A Numerical Simulation for Operating a Dynamic ... - OnePetro

Proceedings of the Twenty-third (2013) International Offshore and Polar Engineering Anchorage, Alaska, USA, June 30–July 5, 2013 Copyright © 2013 by the International Society of Offshore and Polar Engineers (ISOPE) ISBN 978-1-880653-99–9 (Set); ISSN 1098-6189 (Set)

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A Numerical Simulation for Operating a Dynamic Positioned Vessel in Level Ice Quan Zhou and Heather Peng Faculty of Engineering and Applied Science, Memorial University of Newfoundland St. John’s, NL, Canada

submerging and took the influence of speed into consideration. A similar regression model was proposed in (Spencer, 2001) which was based on model-scale/full-scale test of CCGS icebreakers.

ABSTRACT This paper proposed a numerical method to simulate the ice induced forces acting on the hull and the performance of a dynamic positioned vessel advancing in level ice. The interaction between the ship and the ice was considered by adopting a 2-D discretized model. A polygon based detection technique was applied to determine the contact surfaces. The method was validated by comparison with empirical calculation and ship model test of CCGS Terry Fox. Ship maneuvering was simulated by combining the three degree-of-freedom equations of rigid body motion and the Line-of-Sight guidance system.

In attempt to numerically simulate this icebreaking process and time history of ice loads, Wang (S. Wang, 2001) proposed a discrete element method (DEM) for the force of an advancing ice field towards a fixed conical structure. The method was derived from the empirical formula proposed by Kashtelyan (Kerr, 1975). The size of the broken ice floes was calculated based on the speed and character length of the ice. The method was then extended by Sawamura (2009), Nguyen et al. (2009) and Su et al. (2010) to the study of ship maneuvering and hull-ice interaction.

Key Words: Level ice, track keeping, dynamic positioning, continuous icebreaking pattern, contact detection, ice load, ship maneuvering, guidance, feedback control

Lau (2006) presented the detailed results of the PMM model test of Terry Fox, and a different DEM model was proposed in (Liu et al., 2006) which was based on the work done by Spencer. Not only the resistance but also lateral force and yaw moment were measured and further studied in the model test. The relations between turning circle radius and yaw moment and channel width were also studied.

INTRODUCTION Recent increase of hydrocarbon exploration as well as the requirement of transportation in the Arctic region has led to a renewed interest in the ship maneuvering in ice covered area. Vessel operating in ice condition requires a proper dynamic positioning (DP) system to help maintain the ability of course keeping or position keeping which is highly dependent on an accurate model to simulate ice loads. Different ice conditions, e.g., broken ice, level ice, ridge and iceberg, may be encountered by a vessel advancing in Arctic region. Among these, vessel interaction with level ice is of most concern and has been widely studied. There are several methods available for achieving level ice induced resistance in literature, including model tests, empirical and regression formulas and numerical simulations.

Peng (2008) presented the simulation of a dynamic positioning FPSO during offloading operation in open water. The aim of this paper is to propose a method for simulating the behavior of a DP vessel operating in level ice. It is emphasized that the coupling between the vessel motion and icebreaking is taken into consideration. The DEM is applied which is similar to that derived in (Su et al., 2010). A different contact detecting method is introduced herein which requires less fine discretized model. Guidance system and controller are designed on the basis of Line-of-Sight method.

MODELING OF HULL-ICE INTERACTION

The resistance encountered by a vessel when penetrating into ice plate primarily depends on the icebreaking process. This process is in a repeated manner and consists of several events which can be seen in (Enkvist, 1979). Valanto (1989) conducted an experiment to examine those different events. First, ice is crushed, accelerated and turned downward as long as it contacts the hull. Crushing and displacement continues as the hull keeps penetrating into the ice plate. When the vertical force reaches the capacity of the plate, bending failure occurs. A piece of ice then breaks down from the plate and keeps turning until parallels with the hull. The floe then slides down along the hull until it is cleared aside.

