J Mar Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0 D O I 10.1007/S00773-013-0220-7
O R I G I N A L A R T I C L E
Heading control for turret-moored vessel in level ice based
on Kalman filter with thrust allocation
L i Zliou • T o r g e i r M o a n • K a j R i s k a • Biao S u
Received: 19 December 2011/Accepted: 26 March 2013/Published online: 12 A p r i l 2013 © J A S N A O E 2013
Abstract Tliis paper m a i n l y focuses on the heading control of a position-moored vessel operating under level ice regime. A dynamic ice simulator interconnecting the vessel motions w i t h the ice dynamics is used f o r the design of the heading control system. The strategy is to ensure that the vessel is kept at an appropriate position w i t h i n the safe limits. Using a heading controller based on a Kalman filter, the desired control force is computed to counteract the envii-onmental disturbances. A thrust allocation method is developed to go w i t h the heading controller. T o keep the ice forces to a reasonable level, the moored vessel should be aligned w i t h the ice d r i f t direction and small angles up to 15° i n changes on heading against the ice flow c o u l d be possible. Tlrerefore, heading c o n t r o l o f a m o o r e d vessel exposed to l e v e l ice w i t h d r i f t angle 0 ° and 15° is simulated since the d y n a m i c p o s i t i o n i n g system needs to resist ice y a w moments and lateral ice forces on the h u l l . The s i m u l a t i o n indicates that the proposed c o n t r o l strategy p e r f o r m s s a t i s f a c t o r i l y f o r a m o o r e d vessel i n l e v e l ice.
K e y w o r d s Heading control • K a l m a n filter • Thrust allocation • Position mooring • L e v e l ice
L . Zhou ( E l ) • T. M o a n • K . Riska • B . Su Centre f o r Ships and Ocean Structures,
Norwegian University o f Science and Technology, Trondheim, Norway
e-mail: li.zhou@ntnu.no K . Riska
Total SA (DGEF/DEV/TEC/GEG), Paris la Defence, France
1 Station Iteeping metliod
The presence o f sea ice i n the Arctic region poses a chal-lenge f o r keeping the vessel at location w i t h d i f f e r e n t purposes i n industrial activity f o r o i l exploration and exploitation. Station keeping f o r vessels i n both waves and ice-covered areas can be achieved b y three methods: a m o o r i n g system; a dynamic positioning system; or a combination o f the first and second methods.
The pure m o o r i n g method is relatively popular i n industry. Relevant experience w i t h moored structures i n ice is obtained f r o m d r i l l i n g operations i n the Beaufort Sea. A s a conical d r ü l i n g unit, the K u l l u k p l a t f o r m was designed w i t h a variety o f special features to i m p r o v e the perfor-mance under ice conditions i n the shallow water [ 1 ] . The system has a d o w n w a r d sloping circular h u l l near the waterUne that breaks the oncoming ice m a i n l y i n flexure and an outward flare near the bottom that clears the broken ice cusps away f r o m the m o o n pool and m o o r i n g lines. The strong m o o r i n g system could resist ice forces up to 450 tons. I t worked successfully w i t h the d o w n t i m e i n operations o f less than 10 %, but at a cost o f extensive ice management by three icebreakers.
Another method is to use dynamic positioning (DP) sys-tems f o r maintaining the vessels i n position and o n a coiTcct heading. Station keeping w i t h dynamic positioning systems was employed f o r d r i l l i n g and d i v i n g operations i n heavy ice conditions, where the water depth is 30 m [ 2 ] . The operation was supported by t w o icebreakers. The total ice d o w n time was 22 % d u r i n g 6 weeks o f operation i n v a r y i n g ice con-ditions. The related d r i l l i n g operations using dynamic posi-tioning i n ice i n the A r c t i c Ocean was also reported by M o r a n et al. [3]. The site water depth ranges f r o m 1100 to 1300 m . The manual positioning instead of automatic D P operation was applied to acltieve a very successful station keeping w i t h
J Mar Sci Technol (2013) 1 8 : 4 6 0 ^ 7 0 461
Heading controller T h r u s t a l l o c a t l o n ;)l Ship
( ^ M o o r i n g
Fig. 1 B l o c k diagram o f control strategy
nearly no down time w h e n good ice management w i t h ice-breakers is used. Comparing t w o D P operations, it was f o u n d that D P is preferred to be used i n deep water. This was also concluded b y A l l a n et al. [ 4 ] .
