1. Introduction
The unmanned surface vessel (USV) is a small surface platform with the ability of autonomous planning, autonomous navigation, and the ability to independently complete environmental awareness, target detection, and other tasks. Compared with traditional marine ships, USVs are smaller in size, lower in cost, stronger in mobility, and more intelligent. They can perform dangerous tasks without the risk of casualties, and are widely used in military and civil fields. In the military field, USVs can be equipped with communication systems, weapon systems, and sensor systems to perform border patrol, target surveillance, mine clearance, and anti-submarine tasks, greatly improving the military’s combat capability. In the civil field, the USV can carry out fish detection, hydrometeorological detection, map mapping, and other work after being equipped with the corresponding equipment. At the same time, the high mobility and efficiency of the USV allows it to carry out marine survey operations, environment monitoring, underway replenishment, cooperative search and rescue, and other works.
The underactuated unmanned surface vessel, as a typical underactuated system, is disturbed by winds, waves, and currents in the absence of the control input in the sway. The underactuated system means that the number of independent control variables of the system is less than the number of degrees of freedom of the system. This greatly increases the difficulty of control for the underactuated USV. Therefore, it has an important theoretical significance and application value to seek an effective control strategy to realize the trajectory tracking of the underactuated USV.
The purpose of trajectory tracking is to force the vehicle to follow a time-varying trajectory. The trajectory tracking is considered and solved as a stabilization problem for tracking error equations by direct strategies [
1]. For indirect strategies, the first step is to design the desired surge, sway speed, or yaw angle with the guidance laws. Then, by the control methods, the actual input control laws are developed so that the surge, sway speed, or yaw angle follow the desired variables. For the underactuated USV, since the desired sway speed is not forced by the control input directly, it is difficult to guarantee the correct course of the USV [
2]. Therefore, a guidance law for the surge speed and yaw angle should be developed. The yaw angle and the surge speed can be controlled directly by the control inputs [
3,
4].
At present, regardless of the types of trajectory tracking strategy, many research results have been achieved on the USV trajectory tracking by various control methods, including model predictive control [
5,
6], backstepping [
7], dynamic surface control [
8], sliding mode control [
9], prescribed performance control [
1], neural networks [
10], and so on. Because of the complex nature of the environment, controllers of USVs have to be robust against different unknown external disturbances such as winds, waves, and currents. The sliding mode technique, regarded as a remarkable robust control method, has been widely used because of its fast response speed and strong robustness.
However, one disadvantage of a conventional SMC is that the tracking error converges to zero asymptotically. The asymptotic convergence of the desired variable error may make it impossible to achieve the convergence of tracking errors. To achieve the convergence of a desired variable error in a finite time, terminal sliding mode control (TSMC) [
11] and fast TSMC [
12] are adopted in the design of controllers for linear and nonlinear systems. However, one problem of these methods is a singularity that appears when the system state is close to zero. Fortunately, the nonsingular TSMC [
13] and nonsingular fast TSMC [
14] were proposed and they tackle the singularity problem for trajectory tracking. Although the nonsingular fast TSMC realizes the fast convergence, while the avoiding singularity problem, it still needs conservatively large control inputs.
Except for the fact that a tracking error converges slowly, a conventional SMC has the chattering phenomenon. By using a high-order sliding mode algorithm, the chattering can be suppressed and the control accuracy can be improved. The super-twisting method is a kind of high-order sliding mode algorithm with a simple structure [
15], which can effectively suppress chattering and only requires first-order sliding mode information of the system [
16]. This means that there is no need to calculate the high-order derivative of the sliding surface.
