A Control Method for Path Following of AUVs Considering Multiple Factors Under Ocean Currents

Abstract

To improve the path-following performance of autonomous underwater vehicles (AUVs) under ocean currents, a control method based on line-of-sight with fuzzy controller (FLOS) guidance and the fuzzy sliding mode controller (FSMC) is proposed. This method considers multiple factors affecting guidance and adaptively determines the optimal heading angle through the fuzzy controller to enhance guidance capability. Additionally, a novel FSMC based on Lyapunov stability theory is designed to suppress the influence of model uncertainty and external disturbances on the control system. Simulations and experiments of the proposed control method demonstrate that it can maintain precise tracking under disturbances, improving path-following performance metrics by more than 15%.

1. Introduction

The ocean covers about 71% of the Earth’s surface area and is rich in marine mineral resources, biological resources and dynamic resources. As an important tool of exploiting marine resources, the design, manufacture and control technology of AUVs have been widely studied [1]. Path following is a key technology in AUV motion control, applicable to tasks such as seabed exploration, underwater pipeline maintenance and offshore oil development. Accurate following control can significantly improve the operational efficiency of AUVs while enhancing operational safety. However, the types of paths and the moving speed of the vehicle all affect the guidance effect. And environmental disturbances like waves and ocean currents also cause the AUV to deviate from the desired path, increasing the complexity and difficulty of path following. Tracking errors may lead to higher energy consumption, extended operation times or even mission failure. Achieving high-precision path following in complex and dynamic ocean environments has become a research hotspot in the field of AUV motion control.

The path following of AUVs can be divided into two processes: guidance and following. The LOS method is widely used for guidance because it is simple and easy to implement. LOS guidance laws include path-tangent direction, look-ahead distance and drift angle. The path-tangent direction is determined by the path, and the drift angle can be calculated from the velocity measured by the Doppler velocity log installed on the AUV. Therefore, most studies focus on optimizing LOS using the look-ahead distance.

Initially, the look-ahead distance of LOS guidance is a constant value related to the size of the vehicle, but the effect of path following is unsatisfactory. It is found that the look-ahead distance determines the convergence performance of path following. References [2,3] only studied the influence of the cross-track error on the selection of look-ahead distance. Subsequently, it was found in [4] that the selection of look-ahead distance was affected by both the varying cross-track error and the path curvature. On this basis, references [5,6,7] added the research on forward velocity in the choice of look-ahead distance and considered more comprehensively the factors affecting the choice of look-ahead distance. However, the above studies are all based on mathematical models of exponential form, and the maximum and minimum values of the look-ahead distance must be strictly limited. In fact, there is no clear mathematical relation between the look-ahead distance and the three influencing factors, and there is no uniform standard for selecting the maximum and minimum values of the look-ahead distance. In the mathematical model of the look-ahead distance in [2,3,4,5,6,7], even a small change in one of the influencing factors will result in a large change in the range of the look-ahead distance, which will increase the load on the controller of the guidance angle and not be conducive to the stability of the system.

In addition to enhancing the guidance capabilities of AUVs, some studies have focused on improving controller performance to suppress the impact of model uncertainties and external disturbances on the system, thereby achieving precise following. An active disturbance rejection control (ADRC) based on Q-learning for angle control was proposed in [8]. The extended state observer in ADRC was used to estimate and compensate for the internal and external disturbances, which improved the robustness of the control system. However, the internal disturbances only include part of the hydrodynamic coefficient of the AUV, which cannot fully represent the influence of model uncertainty on the system. In [9], a fuzzy approximator was used to identify unknown hydrodynamic parameters and uncertainty models, but external disturbances were ignored. Therefore, fuzzy algorithms [10,11,12] are usually combined with other algorithms in path-following control. References [13,14] designed an adaptive law based on a fuzzy controller to estimate external disturbances and simultaneously solved the influence of the uncertain model and the external disturbances. Reference [15] combined neural networks with reinforcement learning to establish a new actor–model–critic architecture for path following. According to the trajectory samples of AUVs in different states, the neural network model was trained as a state transition function in reinforcement learning, which could effectively cope with the changes in the AUV and the surrounding environment. However, the cost of obtaining trajectory samples in practical engineering is very high.

Sliding mode control is mainly used in systems with unstable structure and strong disturbance, especially in AUV systems. It can suppress the interference of the uncertain model and external interference, but the jitter problem of the sign function in the reaching law needs to be solved urgently. Reference [16] used a continuous function instead of a sign function to reduce chattering, but the approach speed could not be adjusted according to the system state. Therefore, based on the continuous function, the reference [17] used fuzzy control to adjust the approach parameters in the approach law according to the cross-track error of the path following, so as to select the appropriate approach speed. In [18], adaptive estimates of the upper bound of the interference were added to the gain of the continuous function to adjust the reaching law. On the other hand, the chattering can be suppressed by changing the gain of the sign function. In [19], based on the exponential approach law, the approach speed was improved by adjusting the gain of the sign function online through self-learning of the neurons. In addition, the equivalent SMC was used for AUV control in [20,21], where the sign function in its robustness term was replaced by the output value of the fuzzy controller according to the sliding surface, which was also essentially an adjustment of the approach law. At present, most of the methods of suppressing the chattering of the SMC are achieved by modifying the approach law rather than by designing the control law.

Motivated by the aforementioned observations, a control method based on improved LOS guidance and an FSMC is proposed for path following of AUVs with model uncertainty and external disturbances in the horizontal plane. The kinematic and dynamic models of the AUV are derived for the subsequent design of the guidance method and controller. The influence of cross-track error, path curvature and forward velocity (hereafter referred to as three elements) on the selection of the look-ahead distance of the LOS is deduced. As the relationship between them cannot be expressed by an exact mathematical model, a fuzzy controller is selected to adjust the look-ahead distance in real time according to the path-following state to improve the guidance effect, and then the expected heading angle with drift angle compensation is derived. To simultaneously overcome the effects of model uncertainty and external perturbations on the control system, the FSMC is designed based on Lyapunov stability theory. The parameters of the control law of the FSMC can be adjusted online by the fuzzy controller according to the input error of the controller, which effectively improves the stability of the control system and reduces the error of path following. The main contributions are summarized as follows.