Tliruster assisted m o o r i n g provides a third way of station keeping. Considering the water l i m i t a t i o n f o r D P operation and extensive ice management f o r mooring system, an appropriate allowance may be made f o r the effectiveness o f thruster systems i n reducing m o o r i n g loads. I n addition, there is also a potential o f l o w e r i n g f u e l consumption i n shallow and intermediate open water tbrough m o o r i n g systems [ 5 ] . Kjerstad [ 6 ] pointed out that i t w i l l be impor-tant that the vessel's heading is towards the direction o f ice d r i f t . Kuehnlein [ 7 ] identified the main challenges f o r dynamic station keeping i n ice, o f w h i c h he also mentioned that the vessels need to be oriented always against the d r i f t i n g ice w i t h the b o w or the aft end, as the side motions are very l i m i t e d or even not possible due to vertical side walls. Therefore, a heading controller needs to keep the vessel aligned w i t h the d r i f t i n g ice w h i l e a m o o r i n g system needs to provide a reactive force to compensate f o r the mean d r i f t loads o f the environment due to ice. W i l k m a n [ 8 ] summarized some problems that w i l l be challenging f o r D P operations i n ice, such as forces acting on the vessel, forces caused b y ice dynamics, t u r n i n g yaw moment, changes i n ice movement direction, new types o f thruster control allocation and so on. I n this paper, dynamic ice forces acting on the vessel and turning yaw moment under 0 ° and 1 5 ° ice d r i f t angle are simulated based on the mathematical model o f level ice during level i c e - h u l l interaction i n Biao et al. [ 9 ] , Zhou et al. [ 1 0 , 1 1 ] . The aim o f this paper is to propose a method f o r simulating the behavior of a moored vessel w i t h heading control based on a K a l m a n filter i n level-ice cov-ered seas under d i f f e r e n t ice d r i f t angles. The overall schematic of control strategy is shown i n F i g . 1.
2 Mathematical modelling
A c c o r d i n g to S0rensen [ 1 2 ] , the mathematical models o f a vessel may be f o r m u l a t e d i n t w o complexity levels: a
control plant model and a process plant model. The process plant model is a comprehensive description o f the real physics o f plant dynamics w h i c h should be as close as possible. The control plant model is a s i m p l i f i e d mathe-matical description to capture only the m a i n characteristics f r o m the process plant model. The process plant model i n c l u d i n g the kinematics and kinetics is shown i n the f o l l o w i n g .
2 . 1 Kinematics
Let the Earth-fixed position {x, y) and heading i/^ o f the vessel relative to an Earth-fixed f r a m e X^Y-eZb be expres-sed i n vector f o r m hyr\ = [x, y, ij/] and let the vessel-fixed vessel velocities be represented by the state vector V = [ii, V, )•]. The three modes are referred to as the surge, sway and yaw o f a vessel. The o r i g i n of the vessel-fixed f r a m e XYZ is located at the vessel centre Hue i n a distance Xg f r o m the center of gravity.
I n general, the transformation between the vessel- and Earth-fixed coordinate frames can he described b y the three Euler angles; cj) ( r o l l ) , 9 (pitch) and ij/ ( y a w ) . The horizontal m o t i o n o f the vessel is described b y one Euler angle only, w h i c h is the y a w angle. I f there is only surge, sway and yaw ( 3 - D O F ) , the relationship between the E a r t h - f i x e d position vector and the b o d y - f i x e d v e l o c i t y vector is given through a transformation matrix,
ci]/ 0
sip c\(/ 0 0 0 1
( 1 )
( 2 )
where c, s are compact notations f o r cosine and sine, respectively.
2.2 Vessel kinetics
A process plant m o d e l f o r vessel dynamics i n a horizontal plane can be expressed as
( M R B + M A ) V + CRB ( V ) V + D ( V r ) = Tice + tmo + Xth, (3)
where M R B is the system inertia m a t r i x and M A is the added mass, tice is the level ice load vector i n the body-fixed f r a m e , Xj^o is the m o o r i n g force translated f r o m the Earth-fixed coordinate system to b o d y - f i x e d coordinate system b y the rotational matrix described i n E q . 2 , tth is the thruster vector consisting o f forces and moments produced by the propulsion system. The skewsymmetric C o r i o l i s -centripetal m a t r i x C R B ( V ) is given by
462 J M a r Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0 C R B ( V ) 0 0 m{xgr + v) —mu -m{xgr + v) mu 0 (4)
where Xg is the longitudinal position o f the center o f gravity i n b o d y - f i x e d coordinate. When a vessel operates under dynamic positioning where the velocities are small, the Coriolis-centripetal terms C R B ( V ) V can be ignored f o r control design. The term D ( V r ) . i n Eq. 3 represents the damping f o r c e due to the m o t i o n o f the vessel relative to the ambient water. E m p i r i c a l formulas are o f t e n used to calculate the damping forces and moments o n a vessel [13], that is D ( V , ) = 0.075 iSu,\uj\ ( I o g , „ f i „ - 2 ) ^ \p\^j^AxCj,{x)D{x)\v,\v,\ \AXC^{X)D{X)X]V,\V,\ (5)
where = [wr, iv, 'V] is the m o t i o n o f the vessel relative to the ambient water; the integration is over the length L o f the vessel; i?^ is Reynolds number; p is the water density; S is the wetted surface o f the vessel; and C d W is the drag coefficient f o r cross-flow past an i n f i n i t e l y long cylinder w i t h the crosssectional area o f the vessel at the l o n g i t u -dinal coordinate x. D{x) is the sectional draught.
