# A Systematic Pipelaying Control Method Based on the Sliding Matrix for Dynamically Positioned Surface Vessels

Authors: Xinfei Li; Zhongyu Jin; Lihui Wang
Journal: Journal of Sensors
 (2022)

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## Abstract

A pipelaying control method is presented in this paper which includes path planning, path guidance, and path tracking controller for dynamically positioned (DP) surface vessels based on the characteristics of the predefined path in marine pipelaying operation. The pipelaying control method depends on path coding, path selection logic system, and a sliding matrix. The sliding matrix contains a vessel local path and its specified control requirements, which can be updated by sliding down the waypoint table line by line as the vessel is traveling from one path to the next. A line of sight (LOS) algorithm is developed to calculate the desired vessel position and heading on a circular arc path. The motion controller, which can simultaneously control the vessel speed at the directions of surge and sway, is designed by decomposing the desired inertial resulting velocity into the desired body velocity components. A DP simulator for pipelaying operation is developed, and in order to verify the proposed method, a pipelaying simulation is carried out. The simulation results show that the proposed method enables the vessel to move along the desired path while maintaining a set crab angle, a specified speed, and a turning radius. The pipeline can be laid onto the specified waypoints even when the vessel is subjected to drift forces caused by ocean currents, wind, and waves.


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## Body

## 1. Introduction

Over the last two decades, deep-water pipelaying has gone through a spectacular development and the present methods and trends in deep-water pipelay systems are described in [1] by Fossen. Three basic pipelay methods are commonly in use: S-lay, J-lay, and reeling which are described in [2, 3] by Heerema and Ai et al., respectively. Xu et al. [4] proposed a simple and feasible VFIFE model for the static and dynamic analysis of deepwater S-lay and J-lay pipelines. In order to realize the design of continuous pipe laying method of rolling pipe laying method, it is necessary to take into account the characteristics of pipe laying operation and the research of ship path tracking control method and organically combine the two to realize this method. The predecessors have not discussed the specific methods to realize continuous pipe laying, which will be described in this paper.The path planning, path guidance, and path tracking of pipelaying vessels have their own unique characteristics. Firstly, it is difficult to follow the desired pipeline path precisely by indirectly controlling vessel position and heading [5, 6]. Secondly, it is necessary to consider the vessel position, heading, speed, and turning radius simultaneously during pipelaying operation because of pipelaying requirements. Guidance algorithm is an important part of ship path tracking control. It can provide the desired position, heading angle, speed, and other information for ship control and guide the ship to complete the relevant tracking tasks [7, 8]. Therefore, dynamic positioning (DP) control and pipeline dynamics are the two main parts of the deepwater pipelaying simulation model [3].The desired pipe paths consist of a few straight-line paths. In order to achieve a smooth transition between two straight-line paths, a circular arc path is inserted between the adjacent straight-line paths. The methods used to generate such circular arc path have been extensively studied in recent years, and the Dubins path is highly recognized in [9]. The curvature of the Dubins path is usually discontinuous at the intersection of a straight-line path and its adjacent circular arc path. Accordingly, the Pythagorean hodographs [10] and the clothoid arcs [11] had been developed to obtain path with continuous curvature. Lekkas and Fossen [12] proposed the monotonic cubic Hermit spline interpolation (MCHSI) to realize the continuous change of path curvature with low computational cost in ship path planning. However, it is not easy to lay the pipe onto the desired pipeline path by indirectly controlling vessel path. Fu et al. [5, 13] developed a method of path analysis and mapping relation between the desired pipeline path and the desired vessel path and used sliding mode control to study the tracking control of pipe laying operation of pipelaying ship [14]. Rout et al. [15] presents a modified line-of-sight (LOS) guidance law and an adaptive neural network (NN) controller for underactuated marine vehicles in the presence of uncertainties and constraints. Lekkas and Fossen [16, 17] proposed a time-varying lookahead-distance LOS method. When the vessel moves close to the desired path, the heading is gradually adjusted to the desired direction, which is beneficial for reducing the overshoot and oscillation of the vessel position. Among all existing methods, the sideslip motion is easy to occur during path-following due to environmental interferences. Therefore, a nonlinear adaptive path-following controller is proposed by Fossen et al. [18] to compensate for vessel sideslip. For the plane path tracking problem of underactuated ships, Peymani and Fossen [19] designed a dynamic controller by using the idea of integral backstepping, explicitly considering the geometric error and ensuring the path convergence. In addition, aiming at the linear path tracking problem of fully driven ships, they derived the control law using the least square method and adjusted the ship speed according to the geometric distance and the convergence speed of the path [20]. Moe et al. [21] developed a guidance and control system for underactuated surface vessels to solve the control objective of making the vessel to follow a curved path in the presence of unknown ocean currents. Heshmati-Alamdari et al. [22] presented a robust nonlinear model predictive control (NMPC) scheme for the case of underactuated autonomous underwater vehicles (AUVs). Nie and Lin [23] proposed an adaptive fuzzy path following control law, which is based on the improved adaptive integrated line of sight (IAILOS) guidance law for the control of underactuated ship in the case of time-varying ocean current and sideslip. Abdurahman et al. [24] used path-following control and speed allocation to solve the three-dimensional path-following problem of underactuated underwater vehicles. Path-following control was achieved by defining a slip angle relative to the desired heading angle. Lamraoui and Zhu [25] presented an improved active disturbance rejecter control (ADRC) for path-following control of autonomous underwater vehicles under significant fast-varying disturbances caused by waves and sea currents. Du et al. [26] established a robust nonlinear control law by using disturbance observer, auxiliary power system, and dynamic surface control (DSC) technology to keep the ship’s position and heading at the expected value. Yu et al. [27] presented an improved guidance law that transforms 3D path-following errors into controlled speed errors and designed a simplified nonlinear single-input fuzzy controller against unknown disturbances. Huang et al. [28] presented a 2D path-following control method for autonomous underwater vehicles (AUVs) based on dynamic circle heading modification (DCHM). Loginov et al. [29] proposed special control functionality for trailing suction dredgers in dynamic positioning (DP) systems that focused mainly on the dredging track controller, and the method was verified by the simulation and sea-trial results. Shi et al. [5, 30] presented a pipelaying control method for DP vessels, and it was tested in the pipelaying operation. Here, it refers to the control method. Healey and Lienard [31] designed the autopilot of the six degree of freedom underwater vehicle through sliding mode control, which has a good response to the speed, steering, and heave motion of the low-speed AUV. Mu et al. [32] investigated the path-following control problem for an underactuated unmanned surface vehicle (USV) in the presence of dynamical uncertainties and time-varying external disturbances. Rout and Subudhi [33] designed an explicit model predictive controller (Ex-MPC) for implementing a LOS guidance law.The pipeline dynamic model is the other key and important problem in pipelaying operation. Ai et al. [3] proposed a pipelaying nonlinear dynamic model, which integrates the major aspects related to numerical simulation, including coupled pipeline motion and roller contact forces. Jensen [34, 35] proposed a nonlinear dynamic model which was capable of undergoing shearing, twist, and bending for the pipe string, and the model is suitable for pipelaying operations by a dynamically positioned surface vessel. Szczotka [36] studied a dynamic and quasistatic model that showed the high dynamic forces in the pipelaying equipment during pipelaying simulation operated in the sea for analysis of a pipelay spread. Different pipelaying models were studied in the above. However, applications of this method to pipelaying operation, especially for a series of waypoints, are not discussed in detail. In order to study a pipelaying system with DP vessels in time domain, it is necessary to simplify appropriately the pipeline dynamic model.In this study, a systematic method, including path planning, guidance algorithm, and path tracking for pipelaying vessels, is presented from the prospect of the whole DP control system. One of the key steps to achieve path tracking finally is to combine the desired path information and the guidance algorithm with control strategy. The realization of the system depends on the coding method of the pipeline desired path. The method of sliding matrix is proposed to extract the required path information from the waypoint table for continuous pipelaying at specified speed, heading, and turning radius. The sliding matrix contains the vessel local desired path, the desired crab angle, and turning radius and the desired pipelaying speed. A simple and reliable guidance algorithm is also designed for circular arc paths of the pipelaying vessel, which is convenient for the guiding system to execute quickly. Then, the vessel speed and the yaw controllers are also designed. Finally, the effectiveness of the proposed control method of pipelaying control is verified using a DP simulator.


## 2. The Method of Pipelaying Path Based on a Sliding Matrix

### 2.1. Definition of Coordinates

Two coordinate systems adopted to define ship motions are presented in this section first. The North-East-Down (NED) coordinate systemn=xn,yn,zn with origin on defined relative to the Earth’s reference ellipsoid. It is usually defined as the tangent plane on the surface of the Earth moving with the marine craft. xn axis points towards true north, yn axis points towards east, and zn axis points downwards from the Earth surface to its geocentric. This is usually referred to as flat Earth navigation, and it can be simply denoted by n, as shown in Figure 1.Figure 1
NED reference frame.The body-fixed reference frameb=xb,yb,zb with origin ob is a moving coordinate frame that is fixed to the craft, and it can be simply denoted by b. The body axes are chosen to coincide with the principal axes of inertia, as shown in Figure 2. The xb is the longitudinal axis pointing from aft to fore. The yb is the transversal axis pointing to starboard. The zb is the normal axis pointing from top to bottom.Figure 2
Body-fixed reference frame.