Therefore, the super-twisting method is a better sliding mode method to reduce chatting and meet the finite-time stability requirements. Except for unmanned ground vehicles (UGVs) [
17] and unmanned aerial vehicles (UAVs) [
18], the super-twisting algorithm is also adopted for controlling maritime autonomous vessels. The roll suppression of marine vessels subjected to harmonic wave excitations was investigated using an adaptive sliding mode control with a super-twisting algorithm to reduce the chattering phenomenon and guarantee roll control accuracy [
19]. Furthermore, a combined model predictive super-twisting sliding mode control algorithm was proposed for tracking the trajectory of an autonomous surface vehicle in the presence of the time-varying external disturbances [
20]. In [
21], the tracking control method based on a super-twisting sliding mode is proposed for a USV. However, the virtual control laws for the desired surge and sway speeds are designed. The desired sway velocity is not suitable for designing a heading controller, since the complex problem of differential calculation is brought into the controller. In [
22], an improved super-twisting sliding mode control algorithm is proposed for ship heading control, while a sideslip angle compensation is calculated by the estimations of the surge and sway speeds with a finite-time extended state observer.
Aiming at the trajectory tracking problem of an underactuated USV subject to unknown time-varying external disturbances, an effective method is to use a disturbance-observer methodology. In order to eliminate the influence of external disturbance in the navigation of a USV, a finite-time disturbance observer [
23] is proposed to estimate the unknown external disturbances. However, the sliding mode control method is not sensitive to external disturbances, and the anti-disturbance characteristics of the method are realized by a large coefficient of the discontinuous term in the control law. To avoid the difficulty in the actuator control input caused by a large coefficient, the adaptive gain technique is considered [
9] when the upper bound of the disturbance is known. In [
24], an adaptive integral terminal sliding mode technique is designed against bounded disturbances, and it allows a finite-time convergence of the tracking error.
To design better guidance laws for precise tracking, the drift angle has to be considered. The drift, or sideslip angle, is the angle between the heading angle and the course angle. The sideslip angle can be attributed to the sway velocity component and is caused by the lateral acceleration while turning [
25]. The effect of wind and wave currents on the USV increases the variation of the sideslip angle. Therefore, if the navigation direction of the USV is accurately guided, the desired trajectory can be accurately tracked. To compensate for the sideslip angle in the guidance law, the most straightforward way is to measure it by means of sensors and add it to the guidance law. However, in many cases, it is difficult to measure sideslip angle accurately with cheap optical correlation sensors [
25]. Or, the sideslip angle is calculated with the velocities of the USV according to the formula. There are other ways to alleviate the effects of sideslip angles. The integral LOS (ILOS) guidance method was presented [
26], and the influence of a sideslip angle on the USV was eliminated by adaptive law [
27]. Indirect adaptive control methods were proposed. For example, two predictors were developed for the estimation of the tracking error and were proposed to indirectly estimate the sideslip angle [
28]. In [
23,
24,
25,
26], it is assumed that the sideslip angle is quite small (less than 5°), constant, or slowly varying. However, when the USV is disturbed by time-varying disturbances, the sideslip angle is greater than 10°. In this case, the approximate method is unreasonable. Therefore, the accurate direct estimation of the time-varying sideslip angle is very important for USV accuracy tracking.
Inspired by the above methods, in this paper, the main purpose is to design the controller for the trajectory tracking of an underactuated USV in the presence of external disturbances. The main contributions of this paper can be briefly summarized as follows:
A guidance law for the desired surge speed and yaw angle is proposed to obtain a simpler control structure and ensure the correct heading angle and surge speed. In the guidance law, the estimation of the time-varying sideslip angle is considered to guarantee tracking accuracy.
The design of surge speed and heading angle controllers uses the super-twisting second-order sliding mode method. The convergence of the tracking error is analyzed by the Lyapunov stability theory, and it is proved that the proposed controller can ensure the tracking error converges in a finite time.
The comparison of the proposed super-twisting sliding mode control method and adaptive sliding mode control methods is conducted in the simulation. Under the condition that the upper bound of the disturbances is known, results show that the STSMC method has a smaller tracking error and better tracking accuracy.
The paper is organized as follows:
Section 2 gives preliminaries and problem formulation. In
Section 3, the LOS guidance law with the reduced-order extended state observer and the proof of the convergence of trajectory tracking errors are provided. In
Section 4, the procedure of the finite-time super-twisting sliding mode controller is developed, and the finite-time convergence of tracking error is proved. Simulation studies and comparisons are conducted in
Section 5.
