1   Introduction

Maritime accidents usually cause a large loss of human lives, in addition to material losses and damage to the environment. As reported in the safety and shipping review (Allianz Global Corporate and Speciality, 2018), more than 75% of maritime accidents are caused by human errors. It is expected that marine autonomous ships will reduce the number of accidents, by removing human fail‐ ures. In addition, increasing the automation in ships will require fewer onboard seafarers, which is a potential solution for the shortage of workers in the maritime industry in the near future. Meanwhile, as reported in Markets & markets (2021), the global autonomous ships market size is projected to reach 14.2 billion dollars by 2030 with a 9.3% growth rate from 2020 to 2030.

Recent years have seen the rapid development of auton‐ omous ship concepts. To illustrate the situation, RollsRoyce (2015) led a 6.6 million dollar project that brought together universities, ship designers and shipyards to make autonomous ships a reality. It paved the way for autono‐ mous ships. In 2016, Norway opened the World's first offi‐ cial test bed, the Trondheim fjord test bed, for autonomous ships (Maritime, 2016). In 2019, the Minister of the Sea of Portugal presented the first edition of the Bluetech Accel‐ erator Ports & Shipping 4.0, in which, smart shipping and autonomous ships are recognized as the major trends. In 2021, the Maritime Safety Committee of IMO has com‐ pleted the analysis of relevant ship safety conventions, and the next steps for regulating Maritime Autonomous Sur‐ face Ships (MASS) will be assessed (IMO). Recently, the autonomous ship prototype, Yara Birkeland, carried out au‐ tonomous shipping tests in Norway.

One of the basic requirements for the marine autono‐ mous surface ship is to follow the predefined path with small cross-track errors. This property is well known as the path following control system, which mainly consists of the guidance subsystem, navigation subsystem and con‐ trol subsystem. This paper focuses on the state-of-the-art of path following control systems in the published papers, more precisely the guidance and control system, for marine autonomous surface ships. To the authors' best knowledge, few papers give an exhaustive review of the guidance laws and control methodology used for the path following con‐ trol of marine autonomous surface ships. Therefore, it is urgent and necessary to investigate the path following the control system of marine autonomous surface ships and pres‐ ent the statistical distribution of the current guidance technol‐ ogies and control methodologies used in the published pa‐ pers.

This paper is organized as follows: section 2 introduces the path following the control problem, where the error dy‐ namics are presented with the kinematics of marine surface ships. In section 3, the classical guidance laws are firstly in‐ troduced and the state-of-the-art of path following control of marine surface ships is reviewed, as well as the control methodologies. At last, a comparative study is carried out using the defined properties and characteristics based on the selected papers. The conclusions are given in section 4.

2   Path following control problem

The motions of the marine surface ships in waves are usually described in 6 degrees of freedom (DOFs), as presented in Figure 1. They are the surge, sway, and yaw motions, which are described in the horizontal plane, and roll, pitch and heave motion, which is de‐ scribed in the vertical plane. For the path following con‐ trol of marine surface ships, the surge, sway and yaw motions or well known as manoeuvring motions, are typ‐ ically considered, where the hydrodynamic coefficients usually can be approximated using constant values (Faltinsen, 1993). Sometimes roll motion is also augmented to the manoeuvring motions considering its coupling ef‐ fect on yaw and sway motion (Fossen, 2011; Sutulo and Guedes Soares, 2011).

Figure  1  The motions of marine surface ships in waves (Fossen, 2011)

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A typical autonomous path following system usually consists of the three basic subsystems, guidance, naviga‐ tion, and control system, which are also well known as the GNC system, as presented in Figure 2. The main goal of the GNC system is to control the movement of autonomous ships according to the specified task com‐ mands. The task of the navigation system is to determine the state vectors, including the location and orientation, of the au‐ tonomous ships from the onboard sensors. The guidance system generated the control command according to the ship's state vectors at a given time and the requirement of the task. The control system will execute the guidance commands by manipulating the thruster forces and rud‐ der, to achieve the task.

Figure  2  The guidance, navigation, and control system (GNC) for maritime autonomous surface ships (MASS) (Fossen, 2011)

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2.1   Cross-track error of path following control

The path following control problem of maritime auton‐ omous surface ships is defined by assuming that the pre‐ defined path is described by a curved line in the three-di‐ mensional space, $ \boldsymbol{p}_d(\theta)=\left[x_d(\theta), y_d(\theta), z_d(\theta)\right]^{\mathrm{T}}$ and having the control objective to design a control law for the ships that ensures convergence of the ship's position $\boldsymbol{p}(t)=[x(t), y(t), z(t)]^{\mathrm{T}} $ to the path p_d (θ).