Kinematics In this paper, the state variables of the ice sheet and the vessel are defined with respect to three different coordinate systems or reference frames as follow  The North-East-Down (O-NED) frame is consider as inertial and fixed on the earth surface with x-axis pointing north, yaxis pointing east and z-axis pointing downward. The positions and heading of the ship and the ice plate would be integrated in this frame;  The ship-fixed frame, denoted as o-xyz, is attached on the vessel with x-axis pointing the bow, y-axis pointing starboard

Lindqvist (1989) proposed a set of empirical formulas to calculate ship resistance in level ice. He divided resistance into crushing, bending and

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

N H   N r r  N v v  N v v  N r r  N vr vr  N vv v | v |

and z-axis pointing downward. Ship kinetics and ship-hull interaction will be solved in this frame; The ice-fixed frame (oi-xiyizi) is attached on the ice plate which is parallel to O-NED. The node coordinates of the discretized ice edge will be defined with respect to this frame.

 N vvr vvr  N vrr vrr  N r r r r  xG (Yv v  Yr r  Yvr vr  Yvv v | v | Yrr r | r

where u , v, r are surge, sway and yaw velocities relative to the water at the center of gravity. The control forces are calculated by Gong (1993) as

Positions and heading of the ship with respect to O-NED was denoted as

η  ( x E , y E , )

denoted as

. Ship velocity with respect to o-xyz was

v  (u , v, r ) . The vector x hb  ( x, y ) was defined as

X p  1  t n 2 D p2 KT J p 

the x, y-position of hull nodes with respect to o-xyz; and

x ii  ( xii , yii )

x ib  ( xib , y ib )

cos R ( )   sin  0

(7a)

YR  1  aH FN cos  N R  xR  aH xH FN cos 

was defined as the x, y-position

of ice nodes with respect to o-xyz.

η  R ( ) v

(6)

X R  1  t R FN sin 

was defined as the x, y-position of ice nodes with

respect to oi-xiyizi;

(5c)

(7b) (7c)

(1)

 sin cos 0

where t is thrust deduction; n and D p are resolution and diameter of

0 0 1

the propeller, respectively; K T is thrust coefficient; t R is rudder drag

(2)

correction factor; a H and x H are coefficients of lateral force due to rudder deflection; x R is longitudinal position of the rudder;  is the rudder deflection; the normal rudder force, FN is given as

Ship Kinetics Neglecting wave loads and vertical motions, a low frequency, 3Degree-of-Freedom (3DOF) model is adopted in this paper. The equations described in ship fixed coordinate system in a robotic vectorial form (Fossen, 2011) is given by

M RB v  C RB v v  FH  Fc  Fice where MRB is inertia matrix of the ship; Centripetal matrix;

FN 

U R is the effective rudder inflow velocity;  R is the effective rudder inflow angle. Details about propeller thrust and rudder forces can be seen in (Gong, 1993). The ice-induced loads will be calculated numerically by using the method presented below.

Cariolis and

m 0   0 m  0 mx g 

0   M RB mx g  I zz   0  mr  mxg r    C RB   mr 0 0  mxg r 0 0  

(8)

where  is aspect ratio of the rudder; AR is projected rudder area;

(3)

CRB v  is

1 6.13  ARU R2 sin R  2   2.25

Ice-induced Loads (4a)

Several assumptions were made as follow in order to simplify the hullice interaction problem:  The intact ice plate is semi-infinite with uniform thickness and constant velocity with respect to O-NED;  The hull-ice interaction is a continuous process which is repeating the cycle of contacting, crushing and bending;  Vertical displacement of the ice plate is neglected;  Contacting surface remains flat during crushing;  The crushing force is proportional to the area of contacting surface;  The shape of broken ice floe in the bending failure is circular.  The ice load due to bending and submersion is constant and is calculated from empirical equations (Lindqvist, 1989).

(4b)

FH  [ X H , YH , N H ]T are hydrodynamic forces and moments T T acting on the hull; Fc  [ X P ,0,0]  [ X R , YR , N R ] is control forces generated by propeller and rudder; Fice is ice-induced forces.

Based on these assumptions,

Fice

can be calculated as

The hydrodynamic forces are calculate by (Peng , et al., 2008)

X H   X u u  X u   (Yv  X vr )vr YH   X u ur  Yv v  Yr r  Yv v  Yr r  Yv v v v  Yr r r r  Yvr vr

Fice  Fcr  Fsub

(5a)

where

Fcr  [ X cr, Ycr , Z cr ]T is

failure occurs;

(5b)

(9) the crushing force before bending

Fsub  [ X sub,0,0]T

is the force due to bending and

submersion that is calculated from empirical formulas proposed by Lindqvist (1989). The numerical method to calculate as follow.

