Dual-Loop Integral Sliding Mode Control-Based Path Tracking with Reaction Torque for Autonomous Underwater Vehicle

Abstract

Path tracking control is an important method for a Six-Degree-Of-Freedom Autonomous Underwater Vehicle to perform specific underwater tasks. Therefore, this paper investigates a dual-loop integral sliding mode control (DLISMC)-based tracking controller for an AUV with model uncertainties and external disturbances, and introduces a new reaction torque model for static compensation in order to improve the attitude control capability for AUVs when performing path tracking. In addition, the stability of tracking control law based on DLISMC is demonstrated using the Lyapunov function. Finally, numerical simulations are carried out on MATLAB 2016/Simulink and compared with the Proportion–Integral–Differential (PID) control commonly used in industry as well as the dual-loop Proportion–Integral–Differential (DLPID). Simulation results show that the DLISMC has a smaller tracking error, faster convergence speed, and more robustness against external disturbances and reaction torque.

1. Introduction

As the second largest strategic space after land, the ocean has rich biological resources, energy, mineral resources, and so on, which means it is the most realistic and potential space for development [1]. With the continuous development of science and technology, underwater robots have become one of most important tools for human exploration of the ocean. Compared with other types of underwater vehicle platforms, Autonomous Underwater Vehicles (AUVs) have the advantages of high autonomy and large detection range. They are often used to perform some difficult and complex tasks, such as ocean observation, seabed analysis, bathymetry, submarine navigation, recovery equipment, military applications, etc. [2,3,4]. In order to perform the above tasks successfully, it is necessary to design a reasonable and accurate controller for an AUV to achieve trajectory tracking, i.e., control the AUV to reach and follow the time-varying parametric trajectory. However, the trajectory tracking control of the AUV poses a great challenge to controller designers, because the dynamic model of the AUV has fixed multivariable, high-strength nonlinearity and strong coupling and uncertainty [5]. At the same time, due to the complex marine environment, the AUV is subject to time-varying external interference that is difficult to measure or estimate while moving, which also makes the trajectory tracking control of AUVs a challenging task.

Regarding the motion control of autonomous underwater vehicles, many scholars assume that the roll angle effect can be ignored. The advantage of ignoring the roll angle is that the complex 6-DOF dynamic model of an AUV can be simplified to a 5-DOF dynamic model [6,7]. This assumption greatly reduces the coupling of model and the difficulty of controller design. However, in practice, the roll angle has a significant impact on the path tracking of AUVs. Especially when the AUV is engaged in highly difficult and complex movements, the roll angle will be particularly prominent. A large roll angle error will make the AUV appear yaw. Therefore, the optimal control of a 6-DOF AUV’s attitude and position is very necessary.

Nowadays, there has been much research on AUV control methods. Classical control methods include PID control [8,9], backstepping control [10,11], etc. Traditional control methods comprise sliding mode control (SMC) [12,13], adaptive control [14,15], etc. Intelligent control methods comprise fuzzy control [16,17], neural network control [18,19], etc. See

Table 1. SMC-related works.

Directions for Improvement	Degrees of Freedom	References
Sliding mode control	3/5/6	[13,20,21,22,23,24,25]
Chattering effects	3/5/6	[16,17,20,21]
Integration with other methods	3/5/6	[12,13,16,17,21,23,26,27]

.

Sliding mode control is characterized by strong robustness to parameter variations, model uncertainties, and good resistance to external disturbances, so it is very suitable for trajectory tracking control of AUVs under complex environments. At present, many scholars have improved the sliding mode control. Qiao et al. [22] proposed an adaptive fast non-singular ITSMC (AFNITSMC) scheme. If far from the balance point, it will provide a faster convergence rate for tracking error than linear sliding mode (LSM), so it is faster than NITSM [23]. If close to the balance point, it will provide a similar convergence rate for tracking error to NITSM, which improves the convergence rate of the AUV tracking error. Liu et al. [24] introduced the Gaussian function into the integral sliding mode control to reduce the overreaction of the integral term caused by strong disturbances, eliminating the system uncertainty and the steady-state error caused by external disturbances. Zhang et al. [20] addressed the three-dimensional path tracking problem of an underactuated UUV in an unknown external disturbance environment by employing a nonlinear disturbance observer to estimate the unknown external disturbance, so as to simplify the UUV dynamics model. Li [21] addressed the uncertainty and mismatch disturbances in the path-tracking control scheme for underwater robots and the jitter phenomenon of SMC. A robust control scheme combining SMC, fuzzy logic control, neural network, and a disturbance observer is proposed. Where fuzzy logic control and SMC are used to control underwater robots, a neural network and disturbance observer are used to estimate the mismatch disturbance and uncertainty. Zheng [27] investigated the time-varying perturbation trajectory tracking problem based on robust adaptive sliding mode control (RASMC). The singularity problem is solved by designing smoother functions to ensure the continuity of SMC and its derivatives. Moreover, the robust adaptive law is proposed to ensure a good robust performance for a complex underwater environment, and the AUV’s mathematical model is coupled with strong nonlinearity and high uncertainty. A dynamic surface sliding mode controller for trajectory tracking is designed by Li et al. [28]. Aiming at the problem of more control parameters and the poor effect of manual empirical parameter rectification, an Improved Particle Swarm Algorithm (IPSO) is used to optimize the main control parameters and compensate the environmental disturbances with a nonlinear disturbance observer. Li [29] presented a finite time fractional order sliding mode controller (FTFOSMC) for 3D trajectory tracking of underactuated AUVs. The controller employs a second-order sliding mode surface, which ensures fast convergence while guaranteeing that the error converges to zero in finite time. Fractional-order calculus is introduced into the finite-time integral sliding mode control design to compensate for the weakening of the system’s dynamic performance by the fractional-order sliding mode, and meanwhile, to absorb the advantages of the fractional-order sliding mode to enhance the robustness. Compared with traditional integral sliding mode, it reduces the overshooting amount and has better accuracy and control performance. Desai [30] investigated the application of a sliding mode controller based on a linearly extended state observer control strategy (SMC-LESO in short) to a linearized yaw channel model. LESO is designed to estimate the total perturbation/uncertainty of the system state online, and SMC is used to design the controller based on this estimation.

