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Abstract

This article studies the trajectory tracking of an underactuated unmanned surface vehicle. The trajectory tracking controller is divided into two parts: the kinematics controller and the dynamic controller. Taking the hull coordinate system of the unmanned surface vehicle as the datum plane, the kinematics controller is designed based on the backstepping method, and the virtual control input of the speed and the heading deviation of the ship are obtained. A dynamic controller is designed through an exponential sliding surface to stabilize the error of speed and heading deviation, thereby stabilizing the position error. In the process of designing the controller, the stability of the entire closed-loop control system is analyzed using Lyapunov stability theory. Finally, the designed numerical simulation experiment is carried out. The proposed exponential sliding mode controller is compared with the global integral sliding mode controller, and it is verified that the proposed controller has better effectiveness and robustness in the trajectory tracking of the underactuated unmanned surface vehicle.

Keywords

Trajectory tracking unmanned surface vehicle backstepping exponential sliding mode Lyapunov

Introduction

Unmanned surface vehicle (USV) can perform tasks in various unknown and complex surface environments due to its autonomy, intelligence, and modularity. It is widely used in meteorological watermark measurement, island coke surveying and mapping, marine pollution prevention, rescue, subsea exploration, and other dangerous missions with high coefficients and harsh environments.^1–4 Research on trajectory tracking and control of underactuated USV has received wide attention in recent years. Trajectory tracking control means that the ship reaches the target position within a specified time according to the route designed by the planner, so as to realize the safe and efficient operation of the ship. Generally speaking, USV has no lateral propulsion, which makes its motion model underactuated. At the same time, the external interference and the unmodeled part of the dynamic system make the USV have unpredictable nonlinear characteristics, which makes it extremely difficult to carry out trajectory tracking control. At present, the main methods of underactuated USV trajectory tracking control include feedback linearization method, predictive control method, backstepping method, and sliding mode control.

The feedback linearization method can simplify the complex model, which makes it easy to adjust the parameters. Lefeber et al.⁵ proposed a simple state feedback control law and proved that the control law can make the dynamic and global tracking errors exponentially stable and carried out the ship model experiment. Paliotta et al.^6,7 determined a forehand position on the USV and controlled the movement of this point along the trajectory based on the input–output feedback linearization method. However, the feedback linearization method requires a more accurate mathematical model. If the mathematical model cannot be accurately established, the part that has the greater impact on the system will cause the controller to fail to stabilize the system.

The predictive control algorithm is convenient to support various constraint conditions, and predictive fitting is performed as much as possible through rolling optimization, which has a good control effect. Culverhouse et al.⁸ proposed a robust adaptive autopilot based on model predictive controller (MPC), which can handle sudden changes in system dynamics. Liu et al.⁹ used differential homomorphism and state-dependent Riccati equation (SDRE) theory to design the control law for the underactuated surface ship with limited input and used the convex optimization method to solve the constrained control input of the system. Liu et al.¹⁰ proposed a nonlinear model predictive control, where system constraints on inputs, input increments, and outputs are incorporated into the nonlinear MPC framework. Oh and Sun¹¹ proposed an MPC scheme with line-of-sight (LOS) path generation capability using a quadratic programming method to solve the ship’s LOS model linearization. But the predictive control algorithm requires a lot of computing power because it needs a lot of online optimization.

Backstepping is a control method based on the idea of compensation, which decomposes a complex system into simpler subsystems and then reversely controls the system by introducing virtual control variables. It is often used to combine with other controls. Djapic and Nad¹² introduced the design and simulation implementation of a nonlinear controller for underactuated USV based on the command filter backstepping method. Dong et al.¹³ proposed a backstepping control algorithm based on state feedback, adding an overall action to the controller to further enrich the steady-state performance and control accuracy of the USV trajectory tracking control system.

In recent years, the sliding mode control method has received extensive attention from scholars due to its insensitivity and robustness to parameter changes and studies on its application in the industrial field.^14,15 Structure change of sliding mode control has nothing to do with the object parameters and disturbances. Instead, the current state is used to adjust the structure to force the system state to enter the sliding mode surface and gradually stabilize along the sliding mode surface. Ovalle et al.¹⁶ introduced some sliding variables with relative degrees to three different types of underactuated systems. Kim and Lee¹⁷ proposed a robust sliding mode control method with disturbance observer and state observer, which achieved robust control of disturbances by controlling the angle of the water jet system. Jian et al.¹⁸ combines backstepping control and sliding mode control to improve the robustness of the unmanned submarine under system uncertainty and environmental interference. However, due to the sgn function discontinuous switching of traditional sliding mode control, chattering exists in the initial state. In terms of eliminating chattering, Sun et al.¹⁹ introduced a continuous differentiable approximate saturation function, and Sun et al.²⁰ constructed a continuous adaptive term to reduce the system chattering phenomenon. Shen et al.²¹ constructed an integral sliding mode surface by an observer with the quantized output measurement and treated $H_{\infty}$ sliding model control of Markov jump systems with general transition probabilities. Shen et al.²² considered the influence of external disturbances based on their previous study²¹ and established an integral sliding mode output (ISMO) control with anti-interference strategy to suppress the interference.

The traditional sliding mode memory has good robustness, but there is a problem of convergence in unknown time. In order to suppress chattering and overcome the unknown time convergence problem of traditional sliding mode control, Venkataraman and Gulati²³ proposed a terminal sliding mode control. But when the system is far from the equilibrium point, the convergence speed of ordinary terminal sliding mode control is lower than that of a traditional sliding mode. Some scholars have proposed a global terminal sliding mode control, which enables the system to converge quickly under the combined action of linear and nonlinear surfaces. Guan and Ai²⁴ applied a global terminal sliding mode control to the stabilization of USV.