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Fcr

is presented

of the ship-fixed frame, and the other segment was bounded by two adjacent points that at least one of them was within the hull polygon. If even number intersections existed, scenario 1 occurred; otherwise, scenario 2 was identified. To identify scenario 3, an outer frame formed by four points (as shown in Figure 1) was needed. The ice nodes within the outer frame were first detected and then formed a closed polygon with the corner points of the outer frame. After that, the same algorithm used previously could be applied here to detect the hull nodes involving in the contact. In scenario 2 and 3, a new node (the interpolated node in Figure 2) was temporally interpolated on the ship hull and the ice edge, respectively. At each time step, different contact surfaces were detected and expressed as polygons formed by the nodes both on the hull and the ice edge. Those polygons were further used to calculate the areas of these surfaces.

Contact Scenario. A similar 2-D model proposed in (Su et al., 2010) was applied here to simulate the interaction between the ship hull and the ice plate. The water line of the hull and a certain segment of the ice edge were discretized into a number of nodes whose coordinates were defined in ship-fixed frame and ice-fixed frame, respectively. The discretized ship hull was a closed polygon, whereas the discretized ice edge was a broken line. Then, the ice nodes were transformed into shipfixed frame for convenience as shown in Figure 1.

Ice Crushing. As long as the hull had contacted with the ice edge, ice crushing started at the contact point. It continued until the bending failure occurred. The crushing force was assumed normal to the contact surface and is calculated by Su et al. (2010) as

Figure 1 Discretized Ice Edge and Water Line in Ship-fixed Coordinate System

Fcr   cr Acr

Since both the ship hull and the ice edge were discontinuous, the accuracy of the simulation results highly depended on how fine the discretization was (Nguyen et al., 2009; Su et al., 2010). To increase the accuracy of the program as well as saving computing time, different contact scenarios were determined and a polygon based detection technique was then applied. Three different contact scenarios could occur during the simulation as shown in Figure 2:  Scenario 1, nodes both on the ice edge and the ship hull were involved in the contact;  Scenario 2, nodes only on the ice edge were involved in the contact;  Scenario 3, nodes only on the ship hull were involved in the contact.

where

 cr

(10)

is the average crushing strength, and

Acr

is the area of the

contact surface.





Figure 3 Two Cases of Contact Area Calculation Corresponding to ith Triangle Element (Su et al., 2010) To calculate the area, the polygon of contact surface was first divided into a number of triangles as shown in Figure 2. Then, two different cases had to be considered as illustrated in Figure 3 when calculating the area of the contact surface corresponding to a triangle element. Finally, all the areas were summarized to obtain the total area of the whole contact surface. Case 1

1 i Lic A  Lh 2 cos  i cr

(11a)

Case 2

Li  h tan   hi 1  Acri   Lih  Lih c i i 2 Lc  sin  

Figure 2 Three Scenarios of Contact An algorithm which checked the ice nodes whether they were inside the polygon of the hull was adopted to identify different scenarios. If there is, at least, one ice node within the hull polygon, every node on the hull would be investigated to identify scenario 1 or scenario 2. These two scenarios could be distinguished by checking the intersections between two segments: one segment was bounded by a hull node and the origin

where

Lh

and

Lc

are parameters as shown in Figure 3;

(11b)

hi

is the ice

thickness;  is slope angle of the ship hull at contact point. The superscript i denotes the parameters corresponding to ith triangle

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element. where

Besides the crushing force, friction which was related to relative velocity between the hull and the ice should also be considered. Both the crushing force and the friction should be projected onto three axes of the ship-fixed frame. The method proposed in Su et al. (2010) was applied herein to calculate the friction. The directions of friction and crushing force could be denoted as tangential and normal vectors, respectively. These two vectors were determined by the contact point and the shape of the water line.

f V  Fcr v

v2  v n2,1

fV  Fcr vn ,1

v2  vn2,1

while

vn ,1

relative velocity within water plane;

and

Cv

are empirical coefficients,

Cl

has a positive value

has a negative value, both of them are tunable in the

program (

Cl  0.15

and

Cv  0.8 ;

v (

vn , 2

in Figure 4) is

relative velocity that is normal to the contact surface; l is the characteristic length of ice which was given by 14