(1)

Compared with the adaptive LOS (ALOS) and ALOS with proportional guidance law (APLOS) methods proposed in [5], the proposed FLOS method does not need to limit the maximum and minimum values of the look-ahead distance, and the amplitude of the look-ahead distance changing with the three factors is small, which makes the AUV converge to the following path smoothly and reduces the error of following.

(2)

Compared with [13,14,15,16,17,18], the proposed FSMC suppresses sliding mode chattering by designing a new control law based on the characteristics of the sliding surface and Lyapunov stability theory, rather than by adjusting the approach law. This method does not require consideration of the upper and lower bounds of uncertain parameters, nor does it introduce chattering issues caused by the sign function. It offers a novel approach for suppressing chattering in the SMC.

The structure of the paper is as follows. In the Section 2, the mechanism of AUVs is modeled. The Section 3 deduces the influence of three elements on the selection of the look-ahead distance, designs the guidance law based on the fuzzy controller and designs the FSMCs of heading angle and forward velocity based on the Lyapunov stability theory. In the Section 4, the proposed control method is simulated, and the experimental verification is carried out in the Section 5. The conclusion of this paper is shown in Section 6.

2. Mechanism Modeling of AUVs

The mechanism modeling of AUVs requires the establishment of two coordinate systems: body and Earth reference frames, as shown in Figure 1. E-ξηζ is the Earth (fixed) reference frame and O-XYZ is the body (moving) reference frame. The transformation relation between the two reference frames is expressed as

$$\dot{\eta }=J\left(\eta \right)v$$

(1)

where η and v can be expressed as

$$\eta ={\left[\begin{array}{cccccc}x& y& z& \varphi & \phi & \psi \end{array}\right]}^{\mathrm{T}}$$

(2)

$$v={\left[\begin{array}{cccccc}u& v& w& p& q& r\end{array}\right]}^{\mathrm{T}}$$

(3)

This paper focuses on path following in the horizontal plane, so the kinematic equations of the three-degrees-of-freedom AUV are expressed as

$$\left\{\begin{array}{l}\dot{x}=u\cos \psi -v\sin \psi \hfill \\ \dot{y}=u\sin \psi +v\cos \psi \hfill \\ \dot{\psi }=r\hfill \end{array}\right.$$

(4)

The dynamic equations of the AUV are described in detail in [22,23]. Ignoring the high-order damping term of the hydrodynamic coefficient, the dynamic equations of the AUV can be expressed as

$$\left\{\begin{array}{l}{\tau }_u=\left(-m+Y_{\dot{v}}\right)vr+\left(X_u+X_{\dot{v}}\dot{v}+X_{u\left|u\right|}\left|u\right|\right)u+\left(m-X_{\dot{u}}\right)\dot{u}\hfill \\ {\tau }_v=\left(m-X_{\dot{u}}\right)ur+\left(Y_v+Y_{v\left|v\right|}\left|v\right|\right)v+\left(m-Y_{\dot{v}}\right)\dot{v}\hfill \\ {\tau }_r=\left(-Y_{\dot{v}}+X_{\dot{u}}\right)uv+\left(N_r+N_{\dot{v}}\dot{v}+N_{r\left|r\right|}\left|r\right|\right)r+\left(I_z-N_{\dot{r}}\right)\dot{r}\hfill \end{array}\right.$$

(5)

where τ_u = T₁ + T₂, T₁ and T₂ represent the sum of the thrusters on the left and right side of the AUV, respectively. Since the AUV has no lateral thrusters, τ_v = 0. According to the right-hand rule, τ_r = (T₁ − T₂) L, L represents the vertical distance between the thruster and the central axis of the AUV. Other symbols and values are obtained from fluid simulation, as shown in

Table 1. Parameter values for AUV.

Symbols	Value
Mass m (kg) and moment of inertia Iz (kg∙m2)	203/6.1
The linear damping terms $X_u$/$Y_v$/$N_r$	51.67/30.92/14.7
The quadratic damping terms $X_{u\left|u\right|}$/$Y_{v\left|v\right|}$/$N_{r\left|r\right|}$	15.46/12/33.71
Hydrodynamic added mass terms $X_{\dot{u}}$/$X_{\dot{v}}$/$Y_{\dot{v}}$/$N_{\dot{v}}$/$N_{\dot{r}}$	−21/−31.42/−182.41/−75.81/−54.79

.

3. The Design of Improved LOS Guidance and FSMC

3.1. The Guidance Law Based on the Fuzzy Controller

Figure 2 shows a principle diagram of the LOS method, which is employed to obtain the desired heading angle for the optimal guidance strategy. In this figure,

$$U=\sqrt{u^2+v^2}$$

(6)

$$\beta =\mathrm{atan}2\left(v,u\right)$$

(7)

In addition, γ_p is the angle between the tangent line of the point (x_k, y_k) on the path and the ξ axis, expressed as

$${\gamma }_{\mathrm{p}}=\mathrm{atan}2\left({y^{\prime }}_k\left(\theta \right),{x^{\prime }}_k\left(\theta \right)\right)$$

(8)

where x′_k(θ) = ∂x_k/∂θ, y′_k(θ) = ∂y_k/∂θ. According to the principle of the LOS method, the point (x_LOS, y_LOS) is the guidance point, and the best guidance scheme is for the AUV to move towards this point. So, χ_d can be expressed as

$${\chi }_{\mathrm{d}}={\gamma }_{\mathrm{p}}-{\psi }_{\mathrm{LOS}}={\gamma }_{\mathrm{p}}-\arctan \left(y_{\mathrm{e}}/\Delta \right)$$

