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}({\widehat{l}}_i-l_i){\dot{\widehat{l}}}_i+3\epsilon \\ \le {\mathit{z}}_2^T\left(-C_3{k}_{3\min }{\mathit{z}}_2^T{\left|{\mathit{z}}_2\right|}^{1-\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)-C_4{k}_{4\min }{\mathit{z}}_2^T{\left|{\mathit{z}}_2\right|}^{1+\beta }\mathrm{sgn}\left({\mathit{z}}_2\right)\right)-‖{\mathit{z}}_2‖{\displaystyle \sum _{i=0}^2}{\widehat{l}}_i\parallel \mathit{v}{\parallel }^i+{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}({\widehat{l}}_i-l_i){\dot{\widehat{l}}}_i+3\epsilon \end{array}$$

(65)

Substituting (48) into (65) yields by Lemma 2 the following:

$$\begin{array}{ll}{\dot{V}}_2+{\dot{V}}_4& \le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2}}{\mathit{z}}_2^T{\mathit{z}}_2\right)}^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2}}{\mathit{z}}_2^T{\mathit{z}}_2\right)}^{1+{\displaystyle \frac{\beta }{2}}}-‖{\mathit{z}}_2‖{\displaystyle \sum _{i=0}^2}{\widehat{l}}_i\parallel \mathit{v}{\parallel }^i+{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}({\widehat{l}}_i-l_i){\dot{\widehat{l}}}_i\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\mathit{z}}_2^T{\displaystyle \frac{m}{2m_2}}{\displaystyle \frac{{\mathit{M}}_a}{m}}{\mathit{z}}_2\right)}^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\mathit{z}}_2^T{\displaystyle \frac{m}{2m_2}}{\displaystyle \frac{{\mathit{M}}_a}{m}}{\mathit{z}}_2\right)}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \sum _{i=0}^2-C_{\min }({\widehat{l}}_i-l_i){\widehat{l}}_i^{1-\beta }-{\displaystyle \sum _{i=0}^2{C}_{\min }}({\widehat{l}}_i-l_i){\widehat{l}}_i^{1+\beta }}\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_2^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_2^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \sum _{i=0}^2{C}_{\min }\left({\widehat{l}}_i^{2-\beta }-l_i^{2-\beta }\right)-{\displaystyle \sum _{i=0}^2{C}_{\min }}\left({\widehat{l}}_i^{2+\beta }-l_i^{2+\beta }\right)}\end{array}$$

(66)

By Lemma 4 and substituting d, we obtain

$$\begin{array}{l}-{\displaystyle \sum _{i=0}^2{C}_{\min }\left({\widehat{l}}_i^{2-\beta }-l_i^{2-\beta }\right)-{\displaystyle \sum _{i=0}^2{C}_{\min }}\left({\widehat{l}}_i^{2+\beta }-l_i^{2+\beta }\right)}\le -{\displaystyle \sum _{i=0}^2{C}_5{\left({\tilde{l}}_i^2\right)}^{1-\frac{\beta }{2}}-{\displaystyle \sum _{i=0}^2{C}_6}{\left({\tilde{l}}_i^2\right)}^{1+\frac{\beta }{2}}}\\ \le -{\displaystyle \sum _{i=0}^2{C}_5{\left({\tilde{l}}_i^2\right)}^{1-\frac{\beta }{2}}-{\displaystyle \sum _{i=0}^2{C}_6}{\left({\tilde{l}}_i^2\right)}^{1+\frac{\beta }{2}}}\le -{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}{\tilde{l}}_i^2\right)}^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{\left({\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=0}^2\frac{1}{{\gamma }_i}}{\tilde{l}}_i^2\right)}^{1-{\displaystyle \frac{\beta }{2}}}\\ \le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_4^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_4^{1-{\displaystyle \frac{\beta }{2}}}\end{array}$$

(67)

