Three-dimensional path following control of underactuated autonomous underwater vehicle based on damping backstepping

Abstract

This article addresses the problem of three-dimensional path following control for underactuated autonomous underwater vehicles in the presence of ocean current. Firstly, three-dimensional path following error model was established based on virtual guidance method. The control law is developed by building virtual velocity errors and backstepping method, which can simplify the virtual control input and avoid the singular problem induced by initial state constraints. Considering the curvature and torsion characteristics of the three-dimensional desired path, the approaching angle is introduced to guarantee fast convergence of error. Nonlinear damping term is introduced to offset the effects of dynamic uncertainties and external disturbances. The controller stability was proved by Lyapunov stable theory. Finally, simulations were conducted and the results indicate the effectiveness and robustness to parameter uncertainties and external disturbances of the proposed approach.

Introduction

In recent years, autonomous underwater vehicles (AUVs) are employed in a wide range of civilian and military applications, such as long-duration oceanographic sampling, ^1,2 detection and localization of pollutant sources and undersea mines, ^3,4 mine clearance, ⁵ region search for deep-sea wrecks, and so on. ^6,7 To achieve those tasks, AUVs need the ability of accurate tracking control. ^5,6 In order to reduce the vehicle’s weight and cost, save energy consumption, and improve system reliability, it is usually designed to be an underactuated AUV which can achieve the same control effects as a fully actuated AUV with less control inputs. Moreover, a full actuated AUV with actuator failures can be defined as an underactuated one which might be still controlled with an underactuated control scheme. There are not lateral and vertical thrusters for most underactuated AUVs, and only surge velocity, yaw angular velocity, and pitch angular velocity can be controlled directly. Therefore, the underactuated AUV control is much more difficult than the fully actuated AUV control. ⁸

Many researchers have focused on the path following control design for the underactuated AUV and developed some significant results. Approaches on the line-of-sight (LOS) guidance principle are proposed on horizontal path following for underactuated AUVs and marine vehicles. ^{7 –10} Moreover, an integral LOS guidance is introduced for path following of the underactuated AUV in the presence of ocean currents. ⁹ The adaptive robust control and new finite-time disturbance observer are established for trajectory tracking for marine vehicles with unknown disturbances. ¹¹ The direct adaptive fuzzy tracking control scheme with fully unknown parametric dynamics for marine vehicles to deal with uncertainties and unknown disturbances is proposed in the studies by Wang and Er ¹² and Wang et al. ¹³ The neural network and dynamic surface control technique and neurodynamics-based output feedback scheme are proposed in the presence of uncertainties and ocean disturbances. ^{14 –16} Backstepping method has been applied successfully for the path following control of the underactuated AUVs. ^17,18 To avoid the explosion of complexity in backstepping design which is caused by differentiation at each step, researchers have proposed some methods to eliminate the problem in some studies. ^{19 –24} In addition, a nonlinear iterative sliding mode control method was presented for path following of underactuated AUVs. ²⁵ However, much work on path following control of underactuated AUVs has been done in the horizontal plane and vertical plane, respectively, that is, the three-dimensional (3-D) motion model of the underactuated AUV is decoupled into two plane models and the path following controllers are designed, respectively. Due to ignoring the model coupling effects among different degrees of freedom (DOFs), the controllers on decoupling model design cannot achieve accurate 3-D path following for underactuated AUVs. Therefore, to build the 3-D path following error model and design 3-D path following controller have been the key research for AUV motion control.