For typical marine surface ships, it is usually assumed that a marine vessel is assigned to follow a two-dimension‐ al parametrized curved path, as presented in Figure 3. Without loss of generality, the curved path is employed for the demonstration of the cross-track error of the path fol‐ lowing control, since the straight-line path can be consid‐ ered as the path with a constant path tangential angle, γ_p. As presented in Figure 3, a curved path is predefined as (x_p(θ), y_p(θ)), where θ is a variable, and it is assumed to connect the way-points (x_j, y_j) for j = 1, ⋯N.

Figure  3  Path following control of autonomous ships

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To analyse the motion of the ship in the horizontal plane, three geographic reference frames need to be de‐ fined first. The north-east-down (NED) coordinate sys‐ tem { n } = (x_n, y_n, z_n) with the origin point, o_n, is defined as the tangent plane on the surface of the earth (Fossen, 2011). The x-axis points toward to true north, the y-axis points toward the east and the z-axis points downwards. The body-fixed coordinate system { b } = (x_b, y_b, z_b) with the origin point, on, is located on the ship and usually coincides with the centre point of the ship. It is moving with the ship and satisfies the righthand rule, where the x-axis points towards the ship's bow, as described in Figure 3. The last reference frame that needs to be defined is the path-tangential reference frame { p } = (x_p, y_p, z_p), which is located on the pre‐ defined path, and it is rotated by an angle γ_p(θ) from the north-east-down (NED) reference frame using the rota‐ tion matrix (Fossen and Lekkas, 2015):

where the path tangential angle is given as, γ_p (θ) = atan2 (x'_p (θ), y'_p (θ)).

Then, the distance from the ship location (x, y) to the predefined path can be calculated:

where, the y_e is the cross-track error and the x_e is the qa‐ long-track error equals zero by choosing the origin o_p of the path-tangential reference frame to coincide with the vertical projection point of the ship on the path (x_p(θ), y_p(θ)).

By expanding equation (2), the cross-track error y_e is defined as:

Finally, the control objective for the path-following of marine surface ships is defined as:

2.2   Kinematic equations of path following control

Marine surface ships moving in the horizontal plane, as described in Figure 3, are considered rigid bodies, and the kinematic equations are given as:

where, the surge and sway velocity are defined as u and v, respectively. The ψ, r are the yaw (heading) angle and yaw rate, which can be measured using an inertial measurement unit (IMU).

Taking the derivative of Eq. (3) with respect to time and substituting (5) into the results, gives:

where, the ground speed of the ship, U, is defined as U =$\sqrt{u^2+v^2} $. The phase angle, β = atan2 (v, u) is the drift angle, and it is the difference in heading angle ψ and course angle χ.

The dynamic equation of the cross-track error, y_e, can be further simplified as:

3   Path following control in autonomous ships

In this section, the path following control state of art for maritime autonomous surface ships is reviewed. The guidance laws and control methods, which are widely used for the path-following control of marine surface ships are described in this section.

3.1   Guidance system

The guidance system can be considered as the command generator for the path following control of autonomous ships. As described in Figure 2, the input of the guidance system is the predefined path and real-time position and state of the ship. The output is the desired speed and head‐ ing angle for the marine surface ships. To achieve the con‐ trol objective, the guidance law must ensure that the crosstrack error will converge to zero in finite time by steering the rudder and rpm of the propeller, as indicated in Eq (4). As it is well known that the sway dynamics of the marine surface ship are underactuated, therefore, the guidance laws need to be designed considering the underactuated sway dynamic.

The most widely used guidance laws for path following control are usually described as geometric techniques, such as line of sight (LOS) guidance law, pure pursuit (PP), and constant bearing (CB). The line of sight guidance law gives the desired heading angle by imitating an ex‐ perienced captain steering the ship towards the desired point, which is the virtual target point (VTP) located on the path. It is classified as a three-point geometric meth‐ od since it involves the position of the ship, the refer‐ ence point and the virtual target point (Fossen, 2011). The pure pursuit guidance law is a simple method and it only involves two points, the position of the ship and the target. The constant bearing is also a two-point guid‐ ance scheme, where the target velocity is considered.

