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Rudder roll reduction for small ships Martin, Christopher R. 1993-09-15

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RUDDER ROLL REDUCTION FOR SMALLSHIPSbyChristopher Robert MartinB.A.Sc., The University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingTHE UNIVERSITY OF BRITISH COLUMBIAApril 1993© Christopher Robert Martin, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of  /We^/FYI.The University of British ColumbiaVancouver, CanadaDate^?3/607DE-6 (2/88)AbstractRoll motion on a ship can be very uncomfortable, and in theextreme, dangerous. It is possible to use the rudder to reduceroll motion as has been demonstrated by others on naval and coastguard vessels. For this study, five controllers were chosen anddesigned to implement a rudder roll reduction system as part ofan autopilot on a fishing vessel. A numerical model and aphysical model were prepared for testing these controllers. Theresults from the numerical model indicate rudder roll reductiondoes not work for the fishing vessels tested. These results aresuspect for several reasons.Table of ContentsAbstract ^  iiList of Tables ^  viList of Figures ^  viiNomenclature ^  ixPart 1: Introduction ^ 11.1 Problem 31.2 Scope & Assumptions ^ 41.3 General Approach 61.4 Literature Review ^ 7Part 2: Design Process ^ 92.1 System ^ 92.2 Detailed Design ^ 112.3 Parameter Identification ^ 152.4 PID Controller ^ 172.5 Minimum Variance Controller ^ 182.6 GPC Controller ^ 202.7 Discrete-Time Sliding Mode Control ^ 222.8 Pole Placement Control  ^23iiiPart 3: Numerical Simulation ^ 263.1 Numerical Model 263.2 Controller Parameters ^ 313.3 Tests ^ 343.4 Results 363.4 Evaluation of Roll Reduction ^ 48Part 4: Physical Model ^ 534.1 Yaw Sensor 544.2 Roll Sensor ^ 544.3 Telemetry System 554.4 Data Acquisition 554.5 Computer ^ 554.6 Radio Control ^ 564.7 Towing Tank 574.8 Tests ^ 58Part 5: Conclusions ^ 59Part 6: Future Work ^ 60Part 6: Bibliography ^ 61Appendix I: Approximate Maximum Likelihood Identification .^. 64Appendix II: Generalized Predictive Control ^ 66Appendix III: Discrete-Time Variable Structure Control^.  ^. 68ivAppendix IV: Equations of the Transverse Motions ^ 70Appendix V: Wave Excitation ^  79Appendix VI: Numerical Simulation Results ^  81List of TablesTable^ Page1. Kynoc Particulars ^  282. Eastward Ho Particulars. ^  283. Dimensionless Quantities  494. Model Particulars. ^  53viList of FiguresFigure Page1. Ship geometry and motions ^ xi2. System Diagram ^ 93. Sway Excitation Spectrum ^ 304. Yaw Excitation Spectrum 305. Slope Spectrum ^ 316. VSS Impulse Response for Kynoc ^ 377. VSS Roll Reduction Impulse^Response for^Kynoc ^ 378. Pole Placement Impulse Response for Kynoc 389. Pole Placement Rudder Roll Reduction Impulse Responsefor Kynoc ^ 3810. PID Impulse Response for Kynoc ^ 3911. Kynoc 900 Seas ^ 4212. Kynoc 45° Seas 4213. Kynoc 60° Seas ^ 4314. Kynoc 300 Seas 4315. Eastward Ho 90° Seas ^ 4416. Eastward Ho 450 Seas 4417. Eastward Ho 60° Seas 4518. Eastward Ho 30° Seas ^ 4519. VSS Roll Reduction Improvement 15°/s Rudder Rate^    4620. Pole RR Roll Reduction Improvement 150/s Rudder Rate . . 4621. MV_RR Roll Reduction Improvement 15°/s Rudder Rate . . . 4722. SS GPC RR Roll Reduction Improvement 15°/s Rudder Rate . 4723. Example of yaw parameter identification ( random seavi itest - Eastward Ho) ^ 5024. Sample Roll Identification (Random Sea CAse - EastwardHo) ^ 5125. Yaw Parameter Estimates ^ 5226. Model Testing Arrangement 5727. Geomerty of Rudder Moments ^ 7128. 100 Rudder Step Response 7729. 100/100 Zig-zag, rudder and yaw ^ 7830. 100/100 Zig-zag, roll ^ 78viiiNomenclatureThis report covers two engineering areas, each with its ownsymbol standards. Therefore, two nomenclature listings areprovided, one for control and one for naval architecture. Thereis some cross-over, for example 0 is used for roll angle in bothcontrol discussion and naval architecture discussion.Control NomenclatureA — system matrixA(z▪ ) — system polynomialB — input matrixB(z ) — input polynomialAMLS(k) — sliding surfaces(k) - ideal set-pointTD - derivative time constantT/ — integral time constantT0 - roll natural periodC — output matrix^T* - yaw rate time constantG — sliding surface ü — vector of future controlscoefficients^x — state vectorEmu — predicted outputresponse^ a — EFRA factorJ — cost