### 2.2. Waypoints and Its Coding Method for Pipelaying Paths

The desired path of pipeline to be laid is expressed as a set of waypoints. The waypoint table for pipelaying contains the information of all waypoints, including north and east coordinates, crab angle, laying speed, and turning radius of pipeline, as shown in Table1.Table 1
The waypoint table for pipelaying operation.
No.NorthEastCrab angleSpeedTurn radius1N1E1———2N2E2γ1V1—3N3E3γ2V2R1..................kNkEkγk−1Vk−1Rk−2The waypoint table gives a detailed expression of the characteristics of each segment of pipelaying paths, and it also specifies the control requirements of the pipelaying vessel. The pipelaying paths are divided into several straight-line segments which are separated by different waypoints. As shown in Table1, Nk and Ek are the north coordinate and east coordinate of the waypoint Pk, respectively. γk represents the crab angle which is a fixed angle between the desired vessel heading and the path tangential direction. Vk represents the desired pipelaying speed when the vessel is tracking the desired straight-line pipe path PkPk+1 which is also represented by Sk. Rk represents the turning radius of the desired pipe circular arc path Ck, so as to obtain a smooth circular arc transition between a straight-line path to its adjacent next straight-line path, as shown in Figure 3.Figure 3
The coding method of the pipelaying path.In order to make the analysis of pipelaying path easier and achieve automatic identification in path tracking by programming, all the straight-line and circular arc paths in Figure3 need to be coded.As shown in Figure3, Pk, k=1,⋯,n denotes the waypoint k. A straight-line path is denoted as Sk, k=1,⋯,n−1, and S1 represents the first straight-line path P1P2. A circular arc path is denoted as Cl, l=1,⋯,n−2, and C1 represents the first circular arc path between S1 and S2. χ1 is the path-tangential angle between the desired pipe straight-line path P1P2 (S1) and xn axis of n. γ1 is the set crab angle, which is a specified angle between the desired vessel heading and the tangential direction of the straight-line path.


### 2.3. Pipelaying Path Planning Based on the Sliding Matrix

The desired paths of pipeline usually consist of straight-line paths and circular arc paths in sequence. For a straight-line path, it is only necessary to consider the start point and end point whereas for a circular arc path, the circular arc center also needs to be considered. A sliding matrix may be constructed using the information in the waypoint table.As shown in Figure3, the desired vessel heading ψk for straight-line path Sk is obtained by
(1)ψk=χk+γk,whereχk is the path-tangential angle of the desired pipe path Sk relative to χn axis of n. γk is the set crab angle when the vessel is tracking Sk.As shown in Table1, the control information of a sliding matrix is composed of a set of waypoints. The sliding matrix slides down the waypoint table and traverses all waypoints during pipelaying operation, as shown in Table 2.Table 2
The control information of a sliding matrix.
No.NorthEastHeadingSpeedTurn radiuskNkEk———k+1Nk+1Ek+1ψkVk—k+2Nk+2Ek+2ψk+1Vk+1RkThe sliding matrix used to calculate the guidance law is proposed to resolve path-tracking problem in pipelaying operation. The sliding matrix, three rows and five columns, includes position coordinates of three waypoints, the desired heading angle, the desired speed of straight-line path, and the desired turning radius of circular arc path. From Table2, the second column contains north coordinates of three waypoints, the third column contains east coordinates of three waypoints, the fourth column contains the desired heading, the fifth column contains the desired pipelaying speed of straight-line path, and the sixth column contains the desired turning radius.The turning radius of pipelaying is related to characteristics of the pipeline, the gyration radius of the pipelaying vessel, and desired pipelaying speed on circular arc paths. The circular arc path represents the natural circular arc path when the vessel is traveling from one straight-line path to the next one at a specified turning radius and speed. The desired state of the start point of the circular arc path isq1=Nq1,Eq1,ψq1, and the desired state of the end point is q2=Nq2,Eq2,ψq2.The steps for the analysis of the path based on sliding matrix are as follows:Step 1. To calculate the angle between two adjacent straight-line paths.The sketch of desired pipelaying paths is shown in Figure4. Here, χ2 is the path-tangential angle between P2P3 (S2) and the xn axis of n. α1 is the angle between the two straight-line paths S1 and S2.Figure 4
Sketch diagram of the angle between pipe paths.The calculation method ofα1 is related to the rotation direction of circular arc path. When 0≤χ1<π, the angle between P1P2 and P2P3 can be obtained using
(2)α1=π−χ1−χ2rg=00≤χ2<χ1,π−χ2−χ1rg=1χ1≤χ2<χ1+π,χ2−χ1+πrg=0χ1+π≤χ2<2π,whererg=1 implies that the rotation direction of a circular arc path is clockwise and rg=0 implies that the rotation direction is counterclockwise.Whenπ≤χ1<2π, the value of angle α1 between the two straight-line paths can be obtained using
(3)α1=χ1−π−χ2rg=10≤χ2<χ1−π,χ1−χ2rg=0χ1−π<χ2<χ1,π−χ2−χ1rg=1χ1<χ2≤2π.Step 2. To calculate the distance between a waypoint point and its tangent point.To calculate the distance between a waypoint point and its adjacent tangent point, the sketch is shown in Figure5. As shown in Figure 5, O1 is the center of the circular arc path C1, and R1 is the turning radius. The points, q1 and q2, are the tangent points between the circular arc path and its two adjacent straight-line paths, respectively. θOq1 is the angle between the vector O1q1 and xn axis of n. θOq2 is the angle between the vector O1q2 and xn axis of n.Figure 5
Sketch of the desired pipelaying path.It is straightforward to calculate the distance between the waypointP2 and the tangent point q1, which can be expressed as L2q1. Similarly, L1q1 is the distance between the waypoint P1 and q1. Additionally, L2q1 is equal to L2q2. It is assumed that R1 and α1 are known and consequently, L2q1 can be expressed as
(4)L2q1=L2q2=R1tanα1/2.Step 3. To calculate the coordinates of the tangent points.LetL12 denote the distance between the waypoint P1 and P2. Hence, L1q1 can be obtained from
(5)L1q1=L12−L2q1.Here, it is necessary to make sure thatL12>L2q1 when the waypoints are specified.Thereafter, the coordinates of the center ofC1, O1 satisfy
(6)NO1EO1=cosχS1−sinχS1sinχS1cosχS1L1q1LOq1+N1E1.Here,LOq1 is the vertical distance between the tangent point q1 and the center O1. For a clockwise circular arc path, LOq1 equals to R1, and for a counterclockwise circular arc path, LOq1 equals to −R1.The coordinates of the tangent pointq1 satisfy
(7)Nq1Eq1=cosχS1−sinχS1sinχS1cosχS1L1q10+N1E1.Consequently, the coordinates of the tangent pointq2 satisfy
(8)Nq2Eq2=cosχS2−sinχS2sinχS2cosχS2L2q20+N2E2.Step 4. To calculate the angle between two straight lines.The angle range is chosen to represent the part of a circular arc pipe path, as shown in Figure5. Accordingly, it is necessary to calculate the angle between two straight lines connecting two tangent points to the center of the circular arc path. Moreover, it is also used to calculate the LOS point if it is required. In order to ensure angle continuity as a circular arc path passes across xn axis of n, it is necessary to analyze the angle ranges of θOq1 and θOq2 further.Whenθoq2>θoq1, it implies that the angle range of the circular arc path does not contain zero point, and the circular arc path does not pass across xn axis of n. Whereas when θoq2<θoq1, it implies that the angle range of the circular arc path contains zero point, and the circular arc path passes across xn axis of n.Hence, the relationship mentioned above can be expressed as(9)θoq1′=θoq1θoq2′=θoq2rg=1,θoq2>θoq1orrg=0,θoq2<θoq1.Here,θoq1′ and θoq2′ are the updated values of θoq1 and θoq2, respectively.Consequently, Equations (10) and (11) are obtained:
(10)θoq1′=θoq1−2πθoq2′=θoq2rg=1,θoq2<θoq1,(11)θoq1′=θoq1θoq2′=θoq2−2πrg=0,θoq2>θoq1.