3.1.1   Line of sight guidance law

3.1.1.1   Classical LOS

In the path following control of marine surface ships, the LOS guidance law and its modified version are most widely used considering the simple structure and easy implementation in the ship control systems (Fossen et al., 2003; Moreira et al. 2007). In Figure 4, the classical LOS and look ahead based LOS are presented. For the classical LOS guidance law, the LOS vector needs to be defined first, and usually equal to the nL_pp ahead of the ship. As indicated in Figure 4(a), the LOS vector is the vector from the ship position to the virtual target point on the predefined path. Thus, the desired heading angle can be calculated:

Figure  4  The classical line-of-sight guidance law (Fossen et al., 2003)

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Fossen et al., (2003) employed the nonlinear controller for the path following control of a marine craft, and the underactuated control problem was considered, where on‐ ly two controls were used for the 3 DOF motions of the marine surface craft. The control laws for surge and yaw motion were derived using the backstepping control technology and the equilibrium point was proven to be uni‐ formly globally asymptotically stable (UGAS) using the Lyapunov method. The experimental ship model test was carried out for validation.

Fredriksen and Pettersen (2006) studied the way-point manoeuvring of marine surface ships using only two control inputs in surge and yaw motion. The LOS guidance law was used for generating the heading commands. The globally κ exponentially stability of the control law was proved using cascaded control theory, and experimental results using the ship model were presented to validate the proposed control law.

Mu et al., (2018) proposed a fuzzy-based adaptive LOS guidance law for the path-following control of un‐ manned surface vehicles, where the backstepping control technology and neural networks were used for the con‐ trol design. The closed-loop control system was proven to be uniformly ultimately bounded using the Lyapunov stability theory.

An adaptive LOS guidance law and fuzzy heading con‐ trol were used (Niu et al., 2019), for the waypoints-based path following control of unmanned surface vehicles. The experimental tests were carried out to testify to the effectiveness of the proposed control system.

3.1.1.2   Look ahead LOS

A modified version of LOS, look ahead based LOS (Fossen, 2011; Fossen et al., 2015), was proposed for the calculation of the LOS vector, where the look-ahead distance needs to be defined, as described in Figure 5. The heading command is calculated using

Figure  5  The look-ahead based LOS guidance law (Fossen and Lekkas, 2015)

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Moreira et al., (2007) proposed a dynamic look-ahead LOS guidance for the path following control of autono‐ mous marine surface ships. A new method was proposed for the calculation of the LOS vector to improve the con‐ vergence. The proposed method was implemented in the model of the "Esso Osaka" tank and simulations were presented for showing the effectiveness of the proposed system. A similar study was later performed by (Lekkas and Fossen, 2012), where a time-varying equation was used for the look-ahead distance calculation for the LOS guidance law. A nonlinear sliding mode controller was de‐ signed to control the yaw motion and the coupled guid‐ ance and control system was proved to be globally κ-expo‐ nentially stable. Simulation tests were carried out for the validation of the proposed guidance law.

Fossen and Pettersen, (2014) provide a uniform semiglobal exponential stability (USGES) proof for the path fol‐ lowing control of underactuated marine surface ships using the proportional LOS guidance law. The author extends the previous results, which only guarantee the global κ expo‐ nential stability, by using the cascaded stability theory. Though the USGES is slightly weaker than global exponen‐ tial stability, it is still important for the autonomous system, since it guarantees the robustness of the autonomous system in the presentence of environmental disturbance.

Liu et al.(2017, 2016a) presented an extended state ob‐ server (ESO) based LOS guidance law for the path follow‐ ing control of marine surface ships, where a reduced-order ESO was employed to compensate for the sideslip angle due to environmental disturbance. Both simulation and ex‐ perimental tests were presented in this paper. Further work can be found in (Liu et al., 2016b).

Oh and Sun (2010) used a model predictive control (MPC) for the way-point tracking control of an underactu‐ ated marine surface ship, where the input constraints were considered. LOS guidance law was incorporated into the MPC design, and the performance was improved.

Liu et al. (2018) used a model predictive control method and LOS guidance law for the path following control of AS‐ Vs, where the nonlinear Nomoto model with disturbances was employed for the control design. Simulations were car‐ ried out to show the effectiveness of the proposed method.

Moe et al., (2016a) proposed a guidance and control sys‐ tem for the path following control of autonomous surface vessels using absolute velocity measurements. The proposed system does not need to measure the relative velocities using expensive sensors. The cross-track error dynamics of the closed-loop were proved to be UGAS and USGES, and the results were also verified using the simulations. Further work can be found in (Moe and Pettersen, 2017).

Fossen, (2022) presented the course autopilot design for the path following and an extended Kalman filter (EKF) was used to estimate the course over ground (COG) and speed over ground (SOG) efficiently from global naviga‐ tion satellite systems. The path following tests using LOS guidance law was carried out using a high-fidelity model of a MARINER class cargo ship.