function — EFRA factorK — arbitrary gain^6 — EFRA factorK(t) — gain matrix in AML^e — boundary layer thicknessKr — roll weight^n(t) — a posteriori errorsKry — roll velocity weight^e — estimated parametersEy yaw velocity weight^X — EFRA exponential forgettingN — prediction horizon factorNu — control horizon^0 — roll angleP(t) — covariance matrix in^— predicted responseix - predicted free^KH - hydrodynamic roll momentsresponse^ KR - rudder roll moment- yaw angle K - wave excited roll momentL - ship lengthm - ship massNaval Architecture^mx - added mass in the xNomenclature^ directionmy - added mass in the yAR - area of rudder^ directionB - ship's beam NH - hydrodynamic yaw momentsb - linear roll damping^NR - rudder yaw momentCH - block coefficient Nw- wave excited yaw momentD - moulded depth^r - yaw rateF, - Froude Number T,d - mean draughtGMT - effective metacentric^u - surge velocityheight^ V - ship's velocityg - gravitational^v - sway velocityacceleration YH - hydrodynamic sway forcesH/ - significant wave^YR - rudder sway forceheight^ Yw - wave excited sway forceIm - roll mass moment of^zR - vertical distance from c.g.inertia^ to rudder.In - yaw mass moment ofinertia^ aR - effective rudder in-flowJm - added roll mass moment^angleof inertia^ A - ship's displacement- added yaw mass moment^6 - rudder command angleLin of inertia^ - roll damping ratiot — wave slope amplitudeX — rudder aspect ratiop — density of sea waterr — trimx — heading anglerelative to wavesw — wave frequencywe — wave encounterfrequencyFigure 1. Ship geometry and motionsxiPart 1: IntroductionRolling is one of six motions that all ships experience. Asrolling becomes more pronounced, it affects both the crew and thevessel. Rolling reduces task proficiency and both directly andindirectly increases the risk of injury. The ship's fittings,such as machinery mounts and cargo holding structures andsystems, must resist the resulting forces. Excessive motion canlead to capsize, an endangerment of both life and vessel. Bydecreasing roll motion, not only is the comfort level increased,but safety is also increased.There are many methods available to decrease roll motion, thoughall are not currently used. All methods increase roll damping,thereby reducing the dynamic amplification at resonance of theroll motion. They can be divided into two types: passive andactive.The most common passive stabilizing devices are bilge keels,which are fitted on most round bilged boats, but rarely on hardchined vessels. Bilge keels work through viscous effects andvortex shedding at very low speeds. As the speed of the boatincreases, a hydrodynamic lift begins to act, generating anopposing roll moment.Less common passive devices are passive roll tanks. Roll tanksare designed to dampen the roll motion by having water sloshinside them 90 degrees out of phase of the roll motion. However,they add weight, take up cargo space and reduce overallstability, because they reduce the metacentric height. Rolltanks are rarely used on local fishing vessels.Two passive devices that are used locally in the PacificNorthwest are paravanes and batwings.^Paravanes are triangularsheets of wood or metal that are hung in the water from polesextending laterally outward from the boat. They work byalternately diving downward when the pole to which they areattached rises, producing an opposing roll moment. Paravanes areoften found on trawlers because of the existing poles used in thetrawling process.The last passive devices are batwings, which work in much thesame way as bilge keels. Batwings are sheets of metalperpendicular to and extending laterally outward from the keel.They decrease the roll through vortex shedding and, at higherforward speeds, through dynamic lift. However, they alsoincrease the resistance of the vessel.The second type of stabilizers used to decrease roll motion areactive stabilizers. Active systems are characterized bysensors, controllers, and actuators.Active tanks are one type of active stabilizers. Active tankswork on much the same principle as passive tanks; however, asensor is used to tell a pump when and which way to move the2water. An active tank system results in the use of smallertanks, and therefore little loss in stability and active tankscan be used in a wide range of conditions. The disadvantage isthe weight and size of the pump necessary and the energy