Tracking control of USV under conditions of liquid sloshing,input saturation,and external disturbances

Abstract

This paper investigates the attitude tracking control of USVs under conditions of input saturation, external environmental disturbances, and liquid sloshing in fuel tanks. Firstly, the USV is coupled with the moving pulsing ball model (MPBM), and the coupled model is used to address the aforementioned three issues. For external environmental disturbances, a nonlinear disturbance observer is employed to estimate the disturbance values; The forces and torques generated by liquid sloshing are estimated using an radial basis function neural network (RBFNN); The input saturation problem is resolved by using a first-order filter to generate the controller. Finally, the coupled USV-MPBM system is theoretically and practically verified using Lyapunov theory and simulation data. The results show that the system can achieve good attitude control under the proposed controller, and the liquid sloshing is also effectively suppressed.

Introduction

During the transportation of liquid materials, there will inevitably be a phenomenon of sloshing. This is reasonable whether explained from the physical properties of the liquid, such as density and viscosity, or analyzed from the shape, size, and internal structure of the storage tank. Once there is a large-scale and uncontrollable sloshing phenomenon, it will produce undesirable results, including transportation safety and the stability of the system. Therefore, it is necessary to study how to mitigate and suppress sloshing during the transportation of liquids.

In response to the issue of liquid sloshing in storage tanks, researchers have conducted extensive studies and tests, among which the MPBM method is widely applied.^1–3 MPBM is an equivalent model that can regard the liquid in any shaped storage tank as a ball with constant mass and variable radius, thereby accurately describing the phenomenon of liquid sloshing during the operation of USVs. ⁴ Up to now, MPBM has been widely used in spacecraft and has been effectively verified in the application of spacecraft, contributing to the research of spacecraft.^4–6

During common liquid transportation processes, USVs inevitably face various impacts, including the sloshing of liquids within storage pipes.^7,8 Therefore, the sloshing of liquids cannot be overlooked during the operation of USVs. The common impacts include the following aspects: During the sloshing process, a large amount of impact loads are generated. Not only will the motion attitude of the USV change, but the impact force of the load can also damage the structure of the storage tank^9,10; Additionally, if the USV interacts with the sloshing liquid in the storage compartment during operation, it will generate uncontrollable torques, which can greatly affect the stability of the USV system and potentially cause the vessel to capsize and become submerged. ¹¹

During the navigation of the USV, another important aspect is the saturation of the controller. Input saturation has a very significant impact on the performance of USVs, and system stability may be compromised as a result. There are various controllers designed for systems with input saturation, and some common ones currently include: designing a compensation mechanism based on internal auxiliary support that relies on anti-saturation control technology ¹² ; designing an event-triggering mechanism that uses a smooth hyperbolic tangent function to approximate the saturation function^13–15; using self-organizing neural networks to reduce the impact of input saturation,^16–18 etc. After taking certain measures against input saturation, the robustness of the system is improved, ¹⁹ the amplitude of the control input is reduced, the control rate is optimized, ²⁰ the computational complexity is decreased, and both the flexibility and stability of the system are enhanced. ²¹

In recent years, there has been a significant amount of research on fuzzy adaptive fault-tolerant control, such as in references.^22–25 These four articles collectively focus on the fuzzy adaptive fault-tolerant control of MIMO nonlinear systems, aiming to address issues such as actuator and sensor faults, external disturbances, and uncertainties in system dynamics. They all employ adaptive control strategies based on fuzzy logic to enhance the robustness of the systems and verify the effectiveness and stability of the proposed control methods through theoretical analysis and simulation experiments. Different approaches were also used to solve actuator problems, all of which are worth emulating in this article.

So far, most of the research articles on liquid sloshing have focused on spacecraft, with few addressing how to consider suppressing sloshing during the operation of unmanned surface vehicles (USVs). Therefore, this article, drawing on the methods of suppressing sloshing in spacecraft, studies the trajectory tracking control problem of USVs in the presence of liquid sloshing, input saturation, and external disturbances. For a USV system that has both input saturation and liquid sloshing, which needs to consider both aspects simultaneously. The designed controllers and control methods must be comprehensive, effectively addressing the issue of input saturation to prevent the USV control system from exceeding maximum value limits, and also suppressing the sloshing liquid in the storage tank to meet the challenges brought by liquid sloshing, ensuring the entire USV system can operate stably. Considering that the operating environment of USVs is also the ocean, and encountering liquid sloshing inside the cabin during operation is inevitable, it is theoretically feasible to apply MPBM to simulate the liquid sloshing of USVs. Combining USVs with MPBM can help readers better understand and predict the changes in the posture and structure of USVs under liquid sloshing, which is very important for the stability analysis of USVs.

Contribution

In this paper, the attitude control problem of a USV that is simultaneously confronted with input saturation and liquid sloshing is investigated. Initially, the MPBM was employed to equivalently substitute the USV’s liquid storage compartment, resulting in a coupled model of the USV and MPBM. For this coupled model, the designed controller that not only achieved the USV’s attitude control but also effectively mitigated the sloshing within the storage compartment. The main innovative aspects of this paper are as follows:

1. This study comprehensively investigates the trajectory control problem of USVs, specifically addressing the issue of USV trajectory control in the presence of both liquid sloshing and input saturation. This not only provides a new theoretical foundation for the stability control of USVs but also paves the way for future research and applications.

2. In this study, the coupling of the USV with the MPBM is employed. By establishing a coupling model between the two, the trajectory control problem of the USV is effectively resolved. This approach not only deepens our understanding of how to deal with the suppression of liquid sloshing but also significantly enhances the applicability and effectiveness of the solution.

3. This paper systematically investigates the trajectory control problem of USVs under key factors such as external environmental disturbances, liquid sloshing, and input saturation. The simulation results verify the effectiveness of the proposed control method, which can provide theoretical support and practical basis for the stable operation of USVs in complex marine environments to a certain extent.

• Remark 1. Our research outcomes may inspire new directions for research, encouraging subsequent researchers to further explore and innovate based on the achievements we have made. This chain reaction helps to drive technological advancement across the entire field, continuously addressing emerging issues and improving the efficiency and performance of existing technologies.