### 2.4. Mapping from the Desired Pipelaying Path to the Desired Vessel Path

The horizontal projection distance between the pipe touchdown point at seabed and the vessel rotation center during pipelaying operation is defined as the touchdown distance. The desired vessel path needs to be calculated so that the pipe touchdown point can be placed on the desired path during turning in pipelaying operation. It is necessary to estimate the desired vessel path from the pipeline desired path and the touchdown distance. The touchdown distance can be modified anytime if it is necessary during pipelaying operation. The touchdown distance is mainly influenced by the method of pipelaying operation, sea depth, pipe mechanical properties, etc. The desired vessel path can be calculated based on the desired pipelaying path, and their geometric relationship is shown in Figure6.Figure 6
Mapping relationship between the desired vessel path and pipelaying path.V1 is the desired pipelaying speed when tracking S1. ψC1, VC1, and βC1 are the desired heading angle, pipelaying speed, and sideslip angle when tracking the path C1, respectively. RV and R1 are the radius of the desired circular path of the DP vessel and pipeline, respectively. Ltd is the touchdown distance. Ps1 and Ps2 are the start point and end point of the path C1, respectively. PL is the LOS point on the path C1. θOL is the azimuth angle of the straight-line O1PL.The desired pipeline paths are composed of a few straight-line paths and circular arc paths in sequence. It is necessary to control precisely the ship heading angle to make smooth progress of pipelaying operation when the pipelaying vessel runs along the desired path. When the value of the crab angle is set to 0, the desired vessel heading angle is equal to the tangent direction of pipeline path; when the crab angle is a constant value, the desired vessel heading angle is obtained using Equation (1).


#### 2.4.1. To Calculate the Desired Vessel Path from a Desired Straight-Line Pipe Path

Letq1x1,y1 and q2x2,y2 denote the start point and end point of the circular arc path of pipeline, respectively, χ1 denotes the straight path angle, and Ltd denotes the touchdown distance. Let Ps1xs1,ys1 and Ps2xs2,ys2 denote the start point and end point of the circular arc path of vessel, respectively. Accordingly, the coordinates of Ps1xs1,ys1 can be obtained using Equation (12), as shown in Figure 6.
(12)xs1=x1+Ltdcosχ1,ys1=y1+Ltdsinχ1.Consequently, the coordinates ofPs2xs2,ys2 can be obtained using
(13)xs2=x2+Ltdcosχ2,ys2=y2+Ltdsinχ2.In this study, when the crab angle is set to 0, the desired vessel heading angle is equal to the tangent angle of desired pipeline paths.


#### 2.4.2. To Calculate the Desired Vessel Path from a Desired Circular Arc Pipelaying Path

The desired circular arc path of pipe to be laid should be tangent to its adjacent straight-line path. This means that the desired vessel circular path is not tangent to its adjacent vessel straight-line path. Here, all crab angles for the desired straight-line and circular arc paths are set toγ0. Accordingly, the desired heading angle for the first desired straight-line pipeline path is obtained using Equation (14), as shown in Figure 6.
(14)ψ1=χ1+γ0,whereψ1 is the desired heading when tracking the straight-line path S1 and γ0 is the set crab angle.When the vessel is tracking the circular arc pathC1, the desired heading angle ψC1 can be expressed as
(15)ψC1=θOL+δ0+γ0Clockwisepath,θOL−δ0+γ0Counterclockwisepath.Here, the crab angleγ0 is a constant during pipelaying operation.The value ofδ0 can be obtained using
(16)δ0=arcsinR1RV.Hence, the relationship between the desired turning radius of DP vessel and the desired turning radius of pipeline path can be expressed as(17)RV=R12+Ltd2.Here,Ltd is the touchdown distance.


### 2.5. A Guidance Law of Circular Arc Paths for DP Vessels

It is necessary to construct a guidance system that is suitable for DP vessels so that the vessel actual path can converge to the desired path smoothly. The common LOS guidance law is only suitable for path following of straight-line or circular arc path for underactuated water surface ships or underwater vehicles. Therefore, a new LOS guidance algorithm of circular arc paths for DP vessels is presented.


#### 2.5.1. The Guidance Law for a Straight-Line Path

LetPLNLOS,ELOS be the desired LOS point on a straight-line path. The points Nk,Ek and Nk+1,Ek+1 are the start point and end point of the straight-line path, respectively. Let Nt,Et be the real-time position of DP vessels. Thereafter, the following relationship can be obtained.
(18)NLOS−Nt2+ELOS−Et2=RLOS2,tanαk=ΔEΔN=Ek+1−EkNk+1−Nk=ELOS−EkNLOS−Nk,whereRLOS is the radius of the LOS guidance law specified by the practice requirements.


#### 2.5.2. A Guidance Algorithm for Circular Arc Paths

The guidance law of a curved path also takes the vessel position as the center and makes a circle with a radius value based on the principle of LOS guidance law of the straight-line path. One of the intersections of the circle and the circular arc path is chosen as the LOS point. Meanwhile, the direction of the desired resultant speed of the vessel passes through the LOS point. The schematic diagram of the LOS guidance algorithm of the circular arc path is shown in Figure7.Figure 7
A LOS guidance algorithm for a circular arc path.As shown in Figure7, PL denotes the LOS point on the desired vessel circular arc path. ps denotes the real-time position of the vessel set rotation center. O1 denotes the center of C1. θOL denotes the azimuth angle of O1PL, and b denotes the angle between O1PC and xn axis of n. σC is the angle between O1PC and O1PL.It is necessary to know the angle of the straight-line segment connecting the LOS point and the circle centerO1PL for calculating the coordinates of the LOS point on the circular arc path.The following relationship betweenθOL and b can be obtained from Figure 7.
(19)θOL=σC−b.The value of the angleσC is calculated using
(20)σC=π2−δ0.The value of the angleb can be obtained using
(21)b=θOC0<θOC<π2,θOC−π2π2<θOC<π,θOC−ππ<θOC<3π2,2π−θOC32π<θOC<2π.Thereafter, the coordinates of LOS point on a circular arc path can be obtained using
(22)NLOS=NO1+RLcosθOL,ELOS=EO1+RLsinθOL.The LOS point is the desired vessel position on a desired path. Hence, the desired heading angle during the circular path following can be obtained using Equation (15).


## 2.1. Definition of Coordinates

Two coordinate systems adopted to define ship motions are presented in this section first. The North-East-Down (NED) coordinate systemn=xn,yn,zn with origin on defined relative to the Earth’s reference ellipsoid. It is usually defined as the tangent plane on the surface of the Earth moving with the marine craft. xn axis points towards true north, yn axis points towards east, and zn axis points downwards from the Earth surface to its geocentric. This is usually referred to as flat Earth navigation, and it can be simply denoted by n, as shown in Figure 1.Figure 1
NED reference frame.The body-fixed reference frameb=xb,yb,zb with origin ob is a moving coordinate frame that is fixed to the craft, and it can be simply denoted by b. The body axes are chosen to coincide with the principal axes of inertia, as shown in Figure 2. The xb is the longitudinal axis pointing from aft to fore. The yb is the transversal axis pointing to starboard. The zb is the normal axis pointing from top to bottom.Figure 2
Body-fixed reference frame.


## 2.2. Waypoints and Its Coding Method for Pipelaying Paths

The desired path of pipeline to be laid is expressed as a set of waypoints. The waypoint table for pipelaying contains the information of all waypoints, including north and east coordinates, crab angle, laying speed, and turning radius of pipeline, as shown in Table1.Table 1
The waypoint table for pipelaying operation.
No.NorthEastCrab angleSpeedTurn radius1N1E1———2N2E2γ1V1—3N3E3γ2V2R1..................kNkEkγk−1Vk−1Rk−2The waypoint table gives a detailed expression of the characteristics of each segment of pipelaying paths, and it also specifies the control requirements of the pipelaying vessel. The pipelaying paths are divided into several straight-line segments which are separated by different waypoints. As shown in Table1, Nk and Ek are the north coordinate and east coordinate of the waypoint Pk, respectively. γk represents the crab angle which is a fixed angle between the desired vessel heading and the path tangential direction. Vk represents the desired pipelaying speed when the vessel is tracking the desired straight-line pipe path PkPk+1 which is also represented by Sk. Rk represents the turning radius of the desired pipe circular arc path Ck, so as to obtain a smooth circular arc transition between a straight-line path to its adjacent next straight-line path, as shown in Figure 3.Figure 3
The coding method of the pipelaying path.In order to make the analysis of pipelaying path easier and achieve automatic identification in path tracking by programming, all the straight-line and circular arc paths in Figure3 need to be coded.As shown in Figure3, Pk, k=1,⋯,n denotes the waypoint k. A straight-line path is denoted as Sk, k=1,⋯,n−1, and S1 represents the first straight-line path P1P2. A circular arc path is denoted as Cl, l=1,⋯,n−2, and C1 represents the first circular arc path between S1 and S2. χ1 is the path-tangential angle between the desired pipe straight-line path P1P2 (S1) and xn axis of n. γ1 is the set crab angle, which is a specified angle between the desired vessel heading and the tangential direction of the straight-line path.