3.1.1.3   Integral LOS

To handle the drift angle due to environmental disturbance, another well-known version of LOS, integral LOS(Børhaug et al., 2011, 2008; Lekkas and Fossen, 2014), was proposed by extending the classical LOS guidance with integral action:

where, σ is a positive design parameter, and it is an inte‐ gral gain.

Wan et al. (2020), designed the look-ahead distance as a function of the USV's cruising speed and the cross-track‐ ing error for the integral LOS guidance law.

The global asymptotic stability of the proposed control system was proved, and the simulations were carried out to demonstrate its effectiveness.

An integral LOS guidance law was proposed by (Caharija et al., 2012), for the path following control of marine surface vessels, where the integral action was used to com‐ pensate for the slow varying drift angle due to the environ‐ mental disturbance. The closed-loop uniform local expo‐ nential stability was proved, and simulations were carried out to support the theoretical results. The theoretical as‐ pects of the integral LOS guidance law can also be found in (Caharija et al., 2014). An extensive analysis of the inte‐ gral line-of-sight (ILOS) guidance law for path-following control of autonomous surface vessels and underwater ves‐ sels were presented, and both simulation and experiments were used to validate the effectiveness of the proposed sys‐ tem (Caharija et al., 2016).

Fossen et al. (2015), proposed a nonlinear adaptive con‐ troller for the path following control of underactuated ma‐ rine ships by compensating the drift forces due to the envi‐ ronmental disturbance, such as wind, waves and currents. The results were also extended for the path following Du‐ bin's path. The equilibrium points of the proposed sys‐ tem were proven to be USGES, and the simulations show that the cross-track error and the estimated sideslip angle converge exponentially.

A direct and an indirect nonlinear adaptive controller was proposed by Fossen and Lekkas (2015), for the path follow‐ ing control of marine surface ships using LOS guidance law. In this paper, they proposed a disturbance observer to compensate for the drift forces due to ocean currents and proved that the proposed method is globally κ exponentially stable. The simulations for the path following control of ma‐ rine surface vessels were carried out and the case for auton‐ omous underwater vehicles was also validated.

Wiig et al. (2015)presented the uniform semi-global ex‐ ponential stability proof of the integral LOS guidance law for path following control of underactuated marine ves‐ sels. Both the kinematics and dynamics of the system were taken into account, and the simulation results were present‐ ed to support the theoretical analysis.

Su et al. (2021) proposed an adaptive integral LOS guid‐ ance law for the path following control of unmanned sur‐ face vehicles with uncertainties. Three adaptive variables were constructed to eliminate the cross-track errors. The designed guidance subsystem is proved to be uniformly and ultimately bounded by the Lyapunov method.

3.1.1.4   Other works

To avoid collision with the obstacles, Moe and Pettersen, (2016) proposed set-based LOS guidance law for the path following and collision avoidance of autonomous surface vessels. The proposed guidance laws were designed to assure collision avoidance while abiding by the International Regulations for Preventing Collisions at Sea (COLREGs).

Yu et al. (2018) proposed a finite-time predictor LOS guidance law and adaptive neural network controller for the path following of unmanned surface vessels. In this pa‐ per, the author presented the theoretical analysis of the pro‐ posed system, and semi-globally uniformly ultimately bounded was proven by using the Lyapunov method. Rout et al., 2020) also used the adaptive neural network (NN) controller for the path following control of underactuated marine vehicles, where the barrier Lyapunov function (BLF) was used to deal with the system constraints and disturbance. The uniform ultimate boundedness of the closed-loop system was proved and both simulation and experimental tests were carried out for validation.

Miao et al. (2017) used the reduced-order linear extend‐ ed state observer (LESOs) method to estimate the un‐ known sideslip angle. A filtered extended state observer (FESO) was used to estimate the time-varying sideslip an‐ gle(Li et al., 2021a, 2019), where an adaptive fuzzy con‐ troller was used for the heading control of the unmanned surface vehicles. A network-based path-tracking control of an underactuated unmanned surface vehicle was proposed by (Wu et al., 2021), and an extended state observer (ESO) was employed to estimate the unknown disturbances due to the model uncertainties and environmental disturbances. Jiang et al., (2020), used the extended state observer (ESO) to estimate unknown relative velocities together with the ocean currents in real-time.

Wang et al. (2019a) proposed a novel nonlinear adaptive path following controller for the path following of amphibious hovercrafts, where a bounded-gain-forgetting (BGF) adap‐ tive estimator was used to compensate for the drift forces due to the environmental disturbance. A high-gain observ‐ er-based line-of-sight guidance law can be found in (Wang et al., 2019d)

Wang et al., (2019b) proposed a fuzzy observer-based adaptive path following control for an underactuated sur‐ face vehicle (USV). The control laws ensure that the pro‐ posed system was globally asymptotically stable. Simula‐ tion tests were carried out using the proposed method. Similar works can be found in(Wang et al., 2017, 2016).