Unfortunately, to simplify the controller design, most researchers choose to simplify the dynamics model. Incomplete degrees of freedom in just modeling the abstract path tracking algorithm is certainly not a problem, but the actual situation path tracking will allow the AUV to perform various types of underwater tasks such as spot blasting, sonar probing, etc. Although a six degree-of-freedom-based controller is particularly important, few of them consider the effect of reaction moments. Most researchers consider the complexity of the ocean environment and introduce constant or time-varying external disturbances into the system. However, the fact is that the AUV’s own thrusters also affect its attitude control. Nearly all AUV’s thrusters use propellers, and forward thrusters are typically single propellers. In addition to forward thrust, propellers generate reverse torque as they rotate [24]. The presence of the reverse moment affects the roll angle of the AUV itself, causing the body of the AUV to roll. Figure 1a is the initial attitude of the AUV when it is in motion, and Figure 1b is the attitude of the AUV when it is moving along a straight line. It can be clearly seen that the AUV has a rolling motion in addition to moving along a straight line. After multiple experimental observations, we find that the rolling motion of the AUV also changes under different thrusts, and the larger the thrust is, the more obvious the rolling motion is. In order to improve the attitude stability of AUV path-tracking control, this paper proposes a new controller named DLISMC and compares it with PID and DLPID control methods. Simulation experiments consider the reaction torque generated by the AUV itself in addition to the common marine environment disturbance, and introduce the reaction torque model generated by the propeller rotation into the dynamic model to optimize the controller with static compensation.

This paper mainly solves the problem of uncontrolled attitude of six-degree-of-freedom AUV path tracking affected by the reaction torque in the presence of steady-state error caused by model uncertainty and time-varying external interference. Firstly, a complete kinematic and dynamic model of the six-degree-of-freedom AUV and a reaction torque model are introduced. Secondly, the tracking controller based on DLISMC is derived and the stability of this control law is proved using the Lyapunov function. Finally, numerical simulations are carried out with MATLAB/Simulink platform, and the results show that the proposed DLISMC control method overcomes the effect of reaction torque on the attitude and is robust to external disturbances and uncertainties. Compared with previous research, the main contributions of this work are summarized as follows:

(1)

The anti-torsion model is constructed, and the anti-torsion perturbation is introduced into the mathematical model of the six-degree-of-freedom AUV, and the static compensation is performed for the control of the AUV’s roll angle, which enhances the robustness of the total system.

(2)

The control law is designed based on the double-loop structure using an integral sliding mode, and its convergence is proved using Lyapunov theory. Moreover, the feasibility of this control law is verified by simulation experiments.

The remainder of this paper is organized as follows. Section 2 introduces the 6-DOF AUV model, including coordinate system transformation, the kinematics model, the reaction torque model, and the dynamics model. Section 3 introduces the design of double loop integral sliding mode control and proves its stability. Section 4 is the simulation results. Section 5 is the summary of the article.

2. 6-DoF AUV Model

In this section, the six degrees of freedom AUV model is introduced in details. There are four main parts in this section, including the Coordinate system, Kinematic model, Reaction torque, and Dynamic model.

2.1. Coordinate System

Considering the multiple effects of gravity and hydro-dynamic force in underwater environments, the coordinate system for the 6-DOF AUV is described in Figure 2. $E-xyz$ denotes the Inertial frame and $O-uvw$ denotes the Body-fixed frame. Where $\mathit{\eta }={[x,y,z,\varphi ,\theta ,\psi ]}^T$ denotes the position and attitude vector of the AUV in the inertial frame and $\mathit{\upsilon }={[u,v,w,p,q,r]}^T$ denotes the linear and angular velocities vector of the AUV in the body-fixed frame. In Figure 2, x, y and z denote surge, sway, and heave displacement in the inertial frame; $\varphi$, $\theta$, and $\psi$ denote the roll, pitch, and yaw angle in the inertial frame, respectively; u, v, and w denote the surge, sway, and heave velocities in the body-fixed frame, respectively; and p, q, and r denote the roll, pitch, and yaw velocities in the body-fixed frame respectively.