Section 6 concludes this paper.
2. Problem Formulation
The body-fixed frame and earth-fixed inertial frame are two common coordinate systems for the motion control of ship, as shown in
Figure 1. They can be used to describe the motion and attitude of the USVs. The earth-fixed inertial frame is used to describe the positional state of the USVs, and the body-fixed frame is used to describe the linear velocity and angular rate of the USVs [
29]. The center of gravity is consistent with the origin of the body-fixed frame.
represents the longitudinal axis from the stern to the bow,
represents the lateral axis pointing to the starboard, and
represents the vertical axis pointing to the center of the earth
. Similarly,
is the north,
is the east, and
is perpendicular to the stationary horizontal plane and points to the center of the earth. The specific meaning of each symbol in
Figure 1 is given in
Table 1.
For the trajectory tracking problem on the horizontal plane, the heave, pitch, and roll motions are neglected. In addition, the three-degrees-of-freedom (3-DOF) mathematical model for the USV is simplified by applying the following assumptions:
Assumption 1. The whole vehicle is a rigid body with a homogeneous mass distribution and a symmetrical shape and structure.
Assumption 2. The origin of the body-fixed frame is located at the center of gravity of the vehicle.
Assumption 3. The off-diagonal terms of inertial and drag matrices are smaller than the main diagonal terms and can be neglected.
Additionally, considering external environmental disturbances, the kinematic equations and the dynamic equations for an USV can be described as follows [
4]:
where the inertial coordinates of the USV are
and
. The yaw angle is
. The surge and sway velocities are
and
, respectively, and the yaw velocity is
. The control inputs are
and
.
,
,
,
,
,
,
,
, and
are the constant hydrodynamic coefficients and are computed by the equations [
30].
,
, and
are the external disturbances. Obviously, there is no control force in the sway direction; therefore, the USV model is an underactuated system. The external disturbances
,
, and
are continuous and differentiable, and there is a constant
such that
.
In addition, Equation (2) represents the vectors of forces and moments generated by the two thrusters
and
.
where
is the overall beam,
multiplies the starboard thruster to assist a mechanical constrain of the custom vehicle [
4].
According to the USV system (1), the control objective of trajectory tracking is to design control laws
, which ensure that the USV tracks the desired trajectory
. Because of complex algorithms and calculations by the direct control strategy, in this paper, the indirect strategy is introduced, that is, the guidance law is designed to obtain the desired system variables and the controllers are used to track these desired variables. For double-propelled USVs, considering Equation (2), the practical control objective is to design control laws for the two thrusters,
and
, which in a finite time forces the variables,
and
, of the USV (1) to converge to the desired
and
obtained from the proposed guidance law. Thereby, the tracking control laws are designed and they ensure that the USV tracks a desired, time-varying, and smooth trajectory. Control objectives can be expressed as
where
is the settling time and
. Moreover, a diagram of the guidance–control system is illustrated in
Figure 2.
3. Trajectory Tracking Guidance Law
For a USV located at the coordinate point
and the reference trajectory
. The reference trajectory
is a known function and the first order continuous differentiable. The along-track error
and the cross-track error
can be defined in the path tangential reference frame as follows:
where variables
denote the tangent angle of the trajectory, and
.
is a function that returns the angle that has a tangent that is the ratio of two variables.
Taking the time derivative of (3), one achieves the following:
In Equation (4), is the total speed of the USV, and . The sideslip angle .
The speed and can be directly measured by an accelerometer, GPS, or other sensors, and the sideslip angle is calculated. However, most USVs are not equipped with sensors for measuring speed because of the cost or hull space constraints in practical engineering. In addition, the measurement results are inevitably affected by the noise pollution of the sensor or inaccurate measurements. Therefore, the accurate sideslip angle cannot be obtained by the calculation of . To solve the problem of the unknown sideslip angle, a reduced-order extended state observer (ESO) is used to estimate the time-varying sideslip angle.