Preliminary knowledge

Notes

∥·∥ represents the 2-norm; |·| represents the absolute value; $[·{]}^T$ represents the transpose; represents the cross product of a vector; $\mathrm{diag}(·)$ represents a diagonal matrix; $\mathcal{R}$ represents the field of real numbers; ${(·)}^{-1}$ represents the inverse; $\widehat{·}$ represents an estimated value; ${\lambda }_{\max }$ and ${\lambda }_{\min }$ represent the maximum and minimum eigenvalues of a matrix, respectively; $\mathrm{tr}(·)$ represents the trace of a matrix; ★ represents such a matrix, if $\mathbf{e}=[e_1,e_2,\cdots ,e_n{]}^T$ , then

$$\stackrel{★}{e}=\left[\begin{array}{cccc}0\hfill & -e_n\hfill & \cdots \hfill & -e2\hfill \\ \multicolumn{1}{c}{e_n}& 0\hfill & \cdots \hfill & -e_1\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{e_2}& e_1\hfill & \cdots \hfill & 0\hfill \end{array}\right],$$

Assumptions

• Assumption 1. The disturbance caused by the external environment satisfies $\parallel {\tau }_d\parallel \le {\tau }_{d_1}$ , $\parallel \stackrel{·}{{\tau }_d}\parallel \le {\tau }_{d_2}$ , ${\tau }_{d_1}$ and ${\tau }_{d_2}$ are two constants greater than 0.

• Assumption 2. The angular velocity $\mathbf{B}=[B_x,B_y,B_z{]}^T$ and the target angular velocity ${\mathbf{B}}_t=[B_{t1},B_{t2},B_{t3}{]}^T$ are bounded, and satisfy $\left|B_i\right|\le {\left(B_i\right)}_{\max }(i=x,y,z)$ . $\left|B_{\mathrm{ti}}\right|\le {\left(B_{\mathrm{ti}}\right)}_{\max }(i=1,2,3)$ , ${\left(B_i\right)}_{\max }$ is greater than zero, ${\left(B_{\mathrm{ti}}\right)}_{\max }$ is a known constant.

Lemmas

• Lemma 1. When $a>0$ , $b>0$ , and $0\le \alpha \le 1$ , the following inequality holds: $a^{\alpha }{b}^{1-\alpha }\le \alpha a+(1-\alpha )b$ , the equality holds if and only if $a=b$ .(Young’s Inequality) ²⁶

• Lemma 2. $\mathrm{\Omega }\left(x\right)$ is a continuous function, where $x\in F_x$ with $F_x$ is a compact set, it can be approximated to any precision ²⁷ :

$$\mathrm{\Omega }\left(x\right)={\mathbf{W}}^{*\mathbf{T}}\psi \left(x\right)+ϵ\left(x\right),$$

(1)

where ${\mathbf{W}}^{*\mathbf{T}}$ denotes an ideal weight matrix, p is the number of nodes in the hidden layer and $\psi \left(x\right)$ = ${\left[\begin{array}{ccc}{\psi }_1\left(x\right)\hfill & ...\hfill & {\psi }_p\left(x\right)\hfill \end{array}\right]}^T$ with the Gaussian function ${\psi }_j$ being defined as ${\psi }_j\left(x\right)=\exp \left(\frac{{\Vert x-c_j\Vert }^2}{2b_j^2}\right)$ , where $c_j$ is the center vector and $b_j$ is the width of the Gaussian function with $j=1,...,p$ ; $ϵ\left(x\right)$ is the error by approximation and $\Vert ϵ\left(x\right)\Vert \le {ϵ}^{*}$ ( ${ϵ}^{*}>0$ is a normal number).

Kinematics and dynamics equations of USV

The dynamic equations of the USV coupled with the MPBM

As shown in Figure 1, the USV is coupled with the MPBM, resulting in the following models:

$$\mathbf{M}\stackrel{·}{\mathbf{B}}+\mathbf{CB}+\mathbf{DB}=u+f_{\mathrm{sloshing}}+{\tau }_d,$$

(2)

where $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the matrix of Coriolis and centrifugal forces, $\mathbf{D}$ is the damping matrix, $\mathbf{B}=[B_x,B_y,B_z{]}^T$ is the derivative of the USV’s attitude vector (i.e. the angular velocity of the USV), $u$ is the control input of the system, $d$ is the disturbance caused by the external environment, and $f_{\mathrm{sloshing}}$ is the force and moment generated by liquid sloshing, calculated by the MPBM model and dependent on the position and velocity of the pulsating sphere:

$$f_{\mathrm{sloshing}}=(S_r\stackrel{★}{e}-l)H_{L_1}+H_{L_2},$$

(3)

$$H_{L_1}=-m_s[{\stackrel{·}{S}}_p+\stackrel{★}{\stackrel{·}{B}}{r}_s+\stackrel{★}{B}\left(\stackrel{★}{B}{r}_s\right)+2\stackrel{★}{B}{r}_s],$$

(4)

$$\begin{array}{l}{H}_{L_2}=-\frac{2}{5}{m}_s\left(S_r-l\right)\\ \left[\left(-2\stackrel{·}{l}\right)\left({\omega }_s+B\right)+\left(S_r-l\right)\left({\stackrel{·}{\omega }}_s+\stackrel{·}{B}+\stackrel{\ast }{B}{\omega }_s\right)\right]\\ +\left(S_r-l\right)\stackrel{★}{e}{H}_{L_1},\end{array}$$

(5)

where $H_{L_1}$ is the contact force between the pulsating liquid slug and the equivalent spherical cavity, $H_{L_2}$ is the contact torque between the pulsating liquid slug and the equivalent spherical cavity, $m_s$ is the mass of the liquid slug, $S_p$ is the velocity in the coordinate system of the equivalent spherical cavity, $S_r$ is the radius of the equivalent spherical cavity, $l$ is the radius of the pulsating, ${\mathbf{r}}_{\mathbf{s}}$ is the position vector from the center of the equivalent spherical cavity to the center of the pulsating sphere, and $\mathbf{e}$ is the unit vector pointing from the center of one sphere to the center of the other.

Figure 1.

Schematic diagram of MPBM coupled with USV.