Tian ²⁶ decomposed the 3-D path following controller of the underactuated AUV into path following guidance function design and the PID controller design and utilized fuzzy logic theory to improve the guidance function. But the control effects of PID controller are unsatisfactory to system nonlinearity and lack of robustness to external disturbances. Zhou et al. ²⁷ designed three adaptive neural network controllers based on Lyapunov stability theorem to estimate uncertain parameters of the AUV model and unknown current disturbances. These controllers were designed to guarantee that all the error states in the path following system are asymptotically stable. Lapierre and Jouvencel ²⁸ designed a kinematic controller to deal with vehicle dynamics based on backstepping and Lyapunov-based techniques. Borhaug and Pettresen ²⁹ applied cascade system theory and backstepping method to design controllers based on LOS, which can simplify the form of controllers but not discuss path following problem for general desired path. A nonlinear iterative sliding mode controller based on engineering decoupling was designed in the study by Jia et al. ³⁰ to resist the model parameter perturbation and current disturbances, which can only guarantee asymptotic stability of the subsystem but the whole control system. Virtual control variable through the state error model is introduced to compensate nonlinear effects of the model, ³¹ and the feedback gain is introduced to avoid the problem of high-order derivative for virtual variable. To sum up, the path following control methods for underactuated AUVs have been discussed during the past decade. However, the problems of external disturbances and uncertainty of AUV models have not been investigated thoroughly. ³²

Motivated by the above observation, this article investigates the path following control design for the underactuated AUV in the presence of ocean currents. Considering the effect of unknown modeling during AUV actual operations and current disturbances, we establish the 3-D path following error model based on virtual guidance method firstly, then we design the control laws combined backstepping and damping methods according to the curvature and torsion characteristics of the desired path, ³³ which make AUV follow the desired path from any position and guarantee that all state variables converge to the desired value. The impact of dynamic uncertainty and external disturbances is offset by damping term. ³⁴ Stability analysis for the whole control system is performed based on Lyapunov stability theorem, and simulations are conducted to verify the feasibility and robustness to external disturbances.

Problem formulation

AUV dynamic model

The AUV dynamic model in 3-D space is proposed in this section. Assume that the AUV center of gravity is the origin of the kinetic coordinate system and ignore the effects of AUV rolling due to its good symmetrical structure. Therefore, 6-DOF model can be simplified into 5-DOF model. ³⁵

The AUV dynamic equation is

$M\dot{\mathbf{v}}+C_{RB}\left(v\right)v+C_A\left(v_r\right)v_r+D\left(v_r\right)v_r+g\left(\eta \right)=\tau$

1

The AUV kinematic equation is

$\left[\begin{array}{l}\begin{array}{c}\dot{x}\\ \dot{y}\\ \begin{array}{l}\dot{z}\\ \dot{\theta }\end{array}\end{array}\\ \dot{\psi }\end{array}\right]=\left(\begin{array}{cccc}\text{cos}\psi \text{cos}\theta & -\text{sin}\psi & \text{cos}\psi \text{sin}\theta & \begin{array}{cc}0& 0\end{array}\\ \text{sin}\psi c\theta & \text{cos}\psi & \text{sin}\psi \text{sin}\theta & \begin{array}{cc}0& 0\end{array}\\ -\text{sin}\theta & 0& \text{cos}\theta & \begin{array}{cc}0& 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{cc}\begin{array}{c}1\\ 0\end{array}& \begin{array}{c}0\\ 1/\text{cos}\theta \end{array}\end{array}\end{array}\right)\left[\begin{array}{l}\begin{array}{c}u\\ v\\ \begin{array}{l}w\\ q\end{array}\end{array}\\ r\end{array}\right]$

2

where $\eta ={(x,y,z,\theta ,\psi )}^T$ represents the position and attitude angle of AUV in the geodetic coordinate system. $v={(u,v,w,q,r)}^T$ represents the AUV velocities in the kinetic coordinate system; u, v, and w represent the AUV velocities (surge, sway, and heave); and q and r represent the AUV angular velocities (pitch and yaw), respectively. θ and ψ are the AUV pitch angle and yaw angle from the geodetic coordinate system to kinetic coordinate system. $v_r=v-v_c$ , where v _c represents the current velocities in the kinetic coordinate system, which can be described by equation (3). M = M ^T > 0 is the mass and inertial matrix of AUV, C _RB (v) is the rigid-body Coriolis and centripetal matrix, C _A (v _r ) is the hydrodynamic Coriolis and centripetal matrix, and D(v _r ) is the damping matrix. g(η) is a vector containing gravitational or buoyancy forces and moments. τ is a vector containing the control inputs.