(9)

$${\psi }_{\mathrm{d}}={\chi }_{\mathrm{d}}-\beta$$

(10)

It can be seen from (9) and (10) that ψ_d is related to β, γ_p, y_e and Δ, where β and γ_p can be obtained by (7) and (8). In addition, x_e, y_e are the distances from the AUV’s position (x, y) to the normal and tangent lines of the point (x_k, y_k) on the path, respectively, which can be expressed as

$$\left[\begin{array}{c}{x}_{\mathrm{e}}\\ y_{\mathrm{e}}\end{array}\right]=\left[\begin{array}{cc}\cos {\gamma }_{\mathrm{p}}& \sin {\gamma }_{\mathrm{p}}\\ -\sin {\gamma }_{\mathrm{p}}& \cos {\gamma }_{\mathrm{p}}\end{array}\right]\left[\begin{array}{c}x-x_k\left(\theta \right)\\ y-y_k\left(\theta \right)\end{array}\right]$$

(11)

Therefore, the desired heading angle is directly affected by the look-ahead distance Δ. To select the appropriate look-ahead distance, we must deduce the relation between the three elements and the choice of look-ahead distance respectively according to the principle of the LOS method.

(1)

The cross-track error y_e

According to the principle of the LOS method, the guidance diagram of constant and variable look-ahead distance under variable cross-track error is drawn in Figure 3. It can be seen that with the reduction of the cross-track error (from ① to ③), the constant look-ahead distance makes the AUV tend to cross the path, while the variable look-ahead distance makes it approach the path smoothly. Therefore, increasing the look-ahead distance appropriately is more conducive to accurate guidance under a small cross-track error.

(2)

The path curvature c

The path curvature is an index that describes the degree of curvature of a curve, which can be expressed as

$$c={\left|{x^{\prime }}_k\left(\theta \right){y^{″}}_k\left(\theta \right)-{x^{″}}_k\left(\theta \right){y^{\prime }}_k\left(\theta \right)\right|/\left({\left({x^{\prime }}_k\left(\theta \right)\right)}^2+{\left({y^{\prime }}_k\left(\theta \right)\right)}^2\right)}^{3/2}$$

(12)

where x″_k(θ) = d²x_k(θ)/dθ², y″_k(θ) = d²y_k(θ)/dθ². Figure 4 is a guidance diagram of constant and variable look-ahead distance under variable curvature, based on the LOS principle. In the figure, the cross-track error of ① and ②, ③ and ④ is the same, and the curvature of ① and ④ is larger. When the cross-track error is large, whether the look-ahead distance is variable under variable curvature has little effect on the guidance effect, so the look-ahead distance can be selected to be smaller to improve the approach speed. When the cross-track error is small, the constant look-ahead distance makes the AUV too close to the path at ③ and away from the path at ④. Therefore, as the curvature increases, the look-ahead distance should be reduced accordingly to achieve the best guiding effect.

(3)

The forward velocity u

Figure 5 shows the guidance diagram of the AUV under the LOS guidance principle at low and high velocities. It can be seen that the AUV reaches ② first at high velocity. When the cross-track error is large, the guidance effect of different look-ahead distances at different velocities is similar, so the look-ahead distance can be chosen to be smaller to improve the approach speed. When the cross-track error is small, the larger look-ahead distance can give the AUV more time to adjust the heading at high velocity, and the guidance effect is better. It is worth noting that of the three elements, the cross-track error has the greatest influence on the choice of look-ahead distance.

The fuzzy rule table and membership functions are designed according to the inferences drawn from Figure 3, Figure 4 and Figure 5, as shown in

Table 2. The fuzzy rule table of the look-ahead distance.

Δ			c
			VL	L	M	S	VS
ye = VL	u	VS	VS	VS	VS	RS	S
		S	VS	RS	RS	S	M
		M	VS	RS	S	M	L
		L	RS	S	M	M	L
		VL	S	M	L	L	L
ye = M	u	VS	RS	RS	RS	S	M
		S	RS	S	S	M	L
		M	RS	S	M	L	RL
		L	S	M	L	L	RL
		VL	M	L	RL	RL	RL
ye = VS	u	VS	S	S	S	M	L
		S	S	M	M	L	RL
		M	S	M	L	RL	VL
		L	M	L	RL	RL	VL
		VL	L	RL	VL	VL	VL

and Figure 6. Given that the cross-track error exerts the greatest influence, it is necessary to consider the cross-track error first and determine the average level of the fuzzy set of the look-ahead distance according to the fuzzy set of the cross-track error when designing fuzzy rules, as shown by the diagonal of the fuzzy rule. The fuzzy domain of look-ahead distance is [2, 6], divided into very small (VS), relatively small (RS), small (S), medium (M), large (L), relatively large (RL) and very large (VL). The fuzzy domain of the cross-track error is [0, 2], divided into VS, M and VL. The fuzzy domains of path curvature and forward velocity are [0, 1] and [0, 2], divided into VS, S, M, L and VL. The input and output variables of the fuzzy controller adopt triangular and trapezoidal membership functions.

The previous paragraph is the step of setting up the fuzzy controller with the input variables y_e, c, u and the output variable Δ, which is used to calculate the expected heading angle. The flowchart for obtaining the expected heading angle based on the FLOS guidance method proposed in Section 3.1 is illustrated in Figure 7.