The new global terminal sliding mode control currently mainly used in single-input single-output systems is not applied in underactuated systems, especially in high-order underactuated systems. In this article, combined with the exponential terminal sliding mode control surface,^25,26 a new underactuated USV trajectory tracking control method is proposed. The analysis of the controller and the design of the control law about exponential terminal sliding mode control surface is limited to the linear system, and there is no further study on the analysis and application on the nonlinear system. Therefore, this research method has not been applied to nonlinear underactuated systems at present. According to the motion mathematical model of USV, this article first converts the mathematical model of inertial coordinate system into a hull coordinate system so that it facilitates calculation. Second, through the analysis of the Lyapunov stability criterion based on the horizontal and vertical coordinate positions, the virtual control quantity of speed and heading error are determined. Then, analysis and deduction based on the exponential terminal dynamic sliding mode determine the exponential terminal sliding mode control law that satisfies the second-order dynamic system, and finally verify the algorithm effectiveness through numerical experiments compared with the global integral sliding mode controller.

The main contributions of the job in this article can be summarized as follows:

The differential homeomorphism transformation is used to transform the dynamic mathematical model of USV in geodetic coordinate system into that in a hull coordinate system, which then makes it convenient to determine the virtual control quantity for a backstepping method later.

Exponential terminal sliding mode is applied to a multi-input and multi-output underactuated nonliner system. The mathematical model of USV is further transformed to change the original three-input control variables into two-input control variables through numerical derivation. Then the second-order exponential terminal sliding mode controller is constructed, which improves the trajectory tracking ability of the USV.

The parameter uncertainty of the USV and the influence of the external environment are considered, and the feasibility of the designed control law is verified by through Lyapunov criterion. The designed control law is also compared with the global integral sliding mode controller in the numerical test; results show that the proposed control method can not only converge rapidly but also restrain chattering effectively due to no switching surface.

The organization structure of this article is as follows: Section “Mathematical model of USV” introduces the motion mathematical model of USV. Section “Virtual control design based on Lyapunov” designs the virtual control variables through the reverse step method based on the hull coordinate system as the reference coordinate. Section “Exponential sliding mode controller” uses the exponential sliding surface to design control laws. In section “Numerical Simulation,” comparative simulation experiments are conducted to prove the superiority of the controller in this article. Section “Conclusion” makes a simple summary.

Mathematical model of USV

The trajectory control of USV is the control of underactuated system. The main control variables are the propulsion force and steering torque, and the controlled quantities are the position changes of USV in three directions: pitch, yaw, and bow roll. There is a complex nonlinear relationship in the motion process, which cannot be controlled directly by sliding mode control algorithm. The kinematics equation of the USV in the hull coordinate system can be obtained by the differential homeomorphism change, which makes the mathematical model of the position change linearized and facilitates the determination of the virtual control variable. The relationship between the control variable and the three controlled variables can also be obtained intuitively, which is convenient for further analysis.

Kinematics model

The USV has 6 degrees of freedom (DOF). When considering the trajectory tracking control of the USV, this article refers to a study by Sun et al.,²⁷ and only 3-DOF, namely pitch, yaw, and bow roll need to be considered. It is assumed that the USV is symmetric, and the diagram of the 3-DOF plane coordinate system is shown in Figure 1.

Figure 1.

Schematic diagram of 3-DOF coordinate of USV.

$o - xy$ represents the fixed coordinate system of the earth, $G - h_{x} h_{y}$ represents the hull coordinate system of the USV, its moving speed is V, which can decompose in the attached coordinate system, and the component on the $h_{y}$ axis is v, which is the lateral speed of the USV, and the component in the $h_{x}$ axis direction is u, which is the longitudinal speed of the USV. $ψ$ represents the heading angle, and $θ$ represents the angular velocity.

Through analysis, the simplified kinematics model of USV can be obtained as follows

\overset{\cdot}{η} = J (ψ) v

(1)

where $η = [x, y, ψ]^{T}, v = [u, v, r]^{T}$ , $J (ψ)$ represents the rotation matrix from the hull coordinate system to the inertial coordinate system, the expression is as follows

J (ψ) = [\begin{matrix} \cos ψ & - \sin ψ & 0 \\ \sin ψ & \cos ψ & 0 \\ 0 & 0 & 1 \end{matrix}]

(2)

where $J (ψ)^{- 1} = J (ψ)^{T}$ can be derived, and the state space expression of the kinematics model of the USV is expressed as

{\begin{matrix} \overset{\cdot}{x} = u \cos ψ - v \sin ψ \\ \overset{\cdot}{y} = u \sin ψ + v \cos ψ \\ \overset{\cdot}{ψ} = r \end{matrix}

(3)

Dynamics model

The dynamics mathematical model of the USV can be described as

M \overset{\cdot}{v} + C (v) v + Dv + τ_{d} = τ

(4)

where $M = diag {m_{11}, m_{22}, m_{33}}$ is the additional inertia matrix, and $D = diag {d_{11}, d_{22}, d_{33}}$ is the hydrodynamic damping parameter matrix. $η$ and v are consistent as above. $τ = [τ_{u}, 0, τ_{r}]^{T}$ , $τ_{u}$ and $τ_{r}$ are the longitudinal propulsion and steering moment, respectively. $C (v)$ is the Coriolis and centripetal force matrix. $τ_{d} = [τ_{1 dis}, τ_{2 dis}, τ_{3 dis}]^{T}$ , $τ_{d}$ is the disturbances of the unmodeled parts and environment such as wind, waves, currents, and so on. $τ_{1 dis}$ , $τ_{2 dis}$ , and $τ_{1 dis}$ are the disturbance forces and moments in the longitudinal, lateral, and steering angles, respectively. Then, the state space equation expression of the USV’s horizontal dynamics model

{\begin{matrix} \overset{\cdot}{u} = \frac{m_{22}}{m_{11}} vr - \frac{d_{11}}{m_{11}} u + \frac{τ_{u} + τ_{1 dis}}{m_{11}} \\ \overset{\cdot}{v} = - \frac{m_{11}}{m_{22}} ur - \frac{d_{22}}{m_{22}} v + \frac{τ_{2 dis}}{m_{22}} \\ \overset{\cdot}{r} = \frac{m_{11} - m_{22}}{m_{33}} uv - \frac{d_{33}}{m_{33}} r + \frac{τ_{r} + τ_{3 dis}}{m_{33}} \end{matrix}

(5)

Among them, the disturbance is satisfied with $| τ_{1 dis} | \leq τ_{1 max} < + \infty$ , $| τ_{2 dis} | \leq τ_{2 max} < + \infty$ , and $| τ_{3 dis} | \leq τ_{3 max} < + \infty$ .