2.3 L e v e l ice loads
The m o d e l l i n g o f ice load acting on a moored vessel i n level ice depends h i g h l y on the interaction process by w h i c h the h u l l breaks and displaces the ice. Once the ice gets contact w i t h the h u l l , ice is being crushed. T h e n the crusliing f o r c e continues to increase w i t h increasing con-tact area u n t i l its vertical force component gets large enough to cause bending f a i l u r e o f the ice, after w h i c h the broken ice floes start to turn along the vessel's h u l l u n t i l they are parallel to the h u l l . F i n a l l y , the floes submerge and shde along the h u h as they are pushed by the next broken ice floes. The level ice loads rhodelling has been presented i n detail b y Z h o u et al. [10, 11]. The method was also validated b y comparing the simulated results w i t h f u l l scale field measurement o f the K u l l u k . Therefore, the details are neglected i n this paper.
2.4 M o o r i n g systein
C o m p a r e d w i t h the spread m o o r i n g system, the internal tuiTet m o o r i n g system has good weather-vaning capacity to mitigate ice action o n the h u l l since the ice d r i f t direction is changing due to current, w i n d , and so o n i n luost ice-infested waters. I n addition, i t facilitates the operation o f disconnecting and leaving the site q u i c k l y and reliably.
The motions o f a moored vessel i n ice conditions are believed to be significantiy influenced b y the m o o r i n g lines. M o o r i n g systems provide not only time-varying restoring forces but also damping forces, both o f w h i c h should be taken i n t o consideration i n vessel response analysis i n the horizontal plane. A horizontal-plane turret m o o r i n g system m o d e l can be f o r m u l a t e d as
T,™ = - R - i ( i A ) g „ „ ( t l ) - D „ o ( V ) , (6) where gmo('l) and D m o ( V ) are the Earth-fixed restoring
term and the additional damping, respectively. The non-linear mooring line characteristics gn,o(il) can be f o u n d by dedicated software programs f o r slender marine structures, e.g., R I F L E X [14] and others. The m o o r i n g damping t e r m D n i o ( V ) could be obtained based on D N V [ 1 5 ] .
3 Heading controller design
3.1 Control plant m o d e l
The observer and controller are designed through a control plant model o f the D P vessel [ 1 6 ] . I t should be noted that o n l y yaw moment is considered. I t is assumed that the nonlinear damping C R B ( V ) i n yaw is small i n Eq. 3 since the vessel's velocity is small i n station keeping. The control plant model only considers 3DOF. The resulting l o w -frequency model o f y a w i n Eqs. 1 and 3 can be s i m p l i f i e d such that ^ = r, and rth + bice + wi, (7) (8) (9) (10) where Z7ice is the bias vector considering both s l o w l y varying disturbances and unmodelled dynamics f r o m ice disturbance; Tjce is a diagonal m a t r i x o f t i m e constants f o r estimating s l o w l y v a r y i n g y a w moment b y ice; Ei is the ice gain; /«gg is the moiuent o f inertia i n yaw; vi'i and W2 are the zero-mean white noises; is the measured output; and Vy is the measurement noise.
3.2 K a h u a n filter design
Based on the control plant model, the state-model o f a K a l m a n filter f o r heading control design c o u l d be expres-sed as f r o m Eqs. 7 to 10
x = Ax + Bu + Ew, (11)
J Mai- Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0 463
3'm H r + V y , ( 1 2 )
where x = [^,r,bicef is the state vector; u = is the control command, where A^^ is defined i n E q . 2 1 ; w =
1 t ' l , V l ' 2 j represents the process noise vector and
E = 0 1 0 0 0 0 0 66 0 , B = "^66 0 f l 3 ) 0 66 0 0 0 H = [ l 0 0 ] .
The m o d e l given i n E q . 11 f o r m s the basis o f a K a l m a n filter design. I n order to implement the filter on a computer, the m o d e l needs to be discrefized as [17]
x{k + 1) = (!>x{k) + \u{k) + Tw{k), (15) y,^{k)=lLx{k) + Vy{k), , (16) (T) = e x p ( A f t ) , (17) A = A - ^ ( < D - I ) B , (18) and r = A - > ( ( D - I ) E , (19) where h is the sampling time, and the equivalent discrete-time noises w{k) and V y ( ^ ) are Gaussian and w h i t e noises w i t h zero mean. For large offshore vessels and rigs, the sampling time is normally i n the range o f 100-500 ms [ 1 8 ] . Then the sampling time used i n the f o l l o w i n g simulation is 100 ms.
3.3 Reference generation
The simplest f o r m o f a reference m o d e l w i t h a low-pass filter structure is used [ 1 6 ] :
1
(20)
where i/zj denotes the heading command and ifj^ is the generated desired heading; and is the time-constant. I t should be noted that i n order to obtain good tracking per-formance and stability, the bandwidth o f the reference model must be lower than that o f the vessel control system.