$\left[\begin{array}{c}{u}_c\\ v_c\\ w_c\end{array}\right]={\left(\begin{array}{ccc}\text{cos}\psi \text{cos}\theta & -\text{sin}\psi & \text{cos}\psi \text{sin}\theta \\ \text{sin}\psi \text{cos}\theta & \text{cos}\psi & \text{sin}\psi \text{sin}\theta \\ -\text{sin}\theta & 0& \text{cos}\theta \end{array}\right)}^T\left[\begin{array}{c}{U}_c\\ V_c\\ W_c\end{array}\right]$

3

where $V_c={(U_c,V_c,W_c)}^T$ represents the current velocities in the geodetic coordinate system.

Path following error model

In Figure 1, the desired path is the spatial curve described by design parameters. {E}, {B}, and {F} represent the geodetic coordinate system, kinetic coordinate system, and Serret–Frenet coordinate frame, respectively. P is the virtual reference guidance point in the Serret–Frenet coordinate frame. So P is an arbitrary point on the designed path, and Q represents the center of mass. In order to describe the designed path precisely, we define k₁(s) and k₂(s) as the curvature and deflection of the curve. Then, the AUV angular velocities can be expressed as ${\omega }_F={(0,k_2\left(s\right)\dot{s},k_1\left(s\right)\dot{s})}^T$ in the Serret–Frenet coordinate frame. $U_B=\sqrt{u^2+v^2+w^2}$ is the AUV spatial synthetic velocity. The velocity of P is ${\left(\text{d}p/\text{d}t\right)}_F={\left[U_P,0,0\right]}^T$ in the Serret–Frenet coordinate frame. ψ_e and θ_e are yaw angle error and pitch angle error, respectively. Then, equation (4) can be expressed as the AUV velocities in the geodetic coordinate system.

${\left(\frac{\text{d}q}{\text{d}t}\right)}_E={\left(\frac{\text{d}p}{\text{d}t}\right)}_E+R_F^E{\left(\frac{\text{d}l}{\text{d}t}\right)}_F+R_F^E\left({\omega }_F\times l\right)$

4

Figure 1.

Diagram of 3-D path following of AUV. 3-D: three-dimensional; AUV: autonomous underwater vehicle.

With

$R_F^E=\left(\begin{array}{ccc}\text{cos}{\psi }_e\text{cos}{\theta }_e& \text{sin}{\psi }_e\text{cos}{\theta }_e& -\text{sin}{\theta }_e\\ -\text{sin}{\psi }_e& \text{cos}{\psi }_e& 0\\ \text{cos}{\psi }_e\text{sin}{\theta }_e& \text{sin}{\psi }_e\text{sin}{\theta }_e& \text{cos}{\theta }_e\end{array}\right)$

5

Then

${\left(R_F^E\right)}^T{\left(\frac{\text{d}q}{\text{d}t}\right)}_E={\left(\frac{\text{d}p}{\text{d}t}\right)}_F+{\left(\frac{\text{d}l}{\text{d}t}\right)}_F+\left({\omega }_F\times l\right)$

6

where ${\left(\text{d}l/\text{d}t\right)}_F={\left({\dot{x}}_e,{\dot{y}}_e,{\dot{z}}_e\right)}^T$ , ${\left(\text{d}q/\text{d}t\right)}_E={\left[U_B,0,0\right]}^T$ , and $\left({\omega }_F\times l\right)={(0,k_2\left(s\right)\dot{s},k_1\left(s\right)\dot{s})}^T\times {\left(x_e,y_e,z_e\right)}^T$ .