## 2.3. Pipelaying Path Planning Based on the Sliding Matrix

The desired paths of pipeline usually consist of straight-line paths and circular arc paths in sequence. For a straight-line path, it is only necessary to consider the start point and end point whereas for a circular arc path, the circular arc center also needs to be considered. A sliding matrix may be constructed using the information in the waypoint table.As shown in Figure3, the desired vessel heading ψk for straight-line path Sk is obtained by
(1)ψk=χk+γk,whereχk is the path-tangential angle of the desired pipe path Sk relative to χn axis of n. γk is the set crab angle when the vessel is tracking Sk.As shown in Table1, the control information of a sliding matrix is composed of a set of waypoints. The sliding matrix slides down the waypoint table and traverses all waypoints during pipelaying operation, as shown in Table 2.Table 2
The control information of a sliding matrix.
No.NorthEastHeadingSpeedTurn radiuskNkEk———k+1Nk+1Ek+1ψkVk—k+2Nk+2Ek+2ψk+1Vk+1RkThe sliding matrix used to calculate the guidance law is proposed to resolve path-tracking problem in pipelaying operation. The sliding matrix, three rows and five columns, includes position coordinates of three waypoints, the desired heading angle, the desired speed of straight-line path, and the desired turning radius of circular arc path. From Table2, the second column contains north coordinates of three waypoints, the third column contains east coordinates of three waypoints, the fourth column contains the desired heading, the fifth column contains the desired pipelaying speed of straight-line path, and the sixth column contains the desired turning radius.The turning radius of pipelaying is related to characteristics of the pipeline, the gyration radius of the pipelaying vessel, and desired pipelaying speed on circular arc paths. The circular arc path represents the natural circular arc path when the vessel is traveling from one straight-line path to the next one at a specified turning radius and speed. The desired state of the start point of the circular arc path isq1=Nq1,Eq1,ψq1, and the desired state of the end point is q2=Nq2,Eq2,ψq2.The steps for the analysis of the path based on sliding matrix are as follows:Step 1. To calculate the angle between two adjacent straight-line paths.The sketch of desired pipelaying paths is shown in Figure4. Here, χ2 is the path-tangential angle between P2P3 (S2) and the xn axis of n. α1 is the angle between the two straight-line paths S1 and S2.Figure 4
Sketch diagram of the angle between pipe paths.The calculation method ofα1 is related to the rotation direction of circular arc path. When 0≤χ1<π, the angle between P1P2 and P2P3 can be obtained using
(2)α1=π−χ1−χ2rg=00≤χ2<χ1,π−χ2−χ1rg=1χ1≤χ2<χ1+π,χ2−χ1+πrg=0χ1+π≤χ2<2π,whererg=1 implies that the rotation direction of a circular arc path is clockwise and rg=0 implies that the rotation direction is counterclockwise.Whenπ≤χ1<2π, the value of angle α1 between the two straight-line paths can be obtained using
(3)α1=χ1−π−χ2rg=10≤χ2<χ1−π,χ1−χ2rg=0χ1−π<χ2<χ1,π−χ2−χ1rg=1χ1<χ2≤2π.Step 2. To calculate the distance between a waypoint point and its tangent point.To calculate the distance between a waypoint point and its adjacent tangent point, the sketch is shown in Figure5. As shown in Figure 5, O1 is the center of the circular arc path C1, and R1 is the turning radius. The points, q1 and q2, are the tangent points between the circular arc path and its two adjacent straight-line paths, respectively. θOq1 is the angle between the vector O1q1 and xn axis of n. θOq2 is the angle between the vector O1q2 and xn axis of n.Figure 5
Sketch of the desired pipelaying path.It is straightforward to calculate the distance between the waypointP2 and the tangent point q1, which can be expressed as L2q1. Similarly, L1q1 is the distance between the waypoint P1 and q1. Additionally, L2q1 is equal to L2q2. It is assumed that R1 and α1 are known and consequently, L2q1 can be expressed as
(4)L2q1=L2q2=R1tanα1/2.Step 3. To calculate the coordinates of the tangent points.LetL12 denote the distance between the waypoint P1 and P2. Hence, L1q1 can be obtained from
(5)L1q1=L12−L2q1.Here, it is necessary to make sure thatL12>L2q1 when the waypoints are specified.Thereafter, the coordinates of the center ofC1, O1 satisfy
(6)NO1EO1=cosχS1−sinχS1sinχS1cosχS1L1q1LOq1+N1E1.Here,LOq1 is the vertical distance between the tangent point q1 and the center O1. For a clockwise circular arc path, LOq1 equals to R1, and for a counterclockwise circular arc path, LOq1 equals to −R1.The coordinates of the tangent pointq1 satisfy
(7)Nq1Eq1=cosχS1−sinχS1sinχS1cosχS1L1q10+N1E1.Consequently, the coordinates of the tangent pointq2 satisfy
(8)Nq2Eq2=cosχS2−sinχS2sinχS2cosχS2L2q20+N2E2.Step 4. To calculate the angle between two straight lines.The angle range is chosen to represent the part of a circular arc pipe path, as shown in Figure5. Accordingly, it is necessary to calculate the angle between two straight lines connecting two tangent points to the center of the circular arc path. Moreover, it is also used to calculate the LOS point if it is required. In order to ensure angle continuity as a circular arc path passes across xn axis of n, it is necessary to analyze the angle ranges of θOq1 and θOq2 further.Whenθoq2>θoq1, it implies that the angle range of the circular arc path does not contain zero point, and the circular arc path does not pass across xn axis of n. Whereas when θoq2<θoq1, it implies that the angle range of the circular arc path contains zero point, and the circular arc path passes across xn axis of n.Hence, the relationship mentioned above can be expressed as(9)θoq1′=θoq1θoq2′=θoq2rg=1,θoq2>θoq1orrg=0,θoq2<θoq1.Here,θoq1′ and θoq2′ are the updated values of θoq1 and θoq2, respectively.Consequently, Equations (10) and (11) are obtained:
(10)θoq1′=θoq1−2πθoq2′=θoq2rg=1,θoq2<θoq1,(11)θoq1′=θoq1θoq2′=θoq2−2πrg=0,θoq2>θoq1.


## 2.4. Mapping from the Desired Pipelaying Path to the Desired Vessel Path

The horizontal projection distance between the pipe touchdown point at seabed and the vessel rotation center during pipelaying operation is defined as the touchdown distance. The desired vessel path needs to be calculated so that the pipe touchdown point can be placed on the desired path during turning in pipelaying operation. It is necessary to estimate the desired vessel path from the pipeline desired path and the touchdown distance. The touchdown distance can be modified anytime if it is necessary during pipelaying operation. The touchdown distance is mainly influenced by the method of pipelaying operation, sea depth, pipe mechanical properties, etc. The desired vessel path can be calculated based on the desired pipelaying path, and their geometric relationship is shown in Figure6.Figure 6
Mapping relationship between the desired vessel path and pipelaying path.V1 is the desired pipelaying speed when tracking S1. ψC1, VC1, and βC1 are the desired heading angle, pipelaying speed, and sideslip angle when tracking the path C1, respectively. RV and R1 are the radius of the desired circular path of the DP vessel and pipeline, respectively. Ltd is the touchdown distance. Ps1 and Ps2 are the start point and end point of the path C1, respectively. PL is the LOS point on the path C1. θOL is the azimuth angle of the straight-line O1PL.The desired pipeline paths are composed of a few straight-line paths and circular arc paths in sequence. It is necessary to control precisely the ship heading angle to make smooth progress of pipelaying operation when the pipelaying vessel runs along the desired path. When the value of the crab angle is set to 0, the desired vessel heading angle is equal to the tangent direction of pipeline path; when the crab angle is a constant value, the desired vessel heading angle is obtained using Equation (1).


### 2.4.1. To Calculate the Desired Vessel Path from a Desired Straight-Line Pipe Path

Letq1x1,y1 and q2x2,y2 denote the start point and end point of the circular arc path of pipeline, respectively, χ1 denotes the straight path angle, and Ltd denotes the touchdown distance. Let Ps1xs1,ys1 and Ps2xs2,ys2 denote the start point and end point of the circular arc path of vessel, respectively. Accordingly, the coordinates of Ps1xs1,ys1 can be obtained using Equation (12), as shown in Figure 6.
(12)xs1=x1+Ltdcosχ1,ys1=y1+Ltdsinχ1.Consequently, the coordinates ofPs2xs2,ys2 can be obtained using
(13)xs2=x2+Ltdcosχ2,ys2=y2+Ltdsinχ2.In this study, when the crab angle is set to 0, the desired vessel heading angle is equal to the tangent angle of desired pipeline paths.


### 2.4.2. To Calculate the Desired Vessel Path from a Desired Circular Arc Pipelaying Path

The desired circular arc path of pipe to be laid should be tangent to its adjacent straight-line path. This means that the desired vessel circular path is not tangent to its adjacent vessel straight-line path. Here, all crab angles for the desired straight-line and circular arc paths are set toγ0. Accordingly, the desired heading angle for the first desired straight-line pipeline path is obtained using Equation (14), as shown in Figure 6.
(14)ψ1=χ1+γ0,whereψ1 is the desired heading when tracking the straight-line path S1 and γ0 is the set crab angle.When the vessel is tracking the circular arc pathC1, the desired heading angle ψC1 can be expressed as
(15)ψC1=θOL+δ0+γ0Clockwisepath,θOL−δ0+γ0Counterclockwisepath.Here, the crab angleγ0 is a constant during pipelaying operation.The value ofδ0 can be obtained using
(16)δ0=arcsinR1RV.Hence, the relationship between the desired turning radius of DP vessel and the desired turning radius of pipeline path can be expressed as(17)RV=R12+Ltd2.Here,Ltd is the touchdown distance.