By combining (65) and (67), we obtain

$$\begin{array}{l}\begin{array}{ll}{\dot{V}}_5& \le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_1^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_1^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_2^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_2^{1-{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_4^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_4^{1-{\displaystyle \frac{\beta }{2}}}+D\\ & \le -{\displaystyle \frac{\pi }{\beta T_c}}{V}_5^{1+{\displaystyle \frac{\beta }{2}}}-{\displaystyle \frac{\pi }{\beta T_c}}{V}_5^{1-{\displaystyle \frac{\beta }{2}}}\end{array}\\ D=D_2+3\epsilon \end{array}$$

(68)

According to Lemma 1 and (58), (61), and (68), $V$ will converge to the following set of residuals:

$$\left\{\underset{t\to T_{\max }}{\lim }x\left|V⩽\min \left\{{\left({\displaystyle \frac{3\beta T_{c}D}{\pi }}\right)}^{{\displaystyle \frac{2}{2-\beta }}},{\left({\displaystyle \frac{3\beta T_{c}D}{\pi }}\right)}^{{\displaystyle \frac{2}{2+\beta }}}\right\}\right.\right\}$$

(69)

$D=D_2+3\epsilon$ settling time will follow the following equation: $T_{\max }=\sqrt{2}{T}_c,\underset{t\to T_{\max }}{\lim }\left|{\eta }_{ei}\right|\le \xi$, $i=1,2,3,\xi$, whose magnitude is determined by (69).

Proof end □

Remark 4.

Based on the given virtual control law ${\mathit{z}}_1$, the predefined-time adaptive control law, and the sampling data of the underwater robot’s position and velocity states, the radial basis function (RBF) neural network can update its network weights in real time through online learning, thereby effectively compensating for system errors within a certain period. By defining a cost function and deriving the online update rule for the neural network weights, the error compensation effect can be significantly improved. Additionally, using Lyapunov’s first principle, the stability of the system is analyzed, proving that the network weights of the neural network will ultimately converge to a stable value. This convergence guarantees the stability and control accuracy of the closed-loop system, thereby achieving effective compensation for the underwater robot control system”. Generating predefined-time adaptive control laws without model parameters under systems containing unmodeled parts is a difficult task. Thus, decoupling the model into actuated and unactuated subsystems therefore leads the model to be decoupled into driven and non-driven subsystems, and a self-adaptation law is constructed to offset the effect of the unmodeled part of the system, while the radial-based neural network is designed to compensate for the effect of external perturbations on the system to avoid the appearance of steady-state error.

5. Numerical Simulations

5.1. Simulation Results

To validate the effectiveness of the predefined-time backstepping controller, its performance in perturbation estimation and convergence characteristics was evaluated through MATLAB2022a simulations. The parameters of the underactuated underwater vehicle are provided in

Table 1. Underactuated autonomous underwater vehicle parameters.

Parameters	Values	Parameters	Values
$M$	183	$M_{\dot{q}}$	−86
$I_{xx}$	3	$X_{u\left|u\right|}$	−16
$I_{\mathit{z}\mathit{z}}$	95	$Y_{v\left|v\right|}$	−542
$X_u$	−49	$Z_{w\left|w\right|}$	−422
$Y_v$	−243	$M_{q\left|q\right|}$	−62
$Z_w$	−230	$N_{r\left|r\right|}$	−78
$M_q$	−140	$N_{\dot{r}}$	−86
$N_r$	−160	$Z_{\dot{w}}$	−230
$X_{\dot{u}}$	−13	$Y_{\dot{v}}$	−243

and refer to

Table 2. Predefined-time controller parameters.