The matrix form is

${\left(R_F^E\right)}^T\left[\begin{array}{c}{U}_B\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}{U}_P\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}{\dot{x}}_e\\ {\dot{y}}_e\\ {\dot{z}}_e\end{array}\right]+\left[\begin{array}{c}0\\ k_2(s)\dot{s}\\ k_1(s)\dot{s}\end{array}\right]\times \left[\begin{array}{c}{x}_e\\ y_e\\ z_e\end{array}\right]$

7

So the AUV dynamic model in 3-D space is

$\{\begin{array}{l}{\dot{x}}_e=k_1\left(s\right)\dot{s}{y}_e-k_2\left(s\right)\dot{s}{z}_e+U_B\text{cos}{\psi }_e\text{cos}{\theta }_e-U_P\\ {\dot{y}}_e=-k_1\left(s\right)\dot{s}{x}_e+U_B\text{sin}{\psi }_e\text{cos}{\theta }_e\\ {\dot{z}}_e=k_2\left(s\right)\dot{s}{x}_e-U_B\text{sin}{\theta }_e\end{array}$

8

$\{\begin{array}{l}{\dot{\psi }}_e=r/\text{cos}\theta +\dot{\beta }-k_1\left(s\right)\dot{s}\\ {\dot{\theta }}_e=q+\dot{\alpha }-k_2\left(s\right)\dot{s}\end{array}$

9

where α and β are the attack angle and drift angle of AUV, respectively.

$\{\begin{array}{l}\alpha =\text{arctan}\left(\frac{w}{u}\right),u\ge 0;\\ \beta =\text{arctan}\left(\frac{v}{\sqrt{u^2+w^2}}\right)\end{array}$

10

Controller design

Backstepping is to construct Lyapunov function through iterative recursion to get the input of the system, and it is usually presented to deal with the robust control of nonlinear system. ³⁶ In this article, we design the spatial path following controller of the underactuated AUV based on damping backstepping. ³⁴

Consider the following uncertain nonlinear system

$\dot{x}=f\left(x,t\right)+g\left(x,t\right)\left[u+\varphi \left(x\right)\Delta \left(t\right)\right]$

11

Assume that there is a continuous feedback control law ξ = α(x), α(0) = 0 and positive definite unbounded function V and semi positive definite function W. $\varphi \left(x\right)\Delta \left(t\right)$ are dynamic model uncertainties and bounded disturbances.

$\frac{\partial V}{\partial x}\left(x\right)\left[f\left(x\right)+g\left(x\right)\alpha \left(x\right)\right]\le -W\left(x\right)\le 0$

12

Then the stable input of the closed-loop system is as follows ³⁷

$u=\alpha \left(x\right)-k\frac{\partial V}{\partial x}\left(x\right)g\left(x\right){\left|\varphi \left(x\right)\right|}^2,k>0$

13

Pitch and yaw controller

To describe the AUV transient motion approaching to the desired path, the approaching angle ³⁸ is introduced to represent the desired values of yaw angle error and pitch angle error designed as

$\{\begin{array}{l}{\delta }_{\psi }=-{\lambda }_a\frac{e^{2m{y}_e}-1}{e^{2m{y}_e}+1},0<{\lambda }_a<\frac{\pi }{2},m>0\\ {\delta }_{\theta }={\lambda }_b\frac{e^{2n{z}_e}-1}{e^{2n{z}_e}+1},0<{\lambda }_b<\frac{\pi }{2},n>0\end{array}$

14

The goal is to follow the desired path by the guidance laws based on damping backstepping for the underactuated AUV, which makes θ_e and ψ_e converge to δ_θ and δ_ψ, respectively. To follow the desired path, we define the following error ³⁵

$z_1\triangleq \left[\begin{array}{c}{\tilde{\theta }}_e\\ {\tilde{\psi }}_e\end{array}\right]\triangleq \left[\begin{array}{c}{\theta }_e\\ {\psi }_e\end{array}\right]-\left[\begin{array}{c}{\delta }_{\theta }\\ {\delta }_{\psi }\end{array}\right]$