3.4 Heading control design
Pinkster and Nienhuis [19] controlled the x- and }'-positions using a P I D feedback controller w h i l e only derivative action was used f o r the y a w angle, w h i c h is only applicable f o r large tankers. Aalbers et al. [20] used both derivative and proportional action to control yaw m o t i o n . However, Stephens and Meahan [21] proposed that the control o f a
heading requires a f u l l three-term controller i n c l u d i n g proportional, integral and derivative terms i n the design and commissioning o f a new thruster assisted m o o r i n g system ( T A M S ) f o r global producer 111.
The heading control l a w f o r a moored vessel is proposed to be an output P I D w h i c h is given as
0
(21)
(14) where \jj^ = ^ - x j j h = >' - V, ^ and r are the estimated heading and y a w rate b y the K a l m a n filter; if/^ and are the desired heading and yaw rate, w h i c h are the output f r o m the reference model; k^, k, are the P I D controller gains; integrator anti-windup should be implemented i n order to a v o i d that the integrator integrates up beyond the saturation l i m i t s o f the actuators. The controller gains can be designed by the L Q G algorithm [ 1 6 ] .
4 T h r u s t allocation
4.1 Unsaturated thrust allocation
A general relation between the control demand and the i n d i v i d u a l actuator demand thrusts is as f o l l o w s :
a J t h , (22)
where Tc is the vector o f thrust and moment demand f r o m the controller; Tjh is a vector o f thruster demands i n Cartesian coordinates, and is the thruster allocation matrix, defined as f o l l o w s :
and = [h
(23)
(24) where n is the number o f tlmisters. I n 3 D O F (surge, sway and y a w ) , the c o l u m n vectors take the f o l l o w i n g f o r m :
0 1 hx 0" 0 0 o" 1 Ux tunnel thiuster main propeller azimuth thruster (25)
where /,v and denote the l o n g i t u d i n a l and transverse position o f the rth thruster, respectively.
I n general, E q . 22 represents an underdetermined set o f equations since the fact the number o f columns o f thi-uster
464 J Mai- Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0
arrangement matrix is more tlian 3. Tliere w i l l be more variables describing the thruster settings than equations to solve (required forces and moment is solved i n such a way to m i n i m i z e the allocated power. One particular solution to the overdetermined set o f equations is the leastnorm or m i n i -m u -m n o r -m solution. The -m i n i -m u -m n o r -m solution o f Zih could be achieved by finding the Moore-Penrose generalized inverse o f [17]. Then the solution can be expressed as:
(26) (27) and W = 0 Wiy (28)
where Tl is the generalized inverse o f T^;, W is weighting matrix, i n w h i c h the element W;:^ is the cost to use the /th thruster i n the surge axis, and Wiy is the cost to use them i n the sway axis. The higher the cost, the less thrust that w i l l be assigned to the thruster.
T h e n the solved thi-ust vector T^^ can be converted to an azimuth angle command and thrust demand pair f o r each thiuster unit:
arctan
-Ti = xlTl + Tl.
4.2 Saturated thrust allocation
(29)
(30)
I f the solution thrust exceeds the thrust l i m i t f o r any actuator, the solution o f Eq. 26 using the pseudo-inverse technique i n a least-squares sense w i l l no longer hold and; hence, the desired thi-ust and moment demand w i l l not be achieved.
I n this paper, the most important mode o f control is to maintain the vessel's heading i n sense that the bow should be p o i n t i n g into the prevailing weather i n order to mitigate the ice forces acting on the vessel. I f the vessel fails to maintain station w i t h the bow oriented to m i n i m i z e the load, then i t w o u l d certainly be unable to maintain the station f o r other more unfavorable heading angles. There-fore, thrust allocation w i t h the heading p r i o r i t y scheme is o f concern. The main procedure f o r the allocation is given i n the f o l l o w i n g .
A . The first step should be to allocate thrusts as i n Sect. 4 . 1 , and examine the magnitudes o f each demand thrust. I f any thruster is saturated, the demand vector
f o r a heading p r i o r i t y control strategy should be m o d i f i e d , i n w h i c h both the surge and sway demands are taken as zero and only tiie moment is aUocated. B . The magnitudes o f each demand thrust should be
rechecked f o r thi-uster saturation.
B l . I f there is no saturated thruster after allocating the moment, w h i c h means that there is some reserve thrust capacity l e f t i n each thruster, but not enough to allocate the entire demand. I t should be noticed that the azimuth angles and thrust levels are n o w o p t i m u m at this point since meeting the y a w demand is prioritized. W i t h the azimuth angles fixed, the next step is to allocate the thrust required to satisfy the surge or sway. I n addition, a ratio could be chosen between surge and sway, w h i c h reflects the relative importance o f each w i t h respect to the other. B2. I f a l l are thrusters saturated, there is no recourse
except to clamp a l l thrusts.