Right now, the LOS guidance law has been thoroughly studied and well developed, as discussed above. Many re‐ vised versions are proposed to overcome the drawbacks of the classical LOS, for instance, the adaptive LOS, Propor‐ tional LOS, and integral LOS. The main objective of LOS guidance law is to provide the desired heading/course angle for the autonomous ships according to the geometry of posi‐ tions, therefore, it did not account for other information, such as obstacles, collision rules and shipping tasks. In the future, for instance, the LOS guidance law can be extended by considering collision avoidance and COLREGs, as dis‐ cussed in Moe and Pettersen, (2016), however, the stability proof of the whole guidance and control system is a chal‐ lenging problem. Beside the above works, Gu et al., (2022a, 2022b) gave a comprehensive summary of LOS guidance law, disturbance observers and extended state observers for path following of both surface and underwater vehicles.

3.1.2   Pure pursuit guidance law

The pure pursuit (PP) guidance law was typically used for air-to-surface missiles and UAVs (Yamasaki et al., 2009), where the chased targets are usually static or with slow speed. As illustrated in Figure 6, the PP guidance law is the two-point guidance scheme, where only the own ship and the target are considered. In the waypoints following control, the waypoint is usually chosen as the target. The ship aligns the forward speed, vⁿ_a, along the vector between the ship and the waypoint by choosing the desired velocity as (Breivik and Fossen, 2009):

Figure  6  The pure pursuit guidance law for waypoints-based path following

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where the κ is the positive constant. As can be observed, the PP guidance law will control ship travel towards the waypoints by neglecting the cross-track error, y_e. The ad‐ vantage is that the resulted trajectory is short compared with the LOS guidance law. Papoulias (1993) presented a theoretical analysis of the nonlinear dynamic phenomena when using the pure pursuit guidance for marine vehicles. Xu and Guedes(2018, 2015) proposed an opti‐ mized energy-efficient path-following method by combing the PP and LOS guidance law for autonomous ships. A global optimization algorithm was used to search the regularization factors, which play the trade-off between the total cross-track errors and total control energy.

3.1.3   Constant bearing guidance law

The constant bearing (CB) guidance law is also a twopoint guidance method (Fossen, 2011). The geometry of CB guidance law for tracking a moving ship is presented in Figure 7. It can be assumed to be an extended version of PP guidance law by considering the velocity of the target, vⁿ_a. CB guidance law is typically used for air-to-air mis‐ siles. Meanwhile, it was also used for collision avoidance at sea for marine vessels, where the own ship needs to steer away from a situation where another ship approaches at a constant bearing. The desired velocity is calculated us‐ ing the following equation,

Figure  7  The Constant bearing guidance law

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3.1.4   Vector field guidance law

The vector field (VF) guidance law is widely used for the path following control of unmanned aerial vehicles (UAVs) (Cho et al., 2015; Lim et al., 2014; Nelson et al., 2007, 2006; Yamasaki et al., 2009; Yanushevsky, 2011). Unlike the guidance law mentioned above, the virtual target point (or LOS vector point) does not need to be defined in the vector field guidance law. The main idea behind VF guidance law is to generate vectors around the predefined path using the mathematical equation, as illus‐ trated in Figure 8. In this Figure, the vectors indicate the desired heading angle of a ship in the position of the ar‐ row. When the ship tracks each vector, it will converge to the desired path eventually.

Figure  8  Vectors around the path

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Nelson et al., (2007, 2006) developed the vector field guidance law for the path following control of air vehicles, where desired course angles are generated using the pre‐ defined function, as given follows:

where, the χ_p is the path tangent angle, K_p influences the rate of the transition from χ ^∞ to zero. Inspired by the above appli‐ cation, Xu and Guedes Soares, (2016a) proposed the vector field guidance law for the path following control of underac‐ tuated marine surface ships. To improve the robust and con‐ verge rate, the transition region was defined along the pre‐ defined path. The vector field guidance law was defined as:

where the τ is the distance between the path and the edge of the transition region. χ ^e is the entering angle, which defined the desired course angle of the ship enter‐ ing the transition region. The Lyapunov method was used to prove that the proposed guidance law is global asymp‐ totic stable. Further work and extended versions of the vec‐ tor field guidance law can also be found in Xu and Guedes Soares(2020, 2018, 2016b, 2015).