The relationship between the inertial frame and body-fixed frame has been described in [31]. Hence, we introduce the relationship between the AUV’s lineal acceleration and the inertial frame AUV’s position. As shown in Figure 3, there are three basic rotation modes between the body-fixed frame and the inertial frame. Under the premise of the lineal acceleration vector of the AUV relative to the earth represented in the body-fixed coordinate system, we can compute the lineal acceleration vector in the earth-fixed coordinate system by $\varphi$, $\theta$, and $\psi$. In the simulation section, we use the rotation transformation matrix $\mathcal{R}$ to get the error value between the expected speed and actual speed. The rotation transformation matrix $\mathcal{R}$ can be expressed as:

$$\mathcal{R}=\left[\begin{array}{ccc}cos\psi cos\theta & cos\psi sin\theta sin\varphi -sin\psi cos\varphi & cos\psi sin\theta sin\varphi +sin\psi cos\varphi \\ sin\psi cos\theta & sin\psi sin\theta sin\varphi +cos\psi cos\varphi & sin\psi sin\theta sin\varphi -cos\psi cos\varphi \\ -sin\theta & cos\theta cos\varphi & cos\theta cos\varphi \end{array}\right]$$

(1)

2.2. Kinematics Model

The kinematics equation of 6-DOF AUV can be expressed as

$$\dot{\mathit{\eta }}=\mathcal{J}\left(\mathit{\eta }\right)\mathit{\upsilon }$$

(2)

where $\mathcal{J}\left(\mathit{\eta }\right)$ is the Jacobian transformation matrix, which is defined as:

$$\mathcal{J}\left(\mathit{\eta }\right)=\left[\begin{array}{cc}{\mathcal{J}}_1\left({\mathit{\eta }}_1\right)& 0_{3\times 3}\\ 0_{3\times 3}& {\mathcal{J}}_2\left({\mathit{\eta }}_2\right)\end{array}\right]$$

(3)

$${\mathcal{J}}_1\left({\mathit{\eta }}_1\right)=\left[\begin{array}{ccc}cos\psi cos\theta & cos\psi sin\theta sin\varphi -sin\psi cos\varphi & cos\psi sin\theta sin\varphi +sin\psi cos\varphi \\ sin\psi cos\theta & sin\psi sin\theta sin\varphi +cos\psi cos\varphi & sin\psi sin\theta sin\varphi -cos\psi cos\varphi \\ -sin\theta & cos\theta cos\varphi & cos\theta cos\varphi \end{array}\right]$$

(4)

$${\mathcal{J}}_2\left({\mathit{\eta }}_2\right)=\left[\begin{array}{ccc}1& tan\theta sin\varphi & tan\theta \\ 0& cos\varphi & -sin\varphi \\ 0& sin\varphi /cos\theta & cos\varphi /cos\theta \end{array}\right]$$

(5)

The kinematic model of AUVs is widely established [6,7,32,33]. Equation (6) is the kinematic model of the complete six-degree-of-freedom AUV, which shows the relationship between linear and angular velocities in the internal frame and the body-fixed frame.

$$\left\{\begin{array}{c}\dot{x}=ucos\psi cos\theta +v(-sin\psi cos\varphi +cos\psi sin\theta sin\varphi )+w(sin\psi sin\varphi +cos\psi sin\theta cos\varphi )\hfill \\ \dot{y}=usin\psi cos\theta +v(cos\psi cos\varphi +sin\psi sin\theta sin\varphi )+w(-cos\psi sin\varphi +sin\psi sin\theta cos\varphi )\hfill \\ \dot{z}=-usin\theta +vcos\theta sin\varphi +wcos\theta cos\varphi \hfill \\ \dot{\varphi }=p+qsin\varphi tan\theta +rcos\varphi tan\theta \hfill \\ \dot{\theta }=qcos\varphi -rsin\varphi \hfill \\ \dot{\psi }=(qsin\varphi +rcos\varphi )/cos\theta \hfill \end{array}\right.$$

(6)

2.3. Reaction Torque

As we know, when the propulsion propeller of AUV rotates to generate thrust, it will also generate obvious reaction torque. Figure 4 shows the effect of reaction torque during AUV motion from a rear-view perspective. The AUV has a total of seven thrusters, where four thrusters on the flanks and two thrusters on the sides are symmetrically distributed. It is assumed that the reaction torque generated by these six thrusters can cancel each other when the AUV is moving, so only the reaction torque generated by the main thruster will affect the attitude of 6-DOF AUV and cause the AUV to roll sideways.

Since the reaction torque has a great impact on the roll angle of the 6-DOF AUV, the formulation of propeller thrust and reaction torque can be defined as

$$\left\{\begin{array}{c}F=K_T\rho {\varpi }^2\hfill \\ F_Q=K_Q\rho {\varpi }^2\hfill \end{array}\right.$$

(7)

where F is the thrust force, $F_Q$ is the reaction torque, $\rho$ is density, $\varpi$ is the rotor radius, $K_T$ is the propeller thrust coefficient, and $K_Q$ is the propeller reaction torque coefficient. Therefore, the relationship between F and $F_Q$ can be obtained as

$$F_Q=\frac{K_Q}{K_T}F$$

(8)

2.4. Dynamic Model

Regarding the dynamics model of AUVs is common in the control field and can be found in the literature [6,7,32,33]. The dynamic equation of 6-DOF AUV in the body-fixed frame can be expressed as

$$\mathcal{M}\dot{\mathit{\upsilon }}+\mathcal{C}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathcal{D}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathcal{G}\left(\mathit{\eta }\right)={\mathcal{T}}_{\mathit{\upsilon }}+{\mathcal{T}}_d+{\mathcal{T}}_f$$

(9)

where $\mathcal{M}\in {\mathbf{R}}^{6\times 6}$ is the inertia matrix including the added mass, $\mathcal{C}\left(\mathit{\upsilon }\right)\in {\mathbf{R}}^{6\times 6}$ is the matrix of the Coriolis and centripetal forces, $\mathcal{D}\left(\mathit{\upsilon }\right)\in {\mathbf{R}}^{6\times 6}$ is the matrix of the hydrodynamic damping terms, $\mathcal{G}\left(\mathit{\eta }\right)\in {\mathbf{R}}^{6\times 6}$ is the vector of restoring forces (gravity and buoyancy), ${\mathcal{T}}_{\mathit{\upsilon }}$ is the vector of the control forces and moments, ${\mathcal{T}}_d$ is the vector of the time-varying external disturbances, and ${\mathcal{T}}_f$ is the vector of the reaction torque.