However, in practical applications, since $S_p$ and ${\omega }_s$ are not measurable, it is not possible to calculate $H_{L_1}$ and $H_{L_2}$ based on the aforementioned equations. Therefore, inspired by Qi et al., ⁶ the situation has been comprehensively considered, and a controller that does not rely on the unmeasurable states of $H_{L_1}$ and $H_{L_2}$ has been designed. The control purpose is to design an attitude controller for the coupled USV and MPBM system so that the attitude error and related parameters are semi-globally uniformly ultimately bounded.

The kinematic equations of the USV represented by quaternions

To facilitate the design of the controller, the kinematic equation of the USV is represented using the following quaternions:

$$\stackrel{·}{h}=\frac{1}{2}h\otimes \mathbf{B},$$

(6)

$$\{\begin{array}{c}{\stackrel{·}{h}}_{\mathbf{B}}=-\frac{1}{2}{h}_v^T\mathbf{B}\hfill \\ {\stackrel{·}{h}}_v=\frac{1}{2}(\stackrel{★}{h_v}+h_{\mathbf{B}}{I}_3)\mathbf{B},\hfill \end{array}$$

(7)

where $h_{\mathbf{B}}^2+h_v^T{h}_v=1$ , $h=[h_{\mathbf{B}},h_v^T{]}^T=[h_{\mathbf{B}},h_x,h_y,h_z{]}^T$ .

Define the attitude tracking target equation as follows:

$$\{\begin{array}{c}{\stackrel{·}{h}}_{\mathbf{B}t}=-\frac{1}{2}{h}_{\mathrm{vt}}^T{\mathbf{B}}_t\hfill \\ {\stackrel{·}{h}}_{\mathrm{vt}}=\frac{1}{2}(\stackrel{★}{h_{\mathrm{vt}}}+h_{\mathbf{B}t}{I}_3){\mathbf{B}}_t,\hfill \end{array}$$

(8)

where $h_{\mathbf{B}t}^2+h_{\mathrm{vt}}^T{h}_{\mathrm{vt}}=1$ , $h_t=[h_{\mathbf{B}t},h_{\mathrm{vt}}^T{]}^T=[h_{\mathbf{B}t},h_{\mathrm{xt}},h_{\mathrm{yt}},h_{\mathrm{zt}}{]}^T$ .

Then, based on equations (6) and (7), the following error equation can be defined:

$$\{\begin{array}{c}{\stackrel{·}{h}}_{\mathbf{Be}}=-\frac{1}{2}{h}_{v\mathbf{e}}^T{\mathbf{B}}_{\mathbf{e}}\hfill \\ {\stackrel{·}{h}}_{v\mathbf{e}}=\frac{1}{2}(\stackrel{★}{h_{v\mathbf{e}}}+h_{\mathbf{Be}}{I}_3){\mathbf{B}}_{\mathbf{e}},\hfill \end{array}$$

(9)

where

$$h_{\mathbf{Be}}^2+h_{v\mathbf{e}}^T{h}_{v\mathbf{e}}=1,$$

(10)

$h_{\mathbf{e}}=[h_{\mathbf{Be}},h_{v\mathbf{e}}^T{]}^T=[h_{\mathbf{Be}},h_{x\mathbf{e}},h_{y\mathbf{e}},h_{z\mathbf{e}}{]}^T$ , ${\mathbf{B}}_{\mathbf{e}}=G{\mathbf{B}}_t-\mathbf{B}$ , ${\mathbf{S}}_r$ is the transformation matrix between two spherical coordinate system, satisfies $\parallel G\parallel =1,\stackrel{·}{G}=-\stackrel{★}{{\mathbf{B}}_{\mathbf{e}}}G$ .

The design of the controller

In order to address the issue of liquid sloshing suppression, the following virtual angular velocity is defined:

$$\begin{array}{c}{\mathbf{B}}_v={[B_{\mathrm{xv}},B_{\mathrm{yv}},B_{\mathrm{zv}}]}^T\hfill \\ =\frac{{\mathrm{af}}_n^2}{s^2+2p{f}_{n}s+f_n^2}{\mathbf{B}}_{\max }\tanh \left({\mathbf{B}}_d\right),\hfill \end{array}$$

(11)

where in this context, $a$ is a constant greater than 0 and less than or equal to 1, $p$ is the damping ratio, and $f_n$ is the natural frequency, ${\mathbf{B}}_{\max }=\mathrm{diag}(B_x^{\max },B_y^{\max },B_z^{\max })$ , ${\mathbf{B}}_d$ is the ideal angular velocity that needs to be designed, and $\tanh (·)$ is the hyperbolic tangent function.

Let ${\mathbf{B}}_v=v_1$ and ${\stackrel{·}{\mathbf{B}}}_v=v_2$ . Then, according to equation (9), the following system can be obtained:

$$\{\begin{array}{c}{\stackrel{·}{v}}_1=v_2\hfill \\ {\stackrel{·}{v}}_2=-2p{f}_n{v}_2+f_n^2(a{\mathbf{B}}_{\max }\tanh \left({\mathbf{B}}_d\right)-v_1),\hfill \end{array}$$

(12)

Introduce the following auxiliary system for the aforementioned system:

$$\{\begin{array}{c}{\stackrel{·}{m}}_1=-{\beta }_1{m}_1+\frac{1}{2}{h}_{\mathrm{ve}}^T({\mathbf{B}}_d-{\mathbf{B}}_v)\hfill \\ {\mathbf{e}}_1=1-h_{\mathbf{Be}}+m_1,\hfill \end{array}$$

(13)

where $m_1\in \mathcal{R}$ and ${\beta }_1\ge 0$ .

The first Lyapunov function is defined as follows: $V_1=\frac{1}{2}{\mathbf{e}}_1^2$ .

Differentiate $V_1$ with respect to time, and substitute equations (8) and (11) into the derivative of $V_1$ :

$${\stackrel{·}{V}}_1={\mathbf{e}}_1(-{\stackrel{·}{h}}_{\mathbf{B}e}+{\stackrel{·}{m}}_1)={\mathbf{e}}_1[\frac{1}{2}{h}_{v\mathbf{e}}^T\left({\mathbf{e}}_2+{\mathbf{B}}_d-G{\mathbf{B}}_t\right)-{\beta }_1{m}_1],$$

(14)

where ${\mathbf{e}}_2=\mathbf{B}-{\mathbf{B}}_v$ .