requiredto operate the pump.Another type of active damper are fins, which protrude from thehull . A sensor measures the direction and speed of the roll andthe fins' angles are adjusted to produce an opposing roll moment.The disadvantages of fins are they are vulnerable to damage,ineffective at low speeds, and in large motions can stall,producing little lift and little opposing roll moment.The rudder, with which almost every ship is fitted, is usuallyplaced below the centre of gravity of the ship. The lift forceof the rudder acts below the roll centre of the ship, producing aroll moment. Because the ship's speed of response in roll ismuch faster than that in yaw, a control scheme has beenimplemented on some vessels, which reduces the roll withoutaffecting yaw.1.1 ProblemThe primary problem is to develop a rudder roll reductionautopilot with applicability to a broad range of small vessels.Because the autopilot is for small vessels, secondarycharacteristics (problems) that should be considered are:3-the autopilot should be cost effective for the smallvessel's owner.-the sensors necessary should be inexpensive and few.-there should be no major alteration to the steering gear.1.2 Scope & AssumptionsThe control system that was explored in this thesis is a rudderroll reduction system as part of an autopilot. A number ofcontrollers were selected and then evaluated.The system has one input, the rudder, and two outputs, roll andyaw. On all boats the rudder has a maximum angle, in eitherdirection, which it can rotate from the centre line (zerodegrees). For the craft considered here, the rudder has a setconstant slew rate. It is understood that the constant slew rateis slow, which was factored into the design. It was assumed thatthe rudder provides sufficient roll moment at forward speednecessary for roll reduction.This one input, two output system to be designed for is a SingleInput Multi Output (SIMO) system. Some of the major criteria forthe controllers are:- the same controller must be able to be used on differentboats easily.- work on one boat under different loading conditions(parameter variations).4- the controllers must work on real boats, whose motions arenon-linear.- the controllers must work with the presence of wind andwaves affecting the vessel (coloured noise).- the controller must take into account the rudderlimitations (maximum angle and constant slew rate).The project did not cover all the aspects of autopilot design.The autopilot served only as a regulator, that is, for coursekeeping only. No mention will be made of turning or manoeuvring.In addition, the possibility of helmsman steering with a rudderroll reduction system also acting was not explored.All controllers have "tuning knobs" for which adjustments can bemade to improve the controller's response to different stimuli.This project did not address the tuning of each controller fordifferent, specific tasks. All controllers were tuned using thesame test and specified response. The parameters remained fixedthrough all tests.All design was done in discrete time, that is, using differenceequations instead of differential equations and the accompanyinganalytical tools such as z-domain analysis. The process waslinearized for controller design. It was modelled as anAuto-Regressive Moving Average (ARMA) or a state space process.The coloured noise was modelled as an ARMA process driven bywhite noise.5The vessels designed for in this project are in some ways muchdifferent than the vessels dealt with in the literature [l7].Previous controllers were designed for and tested on much largervessels such as naval vessels. Also, most design information andqualitative discussion of transverse motions in standardliterature [20,21] are related to larger vessels.1.3 General Approach.Controllers were chosen and designed. Their parameters were setusing one evaluation criteria for all designs. The firstcontroller designed was of the same form and specifications ofautopilots currently on the market. Its responses were used asthe baseline against which the other controllers' responses weremeasured. All controllers were first designed as autopilotsalone, then as roll reducing autopilots to see what effect theaddition of roll reduction had on the ship response.The controllers were first tested in numerical simulations. Heretheir responses to specific environments and