Motion model in hull coordinate system

According to equation (2), the kinematics model of USV, the formula for transforming the inertial coordinate system into the hull coordinate system can be obtained as follows

[\begin{matrix} h_{x} \\ h_{y} \\ h_{r} \end{matrix}] = [\begin{matrix} \cos ψ & \sin ψ & 0 \\ - \sin ψ & \cos ψ & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ r \end{matrix}]

(6)

According to equation (6), USV’s longitudinal coordinate, transverse coordinate, heading angle, longitudinal speed, transverse speed, and angular speed in the hull coordinate system can be as

\begin{matrix} (h_{1}, h_{2}, h_{3}, h_{4}, h_{5}, h_{6}) = (x \cos ψ + y \sin ψ, - x \sin ψ \\ + y \cos ψ, ψ, u, v, r) \end{matrix}

(7)

The differential homeomorphism transformation is performed on the USV system, and the new state space equation is obtained as

{\begin{matrix} {\overset{\cdot}{h}}_{1} = h_{4} + h_{2} h_{6} \\ {\overset{\cdot}{h}}_{2} = h_{5} - h_{1} h_{6} \\ {\overset{\cdot}{h}}_{3} = h_{6} \\ {\overset{\cdot}{h}}_{4} = w_{1} \\ {\overset{\cdot}{h}}_{5} = w_{2} \\ {\overset{\cdot}{h}}_{6} = w_{3} \end{matrix}

(8)

Among them, $w_{1}$ , $w_{2}$ , and $w_{3}$ are the conversion of control input and disturbance synthesis, the formula is as follows

{\begin{matrix} w_{1} = \frac{m_{22}}{m_{11}} h_{5} h_{6} - \frac{d_{11}}{m_{11}} h_{4} + \frac{(τ_{u} + t_{1 dis})}{m_{11}} \\ w_{2} = \frac{m_{11}}{m_{22}} h_{4} h_{6} - \frac{d_{11}}{m_{22}} h_{5} + \frac{t_{2 dis}}{m_{22}} \\ w_{3} = \frac{(m_{11} - m_{22})}{m_{33}} h_{4} h_{6} - \frac{d_{33}}{m_{33}} h_{6} + \frac{(τ_{r} + t_{3 dis})}{m_{33}} \end{matrix}

(9)

Because $w_{1}$ and $w_{3}$ can be controlled by $τ_{u}$ and $τ_{r}$ directly, respectively. $w_{2}$ is related to $w_{1}$ and $w_{3}$ , let $w_{1} = t_{u}$ , $w_{3} = t_{r}$ , we can get the final-state space equation of USV in the hull coordinate system as

{\begin{matrix} {\overset{\cdot}{h}}_{1} = h_{6} h_{2} + h_{4} \\ {\overset{\cdot}{h}}_{2} = - h_{6} h_{1} + h_{5} \\ {\overset{\cdot}{h}}_{3} = h_{6} \\ {\overset{\cdot}{h}}_{4} = t_{u} \\ {\overset{\cdot}{h}}_{5} = - B h_{5} - A h_{4} h_{6} + T_{2 dis} \\ {\overset{\cdot}{h}}_{6} = t_{r} \end{matrix}

(10)

where $B = m_{11} / m_{11} m_{22} m_{22}$ , $A = d_{11} / d_{11} m_{22} m_{22}$ , and $T_{2 dis} = τ_{2 dis} / τ_{2 dis} m_{22} m_{22}$ .

Virtual control design based on Lyapunov

After obtaining the motion mathematical model of the USV in the hull coordinate system, the model needs to be further changed by combining the tracking trajectory to change the underactuated model into the fully actuated model. In this section, according to the kinematics model of USV, the tracking trajectory is changed by differential homeomorphism to obtain the reference position change of the trajectory on the hull coordinate system. Then, combined with the Lyapunov criterion, two virtual control input can be obtained, which is the three controlled quantities are transformed into two controlled quantities.

Lyapunov in the hull coordinate system

According to the second method of Lyapunov, the asymptotic stability of the system can be proved. In order to make the system eventually uniformly asymptotically stable, define the Lyapunov function in the inertial coordinate system

V_{o} = \frac{1}{2} (x_{e}^{2} + y_{e}^{2})

(11)

where the coordinate error of the USV is $[\begin{matrix} x_{e} \\ y_{e} \end{matrix}] = [\begin{matrix} x - x_{r} \\ y - y_{r} \end{matrix}]$ , $x_{r}$ and $y_{r}$ are the coordinates of the reference trajectory in the x and y axes of the inertial coordinate system, respectively. Equation (11) can be equivalent to equation as below

V = \frac{1}{2} (h_{xe}^{2} + h_{ye}^{2})

(12)

where $h_{xe}$ and $h_{ye}$ represent the longitudinal and lateral coordinate errors of the USV in the hull coordinate system, respectively, and they can be proved as below.