Here, the ideal angular velocity is designed as:

$${\mathbf{B}}_d=-\frac{2{\mu }_1}{1+h_{\mathbf{B}e}}{h}_{v\mathbf{e}}+G{\mathbf{B}}_t,$$

(15)

Therefore, from equations (9), (12), and (13), the following can be obtained:

$${\stackrel{·}{V}}_1=-{\beta }_1{\mathbf{e}}_1^2+\frac{1}{2}{\mathbf{e}}_1{h}_{v\mathbf{e}}^T{\mathbf{e}}_2,$$

(16)

It is observed that if ${\mathbf{e}}_2$ is a zero vector, ${\stackrel{·}{V}}_1$ is less than or equal to zero. In the following, a controller will be designed to drive ${\mathbf{e}}_2$ toward the zero vector.

For $f_{\mathrm{sloshing}}$ , lemma $1$ is utilized and approximated using a RBFNN as ⁶ :

$$f_{\mathrm{sloshing}}=W^{*T}H\left(\mathbf{B}\right)+ϵ\left(\mathbf{B}\right),$$

(17)

where $ϵ\left(\mathbf{B}\right)$ is an activation function, $\parallel ϵ\left(\mathbf{B}\right)\parallel \le \overline{ϵ}$ , $\overline{ϵ}$ is a small positive number, ${\mathbf{W}}^{*}$ the optimal weight matrix. Thus the $f_{\mathrm{sloshing}}$ can be approximated:

$${\hat{f}}_{\mathrm{sloshing}}={\hat{\mathbf{W}}}^{T}H\left(\mathbf{B}\right),$$

(18)

Here, $\hat{\mathbf{W}}$ is denoted as the estimate of ${\mathbf{W}}^{*}$ , and its adaptive rate is provided as follows:

$$\stackrel{·}{\hat{\mathit{w}}}=\alpha (H{{\hat{\tau }}_d}^{T}n-{\beta }_3\stackrel{·}{\mathbf{W}}),$$

(19)

where $\alpha$ and ${\beta }_3$ are constants which greater than $0$ .

• Remark 2. The choice of neural network approximation as the approximation method is mainly based on preliminary exploration and experiments. Neural networks have demonstrated good adaptability and effectiveness in addressing the trajectory control problem of USVs. They are capable of handling complex nonlinear relationships.

For the external environmental disturbance $d$ , the following nonlinear disturbance observer (NOD) is designed:

$$\{\begin{array}{c}{\hat{\tau }}_d=m_2+\mathbf{Q}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{m}}_2=-n(-\mathbf{CB}-\mathbf{DB}+u+{\hat{\mathbf{W}}}^{T}H-{\hat{\tau }}_d),}\end{array}$$

(20)

where $n={\left(\frac{\partial \mathbf{Q}}{\partial (\mathbf{C}+\mathbf{D})}{|}_{\mathbf{B}}\right)}^{-1}$ , $\mathbf{Q}$ is a three-dimensional positive definite diagonal matrix.

Based on the expressions of equations (17) and (19), the ideal control input can be designed:

$$\begin{array}{c}{u}_d=\mathbf{CB}+\mathbf{DB}-{\hat{\tau }}_d-{\hat{\mathbf{W}}}^{T}H\hfill \\ -\frac{1}{2}{h}_{v\mathbf{e}}{\mathbf{e}}_1-{\beta }_2{\mathbf{e}}_2+\mathbf{M}{\stackrel{·}{\mathbf{B}}}_v,\hfill \end{array}$$

(21)

At this point, based on the design mentioned above, it can be obtained that:

$${\stackrel{·}{\mathbf{e}}}_2={\mathbf{M}}^{-1}(-{\beta }_2{\mathbf{e}}_2-\frac{1}{2}{h}_{v\mathbf{e}}{\mathbf{e}}_1+{\tilde{\mathbf{W}}}^{\mathbf{T}}H+ϵ+{\tilde{\tau }}_d+{\mathbf{e}}_3),$$

(22)

where $\tilde{\mathbf{W}}={\mathbf{W}}^{*}-\hat{\mathbf{W}}$ , ${\tilde{\tau }}_d={\tau }_d^{*}-{\hat{\tau }}_d$ , ${\mathbf{e}}_3=u-u_d$ .

To obtain the actual control input, the following first-order filter is proposed:

$$\stackrel{·}{u}+k_{1}u=S\left(u_v\right),$$

(23)

where $k_1>0$ , $u_v=[u_{\mathrm{vx}},u_{\mathrm{vy}},u_{\mathrm{vz}}{]}^T$ is the introduced virtual control input, $S\left(u_v\right)=[S\left(u_{\mathrm{vx}}\right),S\left(u_{\mathrm{vy}}\right),S\left(u_{\mathrm{vz}}\right){]}^T$ is a saturation function. It has the following expression ⁶ :

$$\begin{array}{c}\multicolumn{1}{c}{S\left(u_{\mathrm{vi}}\right)=\{\begin{array}{cc}{u}_{\mathrm{vi}},\hfill & \mathrm{if}\left|u_{\mathrm{vi}}\right|<{\left(u_i\right)}_{\max }\hfill \\ \multicolumn{1}{c}{\frac{u_{\mathrm{vi}}}{\left|u_{\mathrm{vi}}\right|}{\left(u_i\right)}_{\max },}& \mathrm{if}\left|u_{\mathrm{vi}}\right|\ge {\left(u_i\right)}_{\max },\hfill \end{array}}\end{array}$$

(24)

According to the designed first-order filter $\left(23\right)$ , it can be concluded that $u$ and $\stackrel{·}{u}$ can be bounded by given constants $k_1$ and ${\left(u_i\right)}_{\max }$ , for the following reasons:

The differential equation (22) is solved to obtain:

$$u_i\left(t\right)=u_i\left(0\right){\mathbf{e}}^{-k_{1}t}+\frac{1}{k_1}S\left(u_{\mathrm{vi}}\right)(1-{\mathbf{e}}^{-k_{1}t}),$$

(25)

The observation is made on $u$ and the derivative of $u$ :

$$\begin{array}{c}\left|u_i\right|\le \left|u_i\left(0\right)\right|+\left|\frac{{\left(u_i\right)}_{\max }}{k_1}\right|,\hfill \\ \left|{\stackrel{·}{u}}_i\right|\le (1+k_1){\left(u_i\right)}_{\max },\hfill \end{array}$$

Thus, it can be constrained.