15

Differentiate z₁ with respect to time yields

${\dot{z}}_1=\left[\begin{array}{c}{\dot{\theta }}_e\\ {\dot{\psi }}_e\end{array}\right]-\left[\begin{array}{c}{\dot{\delta }}_{\theta }\\ {\dot{\delta }}_{\psi }\end{array}\right]\text{=}\left[\begin{array}{c}q+\dot{\alpha }-k_2\left(s\right)\dot{s}\\ r/\text{cos}\theta +\dot{\beta }-k_1\left(s\right)\dot{s}\end{array}\right]-\left[\begin{array}{c}{\dot{\delta }}_{\theta }\\ {\dot{\delta }}_{\psi }\end{array}\right]$

16

Then

${\dot{z}}_1=A\left(\theta \right)\left[\begin{array}{c}q\\ r\end{array}\right]+\Phi \left(s,\alpha ,\beta ,{\delta }_{\theta },{\delta }_{\psi }\right)$

17

where $A=\left[\begin{array}{cc}1& 0\\ 0& \text{cos}\theta \end{array}\right]$ and $\Phi =\left[\begin{array}{c}q+\dot{\alpha }-k_2\left(s\right)\dot{s}\\ r/\text{cos}\theta +\dot{\beta }-k_1\left(s\right)\dot{s}\end{array}\right]$

Define the following error

$z_2\triangleq \left[\begin{array}{c}\tilde{q}\\ \tilde{r}\end{array}\right]\triangleq \left[\begin{array}{c}q\\ r\end{array}\right]-\left[\begin{array}{c}{q}_d\\ r_d\end{array}\right]$

18

within

$\left[\begin{array}{c}{q}_d\\ r_d\end{array}\right]\triangleq -A^{-1}\left(\Phi +C_1{z}_1\right)$

19

Thus, $\Phi =-A\left[\begin{array}{c}{q}_d\\ r_d\end{array}\right]-C_1{z}_1$ and

${\dot{z}}_1=A{z}_2-C_1{z}_1$

20

where C₁ is a symmetric positive definite matrix.

Furthermore, we get equation (21) from equations (18) and (1)

$\begin{array}{l}{\dot{z}}_2=\left[\begin{array}{c}\dot{q}\\ \dot{r}\end{array}\right]-\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]=\left[\begin{array}{c}\frac{\left(T_q-d_q\right)}{m_q}\\ \frac{\left(T_r-d_r\right)}{m_r}\end{array}\right]-\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]\\ =\left[\begin{array}{cc}\frac{1}{m_q}& 0\\ 0& \frac{1}{m_r}\end{array}\right]\left[\begin{array}{c}{T}_q\\ T_r\end{array}\right]-\left[\begin{array}{c}\frac{d_q}{m_q}\\ \frac{d_r}{m_r}\end{array}\right]-\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]\end{array}$

21

within

$\{\begin{array}{l}{m}_q=I_y-M_{\dot{q}}\\ m_r=I_z-N_{\dot{r}}\\ d_r=-N_{uv}{u}_r{v}_r-N_{v\left|v\right|}{v}_r\left|v_r\right|-N_{ur}{u}_{r}r\\ d_q=-M_{q\left|q\right|}q\left|q\right|-M_{uq}{u}_{r}q-M_{uw}{u}_r{w}_r\\ +\left(z_{g}W-z_{b}B\right)\text{sin}\theta +m{z}_g\left(w_{r}q-v_{r}r\right)\end{array}$

where I_y and I_z are the moments of inertia. X_(⋅), Y_(⋅), Z_(⋅), M_(⋅), and N_(⋅) represent the AUV hydrodynamic coefficients.

Finally, we get the control laws as follows

$\left[\begin{array}{c}{T}_q\\ T_r\end{array}\right]={\left[\begin{array}{cc}\frac{1}{m_q}& 0\\ 0& \frac{1}{m_q}\end{array}\right]}^T\left(\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]+\left[\begin{array}{c}\frac{d_q}{m_q}\\ \frac{d_r}{m_r}\end{array}\right]-A^T{z}_1-C_2{z}_2\right)$

22

where C₂ is a symmetric positive definite matrix.