(12a)

  Ehi3  l   2  12 1    w g 

(12b)

where





(15)

E is Young’s modulus of ice; 

is Poisson’s ratio;

w is water

density. Two intersections between the ice edge and the circumferential crack were then determined as shown in Figure 2. An idealized wedge was then formed, and the open angle was calculated from the coordinates of the apex (contact point) and two intersections. After that, whether the ice wedge would break off was investigated by comparing

was frictional coefficient between ship hull and ice; v was

where

Cl Cv

was relative velocity along

the ship hull in vertical plane as shown in Figure 4.

Pf

with

Fv .

If

Fv  Pf

, no bending failure would occur, the ice

wedge would be just crushed at the cusp; otherwise, bending failure would occur. The circumferential crack would be discretized into nodes, and the ice edge would be updated for the calculation in the next step.



v

v  n





vn,1

FH

vn

FV fV

Based on the analysis, the continuous icebreaking process and icebreaking forces can be determined step by step. First, the hull-ice interaction is detected based on the 2-D discretized model (polygons that represents contact zones are pointed out and stored). Then the real contact area is calculated by either Eq. 11a or Eq. 11b, and crushing force is determined by Eq. 10. An idealized wedge is then assumed, and Eq. 13 can be adopted to obtain bending capacity. If failure occurs, ice edge will be updated and stored for the calculation in the next step. Figure 5 shows the time history of ice-crushing forces in surge, sway and yaw directions.

vn vn,2

Fcr

fH FH

Figure 4 Force and Velocity Components Bending failure. The vertical force increased as the ship was penetrating into the ice plate. As long as it exceeded the load capacity of the ice sheet, bending failure would happen and a circular ice floe would break off. The load capacity is calculated by adopting Kashtelyan’s method (Kerr, 1975).

  Pf  C f    f hi2   2

where



sheet;

Cf

is open angle of a wedge;

(13)

f

Figure 5 Time History of Ice Loads with Ice Thickness of 0.04m, Ship Advancing Velocity 0.3m/s

is flexural strength of the ice

VALIDATION OF THE NUMERICAL MODEL

is an empirical coefficient. Kashtelyan suggested a small NRC-IOT carried out a PMM model test in ice tank with a 1:21.8 scaled model of CCGS Terry Fox (Lau, 2006). The test results were used for validation in (Lau et al., 2006; Sawamura et al., 2010). In this section, this ship model was also used to validate the numerical method proposed above. The shape of the water line is as shown in Figure 6. The flexural and crushing strength of the ice plate are 40.5 kPa (Lau, 2006) and 130 kPa (Sawamura et al., 2010), respectively.

value (around 1) for this constant, while Nguyen et al. (2009) used a value of 4.5 with no explanation. In Su et al. (2010), a study of determining

Cf

was carried out and a value of 3.1 was finally adopted.

In this study this coefficient is assumed as 3.8. To apply Eq. 13, a circular ice floe with the center at the contact point was first assumed to break off from the plate. The radius of circular floe was calculated by the expression given in (Wang, 2001)

R  Cl l 1  Cv v 

(14)

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5

24 .5

80.5

31.5

24 .5

23.3

Figure 6 Discretized Waterline and Slope Angle of the Hull

Figure 9 Relationship between Turning Radius and Yaw Moment. The mean ice resistance is compared with two different empirical methods, as well as the model test results, for different ice thickness and ice velocities as shown in Figure 7. One empirical method was proposed by Lindqvist (1989), and the other was proposed by Spencer (2001). The test was carried out by NRC-IOT (Lau, 2006). The simulation results are well consistent with both empirical calculation and the model test. However, one outlier in the first figure may indicate that the tunable parameters of the program are velocity dependent. The ice broken pattern and yaw moment were also key features to the study. The simulated channel pattern, both in pure sway test and in pure yaw test, is presented in Figure 8. They are similar to the simulation results from (Liu et al., 2006) and the results of PMM model test from (Lau, 2006). Unlike ice resistance and icebreaking pattern, yaw moment was not available in most literature. Lau et al. (2004) proposed an empirical model for the relationship between tuning radius and yaw moment. The simulation results were compared with that from empirical method as shown in Figure 9. These simulation results are the average value of yaw moment in a certain period. A discrepancy between the simulation results of port and starboard at the same turning radius can be seen. This is due to: 1) the new crack is not symmetrically discretized into nodes, so that the channel, as well as the ice loads, may not be the same as the ship turns port or starboard; 2) the ice loads are considered as high frequency cyclical in the program, so that the mean value will be affected by peak loads. However, these stagger simulation points fall around the empirical curve and follow the trend, i.e., yaw moment decreases as the turning radius increases.