The virtual control input is introduced above; Below, the design proceeds as follows. Based on the expression for ${\mathbf{e}}_3$ , the derivative of ${\mathbf{e}}_3$ can be derived as:

$$\begin{array}{c}{\stackrel{·}{\mathbf{e}}}_3=\stackrel{·}{u}-{\stackrel{·}{u}}_d\hfill \\ =S\left(u_v\right)-k_{1}u-{\stackrel{·}{u}}_d\hfill \\ =u_v+O_{u_v}-k_{1}u-{\stackrel{·}{u}}_d,\hfill \end{array}$$

(26)

where $O_{u_v}=S\left(u_v\right)-u_v$ .

Let ${\mathbf{e}}_4={\mathbf{e}}_3-m_3$ , and introduce a new auxiliary system:

$${\stackrel{·}{m}}_3=-{\beta }_4{m}_3+O_{u_v},$$

(27)

Then we can obtain:

$$\begin{array}{c}{\stackrel{·}{\mathbf{e}}}_4=u_v+O_{u_v}-k_{1}u-{\stackrel{·}{u}}_d+{\beta }_4{m}_3-O_{u_v}\hfill \\ =u_v-k_{1}u-{\stackrel{·}{u}}_d+{\beta }_4{m}_3,\hfill \end{array}$$

(28)

Introduce a new filter:

$$g{\stackrel{·}{X}}_d+X_d=v_d,$$

(29)

where, the parameter $g$ is a design element, and $X_d$ represents the output from the filter.

Let ${\mathbf{e}}_5=X_d-v_d$ , then it can be obtained that:

$$\begin{array}{c}{\stackrel{·}{\mathbf{e}}}_5=\frac{v_d-X_d}{g}-{\stackrel{·}{v}}_d\hfill \\ =\frac{{\mathbf{e}}_5}{g}-{\stackrel{·}{v}}_d,\hfill \end{array}$$

(30)

If ${\mathbf{e}}_6={\mathbf{e}}_4-{\mathbf{e}}_5$ is further defined, a virtual control law can be designed based on the obtained results:

$${\stackrel{·}{\mathbf{e}}}_6=-k_{1}u+\frac{{\mathbf{e}}_5}{g}-{\beta }_4{m}_3+u_v,$$

(31)

$$u_v=k_{1}u-\frac{{\mathbf{e}}_5}{g}-{\beta }_4{m}_3-{\beta }_5{\mathbf{e}}_6,$$

(32)

where ${\beta }_5$ is a parameter greater than $0$ .

$${\stackrel{·}{\mathbf{e}}}_6=-{\beta }_5{\mathbf{e}}_6,$$

(33)

Stability analysis

• Theorem 1. Considering the attitude kinematics and dynamics equations of a USV filled with liquid, define $\mathit{N}=[h_{\mathit{Be}},h_{\mathit{ve}}^T,{\mathit{B}}^T{]}^T$ , $F_x$ is a sufficiently large compact set, and let $N\left(t\right)\in F_x$ . Under assumptions, given that the parameters ${\beta }_1$ , ${\beta }_2$ , ${\beta }_3$ , ${\beta }_4$ , ${\beta }_5$ , $g$ , and $n$ satisfy the following conditions: ${\beta }_1>0$ , ${\beta }_2>\frac{5}{2}$ , ${\beta }_3>m\parallel n{\parallel }^2$ , ${\beta }_4>\frac{1}{2}$ , ${\beta }_5>0$ , $0<g<2$ , and $2{\lambda }_{\min }\left(n\right)-\parallel n{\parallel }^2-\frac{1}{2}$ , then the attitude controller $u$ obtained under the RBFNN and NOD can ensure that the closed-loop system has the following properties:

Proof. $\left(a\right)$ Firstly, it will be proven that the parameters ${\tilde{\tau }}_d$ and $\tilde{\mathbf{W}}$ exhibit semi-global uniform boundedness:

Opt for the secondary stability function as follows:

$$V_2=\frac{1}{2}{{\tilde{\tau }}_d}^T{\tilde{\tau }}_d+\frac{1}{2\alpha }\mathrm{tr}\left({\tilde{\mathbf{W}}}^{\mathbf{T}}\tilde{\mathbf{W}}\right),$$

(34)

then, it can be obtained:

$$\begin{array}{c}{\stackrel{·}{V}}_2={\tilde{\mathbf{W}}}^{\mathbf{T}}{\stackrel{·}{\stackrel{~}{\tau }}}_d+\frac{1}{\alpha }\mathrm{tr}\left({\tilde{\mathbf{W}}}^{\mathbf{T}}\stackrel{·}{\stackrel{~}{W}}\right)\hfill \\ \le {{\tilde{\tau }}_d}^T[\stackrel{·}{{\tau }_d}-n({\tilde{\tau }}_d+{\tilde{\mathbf{W}}}^{\mathbf{T}}H+ϵ)]\hfill \\ -\mathrm{tr}[{\tilde{\mathbf{W}}}^{\mathbf{T}}(H{{\hat{\tau }}_d}^{T}n-{\beta }_3\hat{W})]\hfill \\ \le {{\tilde{\tau }}_d}^T\stackrel{·}{{\tau }_d}-{{\tilde{\tau }}_d}^{T}n{\tilde{\tau }}_d-{{\tilde{\tau }}_d}^{T}nϵ-{{\tilde{\tau }}_d}^{T}n{\tilde{\mathbf{W}}}^{\mathbf{T}}H\hfill \\ +{\beta }_3\mathrm{tr}\left({\tilde{\mathbf{W}}}^{\mathbf{T}}\hat{W}\right),\hfill \end{array}$$

(35)