Then

$\left[\begin{array}{c}{\dot{z}}_1\\ {\dot{z}}_2\end{array}\right]=\left[\begin{array}{cc}-C_1& {\rm A}\\ -A^T& -C_2\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]$

23

Note that q_d and r_d are virtual control variables, and the controller is designed based on backstepping. For the dynamic uncertainty, the value is bounded but not measurable, so the nonlinear damping term is introduced to offset the disturbances. We get

$\begin{array}{l}\left[\begin{array}{c}{T}_q\\ T_r\end{array}\right]={\left[\begin{array}{cc}\frac{1}{m_q}& 0\\ 0& \frac{1}{m_r}\end{array}\right]}^T\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]+\left[\begin{array}{c}\frac{d_q}{m_q}\\ \frac{d_r}{m_q}\end{array}\right]-\left[\begin{array}{c}{P}_1{\beta }_1^2\left(q-q_d\right)\\ P_2{\beta }_2^2\left(r-r_d\right)\end{array}\right]-A^T{z}_1-C_2{z}_2)\end{array}$

24

where β₁ = q − q_d, β₂ = r − r _d , and P₁ and P₂ are positive constants.

Consider Lyapunov function

${\dot{V}}_1=\frac{1}{2}\Vert z_1^2\Vert +\frac{1}{2}\Vert z_2^2\Vert$

25

Differentiate V₁ with respect to time yields

${\dot{V}}_1=-z_1^T{C}_1{z}_1-z_2^T{C}_2{z}_2\le -\mu {\Vert z\Vert }^2$

26

where ${\dot{V}}_1$ is bounded by a negative definite quadratic function and μ > 0 is less than the minimum eigenvalue of C₁ and C₂.

Surge velocity controller

Define the following error

$z_3=u-u_d$

27

Consider Lyapunov function

$V_2=\frac{1}{2}{z}_3^2=\frac{1}{2}{\left(u-u_d\right)}^2$

28

Differentiate V₂ with respect to time yields

${\dot{V}}_2=\left(u-u_d\right)\left(\dot{u}-{\dot{u}}_d\right)$

29

To make the following error converge asymptotically, we select

${\alpha }_d={\dot{u}}_d-\lambda \left(u-u_d\right)$

30

where λ is a positive constant.

Nonlinear damping term defined as $-\left[P_3{\beta }_3{ }^2\left(u-u_d\right)\right]$ is to offset the uncertain disturbances. Finally, we get the longitudinal force as follows

$\begin{array}{c}{F}_u=m_u\left[{\dot{u}}_d-\lambda \left(u-u_d\right)\right]-{\text{X}}_{uu}{u}_r{ }^2-{\text{X}}_{vv}{v}_r{ }^2\\ -{\text{X}}_{ww}{w}_r{ }^2-{\text{X}}_{qq}q{q}^2-P_3{\beta }_3^2\left(u-u_d\right)\end{array}$

31

where β₃ = u − u_d and P₃ is a positive constant.

Stability analysis

Considering the 3-D continuous differential bounded desired path and the 3-D path following error model, we investigate dynamic controllers using backstepping method and virtual guiding speed designed by Lyapunov stability theorem. For the stability analysis of kinematics under current disturbances, we consider the following Lyapunov function