Proof

In order to facilitate calculations, we can set $A = [\begin{matrix} \cos ψ & \sin ψ \\ - \sin ψ & \cos ψ \end{matrix}]$ , which is an invertible matrix, $A^{- 1} = A^{T}$ . The transformation of $x_{r}$ and $y_{r}$ in the hull coordinate system is

[\begin{matrix} x_{r} \\ y_{r} \end{matrix}] = A^{T} [\begin{matrix} h_{1 r} \\ h_{2 r} \end{matrix}]

(13)

where $h_{1 r}$ and $h_{1 r}$ represent the coordinates in the inertial coordinate system, respectively. Then, the coordinate error in the inertial coordinate system can be expressed as

[\begin{matrix} x_{e} \\ y_{e} \end{matrix}] = A^{T} [\begin{matrix} h_{1} - h_{1 r} \\ h_{2} - h_{2 r} \end{matrix}] = A^{T} [\begin{matrix} h_{xe} \\ h_{ye} \end{matrix}]

(14)

Therefore, the original Lyapunov function can be transformed into the Lyapunov function in hull coordinates as

\begin{matrix} V_{o} = \frac{1}{2} [h_{xe} h_{ye}] A A^{T} [\begin{matrix} h_{xe} \\ h_{ye} \end{matrix}] = \frac{1}{2} (h_{xe}^{2} + h_{ye}^{2}) \end{matrix}

(15)

Virtual control quantity design

According to the kinematics model in section “Kinematics model,” let the heading of the reference trajectory be $ψ_{r} = \arctan ({\overset{\cdot}{y}}_{r} / {\overset{\cdot}{y}}_{r} {\overset{\cdot}{x}}_{r} {\overset{\cdot}{x}}_{r})$ , and the kinematics model of the reference trajectory is

{\begin{matrix} {\overset{\cdot}{x}}_{r} = u_{r} \cos ψ_{r} - v_{r} \sin ψ_{r} \\ {\overset{\cdot}{y}}_{r} = u_{r} \sin ψ_{r} + v_{r} \cos ψ_{r} \\ {\overset{\cdot}{ψ}}_{r} = r_{r} \end{matrix}

(16)

According to equations (7) and (8), transform the coordinates of the reference point to

{\begin{matrix} h_{1 r} = x_{r} \cos (ψ) + y_{r} \sin (ψ) \\ h_{2 r} = - x_{r} \sin (ψ) + y_{r} \cos (ψ) \end{matrix}

(17)

Carrying out the first-order differentiation on the time of equation (17), we get

{\begin{matrix} {\overset{\cdot}{h}}_{1 r} = {\overset{\cdot}{x}}_{r} \cos (ψ) - x_{r} h_{6} \sin (ψ) + {\overset{\cdot}{y}}_{r} \sin (ψ) + y_{r} h_{6} \cos (ψ) \\ {\overset{\cdot}{h}}_{2 r} = - {\overset{\cdot}{x}}_{r} \sin (ψ) - x_{r} h_{6} \cos (ψ) + {\overset{\cdot}{y}}_{r} \cos (ψ) - y_{r} h_{6} \sin (ψ) \end{matrix}

(18)

Organizing equation (18), we get

{\begin{matrix} {\overset{\cdot}{h}}_{1 r} = h_{4 r} + h_{2 r} h_{6} \\ {\overset{\cdot}{h}}_{2 r} = h_{5 r} - h_{1 r} h_{6} \\ h_{4 r} = {\overset{\cdot}{x}}_{r} \cos ψ + {\overset{\cdot}{y}}_{r} \sin ψ \\ h_{5 r} = - {\overset{\cdot}{x}}_{r} \sin ψ + {\overset{\cdot}{y}}_{r} \cos ψ \end{matrix}

(19)

From $h_{xe} = h_{1} - h_{1 r}$ , $h_{ye} = h_{2} - h_{2 r}$ , then ${\overset{\cdot}{h}}_{xe} = {\overset{\cdot}{h}}_{1} - {\overset{\cdot}{h}}_{1 r}$ , ${\overset{\cdot}{h}}_{ye} = {\overset{\cdot}{h}}_{2} - {\overset{\cdot}{h}}_{2 r}$ , combine equations (7), (8), (19) to get equation (20)

{\begin{matrix} {\overset{\cdot}{h}}_{xe} = H_{4} + h_{ye} h_{6} \\ {\overset{\cdot}{h}}_{ye} = H_{5} - h_{xe} h_{6} \\ H_{4} = h_{4} - h_{4 r} \\ H_{5} = h_{5} - h_{5 r} \end{matrix}

(20)

After taking the time derivative of equation (12), combine equation (20) to obtain

\overset{\cdot}{V} = h_{xe} H_{4} + h_{ye} H_{5}

(21)

After substituting equations (7) and (19) into equation (21), we have

\begin{matrix} \overset{\cdot}{V} = h_{xe} (h_{4} - ({\overset{\cdot}{x}}_{r} \cos ψ + {\overset{\cdot}{y}}_{r} \sin ψ)) \\ + h_{ye} (h_{5} - (- {\overset{\cdot}{x}}_{r} \sin ψ + {\overset{\cdot}{y}}_{r} \cos ψ)) \end{matrix}

(22)

Let $ψ_{e} = ψ - ψ_{r}$ , then $ψ = ψ_{e} + ψ_{r}$ , according to the trigonometric function combination equation (16), we can get

\begin{matrix} \overset{\cdot}{V} = h_{xe} (h_{4} - u_{r} \cos ψ_{e} - v_{r} \sin ψ_{e}) \\ + h_{ye} (h_{5} - v_{r} \cos ψ_{e} + u_{r} \sin ψ_{e}) \end{matrix}

(23)

Because $\tan ψ_{r} = \sin ψ_{r} / \sin ψ_{r} \cos ψ_{r} \cos ψ_{r} = {\overset{\cdot}{y}}_{r} / {\overset{\cdot}{y}}_{r} {\overset{\cdot}{x}}_{r} {\overset{\cdot}{x}}_{r}$ is substituted into equation (16), $v_{r} = 0$ can be obtained; therefore