process parametervariations were examined.The controllers were then to be tested on a physical model. Heretheir responses to a real physical environment with all itsassociated non-linearities , noise from different sources, andNumbers refer to entries in the bibliography .6other peculiarities, were to be examined. The model tests wereto be the more accurate test of whether a rudder roll reductionsystem would work with fishing and other similar sized boats.At the end of each testing section, the responses of eachcontroller are compared and discussed.Finally, conclusions and recommendations are made.1.4 Literature ReviewThe first research into rudder roll reduction was undertaken inthe mid-seventies, using analog controllers. These controllerswere tested using scale model and full scale tests. The types ofvessels were fast container ships and naval craft. They achievedlimited success.In 1981, Kallstrom [7] conducted a detailed study of a rudderand fin roll control system.Throughout the 1980's van Amerongen [2,3] and various coworkerscarried on an extensive project on rudder roll reduction. Theircontroller was based on Linear Quadratic Gaussian (LQG) design.The weighting factors in the LQ design were adjusted on line;however, there was no on line ship parameter estimation. Theydesigned an automatic gain control system to compensate for thelimited slew rate of the rudder. The testing was broad,7encompassing mathematical models, analog modelling, scalemodelling and, full scale trials.In the 1980's, other researchers, such as Katebi et al.[4],investigated rudder roll reduction using LQ control.The amount of roll reduction reported in these papers rangedbetween eight and fifty percent.None of these researchers used on-line parameter estimation.Except for van Amerongen et al., there was no consideration ofrate saturation. All of the vessels investigated were largecontainer vessels or naval vessels.8Part 2: Design Process2.1 SystemThe system consists of four major components, shown in Figure 2.The components are the ship itself (or the plant), the rudder,the controller, including sensors, and the external environment.All of these components affect the ship motions of interest: yawand roll. The ship affects itself through the propeller, whichcan affect yaw through "prop walk," and by the way it is loaded.Figure 2. System DiagramThe autopilot is implemented to keep a steady course. It mustcompensate for external and internal disturbances by using therudder. The ability to damp roll motions is to be added into thedesign of the autopilot. That is, the autopilot not only mustkeep a steady course, but it must also decrease the roll motion.9Important characteristics of the system that will affect thedesign are as follows:- The ship is either neutrally stable or unstable in yaw.Neutrally stable means the ship will hold a course, but theslightest disturbance will send the ship off course, andunstable means the ship cannot hold a course on its own.- The ship is always stable in roll, which means when it isdisturbed it will always settle to a steady value. The rollto rudder response is non-minimum phase. In other wordswhen the rudder is moved to one side of neutral, the shipinitially rolls in one direction but finally settles in theother.- The yaw virtual moment of inertia of a typical west coastfishing boat is five to six times larger than the rollmoment of inertia.- There is cross coupling between the motions, that is, rollaffects yaw and yaw affects roll. It was expected that yawwould be influenced by roll much less than it influencesroll.- The yaw and roll characteristics will change with speed andloading. Speed is a factor because not only does it changethe effectiveness of the rudder, but it also affects thecharacteristics of the ship.10- The rudder is controlled by a hydraulic actuator, whichmoves at a fixed speed. This can cause difficulties andeven instabilities in a roll reduction system, because therudder angle starts to lag behind the desired angle to suchan extent that it starts to increase roll motions.- It is expected that the controllers will be implementedusing a digital processor in commercial production.Although the computational speed of computers have increasedtremendously over the years, there are still economicfactors to be considered. Therefore, some attention will bepaid to the computational effort required.