Caharija et al. (2015) carried out a comparative study for the two popular guidance laws for straight-line path following purposes of underactuated large AUV vehicles: the integral LOS and the vector field guidance law pro‐ posed by Nelson et al.(2007, 2006). The results show that both guidance laws have a good path following control performance, with the VF guidance law performing slight‐ ly better. However, the significant chattering of the hori‐ zontal rudders can be observed, while the ILOS gives a smooth servo command.

To obtain a strong stability property, Xu et al., (2020a) proposed a time-varying vector field guidance law for both straight path and curved path following control of underac‐ tuated marine surface ships. The guidance law is defined as:

where, the θ(t, y_e) is a time-varying function, which con‐ trols the converge rate and ∆ > 0 is a pre-defined constant. The equilibrium point of the guidance and control system is uniform semi-global exponential stable (USGES) using the Lyapunov method and cascaded system theory. The simulation results can be found in Figure 9. Further work can be found in(Hinostroza et al., 2021, 2019; Xu et al., 2021a, 2021b).

Figure  9  Path following control using the time-varying vector field guidance law (Xu et al., 2020a)

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From the above discussion, the VF guidance law is more flexible, since only the mathematical equation for generating vectors needs to be defined without considering the physical definition of each parameter. However, LOS guidance laws are geometric techniques, and their parame‐ ters have a physical definition. For instance, The VF guid‐ ance law can be easily extended for collision avoidance by adding one repelling vector field around the obstacles, as discussed in (Xu et al., 2021a).

3.2   Motion control system

The motion control system for marine vessels is an ac‐ tive research field. Many pioneering works and papers have been published in conferences and journals. Roberts, (2008) provide a review of the early developments of marine control systems and discussed the key developments in ship autopilot design. The classical control methods are usually designed based on the knowledge of the plant mod‐ el, for example, Nomoto model (Nomoto et al., 1957; Xu et al., 2020b), Abkowitz models (Abkowitz, 1980; Xu et al., 2018), just to name a few. Recently, model-free datadriven control methodologies were used for the motion control system of marine ships. Its application in the dy‐ namic positioning of marine ships can be found in (Hassani et al., 2018). Considering the fast development of au‐ tonomous ships in recent years, this paper will briefly re‐ view the motion control methodologies, which were used for the path following of underactuated marine ships.

As discussed above, the motion control system works to‐ gether with the guidance system to achieve accurately pathfollowing control of autonomous ships, where the guidance system works like the experienced helmsman to give the de‐ sired ship's speed and the rudder command, and the motion control system will execute the command by changing the speed and heading angle. Therefore, the two systems are coupled together, and form the cascade structure, as dis‐ cussed in (Fossen and Pettersen, 2014; Xu et al., 2020a).

The most popular control method for marine ships is the PID (proportional integral derivative) controller. It is a typi‐ cally model-based linear control method, and the simplified steering model, such as the Nomoto model, is usually em‐ ployed for the control design. However, it is also possible to generalize to the nonlinear systems by using the nonlinear manoeuvring models. The application for the path following control for marine surface ships can be found in papers (Caharija et al., 2016, 2014; Fossen, 2022; Fossen et al., 2015; Fredriksen and Pettersen, 2006; Jiang et al., 2020; Liu et al., 2017, 2016a; Moe and Pettersen, 2017, 2016; Moreira et al., 2007). The detailed design procedure and parameters tuning method will be omitted since they can be easily found in the control books.

Feedback linearization is one of the well-known meth‐ odologies that was used to control nonlinear systems. The main idea is to nonlinear system dynamics into a linear system (Freund, 1973), then the classical linear control technologies, such as pole-placement, can be used to design the control laws. The feedback lineariza‐ tion is also capable of the path following control of marine surface ships, the related works can be found in (Caharija et al., 2012; Moreira et al., 2007; Skjetne et al., 2011; Xu and Guedes Soares, 2016a, 2016b, 2018, , 2020, 2015), just to name a few.

The backstepping control technology was used to design a stable control law for the nonlinear systems by using the con‐ trol Lyapunov functions (CLF) (Kokotović, 1992). The main idea behind the backstepping control method is to replace the control design problem using a sequence of sub-prob‐ lems on lower-order systems, where the virtual control input is defined to obtain the control laws for each sub-system us‐ ing control Lyapunov functions. The application of the back‐ stepping control method for the path following control can be found in(Do et al., 2004, 2002; Fossen et al., 2003; Li et al., 2021b; Mu et al., 2018; Nie and Lin, 2019; Su et al., 2021).