According to assumptions and lemmas, it follows that:

$$\begin{array}{c}{\stackrel{·}{V}}_2\le -{{\tilde{\tau }}_d}^T({\lambda }_{\min }\left(n\right)-\frac{1}{2}\parallel n{\parallel }^2-\frac{1}{4}){\tilde{\tau }}_d\hfill \\ -(\frac{{\beta }_3}{2}-\frac{m}{2}\parallel n{\parallel }^2)\mathrm{tr}\left({\tilde{\mathbf{W}}}^{\mathbf{T}}\tilde{\mathbf{W}}\right)\hfill \\ +\frac{1}{2}{\left({\tau }_{d_1}\right)}^2+{\left({\tau }_{d_2}\right)}^2+\frac{1}{2}{\overline{ϵ}}^2+\frac{1}{2}{\beta }_3\mathrm{tr}[{\left({\tilde{\mathbf{W}}}^{*}\right)}^T{\tilde{\mathbf{W}}}^{*},\hfill \end{array}$$

(36)

So it can be obtained:

$${\stackrel{·}{V}}_2\le -k_1{V}_2+{\xi }_1,$$

(37)

where $k_1=\min \{2{\lambda }_{\min }\left(n\right)-\frac{1}{2}\parallel n{\parallel }^2-\frac{1}{4},{\beta }_3\alpha -m\alpha \parallel n{\parallel }^2\}$ , if appropriate parameters are chosen, it is possible to make $k_1>0$ .

Next, multiplying both sides of equation (36) by ${\mathbf{e}}^{k_{1}t}$ , and then integrating the resulting expression over the interval $[0,t]$ , it can be obtained:

$$V_2\left(t\right)\le (V_2\left(0\right)-\frac{{\xi }_1}{k_1}){\mathbf{e}}^{-k_{1}t}+\frac{{\xi }_1}{k_1},$$

(38)

So, based on the result from equation (37), it can be concluded that as $t$ approaches infinity, $V_2$ is less than or equal to $\frac{{\xi }_1}{k_1}$ , which implies that $\tilde{d}$ and $\tilde{\mathbf{W}}$ exhibit semi-globally uniform boundedness.

• (b) Secondly, the parameters ${\mathbf{e}}_1$ , ${\mathbf{e}}_2$ , $m_3$ , ${\mathbf{e}}_5$ , and ${\mathbf{e}}_6$ are all semi-globally uniformly bounded:

Select the second Lyapunov function as:

$$\begin{array}{c}{V}_3=V_1+\frac{1}{2}{{\mathbf{e}}_2}^T\mathbf{M}{\mathbf{e}}_2+\frac{1}{2}{m_3}^T{m}_3\hfill \\ +\frac{1}{2}{{\mathbf{e}}_5}^T{\mathbf{e}}_5+\frac{1}{2}{{\mathbf{e}}_6}^T{\mathbf{e}}_6,\hfill \end{array}$$

(39)

Similar to the proof of $\left(a\right)$ , differentiate $V_3$ and then use the lemma to obtain:

$$\begin{array}{c}{\stackrel{·}{V}}_3\le -{\beta }_1{{\mathbf{e}}_1}^2-({\beta }_2-\frac{5}{2}){{\mathbf{e}}_2}^T{e}_2\hfill \\ -({\beta }_4-\frac{1}{2}{m_3}^T{m}_3)-(\frac{1}{g}-\frac{1}{2}){{\mathbf{e}}_5}^T{\mathbf{e}}_5\hfill \\ -{\beta }_5{{\mathbf{e}}_6}^T{\mathbf{e}}_6+\frac{1}{4}{{\tilde{\tau }}_d}^T{\tau }_d+\frac{1}{2}\mathrm{ltr}\left({\tilde{\mathbf{W}}}^{\mathbf{T}}\tilde{\mathbf{W}}\right)+\frac{1}{2}{\overline{ϵ}}^2\hfill \\ +\frac{1}{2}\parallel O_{u_v}{\parallel }^2+\frac{1}{2}\parallel {\stackrel{·}{u}}_d{\parallel }^2+\frac{1}{2}\parallel {\mathbf{e}}_3{\parallel }^2,\hfill \end{array}$$

(40)

By analysis, it can be obtained:

$$\frac{1}{2}\parallel O_{u_v}{\parallel }^2+\frac{1}{2}\parallel {\stackrel{·}{u}}_d{\parallel }^2+\frac{1}{2}\parallel {\mathbf{e}}_3{\parallel }^2\le {\xi }_3,$$

(41)

From equation (37), it follows that :

$$\frac{1}{4}{{\tilde{\tau }}_d}^T{\tau }_d+\frac{1}{2}\mathrm{ltr}\left({\tilde{\mathbf{W}}}^{\mathbf{T}}\tilde{\mathbf{W}}\right)\le {\gamma }_1\frac{{\xi }_1}{k_1},$$

(42)

where ${\gamma }_1=\max \{\mathrm{rl},\frac{1}{2}\}$ .

From equations (40) and (41), it ultimately follows that:

$${\stackrel{·}{V}}_3\le -k_2{V}_3+{\xi }_2+{\xi }_3,$$

(43)

where $k_1=\min \{2{\lambda }_{\min }\left(n\right)-\frac{1}{2}\parallel n{\parallel }^2-\frac{1}{4},{\beta }_3\alpha -m\alpha \parallel n{\parallel }^2\}$ , if appropriate parameters are chosen, it is possible to make $k_2>0$ .

Then, multiplying both sides of equation (42) by ${\mathbf{e}}^{k_{2}t}$ , and then integrating the resulting expression over the interval $[0,t]$ , it can be obtained:

$$V_3\left(t\right)\le (V_3\left(0\right)-\frac{{\xi }_2+{\xi }_3}{k_3}){\mathbf{e}}^{-k_{2}t}+\frac{{\xi }_2+{\xi }_3}{k_2},$$

(44)

So, based on the result from equation (43), it can be concluded that as $t$ approaches infinity, $V_3$ is less than or equal to $\frac{{\xi }_2+{\xi }_3}{k_2}$ , which implies that ${\mathbf{e}}_1$ , ${\mathbf{e}}_2$ , $m_3$ , ${\mathbf{e}}_5$ , and ${\mathbf{e}}_6$ are semi-globally uniformly bounded.