The stability of the proposed control system is analyzed by Lyapunov stability theory, and the Lyapunov function is defined as

$$V={\displaystyle \frac{1}{2}}{y}_{\mathrm{e}}^2$$

(13)

According to (11), the time derivative of V is

$$\begin{array}{l}\dot{V}=y_{\mathrm{e}}{\dot{y}}_{\mathrm{e}}=y_{\mathrm{e}}\left[-\left(\dot{x}-{\dot{x}}_k\left(\theta \right)\right)\sin {\gamma }_{\mathrm{p}}+\left(\dot{y}-{\dot{y}}_k\left(\theta \right)\right)\cos {\gamma }_{\mathrm{p}}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-y_{\mathrm{e}}\underset{x_{\mathrm{e}}}{\underbrace{\left[\left(x-x_k\left(\theta \right)\right)\cos {\gamma }_{\mathrm{p}}+\left(y-y_k\left(\theta \right)\right)\sin {\gamma }_{\mathrm{p}}\right]}}{\dot{\gamma }}_{\mathrm{p}}\end{array}$$

(14)

The projection of (x, y) onto the path is employed as (x_k, y_k), so x_e = 0 [5]. According to (6)–(10), (14) can be simplified as

$$\begin{array}{l}\dot{V}=y_{\mathrm{e}}\left(-\dot{x}\sin {\gamma }_{\mathrm{p}}+\dot{y}\cos {\gamma }_{\mathrm{p}}\right)=y_{\mathrm{e}}U\sin \left(\psi -{\gamma }_{\mathrm{p}}+\beta \right)\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-{\displaystyle \frac{y_{\mathrm{e}}^{2}U}{\sqrt{y_{\mathrm{e}}^2+{\mathsf{\Delta }}^2}}}\le 0\end{array}$$

(15)

Therefore, the proposed guidance system has stability.

3.2. The FSMC Based on Lyapunov Stability Theory

To meet control requirements of AUVs, the FSMCs for forward velocity and heading angle are designed in this section. Taking the velocity controller as an example, let

$$\left\{\begin{array}{l}{g}_1=u_{\mathrm{d}}-u=e\hfill \\ g_2=-\dot{u}\hfill \\ h_{\mathrm{out}}={\dot{\tau }}_u\hfill \end{array}\right.$$

(16)

Taking the derivative of both sides of the equation for the surge force in (5) and considering the ocean current disturbance, we obtain

$${\dot{\tau }}_u=\left(30.92-31.42\ddot{v}\right)u+\left(51.67-31.42\dot{v}\right)\dot{u}+224\ddot{u}-{\displaystyle \frac{\mathrm{d}\left(20.59vr+W\right)}{\mathrm{d}t}}$$

(17)

According to (17), the state equation can be expressed as:

$$\left\{\begin{array}{l}{\dot{g}}_1=g_2\hfill \\ {\dot{g}}_2=a_1{g}_1+a_2{g}_2+b{h}_{\mathrm{out}}+f\hfill \end{array}\right.$$

(18)

$$\begin{array}{l}{a}_1=\left(31.42\ddot{v}-30.92\right)/224\hfill \\ a_2=\left(31.42\dot{v}-51.67\right)/224\hfill \\ b=-1/224\hfill \\ f=\left[\left(30.92-31.42\ddot{v}\right)u_{\mathrm{d}}-{\displaystyle \frac{\mathrm{d}\left(20.59vr+W\right)}{\mathrm{d}t}}\right]/224\hfill \end{array}$$

where a₁ and a₂ are time-varying parameters, f can be considered as the internal and external disturbances of the system, W is the disturbance from waves or ocean currents, e is the error between the expected velocity and the actual velocity. The switching function of the SMC is designed as

$$\begin{array}{ll}s=\mathrm{d}{g}_1+g_2\hfill & \mathrm{d}>0\hfill \end{array}$$

(19)

where d is a constant. The control law is designed as

$$h_{\mathrm{out}}\left(t\right)={\mathrm{\Gamma }}_1{g}_1+{\mathrm{\Gamma }}_2{g}_2+\lambda$$

(20)

$${\Gamma }_1=\left\{\begin{array}{ll}{\alpha }_1,\hfill & g_{1}s>0\hfill \\ {\beta }_1,\hfill & g_{1}s<0\hfill \end{array}\right.\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\Gamma }_2=\left\{\begin{array}{ll}{\alpha }_2,\hfill & g_{2}s>0\hfill \\ {\beta }_2,\hfill & g_{2}s<0\hfill \end{array}\right.\hspace{0.17em}$$

(21)

where Γ₁, Γ_2, α₁, α₂, β₁, β₂ and λ are the parameters of the SMC. To verify the stability of the proposed control system, the Lyapunov function is obtained as

$$V={\displaystyle \frac{1}{2}}{s}^2$$

(22)

To ensure that the sliding mode states g₁ and g₂ converge to the sliding surface, the derivatives of the Lyapunov function must be satisfied

$$\dot{V}=s\dot{s}=s\left(\mathrm{d}{\dot{g}}_1+{\dot{g}}_2\right)<0$$

(23)

Substituting (18) and (20) into (23), we can obtain

$$s\left[\left(a_1+b{\mathrm{\Gamma }}_1\right)g_1+\left(\mathrm{d}+a_2+b{\mathrm{\Gamma }}_2\right)g_2+f+b\lambda \right]<0$$

(24)

Equation (24) is a multivariate inequality, which is difficult to solve. Therefore, each term on the left-hand side of the inequality is

$$\left\{\begin{array}{l}\left(a_1+b{\mathrm{\Gamma }}_1\right)g_{1}s<0\hfill \\ \left(\mathrm{d}+a_2+b{\mathrm{\Gamma }}_2\right)g_{2}s<0\hfill \\ \left(f+b\lambda \right)s<0\hfill \end{array}\right.$$

(25)

where Γ₁ and Γ₂ are adjusted by the fuzzy controller according to the control state of the velocity. The input variables of the fuzzy controller are the velocity error e and the rate of change of the error ec, and the output variables are α₁, α₂, β₁ and β₂. The AUV lacks lateral thrusters and the velocity v and its derivative are both minimal. Consequently, the fuzzy domains of e and ec are [−6, 6], divided into negative big (NB), negative medium (NM), negative small (NS), positive small (PS), positive medium (PM) and positive big (PB). The fuzzy domain of α₁ and α₂ is [1, 6], divided into PS, PM and PB. The fuzzy domain of β₁ and β₂ is [−6, −1], divided into NB, NM and NS. The membership functions of the input and output variables are designed as triangular membership functions, as illustrated in Figure 8.