B3. I f there are s t i l l some thrusters saturated, a new method is necessary to meet the moment c o m m a n d to the best o f actaators' ability. I f there are more than two thrusters unsaturated, then the thrust should be set to the m a x i m u m and the azimuth angle should be fixed f o r each saturated thruster. Then neglecting the saturated thrusters, the next step w i l l allocate the remaining command f o r unsaturated tlii-usters. T h e magnitudes o f each demand thi-ust w i l l be examined again. I f no thi-uster is saturated, the allocation w i l l terminate. Otherwise, the process w o u l d be iterated u n t i l there is only one thi-uster unsaturated l e f t . M o m e n t i n the c o m m a n d vector is also taken as the most important element f o r allocation to the last unsaturated thruster. Surge or sway comes to the second, depending on the relative importance i n the specific control task.
5 Case study
Only ice loads is taken into consideration due to its prominent infiuence on the behavior o f moored vessels compared to other environmental loads i n ice-infested areas. The i n p u t parameters including vessel, m o o r i n g system characteristics and ice f o r the simulation are specified i n this section. Then, corresponding results o f s i m u -lation and the discussions are given.
5.1 O v e r v i e w
A n icebrealang tanker named M T U i k k u as a d o u b l e - h u l l icebreaking motor tanker that was constructed to meet the standards o f the highest Finnish-Swedish Ice Class, l A
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Super, is modeled. The p r i m a r y dimensions o f M T U i k l f u are given i n Table 1.
The m o o r i n g system used i n the simulations is the same as that used i n Zhou et al. [11] consisting o f 12 identical m o o r i n g lines w i t h three lines i n each group. Each 2 m o o r i n g line has an identical chain-wire-chain configura-tion. The angle between adjacent m o o r i n g lines at each corner is 10°.
The M T U i l d a i was equipped w i t h an 11.4 M W azipod unit [ 2 2 ] . The present propulsion system is not able to p e r f o r m the dynamic positioning due to i n e f f i c i e n t thruster allocation and insufficient capacity. A new thruster con-figuration needs to be designed f o r station-keeping opera-tions. The tunnel thruster or the transverse thruster w h i c h produces only transverse force is desired to be used, con-sidering that the moment generation w i l l be more condu-cive to sway force generation rather than surge. I n addition, the azimuth thrusters are attractive i n dynamic positioning systems since they can produce forces i n d i f f e r e n t direc-tions. Thus a new actuator setup o f the moored M T U i k k u is shown i n F i g . 2. There are f o u r thrusters i n a l l w i t h t w o azimuth thrusters (thruster #1 and #2) and two transverse thrusters (thi'uster #3 and #4), where T; and a j are tbrust and azimuth angle f o r the /th thuster. I t should be noted that ^ w i l l always be positive f o r a l l thrusters. I t means that thrust reversal is obtained b y 180° rotation o f the thruster or reversal propeller rotating f o r azimuth thrusters and only reversal propeller rotating f o r transverse thrusters. There-fore, the azimuth angles f o r thrusters #3 (a3) and # 4 (c(4)
T a b l e 1 Primary dimensions o f M T U i k k u Primary dimensions
Length over a l l
Length between perpendiculars Breadth moulded Draught Displacement Deadweight Block coefficient 164.4 m 150.0 m 22.2 m 12 m 22600 ton 15750 ton 0.72
w i l l be either 9 0 ° or - 9 0 ° . The i n d i v i d u a l m a x i m u m thrust delivered by thruster is assumed to be 600 k N .
For the w e i g h t i n g m a t r i x described i n E q . 28, the higher the weighting factors, the less thrust that w i l l be assigned to the thruster i n the selected axis when calculating the solution. I n the preliminary selection, the w e i g h t i n g factors are a l l equal. Considering the interaction the thruster interaction between thruster # 1 and #2 due to direct their propeller wash at each other, the weighting f a c t o r f o r the sway axis is m o d i f i e d to increase the weighting factor to some extent. The m a i n particulars related to the thruster locations and weighting factors are listed i n Table 2.
L e v e l ice is considered w i t h u n i f o r m ice properties shown i n Table 3. The thickness o f level ice is assumed to be constant at 0.6 m i n the case study, considering the possible capacity o f the overall thruster-assisted moored icebreaking tanker. The ice d r i f t velocity is also important i n studying the dynamics o f moored vessels i n ice, where the dynamics o f being moored is the most severe at speed 0.3 m/s [ 1 1 ] . Therefore, an ice d r i f t speed at 0.3 m/s is selected i n the f o l l o w i n g simulations.