## 2.4.1. To Calculate the Desired Vessel Path from a Desired Straight-Line Pipe Path

Letq1x1,y1 and q2x2,y2 denote the start point and end point of the circular arc path of pipeline, respectively, χ1 denotes the straight path angle, and Ltd denotes the touchdown distance. Let Ps1xs1,ys1 and Ps2xs2,ys2 denote the start point and end point of the circular arc path of vessel, respectively. Accordingly, the coordinates of Ps1xs1,ys1 can be obtained using Equation (12), as shown in Figure 6.
(12)xs1=x1+Ltdcosχ1,ys1=y1+Ltdsinχ1.Consequently, the coordinates ofPs2xs2,ys2 can be obtained using
(13)xs2=x2+Ltdcosχ2,ys2=y2+Ltdsinχ2.In this study, when the crab angle is set to 0, the desired vessel heading angle is equal to the tangent angle of desired pipeline paths.


## 2.4.2. To Calculate the Desired Vessel Path from a Desired Circular Arc Pipelaying Path

The desired circular arc path of pipe to be laid should be tangent to its adjacent straight-line path. This means that the desired vessel circular path is not tangent to its adjacent vessel straight-line path. Here, all crab angles for the desired straight-line and circular arc paths are set toγ0. Accordingly, the desired heading angle for the first desired straight-line pipeline path is obtained using Equation (14), as shown in Figure 6.
(14)ψ1=χ1+γ0,whereψ1 is the desired heading when tracking the straight-line path S1 and γ0 is the set crab angle.When the vessel is tracking the circular arc pathC1, the desired heading angle ψC1 can be expressed as
(15)ψC1=θOL+δ0+γ0Clockwisepath,θOL−δ0+γ0Counterclockwisepath.Here, the crab angleγ0 is a constant during pipelaying operation.The value ofδ0 can be obtained using
(16)δ0=arcsinR1RV.Hence, the relationship between the desired turning radius of DP vessel and the desired turning radius of pipeline path can be expressed as(17)RV=R12+Ltd2.Here,Ltd is the touchdown distance.


## 2.5. A Guidance Law of Circular Arc Paths for DP Vessels

It is necessary to construct a guidance system that is suitable for DP vessels so that the vessel actual path can converge to the desired path smoothly. The common LOS guidance law is only suitable for path following of straight-line or circular arc path for underactuated water surface ships or underwater vehicles. Therefore, a new LOS guidance algorithm of circular arc paths for DP vessels is presented.


### 2.5.1. The Guidance Law for a Straight-Line Path

LetPLNLOS,ELOS be the desired LOS point on a straight-line path. The points Nk,Ek and Nk+1,Ek+1 are the start point and end point of the straight-line path, respectively. Let Nt,Et be the real-time position of DP vessels. Thereafter, the following relationship can be obtained.
(18)NLOS−Nt2+ELOS−Et2=RLOS2,tanαk=ΔEΔN=Ek+1−EkNk+1−Nk=ELOS−EkNLOS−Nk,whereRLOS is the radius of the LOS guidance law specified by the practice requirements.


### 2.5.2. A Guidance Algorithm for Circular Arc Paths

The guidance law of a curved path also takes the vessel position as the center and makes a circle with a radius value based on the principle of LOS guidance law of the straight-line path. One of the intersections of the circle and the circular arc path is chosen as the LOS point. Meanwhile, the direction of the desired resultant speed of the vessel passes through the LOS point. The schematic diagram of the LOS guidance algorithm of the circular arc path is shown in Figure7.Figure 7
A LOS guidance algorithm for a circular arc path.As shown in Figure7, PL denotes the LOS point on the desired vessel circular arc path. ps denotes the real-time position of the vessel set rotation center. O1 denotes the center of C1. θOL denotes the azimuth angle of O1PL, and b denotes the angle between O1PC and xn axis of n. σC is the angle between O1PC and O1PL.It is necessary to know the angle of the straight-line segment connecting the LOS point and the circle centerO1PL for calculating the coordinates of the LOS point on the circular arc path.The following relationship betweenθOL and b can be obtained from Figure 7.
(19)θOL=σC−b.The value of the angleσC is calculated using
(20)σC=π2−δ0.The value of the angleb can be obtained using
(21)b=θOC0<θOC<π2,θOC−π2π2<θOC<π,θOC−ππ<θOC<3π2,2π−θOC32π<θOC<2π.Thereafter, the coordinates of LOS point on a circular arc path can be obtained using
(22)NLOS=NO1+RLcosθOL,ELOS=EO1+RLsinθOL.The LOS point is the desired vessel position on a desired path. Hence, the desired heading angle during the circular path following can be obtained using Equation (15).


## 2.5.1. The Guidance Law for a Straight-Line Path

LetPLNLOS,ELOS be the desired LOS point on a straight-line path. The points Nk,Ek and Nk+1,Ek+1 are the start point and end point of the straight-line path, respectively. Let Nt,Et be the real-time position of DP vessels. Thereafter, the following relationship can be obtained.
(18)NLOS−Nt2+ELOS−Et2=RLOS2,tanαk=ΔEΔN=Ek+1−EkNk+1−Nk=ELOS−EkNLOS−Nk,whereRLOS is the radius of the LOS guidance law specified by the practice requirements.


## 2.5.2. A Guidance Algorithm for Circular Arc Paths

The guidance law of a curved path also takes the vessel position as the center and makes a circle with a radius value based on the principle of LOS guidance law of the straight-line path. One of the intersections of the circle and the circular arc path is chosen as the LOS point. Meanwhile, the direction of the desired resultant speed of the vessel passes through the LOS point. The schematic diagram of the LOS guidance algorithm of the circular arc path is shown in Figure7.Figure 7
A LOS guidance algorithm for a circular arc path.As shown in Figure7, PL denotes the LOS point on the desired vessel circular arc path. ps denotes the real-time position of the vessel set rotation center. O1 denotes the center of C1. θOL denotes the azimuth angle of O1PL, and b denotes the angle between O1PC and xn axis of n. σC is the angle between O1PC and O1PL.It is necessary to know the angle of the straight-line segment connecting the LOS point and the circle centerO1PL for calculating the coordinates of the LOS point on the circular arc path.The following relationship betweenθOL and b can be obtained from Figure 7.
(19)θOL=σC−b.The value of the angleσC is calculated using
(20)σC=π2−δ0.The value of the angleb can be obtained using
(21)b=θOC0<θOC<π2,θOC−π2π2<θOC<π,θOC−ππ<θOC<3π2,2π−θOC32π<θOC<2π.Thereafter, the coordinates of LOS point on a circular arc path can be obtained using
(22)NLOS=NO1+RLcosθOL,ELOS=EO1+RLsinθOL.The LOS point is the desired vessel position on a desired path. Hence, the desired heading angle during the circular path following can be obtained using Equation (15).


## 3. Control System Designing for Pipelaying Operation

As shown in Figure8, the pipelaying simulation system of DP vessels is constructed, which includes the waypoint table, path planning module, path guidance module, thrust allocation module, propeller system module, vessel hydrodynamic module, environment force module, state observer module, and motion controller.Figure 8
DP simulation system for a pipelaying vessel.The data transmitted between the modules are represented by the numbers and ①—the coordinates, heading angle, laying speeds, turning radius; ②—the desired vessel path; ③—the desired LOS point, speed, heading; ④—the desired forces; ⑤—the desired thrusts; ⑥—the actual sum forces; ⑦—the estimated wind loads; ⑧—the LOS point.The principle of pipelaying operation of the DP simulation is presented as follows:(1)
The waypoint coordinates, crab angle, pipelaying speed, and turning radius of the pipe are input in the waypoint table according to the operational requirements of a pipelaying task(2)
The desired pipelaying path, the desired vessel path, and heading angle are generated by the path planning module from the waypoint table. Then, they are input into the path guidance module(3)
The vessel real-time position, speed, and heading angle are input into the module of state filter and observer, and the vessel drift (Low Frequency, LF) motion is estimated by filtering out the first-order wave-induced (Wave Frequency, WF) motion(4)
The LOS point is calculated using the path guidance module by inputting the vessel real-time position, heading angle and speed, the desired vessel path, and control requirements. Then, the LOS points are input into the motion controller module(5)
The desired forces in three degrees of freedom are estimated by the motion controller module through calculating the deviation between LOS point and estimated real-time motion. Then, they are input into the thrust allocation module(6)
The desired thrust and the azimuth angle of each thruster are calculated according to the configuration of the propulsion system and thrust allocation method chosen by the DP operator(7)
The actual thrust vector is generated by each thruster according to the desired thrust and azimuth angle(8)
The propeller system produces the sum forces in three degrees of freedom through the actual thrust vectors. Then, the sum forces will be applied to the DP vessel(9)
The DP vessel is moved along the desired path at the given speed and heading angle by the pipelaying controller, even the vessel is exposed to the drift forces caused by ocean currents, wind, and waves