Table 3. The fuzzy rule of α₁ and α₂.

α1/α2		ec
		NS	NM	NB	PS	PM	PB
e	PS	PS	PS	PS	PB	PM	PS
	PM	PS	PS	PS	PM	PM	PM
	PB	PM	PM	PM	PB	PM	PB
	NS	PB	PM	PM	PS	PS	PS
	NM	PB	PM	PS	PS	PS	PS
	NB	PS	PM	PB	PM	PM	PM

and

Table 4. The fuzzy rule of β₁ and β₂.

β1/β2		ec
		NS	NM	NB	PS	PM	PB
e	PS	NS	NS	NS	NS	NM	NM
	PM	NS	NS	NS	NM	NM	NB
	PB	NM	NM	NM	NB	NB	NB
	NS	NS	NM	NM	NS	NS	NS
	NM	NB	NM	NS	NS	NS	NS
	NB	NS	NM	NB	NM	NM	NM

present the fuzzy rules for α₁/α₂ and β₁/β₂, which were obtained based on empirical tuning experience. The above outlines the design step of the velocity controller. The controller of heading angle is analogous to that of the velocity, with the exception of the expression of the equation of state, the switching function and the control law. The equation of state is expressed as

$$\left\{\begin{array}{l}{\dot{g}}_1=g_2\hfill \\ {\dot{g}}_2=g_3\hfill \\ {\dot{g}}_3=a_1{\dot{g}}_1+a_2{\dot{g}}_2+b{h}_{\mathrm{out}}+f\hfill \end{array}\right.$$

(26)

where $g_1={\psi }_{\mathrm{d}}-\psi ,g_2=-r,g_3=-\dot{r},g_4=-\ddot{r},h_{\mathrm{out}}={\dot{\tau }}_r$.

The switching function is expressed as

$$s={\mathrm{d}}_1{g}_1+{\mathrm{d}}_2{g}_2+g_3 {\mathrm{d}}_1>0, {\mathrm{d}}_2>0$$

(27)

The control law of the SMC is expressed as

$$h_{\mathrm{out}}\left(t\right)={\mathrm{\Gamma }}_1{\dot{g}}_1+{\mathrm{\Gamma }}_2{\dot{g}}_2+{\lambda }_1$$

(28)

$${\mathrm{\Gamma }}_1=\left\{\begin{array}{ll}{\alpha }_1,\hfill & {\dot{g}}_{1}s>0\hfill \\ {\beta }_1,\hfill & {\dot{g}}_{1}s<0\hfill \end{array}\right.\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{\Gamma }}_2=\left\{\begin{array}{ll}{\alpha }_2,\hfill & {\dot{g}}_{2}s>0\hfill \\ {\beta }_2,\hfill & {\dot{g}}_{2}s<0\hfill \end{array}\right.\hspace{0.17em}$$

(29)

It is possible for the parameters of the control law of velocity and heading angle to share the same fuzzy rule and membership function. However, it is necessary to adjust the scale factors between the basic and fuzzy domains in practice. Figure 9 presents a schematic diagram of the AUV path-following control system.

4. Simulation Analysis of Proposed Control Method

To verify the FLOS guidance performance and the anti-interference ability of the FSMC in the proposed control method, this paper uses MATLAB R2018b/Simulink software to simulate the path following of the AUV. The simulation is divided into four cases based on the path types: straight-line, curved, polyline and sinusoidal paths. The expressions for the four types of paths and the initial state settings of the AUV are shown in

Table 5. Settings for Four Types of Path in AUV Path-Following Simulation.

Types of Following Paths	Expressions of Following Paths	Initial Conditions of the AUV
Straight-line path	y = x − 2, x∈[2, 9]	[x y ψ] = [0 0 0], [u v r] = [0 0 0], ud = 0.25 m/s
Curved path	x = 50cos θ, y = 50 + 25sin θ	[x y ψ] = [−50 50 0], [u v r] = [0 0 0], ud = 0.5 m/s
Polyline path	y = 1/2x, x∈[0, 180]; y = −2x + 450, x∈[180, 225]	[x y ψ] = [30 0 0], [u v r] = [0 0 0], ud = 1 m/s
Sinusoidal path	x = θ, y = 20 + 5sin (0.02πθ)	[x y ψ] = [0 20 0], [u v r] = [0 0 0], ud = 0.5 m/s

.

4.1. Comparison of Guidance Performance Under Still Water

To compare the guidance performance of the traditional LOS, ALOS and APLOS proposed in [5] and FLOS proposed in this paper, the motion scene of the AUV is selected as a still water environment. The control systems of the four methods are identical, with the exception of the guidance method. The forward velocity and heading angle are regulated by proportional–integral–differential (PID) controllers. The parameters of the velocity controller are set as follows: k_P = 500, k_I = 50, k_D = 0. For the heading angle controller, the parameters for straight-line and polyline paths are configured as k_P = 500, k_I = 12, k_D = 80, while the parameters for the heading angle controller for curved and sinusoidal paths are set to k_P = 30, k_I = 5, k_D = 20. Among the four guidance methods, the guidance law of the traditional LOS method is expressed as

$${\psi }_{\mathrm{d}}={\gamma }_{\mathrm{p}}+\arctan \left(-{\displaystyle \frac{y_{\mathrm{e}}}{\mathsf{\Delta }}}\right)$$

(30)

In the aforementioned formula, the look-ahead distance is a constant. The guidance law of the ALOS method is analogous to that of the traditional LOS method, but the look-ahead distance of the ALOS method is expressed as

$$\left\{\begin{array}{l}{{\mathsf{\Delta }}^{\prime }}_{\min /\max }={\mathsf{\Delta }}_{\min /\max }-{\mathsf{\Delta }}_{\min /\max }\mathrm{sat}({\displaystyle \frac{c}{c_{\max }}})\hfill \\ \mathsf{\Delta }={{\mathsf{\Delta }}^{\prime }}_{\min }+\left({{\mathsf{\Delta }}^{\prime }}_{\max }-{{\mathsf{\Delta }}^{\prime }}_{\min }\right)e^{-{\displaystyle \frac{y_{\mathrm{e}}^2}{u}}}\hfill \end{array}\right.$$