The ice is assumed to move towards the h u l l , illustrated i n F i g . 3, where the ice d r i f t i n g angle and tuiTct location are defined. The optimal tuiret position w i t h L e t = L p p / 4 is assumed. T y p i c a l l y , the angle o f attack is f r o m 0 ° to 15° [15]. Considering the capacity o f actuators, t w o scenarios w i t h ice d r i f t i n g angles at 0 ° and 15° are considered i n the simulation. The i n i t i a l ice edge is defined as a g i v e n pre-broken boundary, w h i c h is not symmetric (shown i n Fig. 4 ) . W h e n the ice sheet keeps intruding the vessel w i t h heading controller, ice force and vessel m o t i o n w i l l get
Value (unit) Xable 2 M a i n particulars f o r tteusters
Thruster no. Location (m) Weighting factor
Fig. 2 The schematic o f thi'uster airangement
kr #1 - 8 5 5 1 10 #2 - 8 5 - 5 1 10 #3 - 7 0 0 1 1 #4 55 0 1 1 T a b l e 3 Ice chaiacteristics .11
Parameter Symbol Value (unit)
Density P' 880 kg/m^ Young's modulus E 5400 M P a Poisson ratio . y 0.33 Crushing strength ac 2.3 M P a Flexural strength q / 0.5 M P a Frictional coefficient 0.15 Ö Springer
466 J Mar Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0
more and more stable and not be substantially affected by the i n i t i a l condition.
5.2 Results and discussion
A s mentioned before, a variable ice d r i f t i n g direction is believed to pose a challenge f o r moored structures since they have to vane against the ice d r i f t i n g direction to m i n i m i z e the impact on the h u l l . Provided that the ice suddenly turns the d r i f t i n g direction towards the tanker side, then the best w a y f o r the tanker is to adjust the heading to rnake sure that the ice is m o v i n g towards the symmetric axis i n surge.
The origin o f the Earth-fixed coordinate is fixed at the turret position. The i n i t i a l heading angle \J/ shown i n F i g . 3
O e Ye (East)
F i g . 3 Definition of turret location and d r i f t i n g angle
North [m]
F i g . 4 I n i t i a l ice boundary in Earth-fixed coordinate
is set at 0 ° . The cases considered i n the simulation are given i n Table 4. I n case H C l , the heading controller tries to keep the i n i t i a l heading angle under the action o f d r i f t i n g l e v e l ice w h i l e f o r case H C 2 , i t simulates the situation that the h u l l suffers f r o m a sudden change i n ice d r i f t direction at 15° and then the heading controller as w e l l as propulsion system acts to rotate the h u l l to bow into the oncoming ice. The other cases simulated here is to demonstrate the effect o f heading control on the w h o l e system.
5.2.7 Ejfect of heading control
A D-controIler as w e l l as an additional P-controlIer i f necessary, rather than integral action, was suggested to be used i n the thruster assisted positioning m o o r i n g system by Fossen [ 1 7 ] . Therefore, only derivative and proportional actions are taken into account i n the control system. The proportional gain /Cp is chosen so that the natural period o f the slowly oscillating ship is f r o m 100 to 200 s i n yaw [ 1 3 ] . The derivative gain fed can be set to be around 60 % o f the critical damping i n y a w based on the proportional gain k^. I n the present simulation, /cp = 8.4 x I 0 ^° N m / r a d and ki = 8.0 X 1 0 " Nms/rad are used.
The coiTcsponding simulated time series o f tun-et posi-tion and heading i n the Earth-fixed coordinate i n l e v e l ice f o r case H C l and N H C l are presented i n F i g . 5. Figure 5 (lower) shows that the heading controller o n the tanker works w e t i i n terms o f keeping the heading o f the h u l l so that the heading is constant at 0 ° except f o r very small variations less than 0 . 5 ° . However, the h u l l w i t h o u t head-i n g controller c o u l d also reshead-ist the dhead-isturbance o f the nonsymmetrical yaw moment induced by i n t r u d i n g level ice sheet w i t h some flexibility ( w i t h i n 2°) due to its o w n capacity when the turret is located at L p p / 4 . I t is also f o u n d f r o m F i g . 5 that the mean o f turret offset i n case H C l is almost the same as that i n case N H C l , but the m a x i m u m o f tuiTet offset i n case H C l is a little larger than that i n case N H C l and thus the corresponding global resultant m o o r i n g force i n case H C l is a Uttie larger than that i n case N H C l , as shown i n F i g . 6. One reason might explain this. The ice breaking force acting on the b o w is not always symmetric about the central longitudinal section during simulation. I n
T a b l e 4 Cases f o r simulation
Case name Control Ice thickness Ice d r i f t D r i f t i n g Desired Simulation Sampling description (m) speed (m/s) angle (°) heading (°) duration (s) time (s)
H C l Heading control 0.6 0.3 0 0 1000 0.1
N H C l No control 0.6 0.3 0
-
1000 0.1H C 2 Heading control 0.6 0.3 15 15 1000 0.1
N H C 2 No control 0.6 0.3 15 - 1000 0.1
J M a r Sci Teclinol (2013) 18:460-470 467 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 0) 1 0) ^ -1 0) X -2
0 . A v 4 - ^ Y \ ^ A ^ ' - ^ A ^ / V . ^ ^
0 100 200 300 400 500 600 700 800 900 1000 Time [s]Fig. 5 Earth-fixed position and heading o f tuiret
" 0 x : t -2 2
I
0 to UJ -2 HC2 NHC2 ^^^^^^ 1/ \P\}^T^\^l r , . ^ i ' ^ 1,'' ^.1 ij ^ r , . ^ i ' ^ 1,'' "O 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Time [s]Fig. 7 Earth-fixed position o f turret and heading relative to field zero point
2000 3000
0 100 200 300 400 500 600 700 800 900 1000 Time [s]
Fig. 6 Earth-fixed horizontal resultant mooring force
the f r e e mode, the minor rotation may be conducive to break ice w i t h less energy. I f the vessel is fixed on the heading, extra e f f o r t may be needed to continuously break ice. Thus, the instantaneous icebreaking force on the vessel w i t h o u t heading control may be l o w e r than that on the constrained vessel w i t h heading control.