The formulation of the dynamic model is

$$\left\{\begin{array}{c}{m}_0\dot{u}+m_0[z_G(\dot{q}+rp)+2(wq-vr)]+(d_{t1}\left|u\right|+d_{t2})u={\tau }_u+{\tau }_{ud}\hfill \\ m_0\dot{v}+m_0[z_G(-\dot{p}+rq)+2(ur-wp)]+(d_{t1}\left|v\right|+d_{t2})v={\tau }_v+{\tau }_{vd}\hfill \\ m_0\dot{u}+m_0[z_G(-p^2-q^2)+2(vp-uq)]+(d_{t1}\left|w\right|+d_{t2})w={\tau }_w+{\tau }_{wd}\hfill \\ I_{xx}\dot{p}+m_0{z}_G(\dot{v}-ur+pw)+I_{r}q-I_{q}r+(d_1\left|p\right|+d_2)p+z_G{m}_{0}gcos\theta sin\varphi =(1+\frac{K_Q}{K_T}){\tau }_p+{\tau }_{pd}\hfill \\ I_{yy}\dot{p}+m_0{z}_G(\dot{u}-rv+qw)+I_{r}p-I_{p}r+(d_1\left|q\right|+d_2)q+z_G{m}_{0}gsin\theta ={\tau }_q+{\tau }_{qd}\hfill \\ I_{zz}\dot{p}+I_{q}p-I_{p}q+(d_1\left|r\right|+d_2)r={\tau }_r+{\tau }_{rd}\hfill \end{array}\right.$$

(10)

Equation (10) represents the matrix expression in Equation (9) through the traditional mathematical expression, which is derived from the analysis and modeling of the specific underwater stress of the AUV.

Furthermore, considering that wind and wave disturbances are bounded in most cases, the following assumptions are given:

Assumption 1. 

The system model estimation error and time-varying external disturbances are bounded, i.e., $max\left(\right|{\mathcal{T}}_d\left|\right)<{\mathcal{T}}_M$, where ${\mathcal{T}}_M$ is the upper bound of external interference.

Assumption 2. 

There is a bound of reaction torque error in the system model, i.e., $max\left(\right|{\mathcal{T}}_f\left|\right)<{\mathcal{T}}_{fM}$, where ${\mathcal{T}}_{fM}$ is the upper bound of reaction torque.

3. Controller Design

ISMC [26] has been widely used in various fields, such as motor control, robot control, and so on. Compared with ordinary sliding mode control, integral sliding mode control has a faster response speed and stronger anti-interference ability, ensuring the robustness and accuracy of the controller [34,35]. This paper proposed a new DLISMC controller for a newly defined AUV model.

As seen in Figure 5, the inner loop and outer loop denote speed loop control and position loop control, respectively. The applied position loop controller combines the traditional ISMC method and the AUV’s kinematics model to achieve accurate position and attitude tracking by inputting the desired position and feedback current position information. Afterwards, the position loop controller generates the virtual control input as the reference speed, which is used as the input of next speed loop. The applied speed loop controller is similar to the previous position loop derived from the traditional ISMC and AUV’s dynamic model, and then introduces disturbances including time-varying ocean disturbances and counter torque disturbances. The whole DLISMC controller generates ideal thrust for each degree of freedom. The investigated 6-DOF AUV gets the real-time speed feedback after the thrust starts to work, so as to achieve accurate tracking of the reference speed. A detailed design for the DLISMC controller will be introduced in the following section.

3.1. Position Loop Controller Design

Due to similarity between $\varphi$, $\theta$, and $\psi$, we take the angle $\varphi$ as an example to express in the following section. ${\varphi }_e$ is defined as the error between the expected angle ${\varphi }_d$ and current angle $\varphi$, which can be expressed as ${\varphi }_e=\varphi -{\varphi }_d$ and $\dot{{\varphi }_e}=\dot{\varphi }-\dot{{\varphi }_d}$.

In order to minimize the buffet phenomenon caused by the sliding mode control, a saturation function is used to design the convergence rate. The sliding surface $S_{\varphi }$ is designed as

$$\left\{\begin{array}{c}{S}_{\varphi }={\alpha }_1^{\varphi }{\varphi }_e+h_{\varphi }\hfill \\ {\dot{h}}_{\varphi }={\alpha }_2^{\varphi }{\varphi }_e,h_{\varphi }\left(0\right)=-{\alpha }_1^{\varphi }{\varphi }_e\left(0\right)\hfill \end{array}\right.$$

(11)

Then, we can obtain the derivation of $S_{\varphi }$ and we define $\dot{S_{\varphi }}=-{\epsilon }_{\varphi }sat\left(S_{\varphi }\right),{\epsilon }_{\varphi }>0$ as the approach rate of $S_{\varphi }$ derivative, so there is the following such equation:

$$-{\epsilon }_{\varphi }\mathrm{sat}\left(S_{\varphi }\right)={\alpha }_1^{\varphi }(\dot{\varphi }-{\dot{\varphi }}_d)+{\alpha }_2^{\varphi }{\varphi }_e$$