(31)

where Δ_min/max represents the maximum or minimum value of Δ. On the basis of (31), the guidance law of the APLOS method is expressed as

$${\psi }_{\mathrm{d}}={\gamma }_{\mathrm{p}}+\arctan \left(-{\displaystyle \frac{y_{\mathrm{e}}}{\mathsf{\Delta }}}-\beta \right)$$

(32)

According to the size and performance of the AUV, the look-ahead distance of LOS is set to 7 m, and the minimum and maximum look-ahead distances of ALOS and APLOS are set to 4 m and 7 m, respectively. In the following sections, simulations, comparisons and analyses of the four guidance methods are conducted for each of the four cases.

• Case 1: The simulation of straight-line path following for an AUV in still water

Figure 10 illustrates the performance of the AUV in straight-line path following, including the path followed in Figure 10a, the cross-track error curve in Figure 10b and the look-ahead distance curve in Figure 10c. The followed path comprises the desired path and the actual motion trajectories of the AUV in the Earth reference frame under the four guidance methods. It is evident that the LOS method approaches the desired path at the slowest rate, resulting in the largest cross-track error. The guidance performance of the ALOS and APLOS methods surpasses that of the LOS method, as their time-varying look-ahead distances are smaller than that of the LOS method, leading to reduced cross-track errors. In contrast, the look-ahead distance of the FLOS method is adjusted in real time by a fuzzy controller, maintaining a smaller value, which enables the AUV to rapidly approach the desired path and facilitates the quick convergence of the cross-track error to zero, demonstrating superior guidance performance.

• Case 2: The simulation of curved path following for an AUV in still water

Figure 11 presents a comparison of the curved-path-following performance of the four methods, with the path followed shown in Figure 11a, the cross-track error in Figure 11b and the look-ahead distance in Figure 11c. It can be observed that the following effect of the LOS method on the curved path is not as good as that of the straight-line path. The constant look-ahead distance allows the AUV to shuttle on both sides of the curve. The following performance of ALOS and APLOS is superior to that of LOS, but the cross-track error changes frequently at positions with large curvature. This phenomenon occurs due to the look-ahead distance reducing to zero in these high-curvature areas for both methods. The FLOS method exhibits the best following effect, with minimal fluctuations in cross-track error and look-ahead distance. Furthermore, the cross-track error of the other three methods except the LOS method is kept within 2 m.

• Case 3: The simulation of polyline path following for an AUV in still water

Figure 12 illustrates the path-following performance of the AUV using four guidance methods for a polyline path. The tracking performance is presented in terms of the following path, cross-track error and look-ahead distance. It can be observed that when the AUV approaches the path from the initial position and the turn, the LOS adopts a constant look-ahead distance to make it approach slowly. The adaptive look-ahead distance allows ALOS and APLOS to approach at a faster rate than LOS, and the following effect is similar for both. The FLOS method is the fastest to approach and stabilize on the path, demonstrating good guidance capability. From the curve of the cross-track error and look-ahead distance, it can be seen that the cross-track error of the start and turn is the largest. At this point, the look-ahead distance of the other three methods is relatively small except that the look-ahead distance of LOS is unchanged, which is consistent with the inference about the selection of look-ahead distance in the third section. In addition, the cross-track error is kept within 2 m when the AUV follows on the path. It is important to note that the look-ahead distances of ALOS and APLOS vary more than that of FLOS during the following process. This is due to the fact that both methods are based on exponential functions and adjust the look-ahead distance according to the maximum and minimum values of the look-ahead distance.

• Case 4: The simulation of sinusoidal path following for an AUV in still water

Figure 13 illustrates the AUV’s tracking capability on a sinusoidal path using different guidance methods. Figure 13a presents the comparison between the actual trajectory of the AUV and the desired path, while Figure 13b,c show the variations in cross-track error and look-ahead distance, respectively. It is evident that the guidance performance of FLOS remains the best among the four guidance methods on the sinusoidal path, exhibiting the smallest cross-track error, which remains within 1 m, and the least fluctuation in look-ahead distance. In contrast, under the guidance of the other three methods, the AUV oscillates on both sides of the desired path, resulting in a significant cross-track error during the tracking process. Furthermore, consistent with the results from Case 2, both ALOS and APLOS exhibit the poorest tracking performance in areas of high curvature, where the look-ahead distance sharply decreases to zero.

Drawing upon the simulation results from the four cases, the curves of the actual trajectories and desired paths presented in Figure 10a, Figure 11a, Figure 12a and Figure 13a are employed to evaluate the path-following performance of the four guidance methods. This assessment is conducted using the mean absolute error (MAE) and root mean square error (RMSE). The formulas for calculating the MAE and RMSE are as follows:

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|e_i\right|}$$

(33)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{e}_i^2}}$$

(34)

where N represents the total number of sampling points and e_i denotes the Euclidean distance between the actual position at the i-th time step and the corresponding target position on the desired path.

Table 6. Path-following performance metrics of four methods under four cases in still water.