### 3.1. Path Selection Method in the Pipelaying Operation

The desired path of the pipelaying vessel can be obtained by making use of the waypoint table and mapping relationship between the pipelaying path and the vessel path. Hence, it is necessary to know the type of pipe path it is traveling along so the corresponding LOS point of the pipelaying vessel can be calculated. The flow chart of the path selection method is shown in Figure9, and the details are explained in the following:
(1)
Specify an index number for each straight-line path and each circular arc path of the pipelaying vessel, as well as the initial LOS point. And the first point in the waypoint table is usually set as the initial LOS point(2)
Letk and l be the indices for a straight-line path and a circular arc path, respectively. And the initial values are set as k=1, l=0(3)
The sliding matrix is generated according to the waypoint table and the initial value ofk(4)
The desired pipelaying path expressed by the sliding matrix is analyzed to determine the end point of straight-line path and circular arc pipe path, the circle center, and rotation direction of the circular arc path of pipeline. The desired vessel path is calculated from the desired pipelaying path(5)
Determine which path type the vessel is going to track when the present path range is exceeded by the pipelaying vessel(6)
The conditionk>l implies that the vessel is going to travel along a straight-line path, and when k=l, it implies that the vessel is going to travel along a circular arc path(7)
When the vessel is traveling along a straight-line path and does not exceed its range, the start and the end points of the straight-line path are output to the LOS point calculating module. When the straight-line path range is exceeded, i.e., the vessel has just reached a circular arc path, then setl=l+1. Consequently, now k=l, and the LOS point on the circular arc path is calculated(8)
When the vessel is traveling along a circular arc path and exceeds its range, it implies that the vessel has just reached the next straight-line path, then setk=k+1. Consequently, now k>l, it is necessary to update the sliding matrix, and it will continue to run from the third stepFigure 9
Execution process of the path selecting method.It can be summarized as follows: when the value ofk is increased by one, the sliding matrix is required to be updated and path analysis is also performed so that the related parameters of the next path are obtained. When the value of l is increased by one, the sliding matrix is not required to be updated. However, it is necessary to determine whether the vessel is sailing on a straight-line path or a circular arc path, and the related LOS point is required to be calculated.


### 3.2. Coordinate Calculation of LOS Point

The LOS point is defined as the desired vessel position on a specified path. It is a dynamically changing point that guides the vessel approach to the desired path along with a specified heading. The vessel path selection is based on the previous LOS point. The flow chart for calculation of LOS points is shown in Figure10 and is explained in the following:
(1)
The initializing parameters are input into LOS point calculating module(2)
Check whether the path is required to be updated by comparing the value ofk and kpre in the process of calculating LOS point(3)
The conditionk=kpre implies that the path does not need to be updated, where k is the index value of the present straight-line path and kpre is the index value of the previous straight-line path(4)
Whenk>kpre, it implies that the path is required to be updated, and the start and the end points of a straight-line path are also required to be recalculated and updated(5)
Determine the type of vessel path by comparing the values ofk and l. If k>l, it implies that it is a straight-line path, and the method used to calculate LOS point of a straight-line path should be chosen. If k=l, it implies that it is a circular arc path, and the method used to calculate LOS point of a circular arc path should be selected(6)
The LOS point of the desired vessel path is obtainedFigure 10
The calculation process of the LOS point.


### 3.3. Multiobjective Motion Controller of DP Vessels

The LOS point is the desired position of the ship moving on the desired path during path following, and its coordinates are changing dynamically when the vessel reaches a new position. The sliding matrix also contains such control requirement information which is obtained by the waypoint table. The diagram of the multiobjective controller of a DP vessel is shown in Figure11.Figure 11
Multiobjective controller of DP vessel.When the vessel real-time position and the LOS point are known, the deviation of the vessel position inn coordinate frame is converted into the deviation in b coordinate frame. Then, the desired surge force, sway force, and yaw moment of the vessel can be calculated in combination with the deviations of the vessel speed and heading. The deviations of position and heading angle in b can be obtained using
(23)N~=Nlos−N,E~=Elos−E,ψ~=ψd−ψ,whereN~ and E~ are the deviations of north and east positions, respectively. ψ~ is the deviation of the heading angle. N and E are the real-time north and east positions, respectively; ψ is the real-time heading angle.Hence, when the vessel is tracking the first desired circular arc path, the desired body velocity components decomposed from the pipelaying speed in the coordinate framen can be expressed as
(24)ud=cosβC1Vd,vd=sinβC1Vd,whereVd is the desired speed in the coordinate frame n; βC1 is the desired sideslip angle; ud and vd are the desired velocity components in the body coordinate frame, respectively.When the vessel is tracking the desired circular arc path, the desired vessel speed is obtained using(25)Vd=VC1.From Figure6, when the rotational direction is clockwise and the circular arc path is clockwise, the value of βC1 is obtained by
(26)βC1=π2−δ0−γ0.The deviations of the vessel position in the coordinate frameb can be expressed as
(27)x~=cosψC1N~+sinψC1E~,y~=−sinψC1N~+cosψC1E~.The deviations of the body velocity components by the vessel position deviation in the coordinate frameb can be expressed as
(28)u~=ud−u+kpxx~,v~=vd−v+kpyy~,r~=rd−r+kpψψ~+kdψψ~̇,wherekpx is the control parameter used to convert the position deviation to the velocity deviation in surge direction. kpy is the control parameter used to convert the position deviation to the velocity deviation in sway direction. kpψ and kdψ are the control parameters used to convert the heading deviation to the rotational speed deviation.Hence, the desired control forces generated by the propulsion system can be obtained using(29)XT=kpuu~+kiu∫0tu~+kduu~̇,YT=kpvv~+kiv∫0tv~+kdvv~̇,NT=kprr~+kir∫0tr~+kdrr~̇,wherekpu,kiu, and kdu are the speed controller parameters of the dynamic positioning system in surge direction. kpv, kiv, and kdv are the speed controller parameters of the dynamic positioning system in sway direction. kpr, kir, and kdr are the speed controller parameters of the dynamic positioning system in yaw direction.


## 3.1. Path Selection Method in the Pipelaying Operation

The desired path of the pipelaying vessel can be obtained by making use of the waypoint table and mapping relationship between the pipelaying path and the vessel path. Hence, it is necessary to know the type of pipe path it is traveling along so the corresponding LOS point of the pipelaying vessel can be calculated. The flow chart of the path selection method is shown in Figure9, and the details are explained in the following:
(1)
Specify an index number for each straight-line path and each circular arc path of the pipelaying vessel, as well as the initial LOS point. And the first point in the waypoint table is usually set as the initial LOS point(2)
Letk and l be the indices for a straight-line path and a circular arc path, respectively. And the initial values are set as k=1, l=0(3)
The sliding matrix is generated according to the waypoint table and the initial value ofk(4)
The desired pipelaying path expressed by the sliding matrix is analyzed to determine the end point of straight-line path and circular arc pipe path, the circle center, and rotation direction of the circular arc path of pipeline. The desired vessel path is calculated from the desired pipelaying path(5)
Determine which path type the vessel is going to track when the present path range is exceeded by the pipelaying vessel(6)
The conditionk>l implies that the vessel is going to travel along a straight-line path, and when k=l, it implies that the vessel is going to travel along a circular arc path(7)
When the vessel is traveling along a straight-line path and does not exceed its range, the start and the end points of the straight-line path are output to the LOS point calculating module. When the straight-line path range is exceeded, i.e., the vessel has just reached a circular arc path, then setl=l+1. Consequently, now k=l, and the LOS point on the circular arc path is calculated(8)
When the vessel is traveling along a circular arc path and exceeds its range, it implies that the vessel has just reached the next straight-line path, then setk=k+1. Consequently, now k>l, it is necessary to update the sliding matrix, and it will continue to run from the third stepFigure 9
Execution process of the path selecting method.It can be summarized as follows: when the value ofk is increased by one, the sliding matrix is required to be updated and path analysis is also performed so that the related parameters of the next path are obtained. When the value of l is increased by one, the sliding matrix is not required to be updated. However, it is necessary to determine whether the vessel is sailing on a straight-line path or a circular arc path, and the related LOS point is required to be calculated.


## 3.2. Coordinate Calculation of LOS Point

The LOS point is defined as the desired vessel position on a specified path. It is a dynamically changing point that guides the vessel approach to the desired path along with a specified heading. The vessel path selection is based on the previous LOS point. The flow chart for calculation of LOS points is shown in Figure10 and is explained in the following:
(1)
The initializing parameters are input into LOS point calculating module(2)
Check whether the path is required to be updated by comparing the value ofk and kpre in the process of calculating LOS point(3)
The conditionk=kpre implies that the path does not need to be updated, where k is the index value of the present straight-line path and kpre is the index value of the previous straight-line path(4)
Whenk>kpre, it implies that the path is required to be updated, and the start and the end points of a straight-line path are also required to be recalculated and updated(5)
Determine the type of vessel path by comparing the values ofk and l. If k>l, it implies that it is a straight-line path, and the method used to calculate LOS point of a straight-line path should be chosen. If k=l, it implies that it is a circular arc path, and the method used to calculate LOS point of a circular arc path should be selected(6)
The LOS point of the desired vessel path is obtainedFigure 10
The calculation process of the LOS point.