- The environmental disturbances are usually caused by windand waves. The disturbances can be constant such as asteady wind, narrow-band random (usually waves), or wide-band random such as wind gusts.2.2 Detailed DesignThe ship will be modelled in a general manner, i.e., usinggeneral coefficients rather than specific values. This will befollowed by the controller design in the same general manner,i.e., using general coefficients.The ship was modelled as a linear discrete time Single InputMulti Output System (SIMO). Each motion, roll and yaw, was11modelled as a second order Auto Regressive Moving Average (ARMA)system using difference equations:A(z1) -,E Bi ( z -1)2(1)where i=1 is the rudder inputi=2 is the cross-coupling of the motions.For yaw( 1 + al z-1 + a2 z-2) Ip. = JD, z-1 u +b2 z-14• ^(2)and for roll( 1 + al z-1 + a2 z-2) . = b1 z-i. u + b2 (z-i _ z-2) ip ^(3)Note that the a; and bi are not the same for the two equations.A standard linear representation in naval architecture for thetransverse motions (roll, yaw, and sway) of vessels is a SIMO12state space system:I = A0.x+.130u^ (4)with states: 2c= Vwhere tk=yaw angle, v=sway velocity, and 0=roll angle and u is therudder angle.Sway is often ignored because it is difficult to measure, as arethe roll and yaw velocities. They could be estimated by usingsuch methods as Kalman Filtering; however, because the plantparameters are estimated, doing both was felt to be toocomputationally expensive.Combining the ARMA difference equations into a state space SIMOsystem, we obtain:x(k+1) = Aa(k)^u(k)^ (5)with states(k)z(k) = (k-1)(k)4)(k-1)13-1The elements of A are the coefficients of A(z-1) and B2(z )polynomials for each of roll and yaw. The B elements consist ofthe B1(z-1) polynomial. That is:aly a2y b2y^0 blyA= 1 0 00 B= 0b2r -b2rair a2r blr0^0^10where the y and r subscripts indicate the co-efficient are fromthe yaw and roll difference equations respectively.Because of the fixed slew rate of the rudder, it is convenient tochange the input to Au and augment the states with u(k). That is:0x(k) = The system matrices are nowA=aly1.1p2ra2y .b2y^0^bly0^000 -b2r air a2r bir B=blyokr0 0 100 o0 0 001 1After the plant was modelled, the controllers were chosen. Acontroller must meet the following criteria:141. be able to compensate non-minimum phase plants.2. be able to compensate open loop unstable plants.3. work with or be extendable to multivariable plants.4. work reasonably well in the presence of "coloured"disturbances.5. handle large parameter variations.6. handle a control input (actuator) that has a fixed rate.7. handle a control input that has amplitude limits.2.3 Parameter IdentificationMost controllers' parameters are based on the plant's parameters.To adapt to different plants and to plants under differentoperating conditions, some controllers use the method of on-lineparameter estimation. In other words they estimate the plants'parameters using previous values of the output and the controlinput, then calculate the next control input.For the controllers in this thesis that require an on-lineparameter estimation algorithm, the algorithm used wasApproximate Maximum Likelihood, with some modifications. Thealgorithm and modifications are summarized here. More detail canbe found in Appendix I.Approximate Maximum Likelihood was chosen because it has goodconvergence properties and is easily extendable to the case of a15plant with a coloured noise disturbance. It has the form ofP(t)x1(t+1)^(t+1)P(t)P(t+1) = P(t)- ^1+4.(t+1)P(t)x1(t+1)P(t+1)x1( t+1)1+4'(ti-up(ox1(t4-3.)e(t+i) = e(t) +K(t+1){37(t+1) -xr(t+1)9(t)]where P is the covariance matrix, x/ is the data vector, and 0 isa vector of the parameters being estimated.The first modification is to add an Exponential Forgetting andResetting Algorithm (EFRA) (8). This algorithm increases thestability of the AML when excitation is poor (i.e., little or nochange in the control input) and tracks time varying parameters.The AML is modified by :aP(t)x.r(t+1)xf(t 1)P(t)P(t+1) -^-  +131-8102(0A ^1+4:(t+1)P(t)x1(t+1)aP( t+1)^ti-1)K( t+1) - 1 +x_r( t+1)Px.r( t+1)The major difference between the EFRA and the usual AMLalgorithms is in the equation for the covariance matrix. Thefirst two terms correspond to the standard AML with exponentialforgetting. The third term provides a lower bound for P, whilethe last