	Case 1		Case 2		Case 3		Case 4
	MAE (m)	RMSE (m)	MAE (m)	RMSE (m)	MAE (m)	RMSE (m)	MAE (m)	RMSE (m)
LOS	0.3279	0.5941	1.1505	1.4554	2.5226	6.0902	0.4916	0.5413
ALOS	0.2993	0.5738	0.5269	0.7375	2.3819	5.9587	0.1817	0.2426
APLOS	0.2845	0.5557	0.5122	0.7258	2.2571	5.9352	0.1756	0.2368
FLOS	0.1905	0.3745	0.2953	0.4068	1.5498	4.0943	0.0893	0.1236
Improvement	33.19%	32.58%	42.33%	43.94%	31.33%	31.05%	49.15%	47.80%

presents the path-following performance metrics of four methods under four different cases in still water. It can be observed that among the four guidance methods under still water, the MAE and RMSE values of FLOS are the lowest, indicating that its followed path is closest to the desired path and that the following error is the most concentrated, and the following performance is the best, while LOS is the opposite. The path-following performance metrics for ALOS and APLOS methods are similar, falling between those of the LOS method and the FLOS method. This indicates that both methods outperform the LOS method in terms of tracking performance, yet their tracking accuracy is inferior to that of the FLOS method. A comparison of the performance metrics of the FLOS and APLOS methods reveals that the FLOS guidance method can effectively improve path-following performance by 30% in four different following paths.

4.2. Comparison of Controller Performance Under Model Uncertainty and External Disturbances

To verify the anti-perturbation capability of the FSMC, this section employs a simulation of the path following of the AUV under the conditions of model uncertainty and external disturbances. The selected reference paths include representative polyline and curved paths, which correspond to linear and varying-curvature path following, respectively. Specifically, the simulation of model uncertainty is achieved by modifying the coefficients in the AUV’s dynamic equations. The resulting modified dynamic equations are as follows:

$$\left\{\begin{array}{l}{\tau }_u=3\left(-m+Y_{\dot{v}}\right)vr+\left(X_u+X_{\dot{v}}\dot{v}+X_{u\left|u\right|}\left|u\right|\right)u+\left(m-X_{\dot{u}}\right)\dot{u}\hfill \\ {\tau }_v=\left(m-X_{\dot{u}}\right)ur+\left(Y_v+Y_{v\left|v\right|}\left|v\right|\right)v+\left(m-Y_{\dot{v}}\right)\dot{v}\hfill \\ {\tau }_r=0.5\left(-Y_{\dot{v}}+X_{\dot{u}}\right)uv+\left(N_r+N_{\dot{v}}\dot{v}+N_{r\left|r\right|}\left|r\right|\right)r+\left(I_z-N_{\dot{r}}\right)\dot{r}\hfill \end{array}\right.$$

(35)

Compared to the original dynamic equations in (5), the coefficient of term $\left(-m+Y_{\dot{v}}\right)vr$ in the equation for τ_u has been tripled, while the coefficient of term $\left(-Y_{\dot{v}}+X_{\dot{u}}\right)uv$ in the equation for τ_r has been halved. The simulation of external disturbances is conducted by augmenting the ocean currents along the positive direction of the η axis by 0.1 m/s. Furthermore, the FLOS method with a PID controller, SMC and FSMC is compared with the APLOS method with a PID controller, and the path-following effects of the four methods are observed. The parameters of the velocity PID controller are k_P = 500, k_I = 90, k_D = 0, the parameters of the heading-angle PID controller for polyline path following are k_P = 100, k_I = 10, k_D = 30 and for curved path following they are k_P = 50, k_I = 20, k_D = 120. The SMC of the velocity uses the exponential reaching law, then the control law is expressed as

$$h_{\mathrm{out}}={\displaystyle \frac{1}{b}}\left[-a_1{g}_1-\left({\mathrm{d}}_1+a_2\right)g_2-f-\epsilon \mathrm{sgn}(s)-ks\right]$$

(36)

The SMC control law of heading angle is analogous to that of velocity, and the stability proofs are omitted here. The parameters of the SMC are d₁ = 4.5, d₂ = 1.5, ε = 8, k = 3. The parameters of the FSMC are d₁ = 0.2, d₂ = 0.5.

• Case 1: The simulation of polyline path following for an AUV under model uncertainty and external disturbances

Figure 14 illustrates a comparative analysis of the polyline path following for the AUV in four states under disturbance conditions, including: (a) path followed, (b) cross-track error, (c) look-ahead distance, (d) heading angle and (e) forward velocity. It can be observed that the APLOS method approaches the path the slowest at the start and overshoots the path too much at the turn. The FLOS method, also with a PID controller, approaches the path slightly faster at the start and turns better than the APLOS method. Due to the influence of disturbances, the cross-track error of the above two methods is within 1.1 m when the AUV follows on the path. The FLOS methods with the SMC and FSMC tend to follow the path the fastest and have the best following effect at the turn, and the cross-error is reduced to 0.7 m on the path. As can be seen from the curve of the look-ahead distance, the look-ahead distance of the APLOS method still varies greatly, whereas the look-ahead distances of the FLOS method are similar under different controllers, and all change smoothly and slowly. From the curve of the heading angle and forward velocity, it can be observed that the tracking effects of heading angle under PID controllers are not optimal, and the velocity adjustments take a considerable amount of time, approximately 75 s. The SMC and FSMC demonstrate superior tracking accuracy for heading angle and velocity, and the velocity can be stabilized at the desired level in about 25 s. Nevertheless, the inherent chattering problem of the SMC results in a deterioration of the static indicators of the curves of heading angle and velocity, which in turn causes damage to the system hardware and increases the energy consumption. The FSMC effectively addresses the chattering problem, and the control effects are optimal, which avoids the influence of model uncertainty and external disturbances on the control system.

• Case 2: The simulation of curved path following for an AUV under model uncertainty and external disturbances

Figure 15 presents a comparative analysis of the curved path following for the AUV in four states under disturbance conditions, with the tracking performance represented by the following path, the trajectory error, the look-ahead distance, the heading angle and the forward velocity. It is obvious that the APLOS method has the worst following effect, which is primarily attributable to the guidance method. The following state of the FLOS method with a PID controller is unstable due to interference at the beginning. In contrast, the FLOS method with the SMC and FSMC can restrain the influence of disturbance on the control system and follow the desired path stably. From the curves of the heading angle and forward velocity, it can be observed that the control curves of the FLOS method with the SMC and FSMC first converge to the expected curve, but the chattering of the SMC still exists. A comparison of the four methods reveals that the FLOS method with the FSMC has the best path-following capability, which proves that both an efficient guidance method and a high-performance controller are essential.