\overset{\cdot}{V} = h_{xe} (h_{4} - u_{r} \cos ψ_{e}) + h_{ye} h_{5} + h_{ye} u_{r} \sin ψ_{e}

(24)

Taking the USV propulsion speed and heading deviation solution as the virtual control input of the kinematics model sub-controller and $ψ_{e} = ψ - ψ_{r}$ , its expected value can be designed as follows

{\begin{matrix} u_{d} = - k_{1} h_{xe} + u_{r} \cos ψ_{e} \\ ψ_{d} = - \arcsin (k_{2} h_{ye} u_{r}) + ψ_{r} \end{matrix}

(25)

where $k_{1}$ and $k_{2}$ are the adjustable gain coefficients, which need to be satisfied with $k_{1} > 0$ , $k_{2} > 0$ . Because the value of the trigonometric sine function is [–1, 1], then require $- 1 \leq k_{2} h_{ye} u_{1} \leq 1$ . When the tracking error $h_{xe}$ , $h_{ye}$ , and $ψ_{e}$ are all 0, the virtual control input $u_{d}$ and $ψ_{d}$ are, respectively, equal to the expected thruster speed $u_{r}$ and the expected ship heading $ψ_{r}$ . Because $sgn (\sin ψ_{e}) = sgn (\tan ψ_{e})$ , so in the case $k_{1} > 0$ , $k_{2} > 0$ , only set the virtual control input as below

{\begin{matrix} u_{d} = - k_{1} h_{xe} + u_{r} \cos ψ_{e} \\ ψ_{d} = - \arctan (k_{2} h_{ye} u_{r}) + ψ_{r} \end{matrix}

(26)

Then, substituting $u_{d}$ and $ψ_{d}$ into equation (24), we can get

\overset{\cdot}{V} = - k_{1} h_{xe}^{2} + h_{ye} h_{5} - k_{2} \overset{2}{\tan} ψ_{e}

(27)

Due to $u_{d}$ and $ψ_{d}$ are not really controllable variables, the dynamic control law and the auxiliary control law should be designed to ensure that $\overset{\cdot}{V}$ is negative, so that $h_{xe}$ and $h_{ye}$ are finally stabilized.

Let $V_{v} = 1 / 12 2 (h_{5}^{2})$ , differentiating in time, then combining the values ${\overset{\cdot}{h}}_{5}$ in equations (7) and (10) to obtain equation (28)

{\overset{\cdot}{V}}_{v} = h_{5} (- B h_{5} - A h_{4} h_{6} + T_{2 dis})

(28)

Because the values of B and A are larger than 0, when $| h_{5} | > (A h_{4} h_{6} - T_{2 dis}) / B$ , $| h_{5} |$ is decreasing, that is, $\overset{\cdot}{V} < 0$ . So when $T_{2 dis}$ is bounded, $h_{5}$ can be regarded as absolutely bounded, that is

| h_{5} | \leq \frac{| (A h_{4} h_{6} - T_{2 dis}) |}{B}

(29)

Since the lateral movement of the USV is mainly produced by the angular velocity r, that is $h_{6}$ . When $ψ_{e}$ tends to 0, $h_{6}$ also tends to 0, then $h_{5}$ also tends to 0, so $h_{ye} h_{5}$ can be stabilized by adjusting $k_{2}$ in the third term of equation (24). And finally make $\overset{\cdot}{V}$ a negative value, so $h_{5}$ can be regarded as $h_{5} \approx 0$ .²⁸

Exponential sliding mode controller

After the virtual controlled quantity is determined as above, the control quantity controls the second derivative of the virtual controlled quantity $ψ_{d}$ . According to the construction method of virtual control quantity, the second-order virtual control quantity is derived, while the second-order exponential terminal sliding mode surface is constructed based on the construction method of exponential terminal sliding mode. Then, the final control law is derived by the new virtual control quantity and sliding mode surface.

Exponential terminal sliding mode

The exponential terminal sliding mode control has a fast asymptotic tracking effect and anti-buffing effect. The first-order dynamic system sliding mode surface^25,26 is

\begin{matrix} S = \overset{\cdot}{x} + \frac{α}{k} (e^{k | x |} - 1) sgn (x) \\ + \frac{β}{k} {(1 - e^{- k | x |})}^{q / p} e^{k | x |} sgn (x) \end{matrix}

(30)

In the equation (30), $x \in R$ , $α > 0$ , $β > 0$ ; p and q are all positive odd numbers, and p is larger than q; $0 < k \leq 1$ .

According to equation (30), when $S = 0$ , the convergence time is

t_{s} = \frac{p}{α (p - q)} \ln \frac{α {(1 - e^{- k | x (0) |})}^{\frac{(p - q)}{p}} + β}{β}

(31)

The convergence time of fast terminal sliding mode control is²⁶

t_{s 1} = \frac{p}{α (p - q)} \ln \frac{α | x {(0) |}^{\frac{(p - q)}{p}} + β}{β}

(32)

On the exponential terminal sliding mode, the system is gradual and stable, and at the same time, it takes a faster time to reach the equilibrium point than $t_{s 1}$ .