The corresponding simulated time series o f turret posi-t i o n and heading i n posi-the Earposi-th-fixed coordinaposi-te i n level ice for case H C 2 and N H C 2 are presented i n F i g . 7. Figure 7 (lower) shows that the h u l l without heading controller rotates s l o w l y f r o m 0 ° to 2 1 ° and tends to oscillate around 15° under the disturbance f r o m the intruding level ice sheet. The heading does not converge to the desired value w i t h i n 1000 s i n the simulation. The heading controller makes the h u l l rotate to the desired heading ( 1 5 ° ) w i t h i n 200 s and then keeps i t constant at 15° except f o r very small variations. The corresponding t i m e series o f global
' 0 100 200 300 400 500 6 0 0 . 700 800 900 1000 Time [s]
Fig. 8 Eai-th-fixed horizontal resultant mooring force
resultant m o o r i n g force i n the horizontal plane f o r b o t h cases are presented i n F i g . 8, where the m a x i m u m global m o o r i n g f o r c e (2690 k N ) f o r the h u l l without heading control are significantiy larger than that (1420 k N ) f o r the h u l l w i t h heading control.
5.2.2 Peiformance of Kalman filter
To implement a K a l m a n filter, the covariance o f the measurement noise and process noise i n the model are necessary. The heading measurement noise is associated w i t h the compass. Balchen et al. [23] used w h i t e heading measurement noise w i t h standard deviation 0.2° i n s i m u -lation experiments. I n the present simu-lation, the K a l m a n filter is designed using the assumption o f a Gaussian, w h i t e
468 J M a r Sci Teclinol (2013) 18:460-^70 Thruster #1 too 200 300 400 500 600 700, 800 900 1000
1 1 1 1 1 1
l_,
1.-JII
-JrJ+j- r-l 1
- 1
1
111
ri
i I
i il
m\
1
5 0.2 O) 0.15 cS
0-1 x : B 0.05 O ^ ^ °0 100 200 300 400 500 600 700 800 900 1000 Time [s]F i g . 9 Simulated data o f Kalman-based filter f o r case H C l
100 200 300 400 500 600 700 800 900 1000
' 0 too 200 300 400 500 600 700 800 900 1000
Time [s]
F i g . 10 Simulated data o f Kalman-based filter f o r case H C 2
heading measurement noise w i t h standard deviation 0 . 1 ° , and a measureinent sample rate o f 100 ms. The t i m e con-stant i n E q . 20 is 100 s. The process noise values can be chosen by applying Bryson's inverse square method [ 2 4 ] . Since the anticipated m a x i m u m values o f the yaw moment due to ice could be very large, the covariance o f the noise W2 is taken as 0. The covariance o f the noise vi'i is chosen as 5 X 1 0 " 2 .
Figure 9 (upper) shows the simulated desired, measured and estimated heading f o r case H C l w h i l e F i g . 9 (lower) shows the absolute eiror between the actual heading and estimated heading. The plots show reasonable performance
0.5 200 400 600 Thruster #2 BOO 1000 0.5 r.'i.-'.AL.'.'ii.i-^.i.-W' 200 400 600 800 1000 Thruster #3 200 400 600 Thruster #4 1000 400 600 Time [s] 1000
F i g . 11 Allocated force i n case H C l
w i t h a heading excursion o f 0.4°. The m a x i m u m estimated heading error is around 0.2°. This may seem a l i t t l e large so that the K a l m a n filter's tracking capability needs to be improved. However, the improved tracking w i l l result i n more aggressive actuator utilization to some extent. This is also mentioned by Jenssen et al. [ 2 5 ] .
Figure 10 (upper) shows the simulated desired, mea-sured and estimated heading f o r case H C 2 w h i l e F i g . 10 (lower) shows the absolute error between the actual head-i n g and esthead-imated headhead-ing. The esthead-imated headhead-ing f o l l o w s the actual heading satisfactorily. I n the rotation stage, there exists a small absolute estimation error i n heading f o r case H C 2 .