term provides an upper bound.In this multivariable case, the two subsystems will be identifiedseparately. That is, the coefficients in (2) will be identified,then the coefficients in (3) will be identified.K(t+1) -16The second modification is a simplification of the coefficientsin the yaw subsystem (2). The fact 0 is the integral of 0, withrespect to time; that is, a pure integrator, will be takenadvantage of. Here al is equal to (-1-a2). Therefore, it isonly necessary to identify a2, which can be done easily (seeAppendix I).2.4 PID ControllerThe Proportional-Integral-Derivative controller would notnormally be chosen as a controller because it is used more intracking applications than for regulation, it is difficult todesign for multivariable systems, and it cannot be modified tohandle a fixed rate control input. However, since it is theindustry standard for autopilots, a PID controller using theindustry standard parameters will be implemented.The control algorithm is:u(k) = qoy(k) +q1y(k-1) +q2y(k-2) +u(k-1)where qo = K(1+ -742T )Tqi = -K(1+2-2- - T—2)To TrTDq2 = Ki--..The standard values [28,29] for the gain K are 1:0.5 to 1:1, thederivative time constant TD is 0.25 to 0.5 seconds, and the17integral time constant T/ is 30 seconds or larger. A sample timeTo of 0.5 seconds was used for all controllers.2.5 Minimum Variance ControllerThe minimum variance controller seeks to minimize the variance ofthe controlled variable:var [y (k)] =^y2 (k))where BO is the expected value operatorThat is, the control is chosen based on the prediction of theoutput.In this case, with the fixed rudder rate, three possible choicesof rudder input exist at any sampling instant:- move the rudder left.- move the rudder right.- keep the rudder at its current position.The input is further restricted when the rudder reaches itsmaximum amplitude. In this case, the possible choices of rudderinput are limited to decrease rudder angle or keep the rudderangle the same.It was found early that minimizing the variance of the outputalone did not produce satisfactory results. It is necessary to18add two terms into the criterion to be minimized:I (k+1) = Eq y2 (k+1) + K[y(k+1) - y(k) 2 + p u2 (k) }The second term is used to represent the velocity of the outputvariable and K is weighting on it, and p is a weighting on thecontrol input.After a large number of simulations were run, it was found thatthe weighting K had a large range of variation for differentoperating conditions.Therefore a new method was found. In this method an "ideal" set-point, s(k), is added:I(k+1) = E [y(k+1) - s (k+1)] 2 + p u2 (k)}Where s(k+1) is the predicted value of an idealized naturalresponse given the previous values of the output. This method issimilar to pole-placement in that a desired response isspecified. The controller minimizes the variance between thedesired response and the actual response.In implementing this type of control, it is convenient to modifythe difference equations (2) and (3) by multiplying them by A( A=(1-z-1) ). The result is :(1 -z-1) ( 1 +^+ a2 2-2) * =^u +b2 z-1 ( 1 -z-1)( 1 -z-1) ( 1 +^z-1 + a2 z-2) 4 = /Di z-1A u + b2 (z-1 - z-2) (1-z-1) 4rThat is, the output is now dependent on the change in u.19The control algorithm then is to:- identify the parameters in the two difference equationsusing AML.- compute the expected values of lk and 0 for the threepossible control inputs (rudder right, rudder left, or norudder movement).- choose the rudder movement which minimizes :^I (k+1) = E{ [tIr (ic+1)^(k+1)) 2 + K1[4)(k+1)^(k+1) 2+ pu2 (k) }The minimum variance controller was chosen because it is thebasic controller for the regulation of systems with stochasticdisturbances.2.6 GPC ControllerThe Generalized Predictive Control controller is an N step aheadpredictor. It predicts the output N steps ahead, given currentvalues of the inputs, outputs, and plant parameters.In this application, two types of GPC control were evaluated.The first is bang-bang GPC control. The cost function^J=^+ To] T ImN[HN + To]where HN = Predicted response of the system based ona = Future controls [ Au(k) Au(k+1) ... u (k+1-Nu) )7'0 = Predicted free response at sample kNu = Control horizonN = Prediction horizon20is calculated explicitly for the three possibilities of Au. Ifthe rudder reaches its amplitude limit, there are only twopossibilities for Au.The control