$V_3=\frac{1}{2}\left(x_e^2+y_e^2+z_e^2\right)$

32

Differentiate V₃ with respect to time yields

${\dot{V}}_3={\dot{x}}_e{x}_e+{\dot{y}}_e{y}_e+{\dot{z}}_e{z}_e$

33

Then

$\begin{array}{l}{\dot{V}}_3=\left[k\left(s\right)\dot{s}{y}_e-\tau \left(s\right)\dot{s}{z}_e+U_B\text{cos}{\psi }_e\text{cos}{\theta }_e-U_P\right]x_e\\ \hspace{1em}+\left[-k\left(s\right)\dot{s}{x}_e\text{+}{U}_B\text{sin}{\psi }_e\text{cos}{\theta }_e\right]y_e\\ \hspace{1em}+\left[\tau \left(s\right)\dot{s}{x}_e-U_B\text{sin}{\theta }_e\right]z_e\\ \hspace{0.17em}=U_B{x}_e\text{cos}{\psi }_e\text{cos}{\theta }_e-U_P{x}_e-U_B{z}_e\text{sin}{\theta }_e\\ \hspace{1em}+U_B{y}_e\text{sin}{\psi }_e\text{cos}{\theta }_e\end{array}$

34

Select virtual control variable

$U_P=\dot{s}=U_B\text{cos}{\psi }_e\text{cos}{\theta }_e+c_3{x}_e,c_3>0$

35

where c₃ is a positive constant to be determined and s represents desired path parameters.

Substitute equation (50) into equation (49), we get

${\dot{V}}_3=-U_B{z}_e\text{sin}{\theta }_e+U_B{y}_e\text{sin}{\psi }_e\text{cos}{\theta }_e-c_3{x}_e^2$

36

Note that ${\theta }_e\in \left(-\pi /2,0\right)$ , then $\text{cos}{\theta }_e>0$ . When the AUV is situated on the right or left side of the desired path, $y_e\ge 0,{\psi }_e\in \left(-\pi ,0\right)$ , then $y_e\text{sin}{\psi }_e\le 0$ ; when the AUV is situated above or below the desired path, $z_e\ge 0,{\theta }_e\in \left(0,\pi \right)$ , then $z_e\text{sin}{\theta }_e\ge 0$ . When the AUV is located in other situations, such as the upper right, we get $y_e\text{sin}{\psi }_e\le 0$ and $z_e\text{sin}{\theta }_e\ge 0$ . Therefore, the inequality ${\dot{V}}_3\le 0$ is satisfied, and the following errors x_e, y_e, and z_e can converge to zero asymptotically ³⁹

$\{\begin{array}{l}\underset{t\to \infty }{\text{lim}}{x}_e=0\\ \underset{t\to \infty }{\text{lim}}{y}_e=0\\ \underset{t\to \infty }{\text{lim}}{z}_e=0\end{array}$

Consider the following Lyapunov function for the closed-loop system

$V_4=V_1+V_2+V_3$

37

Differentiate V₄ with respect to time yields

$\begin{array}{l}{\dot{V}}_4=-z_1^T{C}_1{z}_1-z_2^T{C}_2{z}_2+\left(u-u_d\right)\left(\dot{u}-{\dot{u}}_d\right)\\ \hspace{1em}\hspace{0.17em}-U_B{z}_e\text{sin}{\theta }_e+U_B{y}_e\text{sin}{\psi }_e\text{cos}{\theta }_e-c_3{x}_e^2\\ \hspace{0.17em}\hspace{0.17em}\le -\mu {\Vert z\Vert }^2-\lambda {\left(u-u_d\right)}^2-P_0{\beta }_0^2{\left(u-u_d\right)}^2\\ \hspace{1em}\hspace{0.17em}-U_B{z}_e\text{sin}{\theta }_e+U_B{y}_e\text{sin}{\psi }_e\text{cos}{\theta }_e-k_1{x}_e^2\\ \hspace{0.17em}\hspace{0.17em}\le 0\end{array}$

38

It can be then concluded that the designed control laws based on damping backstepping can guarantee the whole control system steady.