Figure 7 Comparison of Mean Simulated Ice Load, Empirical Method and Experimental Ice Resistance for Different Ice Thickness and Ship Velocities

DESIGN OF CONTROL AND GUIDANCE SYSTEM Depending on the relationship between the number of actuators and the number of DOF, two different control situations can be identified: under-actuation and full actuation (Fossen, 2011). In this paper, an under-actuated situation was considered; thus, only surge and yaw motion were controlled. The two control signals were desired heading and speed. The reference trajectory was given as a number of set-points, and the Line-of-Sight (LOS) guidance method was applied to calculate the desired heading. The other control signal was directly given as an input command.

Figure 8 Channel Pattern and Width for Pure Sway and Pure Yaw Tests

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xk  x 2   yk  y 2  Rk2

LOS Guidance

where

Rk

(19)

was the radius of the circle of acceptance at set-point

pk .

LOS Controller

 los

The purpose of the controller was to implement autopilot to the ship in level ice. Two tasks needed to be finished for that purpose (Breivik, 2003): forcing the ship to converge to a desired path and forcing the surge speed to converge to the desired speed, that is

lim[ (t )   los (t )]  0 t  t 

Figure 10 Illustration of Line-of-Sight Guidance Method (Breivik, 2003).

Hence, two separated conventional PID controllers for heading and surge speed were used. Though an under-actuated control model was used, all three motions would be controlled since the control of sway motion was merged into that of yaw motion. The LOS heading controller ensured the heading converge to the desired angle as well as eliminating the cross-track error. The controllers would only compensate for the low frequency motions, so an observer was needed to filter out the high frequency motion of the ship due to ice loads. The observer was derived from the one for open water excluding the wave frequency model and treating ice loads as bias since no ice load measurement was available. The observer model can be found in (Nguyen et al., 2009; Sorensen, 2005)

LOS methodology is a three-point guidance scheme and has been widely used surface vessel guidance (Fossen, 2011). This method was adopted by Moreira et al. (2005) and (2007). Detailed implementation can be seen in Breivik (2003). In this method a LOS vector was computed from the ship to a point on the path between two way points for heading control as shown in Figure 10.

 los

was called LOS angle. It was the command heading angle for the

ship in next time step.

p  x, y 

T

was the current position of the

plos  xlos , ylos  was a point on the path;  k 1 was the desired heading; pk 1 and pk were two set-points. The LOS angle T

ship;

ηˆ  R ψ υˆ  K 2 ~ y  bˆ  K 3 ~ y M RB υˆ  C RB υˆ  FH  Fc  Fice  R T ψ bˆ  R T ψ K 4 ~ y yˆ  ηˆ

was given by

 los  arctan( ( xlos , y los )

y los  y ) xlos  x

(16)

(21)

was computed as

(17a)

is the estimation error, matrices.

xlos  x 2   ylos  y 2  nLpp 2 nL pp

(17b)

The input signals needed for surge motion control are designed surge speed, estimated surge speed and their higher order time derivatives; the input signals for yaw motion control are designed and estimated heading angle and their time derivatives. The output signals are the propeller revolution, n and rudder deflection angle,  . The PID control laws (Peng et al., 2008) are given as

is an n time ship length. An alternative method was

available by replacing

nL pp

with the sum of the distance between the

current ship position and the path,

yˆ is the estimated motion, ~ y  y  yˆ 33 K 2 , K 3 , K 4  R are observer gain

where y is the actual motion,

ylos  y k 1 y k  y k 1   tan  k 1  xlos  xk 1 xk  xk 1

Where

(20)

lim[u (t )  u d (t )]  0

rmin , and the ship length, L pp .

t n   K p ,1u~  K d ,1u~  K i ,1  u~dt

(22a)