(12)

where $sat\left(s\right)$ denotes the saturation function, which can be expressed as

$$\mathrm{sat}\left(s\right)=\left\{\begin{array}{c}1,s>1\hfill \\ s,\left|s\right|\le 0\hfill \\ -1,s<-1\hfill \end{array}\right.$$

(13)

So far, we bring the parameter $\varphi$ of Equation (6) into Equation (12). The virtual reference speed $p_d$ can be obtained as follows:

$$p_d=\frac{-{\epsilon }_{\varphi }sat\left(S_{\varphi }\right)-{\alpha }_2^{\varphi }(\varphi -{\varphi }_d)}{{\alpha }_1^{\varphi }}-q_{d}sin\varphi tan\theta -r_{d}cos\varphi tan\theta +{\dot{\varphi }}_d$$

(14)

In the same way, we can obtain other virtual reference speeds:

$$\begin{array}{cc}\hfill u_d=& \frac{-{\alpha }_2^x\left(x-x_d\right)-{\epsilon }_{x}sat\left(S_x\right)}{{\alpha }_1^{x}cos\psi cos\theta }-\frac{v_d\left(-sin\psi cos\varphi +cos\psi sin\theta sin\varphi \right)}{cos\psi cos\theta }\hfill \\ \hfill & -\frac{w_d\left(sin\psi sin\theta +cos\psi sin\theta cos\varphi \right)-{\dot{x}}_d}{cos\psi cos\theta }\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill v_d=& \frac{-{\alpha }_2^y\left(y-y_d\right)+{\epsilon }_{y}sat\left(S_y\right)}{{\alpha }_1^y(cos\psi cos\varphi +sin\psi sin\theta sin\varphi )}\hfill \\ \hfill & -\frac{u_{d}sin\psi cos\theta +w_d\left(-cos\psi sin\varphi +sin\psi sin\theta cos\varphi \right)-{\dot{y}}_d}{(cos\psi cos\varphi +sin\psi sin\theta sin\varphi )}\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill w_d=-\frac{{\epsilon }_{z}sat\left(S_z\right)+{\alpha }_2^z\left(z-z_d\right)}{{\alpha }_1^{z}cos\theta cos\varphi }-\frac{-u_{d}sin\theta +v_{d}cos\theta sin\varphi -{\dot{z}}_d}{cos\theta cos\varphi }\end{array}$$

(17)

$$\begin{array}{c}\hfill q_d=-\frac{{\epsilon }_{\theta }sat\left(S_{\theta }\right)+{\alpha }_2^{\theta }\left(\varphi -{\varphi }_d\right)}{{\alpha }_1^{\theta }cos\varphi }+r_{d}tan\varphi +\frac{{\dot{\theta }}_d}{cos\varphi }\end{array}$$

(18)

$$\begin{array}{c}\hfill r_d=-cos\theta \frac{{\epsilon }_{\psi }sat\left(S_{\psi }\right)+{\alpha }_2^{\psi }\left(\psi -{\psi }_d\right)}{{\alpha }_1^{\psi }cos\varphi }+\frac{cos\theta }{cos\varphi }\left({\dot{\psi }}_d-\frac{q_{d}sin\varphi }{cos\theta }\right)\end{array}$$

(19)

3.2. Velocity Loop Controller Design

Due to similarity between p, q and r, we take the speed p as an example to express.

In this section, $p_e$ is defined as the error between virtual reference speed $p_d$ and current speed p, the tracking error of can be expressed as $p_e=p-p_d$ and $\dot{p_e}=\dot{p}-\dot{p_d}$.

Then, we can design the sliding surface of p as:

$$\left\{\begin{array}{c}{S}_p={\alpha }_1^p{p}_e+h_p\hfill \\ {\dot{h}}_p={\alpha }_2^p{p}_e,h_p\left(0\right)=-{\alpha }_1^p{p}_e\left(0\right)\hfill \end{array}\right.$$

(20)

Similarly, we perform the derivation of $S_p$, where we take $\dot{S_p}=-{\epsilon }_{p}sat\left(S_p\right),{\epsilon }_p>0$ as the approach rate of the $S_p$ derivative, hence we can get the following equation:

$$-{\epsilon }_p\mathrm{sat}\left(S_p\right)={\alpha }_1^p(\dot{p}-{\dot{p}}_d)+{\alpha }_2^p{p}_e$$

(21)

In addition, we bring the parameter of Equation (10) into Equation (21). The ideal thrust ${\tau }_p$ can be obtained from Equation (22).

$$\begin{array}{cc}\hfill {\tau }_p=& -\frac{I_{xx}\left[{\epsilon }_{p}sat\left(S_p\right)+{\alpha }_2^p\left(p-p_d\right)\right]}{{\alpha }_1^p}+m_0{z}_G\left(\dot{v}-ur+pw\right)+p\left(d_1\left|p\right|+d_2\right)\hfill \\ \hfill & +I_{r}q-I_{q}r+z_G{m}_{0}gcos\theta sin\varphi +I_{xx}{\dot{p}}_d\hfill \end{array}$$

(22)