• (c) Besides, it will be proved that for any $ϵ>0$ , $\parallel e_2\parallel \le ϵ$ holds when $t\to \infty$ , if there exists $k_2=\min \{2{\beta }_1,\frac{2{\beta }_2-5}{{\lambda }_{\max }\left(J_0\right)},2{\beta }_4-1,\frac{2-g}{g},2{\beta }_5\}$ :

The range of $V_3$ can be obtained from $\left(38\right)$ and $\left(43\right)$ :

$$\frac{1}{2}{{\mathbf{e}}_2}^T\mathbf{M}{\mathbf{e}}_2\le V_3\le \frac{{\xi }_2+{\xi }_3}{k_2},$$

(45)

thus,

$${{\mathbf{e}}_2}^T{\mathbf{e}}_2\le ϵ,$$

(46)

where $ϵ={\left(\frac{2({\xi }_2+{\xi }_3)}{{\lambda }_{\max }\left(\mathbf{M}\right)k_2}\right)}^{\frac{1}{2}}$ .

Therefore, it can be concluded that when $k_2$ exists, as long as appropriate parameters are selected, for any $ϵ>0$ , $\parallel {\mathbf{e}}_2\parallel \le ϵ$ holds when $t\to \infty$ .

• (d) Finally, it will be proved that when $N\left(0\right)$ belongs to $F_x$ , there always exists a $F_x$ such that for all $t\ge 0$ , $N\left(t\right)\in F_x$ : let $m$ be a constant, then $V_3\le m$ . Since $N\in F_x$ , $F_x$ is defined as $F_x=\{h_{\mathbf{Be}},h_{v\mathbf{e}},\mathbf{B}|V_3\le m\}$ . Below, $\left(d\right)$ will be proved in two cases.

  $\left(d_1\right)$ . $V_3\left(0\right)\ne m(V_3\le m)$ , when $N\left(0\right)\in F_x$ .

If the result obtained from equation (43) is less than or equal to $m$ , it can be obtained that:

$$V_3\left(t\right)\le (V_3\left(0\right)-\frac{{\xi }_2+{\xi }_3}{k_3}){\mathbf{e}}^{-k_{2}t}+\frac{{\xi }_2+{\xi }_3}{k_2}\le m,$$

(47)

By solving equation (46) and taking the limit as $t$ approaches infinity on both sides of the resulting expression, it was obtained that:

$$\frac{{\xi }_2+{\xi }_3}{k}\le k_2,$$

Therefore, when $\frac{{\xi }_2+{\xi }_3}{k}\le k_2$ and $V_3\left(0\right)<m$ , it follows that $V_3\le m$ .

• $\left(d_2\right)$ . $V_3\left(0\right)=m$ , when $N\left(0\right)\in F_x$ .

When $v_3\left(0\right)=m$ , taking another look at equation (46) yields the same result:

$$\frac{{\xi }_2+{\xi }_3}{k}\le k_2,$$

Therefore, when $\frac{{\xi }_2+{\xi }_3}{k}\le k_2$ and $V_3\left(0\right)=m$ , it follows that $V_3\le m$ . By combining the conclusions of $\left(d_1\right)$ and $\left(d_2\right)$ , the proof of $\left(d\right)$ can be completed.

Simulation verification

First, the parameters for the liquid-filled USV as follows: $\mathbf{M}=85\mathrm{kg}$ (the value of M is taken as the eigenvalues of the mass matrix.); $m_s=23\mathrm{kg}$ ; $S_r=35\mathrm{cm}$ ; ${\tau }_d=0.02{[\sin \frac{t}{10\mathrm{pai}},\sin \frac{t}{20\mathrm{pai}},\sin \frac{t}{6\mathrm{pai}}]}^{T}N/m$ . Then, the initial state of the USV and the parameters of the equivalent fluid block as follows: ${\mathbf{B}}_0=[-0.3,-0.4,0.5{]}^T$ ; ${\mathbf{h}}_{v0}=0$ ; ${\mathbf{h}}_{\mathrm{vt}0}=[0.1,-0.4,-0.7{]}^T$ ; ${\omega }_{s0}=[2,-2,1{]}^T$ ; $S_{p0}=0$ ; $r_s=[0,0,15{]}^T\mathrm{cm}$ . Besides, the target angular velocity as follows: $B_t=0.02{[\sin \frac{t}{100\mathrm{pai}},\sin \frac{t}{200\mathrm{pai}},\sin \frac{t}{60\mathrm{pai}}]}^T$ . The values for other related parameters are provided as follows: $f_n=60$ ; $\xi =0.6$ ; $m=20$ ; $a=0.5$ ; $\alpha =1$ ; $k_1=3$ ; ${\beta }_1=0.5$ ; $\beta 2=600$ ; ${\beta }_3=12$ ; ${\beta }_4=0.4$ ; ${\beta }_5=30$ ; $g=0.005$ ; $\mathbf{T}=[5B_x-0.03B_y+0.08B_z,-0.03B_x+6B_y+0.1B_z,0.08B_x-0.1B_y+5B_z{]}^T$ . Additionally, since the article requires constraints on $B$ , $u$ , and $\stackrel{·}{u}$ to suppress liquid sloshing, the limiting values are set as ${B_i}^{\max }=0.9(i=x,y,z)$ and ${u_i}^{\max }=6(i=x,y,z)$ . Based on the parameters provided above, simulation comparisons for $h_{\mathbf{e}}$ , $\mathbf{B}$ , $S_p$ , and ${\omega }_s$ under conditions with and without restrictions on $\mathbf{B}$ , $u$ , and $\stackrel{·}{u}$ , as shown in Figures 3 to 10. Simulation experiments on the changes of ${\mathbf{B}}_d$ , $m_2$ , ${\tilde{\tau }}_d$ , $\Vert \hat{\mathbf{W}}\Vert$ , $u_d$ , $u$ , $\stackrel{·}{u}$ , and $m_3$ were also performed, as shown in Figures 11 to 17.

• Remark 3. In Theorem 1, we assume ${\beta }_4>\frac{1}{2}$ to ensure the rigor of the theoretical analysis and the universality of the conclusions. This condition is derived from mathematical derivations, aiming to ensure that the theoretical results are valid in all possible cases. In the simulation part, we choose ${\beta }_4=0.4$ based on specific needs in practical applications. Specifically, this choice is made to better reflect certain characteristics in real-world scenarios. In practice, we have found that even when ${\beta }_4$ is less than $\frac{1}{2}$ , the system can still perform well. This indicates that in practical applications, some theoretical conditions can be appropriately relaxed to accommodate specific situations.