## 3.3. Multiobjective Motion Controller of DP Vessels

The LOS point is the desired position of the ship moving on the desired path during path following, and its coordinates are changing dynamically when the vessel reaches a new position. The sliding matrix also contains such control requirement information which is obtained by the waypoint table. The diagram of the multiobjective controller of a DP vessel is shown in Figure11.Figure 11
Multiobjective controller of DP vessel.When the vessel real-time position and the LOS point are known, the deviation of the vessel position inn coordinate frame is converted into the deviation in b coordinate frame. Then, the desired surge force, sway force, and yaw moment of the vessel can be calculated in combination with the deviations of the vessel speed and heading. The deviations of position and heading angle in b can be obtained using
(23)N~=Nlos−N,E~=Elos−E,ψ~=ψd−ψ,whereN~ and E~ are the deviations of north and east positions, respectively. ψ~ is the deviation of the heading angle. N and E are the real-time north and east positions, respectively; ψ is the real-time heading angle.Hence, when the vessel is tracking the first desired circular arc path, the desired body velocity components decomposed from the pipelaying speed in the coordinate framen can be expressed as
(24)ud=cosβC1Vd,vd=sinβC1Vd,whereVd is the desired speed in the coordinate frame n; βC1 is the desired sideslip angle; ud and vd are the desired velocity components in the body coordinate frame, respectively.When the vessel is tracking the desired circular arc path, the desired vessel speed is obtained using(25)Vd=VC1.From Figure6, when the rotational direction is clockwise and the circular arc path is clockwise, the value of βC1 is obtained by
(26)βC1=π2−δ0−γ0.The deviations of the vessel position in the coordinate frameb can be expressed as
(27)x~=cosψC1N~+sinψC1E~,y~=−sinψC1N~+cosψC1E~.The deviations of the body velocity components by the vessel position deviation in the coordinate frameb can be expressed as
(28)u~=ud−u+kpxx~,v~=vd−v+kpyy~,r~=rd−r+kpψψ~+kdψψ~̇,wherekpx is the control parameter used to convert the position deviation to the velocity deviation in surge direction. kpy is the control parameter used to convert the position deviation to the velocity deviation in sway direction. kpψ and kdψ are the control parameters used to convert the heading deviation to the rotational speed deviation.Hence, the desired control forces generated by the propulsion system can be obtained using(29)XT=kpuu~+kiu∫0tu~+kduu~̇,YT=kpvv~+kiv∫0tv~+kdvv~̇,NT=kprr~+kir∫0tr~+kdrr~̇,wherekpu,kiu, and kdu are the speed controller parameters of the dynamic positioning system in surge direction. kpv, kiv, and kdv are the speed controller parameters of the dynamic positioning system in sway direction. kpr, kir, and kdr are the speed controller parameters of the dynamic positioning system in yaw direction.


## 4. Simulation Analysis of the Pipelaying Operation

The method presented in this study is a systematic control method, including pipeline path coding, pipeline path analysis based on the sliding matrix, mapping conversion between the desired pipeline path and the desired vessel path, the LOS guidance law of circular arc paths, and pipelaying controller design. To verify the effectiveness of the proposed method, a simulation is performed by setting a series of waypoints, pipelaying speed, crab angle, and turning radius.A DP simulator for pipelaying operation is built using Qt software. As shown in Figure8, the simulator consists of the waypoint table, path planning module, path guidance module, thrust allocation module, propeller system module, hydrodynamic module, environment force module, state observer module, and motion controller. It is assumed that the pipelaying water depth is a constant, and the tension of pipe tensioner is also a constant. For this simulation, the pipeline is assumed to be rigid without deformation in the pipelaying operation. Hence, the touchdown distance is also a constant.The relevant parameters of the waypoint table need to be specified before starting pipelaying operation. All coordinates of six waypoints are set as shown in Table3. All crab angles of the desired pipe path are set to 0 degree which means that the desired vessel heading angle is specified to be the same as the tangential angle of the desired pipeline path. The desired pipelaying speed on straight-line paths is set to 2 m/s, and the turning radius of the pipeline is set to 300 m. In addition, the touchdown distance is set to 200 m, and the desired speed on circular arc paths is set to 0.3 m/s.Table 3
The waypoint table of the pipelaying operation.
No.N (m)E (m)Crab angle (degree)Speed (m/s)Turn radius (m)100020210001000023003100020000230042002600023005500400002300615004100020Simulation environment parameters of the pipelaying operation are set as follows: the wind speed is 2 m/s, the current velocity is 1 m/s, and the wave height is 1.5 m. All environmental disturbances are assumed collinear with a direction of 40°.


### 4.1. Pipelaying Precision Analysis

The time step is set to 0.5 s. The results from the pipelaying simulation are shown in Figures12–16.Figure 12
The specified waypoints, desired pipe path, and desired vessel path.Figure 13
Path following of the vessel.
(a)
The desired and the actual vessel path(b)
The cross-track errorFigure 14
The pipelaying results of the pipeline.
(a)
The desired and the actual pipelaying path(b)
The cross-track errorFigure 15
The vessel speed and pipelaying speed.Figure 16
The vessel heading angle and sideslip angle.
(a)
The desired and the actual heading(b)
The sideslip angleFor the results plotted in Figure12, the desired pipelaying path is derived from the specified waypoints in Table 3 via the proposed method of path planning. Then, the desired vessel path is generated from the desired pipelaying path based on the parameters such as touchdown distance and turning radius in Table 3 using the presented method of pipeline path mapping. It can be seen that the vessel takes a turn at the outside of the pipeline circular arc path due to the touchdown distance and this is consistent with the theoretical results of the pipelaying path. From the simulation results shown in Figure 12, it is found that the pipelaying method of path analysis and path mapping works as expected.The results of the vessel path obtained from the simulation and its cross-track error against the desired path are plotted in Figure13.As shown in Figure13(a), the vessel path from the simulation matches well with the desired vessel path on the segments of straight-line paths and circular arc paths, even when the vessel is subjected to drift forces caused by ocean waves, wind, and currents. Therefore, these confirm that the vessel can track the desired path well through the application of the proposed method that combines path analysis, path selection, path guidance, and control technology.As shown in Figure13(b), the cross-track error of the vessel path is less than 0.1 m on straight-line paths. However, the vessel cross-track error suddenly increases at the junction of a straight-line path and its adjacent circular arc path where the error reaches to 8.3 meters. This increased error may be the result of the continuous changing of the desired heading angle on circular arc paths, especially a sudden change of desired heading on the transition between a straight-line and its adjacent circular arc path. Then, the cross-track error gradually reduces, and finally, the path from simulation converges to the desired vessel path. Therefore, Figure 13(b) demonstrates that the proposed pipelaying controller method can make the vessel continuously track the desired vessel path.The results of the pipe path from the simulation and its cross-track error against the desired pipe path are plotted in Figure14.As shown in Figure14(a), the pipeline path from simulation matches well with the desired pipeline path during straight-line path following, and the simulated turning radius of the pipeline is the same as the specified turning radius. Therefore, it is reasonable to believe that the actual pipeline can be laid onto the desired pipelaying path by indirectly controlling the vessel motion. Additionally, Figure 14(a) also demonstrates that continuous pipelaying operation can be achieved via the proposed pipelaying control method. Consequently, the proposed method can be applied to various laying paths of pipelines and it is robust.The cross-track error of the pipe path is shown in Figure14(b). The cross-track error is less than 0.5 m on the segments of straight-line paths. However, the cross-track error suddenly increases at the junction of a straight-line path and its adjacent circular arc path. The maximum value of the cross-track error of the pipeline is approximately 9.5 m, which may be caused by a sudden change of the desired heading when the vessel is traveling between a straight-line path to its adjacent circular arc path. Besides, it may also be caused by the assumption that the pipe is rigid, and the cross-track error of the pipeline should be less than this value if the pipeline dynamic model [3, 6, 36] is considered. However, the cross-track error is gradually converged to zero when the vessel travels past the transition points.


### 4.2. Speed Analysis in Pipelaying Operation

The curves of speed performance during pipelaying operation are shown in Figure15.The pipelaying speed is 2 m/s on straight-line paths, and the speed is decreased to 0.3 m/s on the circular arc paths, which are consistent with the specified speed in the waypoint table. As shown in Figure15, the vessel speed along the surge direction is close to the pipelaying speed when the vessel is moving along a straight-line path. However, there is a relatively constant offset between the vessel speed along the surge direction and the pipelaying speed when the vessel is moving along a circular arc path, which is due to the vessel sway speed on the circular arc paths. Figure 15 also shows that the vessel speed along the sway direction is up to 0.18 m/s; this is the result of continuous changing of the desired heading angle which is consistent with the tangent of the desired pipe path despite that the predefined pipelaying speed on circular arc paths is reduced to 0.3 m/s. It is noticed that the vessel speed along the sway direction is decreased to zero on the straight-line paths. Hence, it is proved that the pipelaying controller designed in this study can control the vessel position and speed simultaneously during pipelaying operation.