In the same way, we can obtain the ideal thrust of other degrees of freedom:

$$\begin{array}{cc}\hfill {\tau }_u=& \frac{-m_0{\epsilon }_{u}sat\left(S_u\right)}{{\alpha }_1^u}+m_0\left[z_G\left(\dot{q}+rp\right)+2\left(wq-vr\right)\right]\hfill \\ \hfill & +u\left(d_{t_1}\left|u\right|+d_{t_2}\right)+m_0{\dot{u}}_d-\frac{m_0{\alpha }_2^u}{{\alpha }_1^u}\left(u-u_d\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\tau }_v=& \frac{-m_0{\epsilon }_{v}sat\left(S_v\right)}{{\alpha }_1^v}+m_0\left[z_G\left(-\dot{p}+rq\right)+2\left(ur-\omega p\right)\right]\hfill \\ \hfill & +v\left(d_{t_1}\left|v\right|+d_{t_2}\right)+m_0{\dot{v}}_d-\frac{m_0{\alpha }_2^v}{{\alpha }_1^v}\left(v-v_d\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\tau }_w=& \frac{-m_0{\epsilon }_{w}sat\left(S_w\right)}{{\alpha }_1^w}+m_0\left[z_G\left(-p^2-q^2\right)+2\left(vp-uq\right)\right]\hfill \\ \hfill & +w\left(d_{t_1}\left|w\right|+d_{t_2}\right)+m_0{\dot{w}}_d-\frac{m_0{\alpha }_2^w}{{\alpha }_1^w}\left(w-w_d\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\tau }_q=& -\frac{I_{yy}\left[{\epsilon }_{q}sat\left(S_q\right)+{\alpha }_2^q\left(q-q_d\right)\right]}{{\alpha }_1^q}+m_0{z}_G\left(\dot{u}-rv+qw\right)+q\left(d_1\left|q\right|+d_2\right)\hfill \\ \hfill & -I_{r}p+I_{p}r+z_G{m}_{0}gsin\theta +I_{yy}{\dot{q}}_d\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\tau }_r=& -I_{zz}\frac{{\epsilon }_{r}sat\left(S_r\right)+{\alpha }_2^r\left(r-r_d\right)}{{\alpha }_1^r}+I_{zz}{\dot{r}}_d+I_{q}p-I_{p}q\hfill \\ \hfill & -r\left(d_1\left|r\right|+d_2\right)\hfill \end{array}$$

(27)

3.3. Stability Analysis

When the whole control system uses Equations (22)–(27) as input for position and attitude control and satisfies Assumption 1, the tracking error of the 6-DoF AUV converges to zero. The stability of the position loop and the speed loop is demonstrated below.

Herein, we take the speed loop as an example. Select the following Lyapunov function:

$$V_1=\frac{1}{2}{S}_p^2$$

(28)

Deriving $V_1$, Equations (10), (20) and (22) are substituted into ${\dot{S}}_p$:

$$\begin{array}{cc}\hfill {\dot{V}}_1=& {\dot{S}}_p{S}_p\hfill \\ \hfill =& S_p[{\alpha }_1^p(\dot{p}-{\dot{p}}_d)+{\alpha }_2^p(p-p_d)]\hfill \\ \hfill =& S_p\left\{{\alpha }_1^p\right[\frac{(1+\frac{K_Q}{K_T}){\tau }_p+{{\tau }_p}_d-m_0{z}_G(\dot{v}-ur+pw)}{I_{XX}}\hfill \\ \hfill & -\frac{-I_{r}q+I_{q}r-p(d_1\left|p\right|+d_2)-z_G{m}_{0}gcos\theta sin\varphi }{{I_x}_x}-{\dot{P}}_d]+{\alpha }_2^p(u-u_d)\}\hfill \\ \hfill =& S_p\{{\alpha }_1^p[\frac{\frac{K_Q}{K_T}{\tau }_p+{{\tau }_p}_d}{I_{xx}}-\frac{{\epsilon }_p\mathrm{sat}\left(S_p\right)+{\alpha }_2^p(p-p_d)}{{\alpha }_1^p}]+{\alpha }_2^p(p-p_d)\}\hfill \\ \hfill & \le S_p{\alpha }_1^p\frac{T_M+T_{fM}}{I_{xx}}-{\epsilon }_p{\alpha }_1^p\left|S_p\right|\hfill \\ \hfill & \le \left|S_p\right|{\alpha }_1^p(\frac{T_M+T_{fM}}{I_{xx}}-{\epsilon }_p)\hfill \end{array}$$

(29)

Since the famous Lyapunov theory ${\epsilon }_p$ needs to meet the following conditions, and the sliding surface $S_p$ mentioned in Equation (20) will converge to zero.

$${\epsilon }_p>{I_{xx}}^{-1}(T_M+T_{fM})$$

(30)

Remark 1. 

$I_{xx}$ is the nonzero moment of inertia of the 6-DoF AUV. The coefficient ${\alpha }_1^p$ is also nonzero constant. Therefore, ${\epsilon }_p$ exists and is a bounded positive constant.