Considering
in (4) is rewritten,
where
. As in [
14], the reduced-order ESO is proposed to estimate
, which contains the unknown sideslip angle
.
where variable
is the auxiliary state of the observer, parameter
is the observer gain, and
denotes an estimate of
. We assume that the rate of change in the unknown
is bound, which satisfies
and
is a positive constant. The initial values of (5) are set as
and
.
At present, with the support of the ESO, the estimation of the sideslip angle
can be calculated as
Additionally, the estimation error of
is defined as
. Taking the derivative of
Lemma 1. Subsystem viewed as a system with the states being , the input being , is ISS.
Remark 1. For the detailed proof of Lemma 1, using the Lyapunov function . In Section 3 of [31], the detailed proof is provided. Additionally, the relationship between the bound of this error and the bandwidth of the reduced-order ESO is given. The estimation error can be tuned arbitrarily close to zero by increasing the bandwidth of the ESO.
So far, the desired variables and are designed to make the tracking errors and converge to zero.
Proposition 1. Let the desired yaw angle and surge speed be such thatwhere the desired total speed , is the look-ahead distance, is the positive controller gain. Lemma 2. If the surge speed error , and the yaw angle error converge to zero, the convergence of the position error and is guaranteed.
Proof. Consider the following Lyapunov function candidate,
By differentiating and substituting (4) with
and
into it, it yields the following:
By setting , it is obvious that . Therefore, it can be concluded that and converge to zero. □
5. Simulation Results
This section presents the simulation results to validate the effectiveness and robustness of the proposed sliding mode control strategies. The VTec S-III USV [
33] is selected as the simulation USV. The value of each thruster has a saturation of +/− 36.5 N that represents the thruster model in the simulation.
To verify the robustness of the proposed controller, under the same external disturbance, the trajectory tracking results of straight and sinusoidal trajectories are demonstrated, respectively. According to [
3], and considering the physical parameters of the USV, the external disturbances are assumed to be
The control parameters are , ,, , , , , , , and .
To demonstrate the advantage of the proposed control law, the method ASMC presented in [
4] is selected for comparison and the same parameters of ASMC are used.
The desired reference trajectories of the two scenarios are as follows:
The initial velocities are . The initial position and yaw angle are .
The tracking results for a straight line are shown in
Figure 3,
Figure 4,
Figure 5,
Figure 6 and
Figure 7. As is shown in
Figure 3 and
Figure 4, good performance is achieved in the case of initial errors by the two methods.
Figure 3b and
Figure 4 show that the control of STSMC has smaller tracking errors. The performance of the surge speed and heading angle controllers is shown in
Figure 5. Th results show that the designed controller can quickly and accurately track the desired value of the guidance law. The observer performance for the reduced-order ESO is shown in
Figure 6. It demonstrates that the designed observer can quickly and accurately estimate the lumped disturbance. The control inputs,
and
, are shown in
Figure 7.
Table 2 contains the metric comparison between controllers, including the mean squared error and Euclidean norm [
4]. The mean squared error is the average of the squared difference between the desired and the actual value of a variable. Thus, the MSE is used to quantify the controller performance according to how small the steady-state error is. The Euclidean norm is the square root of the sum of the squares of the vector values. Hence, the Euclidean norm is used to quantify the amount of control effort used by each controller. The STSMC does not consume more control input energy and it improves the tracking accuracy.
- (2)
Sinusoidal trajectory:
The initial velocities are . The initial position and yaw angle are .
The tracking results for a sinusoidal trajectory are shown in
Figure 8,
Figure 9,
Figure 10,
Figure 11 and
Figure 12. Using the proposed STSMC method, the tracking errors are sufficiently small and can be maintained even in the presence of external disturbances. Moreover, the tracking errors have a higher convergence speed and convergence accuracy.
Table 3 contains the metric comparison between controllers for sinusoidal trajectory. Similar to straight trajectory tracking, STSMC does not consume more control input energy and the tracking accuracy is improved for sinusoidal trajectory.