5.2.3 Peiformance of thrusters
The m o d e l i n g o f thi'uster dynamics is not i n c l u d e d i n the simulation. I n order to study the dynamics o f heading introduced b y ice, the m a x i m u m rate that each thruster can change thrust as a result o f inertia o f the motor, the pro-peller together w i t h then other components i n the drive system and the m a x i m u m slew rate f o r changing thrust direction are not taken into consideration i n the present simulation o f thrust allocation. The ratio o f the delivered thrust to the m a x i m u m thrust f o r each thruster i n time
J M a r Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0 469 Thruster #1 400 600 Thruster #2 1000 200 100 400 600 Thruster #3 800 1000 0 ) c < < •100 100 -100 400 600 Thruster # 4 1000 400 600 Time [s]
F i g . 12 Allocated angle i n case H C l
Thruster #1 1000 1 >< as E 0.5 E 0.5 ' 1 1 1 1 1 1
1
Thruster #2 ] .l,V.>-J'-'/; Thruster #3 CO E 0.5 0UI
Thruster #4 0 100 20O 300 400 500 600 700 800 900 1000 Time [s]F i g . 13 Allocated force i n case H C 2
^ 200 -g 100 ^ 0 Thruster #1 -100 „ too UI (D ra-100 0 too 200 300 400 500 600 700 800 900 1000 Thruster #2 -200
VIA ,
100 0 too 200 300 400 500 600 700 800 900 1000 Thruster #3 -100 CD a> c<
too 0 too 200 300 400 500 600 700 800 900 1000 Thruster #4 •100 0 too 200 300 400 500 600 700 800 900 1000 Time [s]F i g . 14 Allocated angle i n case H C 2
d o m a i n f o r case H C l is s l i o w n i n F i g . 11. F r o m F i g . 11 i t is seen that the allocated forces display large v a r i a -tions and the transverse thrusters (thruster #3 and # 4 ) are m u c h m o r e used than the m a i n a z i m u t h i n g thrusters (thruster # 1 and # 2 ) . The corresponding a z i m u t h angle is s h o w n i n F i g . 12. F r o m F i g . 12 i t is f o u n d that the signs o f allocated angles change f r e q u e n t l y due to ice dynamics.
The time history o f the ratio o f the delivered thrust to the m a x i m u m thrust and the corresponding azimuth angle f o r each thruster i n case H C 2 are shown i n Figs. 13 and 14, respectively. F r o m F i g . 13, i t is seen that compared to the case H C l , a l l thrusters are m u c h more dependent and even get saturated during the transient rotation process i n order to f o l l o w the desired heading and finally make the h u l l head up towards the d r i f t i n g ice. S i m i l a r l y to case H C l , the allocated azimuth angles i n case H C 2 displays large oscillations f r o m F i g . 14.
6 Conclusions
This paper simulates the dynamic behavior o f a heading controlled moored icebreaking tanker i n level-ice and A designed heading controller based on a K a l m a n filter. I n addition, a thrust allocation method suitable f o r the b e a d i n g controller is proposed. F r o m the simulation, there are some
470 J M a r Sci Teclinol (2013) 1 8 : 4 6 0 ^ 7 0
significant findings w h i c h are w o r t h p o i n t i n g out as f o l l o w s :
• Heading controller based on a K a l m a n filter is feasible to keep the moored vessels aligned w i t h the encoun-tering l e v e l ice w i t h changing d r i f t direction. H o w e v e r , when the actual heading does not deviate f r o m the expected one, the u t i l i z a t i o n o f heading controller m i g h t deteriorate the p e i f o r m a n c e o f the system and thus increase the m o o r i n g forces. Therefore, i t is preferable to switch on and o f f to reduce the f u e l consumption o f p o s i t i o n moored vessels. I t is noted that to have good ice-vaning capacity, the location o f the turret is important. W h e n the tuiret gets close to the center o f the gravity o f the floating tanker i n level ice, i t is fikely that the huU w i f l rotate due to the n o n -symmetric ice action i n yaw direction and lack o f stability capacity. The ideal location seems to be around one quarter o f the length between two perpen-diculars f o r w a r d the center o f the gravity.
• Based on n u m e r i c a l simulation i n the present paper, the turn m o m e n t due to ice disturbance is changing r a p i d l y . The designed K a l m a n filter c o u l d track the changing o f heading. I m p r o v i n g the t r a c k i n g capacity is possible, but at a cost o f heading filtering and aggressive thruster usage. H o w to balance the p r o b l e m is s t i l l a challenge. • A l l thrusters are used extensively i n the turning process. The requuement f o r a p r o p u l s i o n system is h i g h i n terms o f increased thi-uster usage and additional wear and maintenance w h e n the heading control system coiupensates the q u i c l d y changing ice disturbance. Therefore, the dynamic characteristics o f actuators need to be considered i n the design o f the D P c o n t r o l system f o r ice conditions.
A l l the above conclusions are based specifically on the simulations. M o r e data are desired to validate this m o d e l . The effects of ice ridges and p i l e up should be studied, w h i c h are believed to have an influence on the actual design of station keeping systems. A t present, most industrial sta-t i o n keeping operasta-tions are conducsta-ted under sta-the ice man-agement w i t h extensive u t i l i z a t i o n o f icebreakers. The combined heading control and m o o r i n g system is s t i l l at an early stage, but i t is p r o m i s i n g . I f i t comes true, then the cost of ice management w i l l be reduced significantly.
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