Therefore, the designed virtual control input Equation (22) can guarantee that the 6-DOF AUV can achieve stability in the velocity loop. The position loop $\varphi$ selects the following Lyapunov function:

$$V_2=\frac{1}{2}{S}_{\varphi }^2$$

(31)

Deriving $V_2$, Equations (6), (11), and (14) are substituted into ${\dot{S}}_{\varphi }$:

$$\begin{array}{cc}\hfill {\dot{V}}_2=& {\dot{S}}_{\varphi }{S}_{\varphi }\hfill \\ \hfill =& S_{\varphi }[{\alpha }_1^{\varphi }(\dot{\varphi }-{\dot{\varphi }}_d)+{\alpha }_2^{\varphi }(\varphi -{\varphi }_d)]\hfill \\ \hfill =& S_{\varphi }\{{\alpha }_1^{\varphi }(p+qsin\varphi tan\theta +rcos\varphi tan\theta -{\dot{\varphi }}_d)+{\alpha }_2^{\varphi }(\varphi -{\varphi }_d)\}\hfill \\ \hfill =& S_{\varphi }\left\{{\alpha }_1^{\varphi }\right(\frac{-{\epsilon }_{\varphi }sat\left(S_{\varphi }\right)-{\alpha }_2^{\varphi }(\varphi -{\varphi }_d)}{{\alpha }_1^{\varphi }}-qsin\varphi tan\theta -rcos\varphi tan\theta \hfill \\ \hfill & +{\dot{\varphi }}_d+qsin\varphi tan\theta +rcos\varphi tan\theta -{\dot{\varphi }}_d)+{\alpha }_2^{\varphi }(\varphi -{\varphi }_d)\}\hfill \\ \hfill =& -S_{\varphi }{\epsilon }_{\varphi }sat\left(S_{\varphi }\right)\le -{\epsilon }_{\varphi }\left|S_{\varphi }\right|\hfill \end{array}$$

(32)

Finally, it can be deduced from Equation (32) that ${\dot{V}}_2\le 0$ is established. Therefore, the virtual reference speed designed in Equation (14) can ensure that the AUV can track the designed path curve and achieve stability in the position loop. The speed loop and position loop have both reached stability, thereby proving the stability of the entire system.

4. Simulation

4.1. Simulation Step

In this section, the verification of the sliding mode position and attitude control method based on double-loop integration is introduced. The MATLAB Simulink platform is used to for numerical verification, as shown in Figure 6. In order to better represent the performance advantages of the DLISMC method, we compare it with the PID control method and DLPID control method. The traditional PID method directly takes the position and angle errors of the AUV as an input, and obtains the thrust output of the AUV through proportion, integration, and differentiation. The DLPID method is similar to the DLISMC, which can be divided into speed loop control and position loop control. The controller of each loop is composed of traditional PID controllers.

The parameters of the 6-DoF AUV model in [24] are also adopted in this paper, and the saturation limit of the 6 propellers is set to ±200 N.

Without losing generality, time-varying external interference with different amplitudes and periods is introduced:

$$\left\{\begin{array}{c}{{\tau }_u}_d=-10sin\left(0.6t\right)\hfill \\ {{\tau }_v}_d=-9sin\left(0.5t\right)\hfill \\ {{\tau }_w}_d=-3sin\left(0.4t\right)\hfill \\ {{\tau }_p}_d=-4sin\left(0.45t\right)\hfill \\ {{\tau }_q}_d=-2sin\left(0.3t\right)\hfill \\ {{\tau }_r}_d=-6sin\left(0.1t\right)\hfill \end{array}\right.$$

(33)

The propeller thrust coefficient $K_T$ and the propeller reaction torque coefficient $K_Q$ in the reverse torque model are set according to [24].

In order to verify the feasibility of the proposed control method, two kinds of trajectory-tracking situations are realized. One is the three-dimensional plane cosine trajectory, and the other is the space spiral expected trajectory.

The initial conditions of the vehicle are set as $x\left(0\right)=10 \mathrm{m},y\left(0\right)=5 \mathrm{m},z\left(0\right)=10 \mathrm{m},\varphi \left(0\right)=0.1 \mathrm{rad},\theta \left(0\right)=0 \mathrm{rad},\psi \left(0\right)=0.1 \mathrm{rad},u\left(0\right)=v\left(0\right)=w\left(0\right)=0 \mathrm{m}/\mathrm{s},p\left(0\right)=q\left(0\right)=r\left(0\right)=0 \mathrm{rad}/\mathrm{s}$. The dynamic model parameters of the AUV in the simulation are set according to [25]. The method of the parameter selection for the controller refers to [36,37]. By using the control variable method, the influence of different parameters on different errors is analyzed. After multiple parameter adjustments, the control parameters that meet the experimental requirements are selected.

Table 2. Controller Parameters.

Method	Controller Parameters
PID	$P_x=435,I_x=100,D_x=-10,P_y=1000,D_y=-10,P_z=-1000$
	$P_{\varphi }=-10,000,P_{\theta }=-10,000,P_{\psi }=-10,000$
DLPID	$P_x=800,P_y=800,P_z=50,I_z=1,P_{\varphi }=200,I_{\varphi }=10,D_{\varphi }=10,P_{\theta }=100,I_{\theta }=10,P_{\psi }=130,I_{\psi }=10$
	$P_u=3,I_u=0.005,P_v=800,I_v=1,P_w=50,I_w=5,P_p=5,I_p=10,D_p=2,P_q=500,I_q=10,P_r=20,I_r=10$
DLISMC	${\alpha }_x^1={\alpha }_y^1=0.00001,{\alpha }_x^2={\alpha }_y^2={\alpha }_z^2=100,000,{\alpha }_z^1=0.0001,{\alpha }_{\varphi }^1={\alpha }_{\theta }^1=0.01,{\alpha }_{\varphi }^2={\alpha }_{\theta }^2=5,{\alpha }_{\psi }^1=0.0099,{\alpha }_{\psi }^2=1.2$
	${\alpha }_u^1={\alpha }_v^1={\alpha }_w^1=0.001,{\alpha }_u^2={\alpha }_v^2={\alpha }_w^2={\alpha }_q^1={\alpha }_q^2=1,{\alpha }_p^1=0.01,{\alpha }_p^2=0.5,{\alpha }_r^1=0.099,{\alpha }_r^2=3$

gives the parameters of the proposed controller in detail.