0

nLpp  rmin  L pp

(18)

   K p,3 R T ˆ ~  K d ,3 ~ r  K i ,3 R T ˆ  ~dt t

0

As the ship approaches the set-point

(22b)

pk , an algorithm for switching

between set-points was needed. The next set-point,

pk 1 ,

where

would be

K p , Kd , Ki

are control gains, subscript 1 denotes surge

direction, subscript 3 denotes yaw direction; the tilde on top of the terms denotes the difference between the estimated value and the

selected when the ship entered the circle of acceptance at the set-point,

pk , as shown in Figure 10. The criteria can be expressed as

designed value; ˆ is the estimated heading angle. After achieving the propeller revolution and the rudder deflection angle, control forces can be obtained.

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The initial states of the ship for path-following investigation are given as:

Simulation Results

  0, 0, 0T ,   3.9, 0, 0T

Full-scale R-Class icebreaker was used in this section for validation. The turning circle tests both in open water and in level ice were carried out with the developed program and the results were compared with full-scale experiments (Williams, et al., 1992). The level ice is assumed as 0.51m thickness with ice strength 200kpa. After that, the pathfollowing ability was investigated both in open water and in level ice. The desired path was defined by a set of way-points as shown in Table 1. The turning circles in open water and level ice are shown in Figure 11, and the comparison is listed in Table 2.

The radius of acceptance in LOS guidance method for all set-points was set to twice the ship lengths. The path-following performance is shown in Figure 12. The advancing speed is shown in Figure 13.

Table 1 Desired Set-Points and Surge Velocities Coordinates (x, y) (0, 0)

Point Number 1

Surge Velocity knots 8.0

2

(1000, 100)

8.0

3

(2000,-1500)

8.0

4

(3000, -2000)

8.0

5

(5500, -800)

8.0

6

(5500, 500)

8.0

7

(3500, 1500)

8.0

Figure 12 Desired Path and Real Tracks both in Water and in Ice

Figure 13 Advancing Speed Time History both in Water and in Ice The simulated tactical diameter and the steady speed of the turning circle test are consistent with the result of the experiment as shown in Table 2. From the comparison of turning test both in water and in ice, we conclude that the presence of ice will significantly affect the turning ability of the ship. The path-following ability of the ship in ice is not as good as that in open water which can be seen from Figure 12.

Figure 11 Turning Circle both in Open Water (12.4 knots, 35o starboard) and in Level Ice (12 knots, 30o starboard) Table 2 Comparison between Simulation and Experiment Results

Entry Simulation in open water Experiment in open water Simulation in level ice Experiment in level ice

Entry speed, knots

Helm angle, deg

Turning diameter, m

Steady speed, knots

12.4

35

379

7.67

12.4

35

390

7.5

12.0

30

1294

10.2

12.0

30

1220

10.2

The LOS guidance method, which is often applied to the pathfollowing in open water, can also be used in level ice condition. The under-actuated ship can pursue the desired heading and eliminate the cross track error simultaneously with the help of LOS guidance. The PID control law is sufficient for a ship to follow a pre-defined path and maintain the advancing speed.

CONCLUSIONS A numerical method for simulating the ice induced forces acting on the hull and the performance of a dynamic positioned vessel advancing in

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level ice is proposed in this paper. The icebreaking process is assumed in a cycle pattern which involves crushing, bending and submersion. The crushing force is calculated numerically at each time step. The coupling of icebreaking and ship dynamics are considered. The bending and submerging forces are assumed time-independent but related to advancing velocity, ice thickness and the shape of the hull. They are determined from the empirical calculation. The controller is designed based on LOS guidance model. A LF control model is applied by filtering out ice-induce motions. Simulation results show that the mean ice loads agrees with the empirical and experimental ice resistance, that the channel pattern agrees with other’s work, that the relationship between turning radius and yaw moment agrees with empirical method. The simulation also shows that the LOS method based guidance and controller performs well for a DP vessel operating in level ice conditions.

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ACKNOWLEDGEMENTS This research was financially supported by the NSERC CREATE research program for Offshore Technology Research, MITACS Accelerate Fund and NSERC Discovery Grant. The authors would like to express their thanks to the ABS Harsh Environment Technology Centre in Memorial University as well.

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