In addition to the control methods mentioned above, there are also many popular control methods for the path following control of marine surface ships, such as the sliding model control (Fossen and Lekkas, 2015; Lekkas and Fossen, 2012; Li et al., 2021a, 2021b, 2019; Moe et al., 2016b), the model prediction control (Liu et al., 2018; Oh and Sun, 2010), the L1 adaptive control method (Breu and Fossen, 2011; Ren et al., 2018; Svendsen et al., 2012; Xu et al., 2021b), the dynamic surface control(Huang et al., 2021a, 2021c; Liu et al., 2016c, 2016b; Wan et al., 2020) and the fuzzy method (Li et al., 2021a; Wang et al., 2019b). Recently, the machine learning methods, such as neural networks, are also employed for the marine control system, as published in(Huang et al., 2021a, 2021b; Liu et al., 2016d; Mu et al., 2018; Rout et al., 2020; Y. Wang et al., 2019b; Wu et al., 2021).

3.3   Comparison

In this part, the comparative study of the path follow‐ ing control system for marine surface ships will be pre‐ sented. There are 80 papers published in journals and in‐ ternational conferences, summarized in Table 1, using the different properties, such as control structure, results (i. e., simulation or experiment), guidance law, control‐ ler, environmental disturbance, vessel dynamics, stabili‐ ty properties, and underactuated control problem.

There are two typical approaches for the design of the guidance and control system to solve the path fol‐ lowing problem: The separated Guidance and Control (SGC) approach and the Integrated Guidance and Control (IGC) approach (Rubí et al., 2020). The Separated Guidance and Control (SGC) approach, consists of an outer-loop guidance law for the generation of the desired heading an‐ gle and an inner-loop controller to track the desired head‐ ing angle and generated the output rudder angle. The sta‐ bility property of the whole system is carried out for the outer and inner loop separately, and the coupled term is ne‐ glected. The Integrated Guidance and Control (IGC) ap‐ proach combines the guidance and control system, and the whole system is a cascade where the heading control sys‐ tem is the driving system and the guidance system is the driven system, as presented in Figure 10. As can be ob‐ served, the heading control error will affect the conver‐ gence of the guidance system, which is to minimize the cross-track error, therefore, the coupled term is considered in the Integrated Guidance and Control approach. From ta‐ ble 1. the distribution of the control structure using the SGC and IGC approach is presented in Figure 11. About 62% of papers choose the SGC approach for simplifying the stability analysis, and 38% use the IGC approach.

Figure  10  The cascaded path following control system (Lekkas and Fossen, 2014, 2012)

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Figure  11  The distribution of the control structure

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The results are usually validated using the simulations and experiments using the free-running ship model (Xu et al., 2020b). Right now, a large number of papers (81%) are using the simulation for the validation of the path following control, whereas only 19% of papers, the experimental studies were carried out, as indicated in Figure 12. Currently, most experimental studies are carried out using scale ship models in the lab or outdoor water area. The experimental tests in the lab can provide high accuracy position and ve‐ locity with the aid of an advanced position measurement system, meanwhile, the environmental disturbance (wind, wave and current) can also be scaled according to the ship model and can be controlled during the tests. There are some experimental works carried out in the laboratory, such as (Fossen et al., 2003; Fredriksen and Pettersen, 2006). The most convenient way to validate the path-following control system for Maritime Autonomous Surface Ships is using the scaled ship models in the outdoor water area. The sensors for the measurement of the position and velocity and actuators are needed to installed on the ship model, and a data acquisi‐ tion system and remote communication system is also need‐ ed for the data measurement and storage. Some interested works on the autonomous ship model can be found in (Liu et al., 2017; Mousazadeh et al., 2018; Niu et al., 2019; Rout et al., 2020; Sans-Muntadas et al., 2019; Wen et al., 2020; Wu et al., 2021; Xu et al., 2021a, 2020b; Zhu et al., 2020), just to name a few.

Figure  12  The validation methods

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As discussed above, there are 4 typical guidance laws, but only LOS and VF guidance laws are widely used for the path following control of marine surface ships, as presented in Figure 13. Right now, LOS and its revised versions take the leading position (about 63%) in the published papers due to the simple structure and easy implementation. The in‐ tegral LOS (18%) is also widely used for cancelling the drift angle due to environmental disturbance. The VF guid‐ ance law (13%) is a novel method for the marine surface ship and it is still under fast development and provides good control performance (Caharija et al., 2015).

Figure  13  The distribution of the guidance laws

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Right now, there are many controllers used to solve the path following problem, both linear controllers and non‐ linear controllers, such as proportional-integral-derivative (PID) controller, Linear control method, backstepping con‐ trol, sliding mode control, feedback linearizing control, sliding mode control, adaptive control, as presented in Figure 14. They are usually designed based on linear or nonlinear ship dynamics, well-known as model-based con‐ trol methodology. PID controller is still popular in the re‐ viewed papers (27%), considering its simple structure and easy implementation.