4.2. Simulation Results in Scene 1

In Scene 1, the 6-DoF AUV is commanded to track the 3-dimensional plane cosine trajectory for 100 s to test the tracking effect of the AUV controlled by the controller under common conditions. The desired trajectory in the inertial-frame is described as

$$\left\{\begin{array}{c}{x}_d\left(t\right)=10sin\left(0.025\pi t\right)\hfill \\ y_d\left(t\right)=10cos\left(0.025\pi t\right)\hfill \\ z_d\left(t\right)=0.25t+1\hfill \\ {\varphi }_d\left(t\right)=0\hfill \\ {\theta }_d\left(t\right)=-arctan\left(\frac{1}{\pi }\right)\hfill \\ {\psi }_d\left(t\right)=0.025\pi t\hfill \end{array}\right.$$

(34)

The derived curves for two control schemes are shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. Figure 12 shows the control input for AUV and Figure 13 gives the curve of reaction torque.

4.3. Simulation Results in Scene 2

In Scene 2, the AUV is commanded to track the space spiral expected trajectory for 100 s to test whether the controller can drive the AUV well when the AUV meets complex motion requirements. The desired trajectory in the inertial frame can be described as

$$\left\{\begin{array}{c}{x}_d\left(t\right)=0.25t\hfill \\ y_d\left(t\right)=10sin\left(0.025\pi t\right)\hfill \\ z_d\left(t\right)=0\hfill \\ {\varphi }_d\left(t\right)=0\hfill \\ {\theta }_d\left(t\right)=0\hfill \\ {\psi }_d\left(t\right)=cos\left(0.025\pi t\right)\hfill \end{array}\right.$$

(35)

The derived curves for two control schemes are shown in Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

4.4. Discussions

Figure 7 demonstrates the 3D plots of the AUV tracking the cosine curve under different control methods. From these results, it can be seen that the DLISMC algorithm tracks the desired trajectory earlier than the PID or DLPID. Figure 8 and Figure 9 show the positional and angular errors of the AUV during tracking. In terms of position error, there is not much difference between the two methods in x and y directions from these results. In terms of the convergence time in the x and y directions, the proposed DLISMC is 16% and 23% faster than the DLPID algorithm, and 70% and 72% faster than the PID algorithm, but all of them eventually ensure that the errors are within a certain range. In terms of convergence time in the z direction, the DLISMC is 0.1% faster than the PID and it is clear that the DLISMC is 74% faster than the DLPID, and that the error saturation of the DLISMC is smaller than that of the DLPID algorithm. This is due to the integral term of the PID controller. Figure 10 and Figure 11 show the linear velocity error and angular velocity error of the AUV during tracking, respectively. In terms of angular velocity error, both control methods start with a significant buffer, which is related to the initial position of the AUV, indicating that the AUV is adjusting the attitude of the angle-tracking profile. Figure 12 shows the control input for AUV and Figure 13 gives the curve of reaction torque.

Figure 14 represents a three-dimensional view of the AUV-tracking spiral curve under different control methods. It is obvious from these results that, after a period of time, the controllers designed by the two methods can track the desired trajectory well. Figure 15 and Figure 16 represent the position error and angle error of the AUV during path tracking. It can be seen that the curves of the DLISMC and DLPID are stable and the error is within a certain range, which means that the whole position loop is convergent and stable. Moreover, the curve of the DLISMC fluctuates less than that of DLPID. In addition, the DLISMC converges 35%, 68%, and 62% faster than the DLPID and 50%, 60%, and 27% in the x, y, and z directions, respectively. The large amplitude of the DLPID in these results is caused by the integral term of the original PID method. Figure 17 and Figure 18 display the linear velocity error and angular velocity error of the AUV during trajectory tracking. It is obvious that results of the DLPID curve and DLISMC curve are stable at zero, which means that the entire speed loop is stable. Figure 19 and Figure 20 give the control input of AUV’s thrust and reaction torque moments, respectively. From these results, it can be seen that the thrusts during tracking are all within the +/−200 N limit, and the curve of the thrusts changes smoothly. The variation of reaction torque is also consistent with the above model, and the DLISMC controller can stably control the AUV for tracking even with reaction torque disturbances. In the presence of interference, the performance of the DLPID is significantly lower than that of the DLISMC, which has stronger anti-interference stability. Under the influence of reaction torque, the DLPID algorithm is unstable in the roll angle and roll speed, and cannot stabilize the attitude of the AUV. These results verify that the proposed DLISMC algorithm can overcome these problems perfectly.

5. Conclusions

In this paper, a six-degree-of-freedom AUV path-tracking control method DLISMC algorithm is investigated. The effect of reaction torque on the AUV attitude during path tracking is considered, and a reaction torque model is introduced to statically compensate the attitude control of the AUV, so as to effectively improve the stability of the AUV’s attitude. The stability condition of the DLISMC controller is obtained by utilizing the Lyapunov function. Simulation results verify the effectiveness and better robustness of the proposed method. In the future work, we will introduce the adaptive rate to solve the parameter complexity problem and design new convergence rate to improve the sliding mode jitter phenomenon as well as introducing an event-driven mechanism to reduce the updating of the AUV control signals.