• Remark 4. Parameter uncertainties can cause the system’s dynamic behavior to deviate from the expected model, while inaccurate calibration may lead to improper controller settings. Both of these factors have the potential to undermine the overall stability and robustness of the closed-loop system.

• Remark 5. In fact, both parameter uncertainties and inaccurate calibration can significantly impact the stability of the system. Parameter uncertainties can cause the system’s dynamic behavior to deviate from the expected model, narrow the stability margins, and degrade system performance; while inaccurate calibration can lead to a decline in controller performance, system instability, and reduced adaptability. To ensure the stability and performance of the system, it is necessary to adopt effective strategies to address these impacts, such as robust control design, adaptive control, and methods of parameter estimation with online adjustment. These approaches are capable of effectively dealing with the issues of parameter variations and inaccurate calibration, thereby ensuring the stability and performance of the system.

Figures 2 and 3 are images depicting the attitude tracking error $q_{\mathbf{e}}$ of the USV. From Figure 2 that $q_{\mathbf{e}}$ converges at around $22$  s. Although in Figure 3 can converge faster, the control precision is poor due to the lack of restrictions on $\mathbf{B}$ , $u$ , and $\stackrel{·}{u}$ , and the excessive response is very high at the start. Figures 4 and 5 are images depicting the attitude angular velocity $B$ of the USV. From Figure 4 that the convergence speed and control precision of $\mathbf{B}$ are both well balanced. $B$ in Figure 5 has the same problem as $h_{\mathbf{e}}$ in Figure 3. Figures 6 to are images obtained from the simulation of the velocity and angular velocity of the liquid slug. It can be seen from Figures 6 and 8 that the motion state of the liquid slug can slowly converge to a very small value during the USV’s movement. However, in Figures 7 and 8, the liquid mass’s motion is activated from the beginning and eventually converges to a large value. Based on the simulation analysis of Figures 2 to 9, we can draw the following conclusion: The restrictions on the values of $\mathbf{B}$ , $u$ , and $\stackrel{·}{u}$ not only help achieve ideal attitude tracking performance but also effectively suppress the oscillation of the USV filled with liquid during the task completion process.

Figure 2.

The law of change of velocity error $q_e$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are limited).

Figure 3.

The law of change of velocity error $q_e$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are not limited).

Figure 4.

The law of change of $B$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are limited).

Figure 5.

The law of change of $B$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are not limited).

Figure 6.

The law of change of $V_s$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are limited).

Figure 7.

The law of change of $V_s$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are not limited).

Figure 8.

The law of change of ${\omega }_s$ with time (, $u_c$ , ${\stackrel{·}{u}}_c$ are limited).

Figure 9.

The law of change of ${\omega }_s$ with time ( $B$ , $u_c$ , ${\stackrel{·}{u}}_c$ are limited).

Figure 10.

The law of change of $B_d$ with time.

Figure 11.

The law of change of $z_2$ with time.

Figure 12.

The law of change of $\tilde{d}$ with time.

Figure 13.

The law of change of $\Vert \hat{\mathbf{W}}\Vert$ with time

Figure 14.

The law of change of $u_d$ with time

Figure 15.

The law of change of $u$ with time

Figure 16.

The law of change of $\stackrel{·}{u}$ with time.

Figure 17.

The law of change of $z_3$ with time.

Figure 10 is the simulation image of the ideal angular velocity. When comparing Figure 10 with Figure 4, it is evident that the custom-designed filter is capable of confining the actual angular velocity to a specific range. In Figure 11, a simulation of the internal state error variables, and it can be seen that the error $z_2$ is bounded. A simulation of the disturbance error is also conducted. The disturbance ${\tilde{\tau }}_d$ converges to a very small value in about $6$  s, which also demonstrates that the designed disturbance observer has a good performance. The results are shown in Figure 12. Figure 13 is the simulation image of the second-order norm of the adaptation rate $\hat{\mathbf{W}}$ . It can be seen from the figure that the estimation speed of $\hat{\mathbf{W}}$ is very fast and it can converge to a very small value. Figures 14 and 15 are the simulation images of the ideal input and the actual input. Since the input involves a saturation function, the actual input is limited. It can also be seen from Figures 16 and 17 that $\stackrel{·}{u}$ and $m_3$ are bounded. To sum up, the proposed scheme is effective, the USV can not only complete the task, but also reduce the sway during task execution.

Conclusion

This paper investigates the attitude control problem of USVs under conditions of external disturbances, input saturation, and liquid sloshing in fuel tanks. It also addresses the suppression of forces and torques generated by liquid sloshing, and finally verifies the results under Lyapunov theory. We found that the coupled model significantly improves the attitude control accuracy of USVs in complex environments; the nonlinear disturbance observer can accurately estimate external disturbances, enhancing the system’s robustness; the RBFNN effectively suppresses liquid sloshing; and the first-order filter successfully resolves the input saturation problem, improving the system’s anti-saturation capability. We are also aware that there will be many challenges in practical applications. We plan to delve into these challenges and propose corresponding solutions. Specifically, we will consider the following aspects in our subsequent research: computational demands, sensor noise, hardware limitations. In the future, we briefly outlined the potential directions for future research, which aims to reinforce the contributions of the current study and lay the foundation for its subsequent development. The future research directions include: more complex liquid sloshing models; robust control strategies under multiple disturbances; integration with other control issues. These directions will deepen the understanding of the liquid sloshing control problem in USVs and provide valuable references for follow-up research. Another, investigating the recovery capability under potential failure scenarios will be an extension of our future work. This includes developing a fault detection and diagnosis system that can monitor the status of sensors and actuators in real time and promptly issue alerts when failures occur; and researching and designing fault-tolerant control strategies to ensure the system can still operate stably and complete tasks in the event of sensor degradation or actuator failure. Explicitly stated is that future research will consider testing on actual USV hardware. Cooperation with laboratories that have USV hardware will be sought, or relevant hardware resources will be applied for if possible. The algorithm will be further optimized in future research to ensure its real-time performance and reliability on actual hardware.