### 4.3. Analysis of the Heading and Sideslip Angle

The vessel attitude performance in the pipelaying simulation, including the heading angle and sideslip angle, is shown in Figure16.As shown in Figure16(a), the desired heading angle on the segments of the straight-line path is a constant, which is equal to the direction of the straight-line path. The desired heading angle on the segments of circular arc paths is a straight line with a constant slope, which is equal to the tangent direction of circular arc paths. Due to the limitation of the response speed of the vessel propulsion system, the propellers cannot immediately produce the desired thrusts, and there is always a time lag between the actual heading angle and the desired heading angle during path following on circular arc paths. However, the time lag is quite small. Therefore, it proves that the actual heading angle can track the desired value during operation using the designed pipelaying controller.As shown in Figure16(b), the sideslip angle fluctuates around zero on the segments of straight-line paths, which may be caused by the environmental forces from wind, wave, and currents. However, the vessel actual heading angle converges to the tangential direction of the desired circular arc paths and it is obtained by allowing a certain sideslip angle. The sideslip angle increases from 0° to 33° or decreases from 0 to -33°when the vessel is traveling along a circular arc path.The proposed method, including path planning, path guidance, path tracking, and controlling, based on the sliding matrix for pipelaying operation in the study is effective. It is also proved by pipelaying simulation and analysis that the method is accurate in terms of the actual path, pipelaying speed, heading angle, and sideslip angle. The pipeline can be continuously laid onto the predefined path according to the specified pipe path, crab angle, pipelaying speed, and turning radius.


## 4.1. Pipelaying Precision Analysis

The time step is set to 0.5 s. The results from the pipelaying simulation are shown in Figures12–16.Figure 12
The specified waypoints, desired pipe path, and desired vessel path.Figure 13
Path following of the vessel.
(a)
The desired and the actual vessel path(b)
The cross-track errorFigure 14
The pipelaying results of the pipeline.
(a)
The desired and the actual pipelaying path(b)
The cross-track errorFigure 15
The vessel speed and pipelaying speed.Figure 16
The vessel heading angle and sideslip angle.
(a)
The desired and the actual heading(b)
The sideslip angleFor the results plotted in Figure12, the desired pipelaying path is derived from the specified waypoints in Table 3 via the proposed method of path planning. Then, the desired vessel path is generated from the desired pipelaying path based on the parameters such as touchdown distance and turning radius in Table 3 using the presented method of pipeline path mapping. It can be seen that the vessel takes a turn at the outside of the pipeline circular arc path due to the touchdown distance and this is consistent with the theoretical results of the pipelaying path. From the simulation results shown in Figure 12, it is found that the pipelaying method of path analysis and path mapping works as expected.The results of the vessel path obtained from the simulation and its cross-track error against the desired path are plotted in Figure13.As shown in Figure13(a), the vessel path from the simulation matches well with the desired vessel path on the segments of straight-line paths and circular arc paths, even when the vessel is subjected to drift forces caused by ocean waves, wind, and currents. Therefore, these confirm that the vessel can track the desired path well through the application of the proposed method that combines path analysis, path selection, path guidance, and control technology.As shown in Figure13(b), the cross-track error of the vessel path is less than 0.1 m on straight-line paths. However, the vessel cross-track error suddenly increases at the junction of a straight-line path and its adjacent circular arc path where the error reaches to 8.3 meters. This increased error may be the result of the continuous changing of the desired heading angle on circular arc paths, especially a sudden change of desired heading on the transition between a straight-line and its adjacent circular arc path. Then, the cross-track error gradually reduces, and finally, the path from simulation converges to the desired vessel path. Therefore, Figure 13(b) demonstrates that the proposed pipelaying controller method can make the vessel continuously track the desired vessel path.The results of the pipe path from the simulation and its cross-track error against the desired pipe path are plotted in Figure14.As shown in Figure14(a), the pipeline path from simulation matches well with the desired pipeline path during straight-line path following, and the simulated turning radius of the pipeline is the same as the specified turning radius. Therefore, it is reasonable to believe that the actual pipeline can be laid onto the desired pipelaying path by indirectly controlling the vessel motion. Additionally, Figure 14(a) also demonstrates that continuous pipelaying operation can be achieved via the proposed pipelaying control method. Consequently, the proposed method can be applied to various laying paths of pipelines and it is robust.The cross-track error of the pipe path is shown in Figure14(b). The cross-track error is less than 0.5 m on the segments of straight-line paths. However, the cross-track error suddenly increases at the junction of a straight-line path and its adjacent circular arc path. The maximum value of the cross-track error of the pipeline is approximately 9.5 m, which may be caused by a sudden change of the desired heading when the vessel is traveling between a straight-line path to its adjacent circular arc path. Besides, it may also be caused by the assumption that the pipe is rigid, and the cross-track error of the pipeline should be less than this value if the pipeline dynamic model [3, 6, 36] is considered. However, the cross-track error is gradually converged to zero when the vessel travels past the transition points.


## 4.2. Speed Analysis in Pipelaying Operation

The curves of speed performance during pipelaying operation are shown in Figure15.The pipelaying speed is 2 m/s on straight-line paths, and the speed is decreased to 0.3 m/s on the circular arc paths, which are consistent with the specified speed in the waypoint table. As shown in Figure15, the vessel speed along the surge direction is close to the pipelaying speed when the vessel is moving along a straight-line path. However, there is a relatively constant offset between the vessel speed along the surge direction and the pipelaying speed when the vessel is moving along a circular arc path, which is due to the vessel sway speed on the circular arc paths. Figure 15 also shows that the vessel speed along the sway direction is up to 0.18 m/s; this is the result of continuous changing of the desired heading angle which is consistent with the tangent of the desired pipe path despite that the predefined pipelaying speed on circular arc paths is reduced to 0.3 m/s. It is noticed that the vessel speed along the sway direction is decreased to zero on the straight-line paths. Hence, it is proved that the pipelaying controller designed in this study can control the vessel position and speed simultaneously during pipelaying operation.


## 4.3. Analysis of the Heading and Sideslip Angle

The vessel attitude performance in the pipelaying simulation, including the heading angle and sideslip angle, is shown in Figure16.As shown in Figure16(a), the desired heading angle on the segments of the straight-line path is a constant, which is equal to the direction of the straight-line path. The desired heading angle on the segments of circular arc paths is a straight line with a constant slope, which is equal to the tangent direction of circular arc paths. Due to the limitation of the response speed of the vessel propulsion system, the propellers cannot immediately produce the desired thrusts, and there is always a time lag between the actual heading angle and the desired heading angle during path following on circular arc paths. However, the time lag is quite small. Therefore, it proves that the actual heading angle can track the desired value during operation using the designed pipelaying controller.As shown in Figure16(b), the sideslip angle fluctuates around zero on the segments of straight-line paths, which may be caused by the environmental forces from wind, wave, and currents. However, the vessel actual heading angle converges to the tangential direction of the desired circular arc paths and it is obtained by allowing a certain sideslip angle. The sideslip angle increases from 0° to 33° or decreases from 0 to -33°when the vessel is traveling along a circular arc path.The proposed method, including path planning, path guidance, path tracking, and controlling, based on the sliding matrix for pipelaying operation in the study is effective. It is also proved by pipelaying simulation and analysis that the method is accurate in terms of the actual path, pipelaying speed, heading angle, and sideslip angle. The pipeline can be continuously laid onto the predefined path according to the specified pipe path, crab angle, pipelaying speed, and turning radius.


## 5. Conclusions

Through the operation simulation of pipelaying by DP ship and the analysis of the simulation results of pipeline trajectory, ship trajectory, pipelaying accuracy, pipelaying speed, and heading angle, the following conclusions can be obtained:(1)
A systematic method of pipelaying motion planning, guidance, and control of DP ship is proposed in this paper, which can control the pipelaying ship to switch smoothly between straight line and curve path, automatically update the path and relevant control information along the waypoint table, and realize the continuous laying of offshore pipeline at a specified speed(2)
A pipeline path analysis and coding method based on a sliding matrix are proposed, which can realize automatic pipeline path analysis, automatic coding storage, and path mapping. With the progress of pipelaying, the relevant path and control information are automatically updated, which simplifies the preprocessing of the whole waypoint table and improves the automatic analysis ability of DP pipelaying path(3)
A circular path guidance method is proposed, and a multiobjective motion controller of DP ship is designed, which can effectively control the position, turning radius, speed, and attitude of DP pipelaying shipRelevant simulation results show that the control method of pipelaying is effective and works reasonably well. This method is adopted in this paper to achieve the continuous pipelaying of the DP vessel, and it can also provide a valuable reference for path planning, guidance, and path tracking control of other surface vessels and underwater vehicles.



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*Source: 9702532-2022-01-25.xml*