Control law design

In order to make the USV’s thruster speed and heading angle reach the desired value, it is necessary to design the auxiliary control laws so that the thruster speed tracking error and the heading angle tracking error eventually approach 0. The thruster speed error and heading angle error are defined as follows

u_{E} = u - u_{d}, ψ_{E} = ψ - ψ_{d}

(33)

Propulsion control law $τ_{u}$ design

In order to stabilize the thruster speed error $u_{E}$ , the exponential global terminal sliding mode dynamic surface is selected as below

\begin{matrix} S_{u 1} = {\overset{\cdot}{u}}_{E} + \frac{α_{0}}{k_{u 1}} (e^{k_{u 1} | u_{E} |} - 1) sgn (u_{E}) \\ + \frac{β_{0}}{k} {(1 - e^{- k_{u 1} | u_{E} |})}^{\frac{q_{0}}{p_{0}}} e^{k_{u 1} | u_{E} |} sgn (u_{E}) \end{matrix}

(34)

Taking the time derivative of $u_{E}$ , combined with equation (5), we can get

{\overset{\cdot}{u}}_{E} = \frac{(m_{22} vr - d_{11} u + τ_{u} + τ_{1 dis} - {\overset{\cdot}{u}}_{d} m_{11})}{m_{11}}

(35)

Let $S_{u 1} = 0$ , suppose $| τ_{1 dis} - {\overset{\cdot}{u}}_{d} m_{11} | < + \infty$ , combined with equation (35), obtain the equivalent control law

\begin{matrix} τ_{u} = - m_{22} vr + d_{11} u - m_{11} \frac{α_{0}}{k_{u 1}} (e^{k_{u 1} | u_{E} |} - 1) sgn (u_{E}) \\ - m_{11} \frac{β_{0}}{k_{u 1}} {(1 - e^{- k_{u 1} | u_{E} |})}^{\frac{q_{0}}{p_{0}}} e^{k_{u 1} | u_{E} |} sgn (u_{E}) \end{matrix}

(36)

For proof $τ_{u}$ makes $u_{E}$ the final stabled, design the Lyapunov function

V_{u 1} = \frac{1}{2} m_{11} u_{E}^{2}

(37)

Taking the time derivative of $V_{u 1}$ , combined with the equations (35) and (36), we get

\begin{matrix} {\overset{\cdot}{V}}_{u 1} = - (α_{0} + β_{0} {(e^{k_{u 1} | u_{E} |} - 1)}^{\frac{q_{0} - p_{0}}{p_{0}}} e^{\frac{k_{u 1} | u_{E} | (q_{0} + p_{0})}{q_{0}}}) \\ \frac{| u_{E} | (e^{k_{u 1} | u_{E} |} - 1)}{k_{u 1}} + (τ_{1 dis} - {\overset{\cdot}{u}}_{d} m_{11}) u_{E} \end{matrix}

(38)

Because ${\overset{\cdot}{u}}_{d} = - k_{1} {\overset{\cdot}{h}}_{xe} + u_{r} r \cos (\overset{\cdot}{ψ} - {\overset{\cdot}{ψ}}_{r})$ , suppose that ${\overset{\cdot}{h}}_{xe} < + \infty, {\overset{\cdot}{ψ}}_{r} < + \infty$ , so $| τ_{1 dis} - {\overset{\cdot}{u}}_{d} m_{11} | < + \infty$ , and

e^{k_{u 1} | u_{E} |} - 1 \geq 0

(39)

Therefore, only need to adjust the appropriate $α_{0}$ and $β_{0}$ to make ${\overset{\cdot}{V}}_{u 1} < 0$ , which is making $u_{E}$ asymptotically stable.

Steering control law $τ_{r}$ design

Taking the second derivative of $ψ_{E} = ψ - ψ_{d}$ in time, we get

{\begin{matrix} {\overset{\cdot}{ψ}}_{E} = r - {\overset{\cdot}{ψ}}_{d} \\ {\overset{\cdot\cdot}{ψ}}_{E} = \overset{\cdot}{r} - {\overset{\cdot\cdot}{ψ}}_{d} \end{matrix}

(40)

where r is not directly controllable, only $\overset{\cdot}{r}$ is directly controllable, so the second-order exponential dynamic sliding mode surface is selected as below

{\begin{matrix} S_{r 0} = {\overset{\cdot}{ψ}}_{E} + \frac{α_{1}}{k_{r 1}} (e^{k_{r 1} | ψ_{E} |} - 1) sgn (ψ_{E}) \\ + \frac{β_{1}}{k_{r 1}} {(1 - e^{- k_{r 1} | ψ_{E} |})}^{\frac{q_{1}}{p_{1}}} e^{k_{r 1} | ψ_{E} |} sgn (ψ_{E}) \\ S_{r 1} = k_{3} {\overset{\cdot}{S}}_{r 0} + \frac{α_{1}}{k_{r 1}} (e^{k_{r 1} | S_{r 0} |} - 1) sgn (S_{r 0}) \\ + \frac{β_{1}}{k_{r 1}} {(1 - e^{- k_{r 1} | S_{r 0} |})}^{\frac{q_{1}}{p_{1}}} e^{k_{r 1} | S_{r 0} |} sgn (S_{r 0}) \end{matrix}

(41)

Combining ${\overset{\cdot\cdot}{ψ}}_{E} = \overset{\cdot}{r} - {\overset{\cdot\cdot}{ψ}}_{d}$ with equation (5), we can get

{\overset{\cdot\cdot}{ψ}}_{E} = \frac{((m_{11} - m_{22}) uv - d_{33} r + τ_{r} + τ_{3 dis} - {\overset{\cdot\cdot}{ψ}}_{d} m_{33})}{m_{33}}

(42)

Let $S_{r 1} = 0$ , according to the previous section, we can also prove $| τ_{3 dis} - {\overset{\cdot\cdot}{ψ}}_{d} m_{33} | < + \infty$ . Through the joint equations (41) and (42), we can get the equivalent control law

\begin{matrix} τ_{r} = (m_{22} - m_{11}) uv + d_{33} r - m_{33} \frac{α_{1}}{k_{r 1}} (e^{k_{r 1} | ψ_{E} |} - 1) sgn (ψ_{E}) \\ - m_{33} α_{1} e^{k_{r 1} | ψ_{E} |} - m_{33} \frac{β_{1}}{k_{r 1}} {(1 - e^{- k_{r 1} | ψ_{E} |})}^{\frac{q_{1}}{p_{1}}} \\ (e^{- k_{r 1} | ψ_{E} |} sgn (ψ_{E}) + m_{33} k_{r 1} e^{k_{r 1} | ψ_{E} |}) \\ - m_{33} \frac{q_{1} β_{1}}{p_{1}} e^{k_{r 1} | ψ_{E} |} sgn (ψ_{E}) {(1 - e^{- k_{r 1} | ψ_{E} |})}^{\frac{(q_{1} - p_{1})}{p_{1}}} \\ (sgn (ψ_{E}) e^{- k_{r 1} | ψ_{E} |}) - m_{33} \frac{α_{1}}{k_{r 1}} (e^{k_{r 1} | S_{r 0} |} - 1) sgn (S_{r 0}) \\ - m_{33} \frac{β_{1}}{k_{r 1}} {(1 - e^{- k_{r 1} | S_{r 0} |})}^{\frac{q_{1}}{p_{1}}} e^{k_{r 1} | S_{r 0} |} sgn (S_{r 0}) \end{matrix}