Using the actual trajectories and expected paths depicted in Figure 14a and Figure 15a, the path-following performance of the AUV under disturbances is assessed using MAE and RMSE. The performance metrics for the AUV in four states are summarized in

Table 7. Path-following performance metrics of the AUV in four states under disturbance.

	Case 1		Case 2
	MAE (m)	RMSE (m)	MAE (m)	RMSE (m)
APLOS + PID	2.6297	7.0756	0.7684	1.0131
FLOS + PID	1.9122	4.9004	0.4872	0.6386
FLOS + SMC	0.9802	2.3645	0.3895	0.5168
FLOS + FSMC	0.9388	2.2389	0.3625	0.4673
Improvement	50.94%	54.28%	25.60%	26.85%

. The MAE and RMSE values of the FLOS method with a PID controller are lower than those of the APLOS method, indicating that the FLOS method has a superior guidance effect. The MAE and RMSE values of the FLOS method with the SMC and FSMC are significantly smaller than those of the two methods under the PID controller, indicating that the above method is capable of mitigating the impact of model uncertainty and external disturbances on the control system. The values of the two indexes of the method with the FSMC are lower than those of the method with the SMC, which indicates that the proposed FSMC effectively solves the chattering problem. A comparison of the performance metrics of the FLOS method with a PID controller and FSMC under disturbances reveals that the FSMC can effectively enhance path-following performance by 25%, demonstrating its superior disturbance rejection capability.

5. Experimental Verification of Proposed Method

To verify the feasibility of the proposed improved LOS guidance and FSMC, an underwater path-following experiment was carried out on an AUV with a length of 1.7 m and a mass of 203 kg. The control system adopts the EMB3500 industrial controller based on a Linux system and STM32 single-chip microcomputer as the controller of the driving mechanism, and the sampling period is 1 s. Figure 16 shows the AUV in water trials. The AUV actually operates in an environment with external disturbances due to surges generated by other vehicles operating in the water. Additionally, to simulate the influence of model uncertainty on AUV motion, a cylindrical float is mounted on the left at the front of the body. In the experiment, APLOS with better performance in simulation was used for comparison with the proposed method. The straight-line path followed by the AUV is {y = x − 2, x∈[2, 9]}. The initial states are as follows: [x y ψ] = [0 0 0], [u v r] = [0 0 0], and u_d = 1 m/s. The curved path is expressed as {x = 5cos θ, y = 7 + 7sin θ}. The initial states are as follows: [x y ψ] = [0 0 0.4], [u v r] = [0 0 0], and u_d = 0.2 m/s.

Figure 17 and Figure 18 respectively depict the experimental results of straight-line and curved path following for the AUV under different states with disturbances, including the path followed, heading angle, look-ahead distance and forward velocity. As illustrated in the figures, the FLOS method with a PID controller exhibits superior following performance in comparison to APLOS. However, both methods exhibit suboptimal tracking accuracy for heading angle and velocity due to interference. The path following of the FLOS method with the FSMC is the most effective, benefiting from the superior FLOS guidance and the anti-interference capability of the FSMC. This method permits the smooth adjustment of the look-ahead distance and the rapid stabilization of the heading angle and velocity to the desired curve.

The experimental trajectories and desired paths under disturbances, as shown in Figure 17a and Figure 18a, were used to evaluate the AUV’s path-following performance. Metrics such as MAE and RMSE were applied for this assessment. The performance metrics for the AUV under different control strategies are presented in

Table 8. Performance Metrics of Path-Following Experiments with Disturbances.

	Figure 15		Figure 16
	MAE (m)	RMSE (m)	MAE (m)	RMSE (m)
APLOS + PID	0.9009	1.2383	0.1255	0.1655
FLOS + PID	0.8249	1.1389	0.1139	0.1516
FLOS + FSMC	0.7504	1.0412	0.1005	0.1363
Guidance improvement	8.43%	8.03%	9.25%	8.39%
Controller improvement	9.02%	8.59%	11.77%	10.11%
Overall improvement	16.59%	15.91%	19.94%	17.66%

. It can be seen that the MAE and RMSE values of the FLOS method with the FSMC are smaller than those of other methods, indicating that the following error is minimized. A comparative analysis of the path-following experimental results of three methods reveals that the FLOS guidance method can achieve a maximum improvement of 9.25% in tracking performance. Additionally, the disturbance rejection capability of the FSMC can enhance tracking performance by up to 11.77%. Overall, the proposed control method results in a total improvement of over 15% in tracking performance.

6. Conclusions

This paper proposes a FLOS method with an FSMC to improve the guidance capability and suppress the influence of model uncertainty and external disturbances on path following. The FLOS method was compared with the proposed APLOS from the literature [5] in still water through simulation. The results indicate that the tracking performance metrics of the FLOS guidance method improved by over 30% compared to the APLOS method across straight-line, curved, polyline and sinusoidal paths, demonstrating its ability to rapidly and accurately approach the desired path. The paper also conducted path-following simulations for an AUV based on the FLOS guidance method under model uncertainties and current disturbances using three types of controller: PID, SMC and FSMC. The results indicate that the FSMC can effectively solve the inherent chattering of the SMC, prevent the influence of disturbances on the control system and reduce the error of path following. Compared to the PID controller, the tracking performance using the FSMC method can be improved by 25%. In addition, experiments on path following of an AUV under disturbances were conducted to validate the feasibility and effectiveness of the proposed control method, resulting in an overall improvement of 15% in path-following performance.

It is important to note that the proposed control method is intended for use on AUVs equipped with speed sensors and is specifically limited to path following in the horizontal plane. This work will be further extended to the three-dimensional path following of AUVs in the future, with applications in the maintenance of underwater pipelines and the exploration of marine resources, highlighting its significant economic value.