Parameters	Values	Parameters	Values
$\beta$	0.4	${\mathit{K}}_1$	$\mathrm{diag}\left(\right[5,5,5])$
${\mathit{K}}_2$	$\mathrm{diag}\left(\right[5,5,5])$	$T_c$	20
${\rho }_1$	$\exp \left(-5t\right)+0.1$	${\rho }_2$	0.01
${\rho }_3$	0.01	${\mathit{K}}_3$	$\mathrm{diag}\left(\right[0.5,0.5,0.5])$
${\mathit{K}}_4$	$\mathrm{diag}\left(\right[0.4,0.3,0.8])$	${\gamma }_0$	0.01
${\gamma }_1$	0.1	${\gamma }_2$	0.01
$\mathit{D}$	$\mathrm{diag}\left(\right[0.1,0.1,0.1])$	$\mathit{R}$	$\mathrm{diag}\left(\right[0.001,0.001,0.001])$
$\psi$	1	$\delta$	0.01
$\overline{\mathrm{W}}$	5	$b$	0.1

for parameter settings. [5]. The initial position was set as $\mathit{\eta }={\left[\begin{array}{c}x,y,\mathit{z},\theta ,\psi \end{array}\right]}^T={[14,6,4,0,0]}^T$. Considering the operational conditions, the unknown perturbation was modeled as $\mathit{f}={\left[\exp (-7/5t)+\exp (-6\ast 7t)+3,0,0,\exp (-3/2t)+5,\exp (-t)+4\right]}^T$. The desired trajectory was designed to follow: ${\mathit{\eta }}_{1d}={\left[10\sin \left(0.06t\right)+10\cos \left(0.03t\right),10\sin \left(0.03t\right)+10\cos \left(0.06t\right),0.1t\right]}^T$. The system parameters of the underwater robot are presented in Table 1. For the control strategy, the predefined-time virtual control law was implemented as provided in Equation (43), while the backstepping controller was applied according to Equation (55). The adaptive control law was detailed in Equation (48), and the iterative neural network strategy was outlined in Equation (54). The constrained parameters of the predefined-time controller were selected as ${\mathit{\tau }}_{Lu}=-300,$ ${\mathit{\tau }}_{Lq}={\mathit{\tau }}_{Lr}=-500,{\mathit{\tau }}_{Hu}=500,{\mathit{\tau }}_{Hq}={\mathit{\tau }}_{Hr}=700$. The RBFNN consisted of 12 neurons, each employing a Gaussian radial basis function. The center of each Gaussian function was randomly selected within the range (0, 10) for each of the three dimensions, and the variance for all the Gaussian functions was set to 0.1.

Figure 3 and Figure 4 illustrate the three-dimensional trajectory tracking results and the tracking performance in three directions for the underactuated AUV, respectively. The experimental results indicate that, considering both unmodeled dynamics and time-varying external disturbances, the underactuated AUV is capable of accurately tracking the given three-dimensional trajectory, with tracking errors converging within 10 s. The significant deviation between the initial position and the desired trajectory is the primary cause of large control errors in the early stages of underwater robot operation. During this process, the control system requires a certain amount of time to adjust the robot’s position and orientation, gradually reducing the initial error through corrective actions, ultimately steering the robot towards the desired trajectory. Figure 5 further validates the rationality of the predefined-time controller design, showing that no chattering phenomena occurred during the entire control process. Figure 6 compares the external disturbances with the output of the RBF neural network (RBFNN). The results demonstrate that after 5 s of iteration, the RBFNN output ${\widehat{\mathit{f}}}_{NN}$ closely approximated the assumed time-varying disturbance ${\mathit{f}}_a$. Figure 7 depicts the dynamic evolution of the adaptive laws, showing that the adaptive signals exhibited smooth and bounded behavior throughout the control process.

To further evaluate the compensation performance of the RBFNNs, two sets of simulation experiments were conducted on the AUV: one with RBFNN-based compensation and the other without any compensation. Figure 8 presents the comparison of tracking errors in one direction for the underactuated AUV between the two cases. The analysis reveals that the use of RBFNN compensation significantly reduced the steady-state error compared to the uncompensated scenario. Furthermore, RBFNN compensation accelerated the system’s response speed and effectively mitigated the impact of external disturbances on system performance.