(43)

Proving that $τ_{r}$ makes $ψ_{E}$ finally stable, and designing the Lyapunov function

V_{r 1} = \frac{1}{2} m_{33} S_{r 0}^{2} + \frac{1}{2} ψ_{E}^{2}

(44)

Taking the time derivative of $V_{r 1}$ and combine the equations (41), (42), and (43) to get

{\begin{matrix} {\overset{\cdot}{V}}_{r 1} = - D (α_{1} + β_{1} {(e^{k_{r 1} | S_{r 0} |} - 1)}^{\frac{q_{1} - p_{1}}{p_{1}}} e^{\frac{k_{r 1} | S_{r 0} | (q_{1} + p_{1})}{q_{1}}}) \\ - E (α_{1} + β_{1} {(e^{k_{r 1} | ψ_{E} |} - 1)}^{\frac{q_{1} - p_{1}}{p_{1}}} e^{\frac{k_{r 1} | ψ_{E} | (q_{1} + p_{1})}{q_{1}}}) \\ + (τ_{3 dis} - {\overset{\cdot\cdot}{ψ}}_{d} m_{33} + ψ_{E}) S_{r 0} \\ D = | S_{r 0} | (e^{k_{r 1} | S_{r 0} |} - 1) / k_{r 1} \\ E = | ψ_{E} | (e^{k_{r 1} | ψ_{E} |} - 1) / k_{r 1} \end{matrix}

(45)

Because $| ψ_{E} | < + \infty$ , combined with equation (39), and adjust the appropriate $α_{1}$ , $β_{1}$ , and $k_{r 1}$ to make $S_{r 0}$ and r finally stabilized according to the derivation process in the previous section, then ${\overset{\cdot}{V}}_{r 1} < 0$ , that is the system will eventually be uniformly and gradually stabilized.

Numerical simulation

This chapter verifies the effectiveness and robustness of the underactuated trajectory tracking controller proposed in this article through a series of designed numerical simulation experiments. The motion model of underactuated USV, CyberShip II, is selected as the test bench. CyberShip II is a model experimental USV developed by the Norwegian University of Science and Technology. The hydrodynamic parameters of the ship are published by the Norwegian University of Science and Technology. Many scholars, such as Qiu et al.,²⁹ Wang et al.,³⁰ and Hu et al.,³¹ who study USV verify their theoretical models based on the motion mathematical model of CyberShip II. The structure of CyberShip II is as shown in the Figure 2.³²

Figure 2.

The USV structure employed in numerical test.

According to Qiu et al.,²⁹ the dynamic parameters of the mathematical model of USV, CyberShip II, are set as follows

\begin{matrix} m_{11} = 25.8 kg, m_{22} = 33.8 kg, m_{33} = 2.76 kg; \\ d_{11} = 0.72 kg / s, d_{22} = 0.8896 kg / s, d_{33} = 1.9 kg / s \end{matrix}

Set the disturbance of the unmodeled part as below

{\begin{matrix} f_{1} = - 0.2 d_{11} u^{2} - 0.1 d_{11} u^{3} \\ f_{2} = - 0.2 d_{22} v^{2} - 0.1 d_{22} v^{3} \\ f_{3} = - 0.2 d_{33} r^{2} - 0.1 d_{33} u^{3} \end{matrix}

(46)

The disturbance of the external environment is

{\begin{matrix} d_{1} = 1 + 1.5 \sin (0.2 t) + 0.5 \cos (0.5 t) \\ d_{2} = 1 + 1.2 \sin (0.1 t) + 0.1 \cos (0.4 t) \\ d_{3} = 1 + 1.3 \sin (0.5 t) + 0.2 \cos (0.3 t) \end{matrix}

(47)

The resultant force of the total disturbance can be obtained as follows

{\begin{matrix} τ_{1 dis} = f_{1} + d_{1} \\ τ_{2 dis} = f_{2} + d_{2} \\ τ_{3 dis} = f_{3} + d_{3} \end{matrix}

(48)

Simulation is carried out through Simulink in MATLAB platform, and the block diagram of Simulink is designed as shown in Figure 3. The parameters received by the controller of the underactuated USV include the change rate of the reference trajectory, the errors between the motion state of the reference trajectory and that of the agent. The motion state includes dynamic state and kinematic state. The outputs $τ_{u}$ and $τ_{r}$ of the controller are calculated by the control algorithm proposed above. The new motion state of the agent is obtained after $τ_{u}$ and $τ_{r}$ are input into the motion mathematical model of the agent. The coordinates of x and y axes in the inertial coordinate system are selected for visualization.

Figure 3.

Simulink block diagram of the numerical test.