Simulation

In this section, to verify the feasibility and robustness of the path following controller proposed in this article, we conduct simulations on the underactuated AUV WL-II (Figure 6) developed by Harbin Engineering University in China. ⁸ The model is represented using equation (1) and detailed parameters can be found in the studies by Liang et al. ^40,41

Firstly, the 3-D desired path is defined as a spiral in the following form

$\{\begin{array}{l}{x}_d=20\text{cos}\left(\frac{\pi }{24}s\right)\\ y_d=20\text{sin}\left(\frac{\pi }{24}s\right)\\ z_d=-s\end{array}$

Then, the initial position and attitude angle of AUV are $\left(x,y,z\right)=\left(10,-5,0\right)$ and $\left(\psi ,\theta \right)=\left(0,0\right)$ , respectively. The initial velocity matrix is $\left(u,v,w\right)\text{=}\left(0,0,0\right)$ , and the desired velocity is u_d = 1 m/s. The control law parameters are chosen as follows

$\lambda =5,P_1=P_2=P_3=0.5,{\lambda }_a=\pi /4,{\lambda }_b=\pi /20,m=0.2,n=1.5,C_1=\left[\begin{array}{cc}0.2& 0\\ 0& 10\end{array}\right],C_2=\left[\begin{array}{cc}1.5& 0\\ 0& 0.4\end{array}\right],c_3=0.4$

In deep sea, the current is assumed irrotational and constant (or slowly changing), so the current disturbances in this article can be defined as fixed value and the current velocities are chosen as $V_c={(0.3,-0.2,0)}^T$ in the geodetic coordinate system. ³⁵

To verify the performance of the controller design in this article, we compare and analyze the simulation results of AUV path following control using damping backstepping with traditional backstepping.

The simulation results are shown in Figures 2 to 9. Figure 2 shows the path following of AUV in 3-D space in the presence of current disturbances, and Figures 3 and 4 are the projection curves on the horizontal and vertical plane, respectively. As can be seen, the proposed controller based on virtual guiding speed and damping backstepping method shows its high performance, and the AUV trajectory can converge on the desired spatial path precisely compared to the traditional backstepping. Furthermore, the following errors converge faster using damping backstepping than traditional backstepping from Figure 5. Compared with the traditional backstepping, Figure 7, 8 and 9 can be clearly seen that the damping backstepping control method has better ability and stronger robustness of current disturbances and bounded dynamic uncertainty. In addition, the AUV control force and moments are relatively stable without chattering under the damping backstepping control and the effectiveness of the proposed control method is verified.

Figure 2.

3-D path following of AUV. 3-D: three-dimensional; AUV: autonomous underwater vehicle.

Figure 3.

Vertical projection for 3-D path following of AUV. 3-D: three-dimensional; AUV: autonomous underwater vehicle.

Figure 4.

Horizontal projection for 3-D path following of AUV. 3-D: three-dimensional; AUV: autonomous underwater vehicle.

Figure 5.

Path following errors of AUV. AUV: autonomous underwater vehicle.

Figure 6.

WL-II underactuated AUV. AUV: autonomous underwater vehicle.

Figure 7.

Velocity response of AUV. AUV: autonomous underwater vehicle.

Figure 8.

Angle response of AUV. AUV: autonomous underwater vehicle.

Figure 9.

Control force and moments of AUV. AUV: autonomous underwater vehicle.

Conclusion

This article considers the problems of 3-D path following for the underactuated AUV in the presence of current disturbances. Path following error model of AUV in Serret–Frenet coordinate frame is established, and the virtual guiding speed is designed based on Lyapunov stability theory and backstepping to make the AUV follow the desired path and guarantee the closed-loop control system to converge finally. Moreover, we simplify the virtual variable design of the controller, which can avoid the singular value problems of the traditional backstepping in solving the controller and dynamic model. Nonlinear damping function is introduced to offset the effects of dynamic uncertainty and current disturbances. Considering the curvature and torsion of the following curve, the approaching angle is introduced to guarantee fast convergence of error. Simulation results show that the 3-D path following controller for underactuated AUVs proposed in this article is effective and robust to current disturbances greatly, which indicates high accurate tracking capacity and good robustness.

Future research will address the extension of these results, and path following controller based on nonlinear current observer should be developed to reconstruct the velocities based on kinematic and dynamic models, so the disturbances can be counteracted in the controller. Thus, how to design disturbance control law effectively for constant unknown current disturbances is the key work to achieve AUV precise path following control.