5.2. Comparison Study

To demonstrate the superiority of the predefined-time control strategy proposed in this study, this subsection presents a comparative analysis against the conventional fixed-time control law [25].

$${\mathit{\alpha }}_c={\Lambda }^{-1}\left(-{\mathit{K}}_1\tanh \left({\mathit{z}}_1\right)-{\mathit{K}}_{2}diag\left({\left|{\mathit{z}}_1\right|}^{2\beta -1}\right)\mathrm{sgn}\left({\mathit{z}}_1\right)-{\mathit{v}}_1^{\ast }+{\mathit{J}}_1^T{\dot{\mathit{\eta }}}_g-\dot{\mathit{M}}\right.)$$

(70)

$${\mathit{\tau }}_c=m\left(-{\Lambda }^T{\mathit{z}}_1-\sigma \tanh \left({\mathit{z}}_2\right)-{\mathit{K}}_3\tanh \left({\mathit{z}}_2\right)-{\mathit{K}}_4{\left|{\mathit{z}}_2\right|}^{2\beta -1}\mathrm{sgn}\left({\mathit{z}}_2\right)\right)$$

(71)

$$\sigma ={\widehat{l}}_{c0}+{\widehat{l}}_{c1}\parallel \mathbf{\upsilon }\parallel +{\widehat{l}}_{c2}\parallel \mathbf{\upsilon }{\parallel }^2$$

(72)

$${\dot{\widehat{l}}}_{ci}={\gamma }_i‖{\mathit{z}}_2‖{‖\mathbf{\upsilon }‖}^i-C_{\min }2\beta {\gamma }_i^{2-\beta }{\widehat{l}}_{ci}^{1-\beta }-C_{\min }2\beta {\gamma }_i^{2+\beta }{\widehat{l}}_{ci}^{1+\beta },i=1,2,3$$

(73)

The definitions of all symbols in the equations remained consistent with those in the proposed control law, with the subscript “c” denoting the comparison scheme. The simulation was conducted under the same conditions outlined in Section 3.1, and the results are presented in Figure 9. The figure illustrates that while the actual trajectory of the comparison scheme can follow the desired trajectory, the proposed control strategy demonstrated a faster convergence, reaching a small neighborhood of zero within 10 s.

Furthermore, the adoption of the RBF neural network in the proposed approach effectively approximated the time-varying error, thereby reducing the impact of perturbations in the steady state. In contrast, the error comparison graphs in the cross-swing and pendant-swing directions revealed an overshooting phenomenon in the comparison scheme, underscoring the advantages of the proposed method in terms of trajectory tracking accuracy and response speed.

This section compares the radial basis function neural network (RBFNN) with the feedforward neural network (FNN). To control for variables, both networks were configured with consistent parameters and evaluated under the conditions outlined in Section 3.1. In Figure 10. The results demonstrate that both the proposed RBFNN and the FNN were able to effectively compensate for errors after 5 s. However, the proposed RBFNN provided more efficient compensation when there were significant changes in the error, whereas the FNN exhibited suboptimal compensation at the inflection points. Although the error compensation signal proposed in this study showed some oscillations at the inflection points, the amplitude of these oscillations remained within the permissible range of the controller, making it reasonable for practical applications. Therefore, the RBFNN-based error compensation scheme proposed in this paper showed a clear advantage over the FNN.

6. Conclusions

To address the challenge of trajectory tracking control for underactuated AUVs in the presence of external disturbances and unmodeled dynamics, this paper proposes an adaptive predefined-time control method based on the backstepping approach, incorporating an RBFNNs. By decoupling the system into a driving submodel and a non-driving subsystem, a controller independent of model parameters was designed, effectively compensating for the unmodeled dynamics through an adaptive algorithm. To enhance convergence performance, a predefined-time control strategy was introduced to accelerate error convergence. Additionally, the RBF neural network was employed to mitigate the effects of external disturbances on the system. The effectiveness and superiority of the proposed control scheme were demonstrated through stability analysis and comparative simulations.