The parameters of the exponential dynamic sliding mode controller designed for $τ_{u}$ are

k_{1} = 0.5, k_{u 1} = 0.3, α_{0} = 3, β_{0} = 3, q_{0} = 5, p_{0} = 7

The parameters of the exponential dynamic sliding mode controller designed for $τ_{r}$ are

k_{2} = 2, k_{3} = 0.5, k_{r 1} = 0.3, α_{1} = 6, β_{1} = 6, q_{1} = 5, p_{1} = 7

The limit values of the output control laws $τ_{u}$ and $τ_{r}$ are set as follows

\begin{matrix} τ_{umax} = 250 N, τ_{umin} = - 250 N \\ τ_{rmax} = 450 Nm, τ_{rmin} = - 450 Nm \end{matrix}

In the simulation, in order to verify the trajectory tracking effect of the controller, a linear trajectory and a sine trajectory are set for verification. Their mathematical expressions are equations (49) and (50), respectively

\begin{matrix} {\begin{matrix} x_{r} = t \\ y_{r} = 0.5 t \end{matrix} \end{matrix}

(49)

{\begin{matrix} x_{r} = t \\ y_{r} = 40 \sin (0.01 t) \end{matrix}

(50)

In order to prove that the designed exponential dynamic sliding mode controller has a better convergence effect, compare it with the global integral sliding mode controller designed by Sun et al.,²⁷ and set the initial state of the USV

\begin{matrix} x (0) = 0 m, y (0) = 0 m, ψ = \frac{pi}{2} rad \\ u (0) = 0 m / s, v (0) = 0 m / s, r = 0 rad / s \end{matrix}

The tracking trajectory of the exponential sliding mode controller used in this article and the ordinary sliding mode controller on the linear trajectory under the condition of no disturbance are shown in Figure 4. It can be seen from the figure that the exponential sliding mode controller and global integral sliding mode controllers all have good tracking effect on straight trajectories under the condition of no disturbance. However, it can be seen from the tracking error diagram in Figure 5 that compared with the global integral sliding mode controller, the exponential sliding mode controller has a better tracking effect.

Figure 4.

Tracking performance results of line trajectory without disturbances.

Figure 5.

Tracking error result of line trajectory without disturbances.

The tracking trajectory of the linear trajectory by the exponential sliding mode controller and the ordinary sliding mode controller with the disturbance situation setting as mentioned above are shown in Figure 6. After the disturbances setting, compared with the exponential sliding mode controller and the global integral sliding mode controller under the condition of no disturbance, the agent suffers different degrees of influence on the tracking of the trajectories. No matter the trajectory diagram in Figure 6 or the tracking error diagram in Figure 7, all can be clearly seen that the exponential sliding mode controller has better anti-interference ability than the global integral sliding mode.

Figure 6.

Tracking performance results of line trajectory without disturbances.

Figure 7.

Tracking error result of line trajectory without disturbances.

It can be seen from Figures 8 and 9 that both the exponential sliding mode controller and the global integral sliding mode controller have good tracking effects on the sine curve in the absence of disturbances. But the exponential sliding mode controller shows the obvious advantages on tracking ability.

Figure 8.

Tracking performance results of sine trajectory without disturbances.

Figure 9.

Tracking error results of sine trajectory without disturbances.

After setting the disturbances, it can be seen from Figures 10 and 11 that the global integral sliding mode controller has a large error in the tracking of the sine curve, while the exponential sliding mode controller makes a rapid adjustment after a large error to make the error quickly converges to 0. For exponetial sliding mode, the error on the x axis is not dominant, but the error on the y axis and the total error are significantly better than the global sliding mode. The exponential sliding mode controller shows a better robust effect.

Figure 10.

Tracking performance results of sine trajectory with disturbances.

Figure 11.

Tracking error results of sine trajectory with disturbances.

Figures 12 and 13, respectively, show the output control force and torque changes of the USV under the disturbance condition on the line trajectory and sine trajectory. The results show that the control outputs $τ_{u}$ and $τ_{r}$ of the two controllers have reached saturation at the initial stage of control and then dropped rapidly. In the tracking process, the two control variables are basically kept within 10 units. The initial surge is mainly because a larger gain or difference produces a larger initial value that exceeds the limit of the propulsion system, but does not affect the controller stability. In short words, the trajectory tracking controller designed in this article has good tracking convergence performance.

Figure 12.

The output of the line trajectory tracking controller with disturbances.

Figure 13.

The output of the sine trajectory tracking controller with disturbances.

Conclusion

Although the tracking controller proposed in this article is utilized, the trajectory tracking control of the USV can be realized. Based on the Lyapunov function, the virtual control variables are determined by the backstepping method to be the propulsion control and the heading deviation, respectively, then the exponential mode sliding controller is analyzed, and the second order is designed by deriving the second order of the exponential mode sliding controller. The exponential sliding mode surface is solved to obtain the control laws, and their stability is analyzed through the Lyapunov function. In order to further prove the effectiveness and robustness of the controller, the tracking simulation experiment of the linear trajectory and the sinusoidal trajectory with two different curvature trajectories are carried out, and the simulation is also performed under the unmodeled disturbance and the external environment disturbance. By comparing with the global integral sliding mode controllers, the simulation results prove that the trajectory tracking controller of the USV proposed in this article is more superior, which supplements the related research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed the receipt of the following financial support for the research, authorship, and/or publication of this article: This article is supported by the results of the research project funded by the Shanghai High-level local University Innovation team (Maritime safety and security)/Shanghai Municipal Science and Technology Commission Local College Capacity Building Project (22010502000, 23010501900).

ORCID iD

Tengbin Zhu

Data availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study

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Trajectory tracking control of unmanned surface vehicle based on exponential global fast terminal sliding mode control

Abstract

Keywords

Introduction

Mathematical model of USV

Kinematics model

Dynamics model

Motion model in hull coordinate system

Virtual control design based on Lyapunov

Lyapunov in the hull coordinate system

Proof

Virtual control quantity design

Exponential sliding mode controller

Exponential terminal sliding mode

Control law design

Propulsion control law τ u design

Steering control law τ r design

Numerical simulation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data availability

References

Propulsion control law $τ_{u}$ design

Steering control law $τ_{r}$ design