Figure  14  The distribution of the control design technologies

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The nonlinear controllers, such as backstepping control (12%), feedback linearizing control (15%) and sliding mode control (20%) rank second in the reviewed papers. It is worth mentioning that there are some mode-free control methods, such as Neural network control (5%), Fuzzy log‐ ic control (8%). They also provide a good tracking perfor‐ mance, and the stability properties are still a challenge for those control methods.

As can be observed from Table 1 and Figure 15, most of the papers provide the stability proof for their control, such as uniformly ultimately bounded (22%), Finite-time stability (6%), Input-to-state stability (7%), global asymptotic stability (37%), uniformly local exponential stability (4%), globally κ -exponentially stability (7%), and Uniform semi-global expo‐ nential stability (17%).

Figure  15  The distribution of the stability of the closed-loop system

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Stability is the most important property of any control sys‐ tem from a control-theoretic perspective since it is closely re‐ lated to the safety, robustness, and reliability of robotic sys‐ tems. Global exponential stability (GES) is usually the most desired quality of a closed-loop control system (Khalil, 2002; Loría and Panteley, 2005), but, it cannot be achieved due to the structural properties of the cross-track error (Fossen and Pettersen, 2014). The uniform semi-global exponential stabil‐ ity (USGES) is slightly weaker than GES and it is important for the performance and robustness of the control system un‐ der environmental disturbance. In Table 1 and Figure 16, most papers will consider the environmental disturbances (81%), such as wind, waves, and currents, the vessel dynamic (81%) and underactuated motions (75%) of marine surface ships.

Figure  16  The distribution of the environmental disturbance, vessel dynamics and underactuated dynamics

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4   Conclusions

The path following control system is one of the most crit‐ ically important systems for marine autonomous surface ships since it guarantees that the autonomous ship can fol‐ low the predefined path with small cross-track errors. This paper reviews the path following control systems, more pre‐ cisely, guidance systems and control systems, for marine au‐ tonomous surface ships. The path following control prob‐ lem and control objectives are firstly defined and the crosstrack errors function is derived using the kinematic equa‐ tions of marine surface ships. Then, the paper describes the most widely used guidance laws, such as LOS and modified version, PP, constant bearing and VF guidance law. The con‐ trol methodologies, such as PID, feedback linearization, backstepping method, sliding mode control and neural net‐ work-based control are briefly introduced.

To compare the state-of-the-art of path-following control for autonomous ships, 80 papers published in journals and inter‐ national conferences are carefully selected and analysed. Several properties and characteristics are defined, such as the control structure, guidance laws, control method, stability results, envi‐ ronmental disturbance and vessel dynamics, just to name a few.

The conclusions can be summarized:

(1) Most of the papers (62%) choose the separated guidance and control (SGC) for the design of the path following control system.

(2) Simulations are still the main means for validation, tak‐ ing up almost 81% of the selected papers. Experimental stud‐ ies are still very challenging due to many factors, such as the laboratory facility, sensor technology, model scale effect, test location and environmental disturbance, just to name a few.

(3) LOS and its extended versions take the leading posi‐ tion (about 63%) in the selected papers due to the simple structure and easy implementation.

(4) The linear model-based PID controller is still popular in the reviewed papers (27%), and nonlinear control technologies, such as Feedback linearizing, backstepping control and sliding mode control are also widely used in the selected papers.

(5) Most papers provide stability proof for the proposed system, and the global asymptotic stability proof (37%) was proved in most papers.

(6) The environmental disturbance (88%), vessel dynamics (84%) and underactuated motion (79%) are considered for most papers. The wind, wave and current are the typical distur‐ bance considered in the selected paper. The vessel dynamic is typically considered for the model-based control design, and the model-free control method, such as fuzzy logic control, and neutral network control, usually neglected the vessel dynamics.

In the future, the following works on path following control of underactuatd marine surface ships are recom‐ mented. The integrated Guidance and Control (IGC) approach is recommended for the design of the guidance and control for autonomous surface ships, especially the low speed commercial ships, because the coupled terms are ne‐ glected in the separated guidance and control (SGC) ap‐ proach. LOS guidance law can provide a good performance and is still recommented. The vector field guidance law is ex‐ pansible and easy to implement for collision avoidance, but its stability still needs to be investigated. The autonomous ship model is recommended for validation, but the sea trials of autonomous ships are still necessary for the development of automatic control systems.