horizon used was one. The prediction horizon isbased on the slowest rise time of the system, usually the yawrate. A prediction horizon of 10 was used. The algorithm isthen to identify the parameters of the system using AML, predictthe output for the three possible inputs, and implement the onewhich minimizes J.The second type of GPC control was rate constrained GPC. Herethe control was calculated to be:a = {{H;UATT' H;:,, To}Because the control horizon is one, the rate constraint isimplemented easily. The algorithm is check if Au exceeds therudder rate, if yes then set Au equal the rudder rate, otherwiseimplement Au calculated. Setting Au equal to the rudder rate isnot arbitrary. It is assumed the constrained optimum lies on theboundary of the cost function and the constraint. A predictionhorizon of 10 was also used for the rate constrained GPC.GPC was chosen for rudder roll reduction because it is both agood regulator and tracker. Extensive research done recently onthis type of adaptive control has shown it to be robust andstable [9-12]. It can be extended to multivariable systems.212.7 Discrete-Time Sliding Mode ControlSliding Mode control is based on a discontinuous control functionthat switches when it crosses a boundary in the state space.When the states repeatedly cross this boundary and move towardthe origin of the space, the system is said to be sliding.During this time the system is remarkably robust in the presenceof disturbances and parameter variation. The resulting controlstructure is usually non-linear and results in a VariableStructure System (VSS), where the control is different indifferent parts of the state space. The simplest discontinuouscontrol device is a device with two control states, e.g., arelay. For rudder roll reduction, the two control states aremove the rudder right and move the rudder left.The design procedure is to specify a nominal A and B, that is ageneral or average quantitative expression of the system. Solvethe following matrix Ricatti equation for P :P = Q + ATPA - ATPBRRBTP - PBRRBTPA + PBRR13TPBRRBTPwhere RR = RudderRateQ = Positive Definite MatrixThe sliding sub-space coefficients are then:G = BTP22and the control input is:Au = -RRsign ( Gx )The preceding control input can lead to chattering, or the rapidswitching of the control between two values. The fact thecontrol input can be zero movement, besides moving right or left,and the idea of a boundary layer will be used to smooth out thechattering.Instead of using a discontinuous surface, a small region will beconstructed around the discontinuous surface. If the states arewithin this region, the control input used will be no change inrudder angle. In other words,If I S(k) I =I Gx(k) I < e then u(k) = 0where e is the boundary layer width.Sliding Mode control was chosen because it has been used in asituation similar to rudder roll reduction of a ship, the controlof transverse motions of aircraft [16]. Sliding mode control canbe used with multivariable control and uses a discontinuouscontrol input, as the fixed rudder rate is.2.8 Pole Placement ControlThe pole placement controller attempts to place the closed looppoles of the system at some arbitrary location. Given the state23space system:x(k+1) = Az(k) +Bu(k)and assuming the input has the form:u(k) = -1Cx(k)then the closed loop response of the system is:.r(k+1) = [A - BIC]x(k)and by setting K, the "natural" response of the system [A-BK] canbe set arbitrarily. The order of the "natural" response is thesame order of the original system A.For rudder roll reduction, the yaw response was set with twostable exponential decay terms and the roll response was set ashaving the same natural frequency as, but a much larger dampingratio than, the open-loop roll response.The control algorithm is:- identify the ship parameters in roll and yaw using AML.- calculate the gains K necessary to give the desired closedloop response.- calculate the rudder angle given the current states and thegains calculated.Pole placement was implemented as a rudder roll reductioncontroller because of the explicit specification of the closedloop response. For example, the roll response is specified tokeep the same natural frequency but increase the damping. The24possible problem of this controller is if the rudder rate is tooslow, the rudder angle may begin to lag the control signal andstart exciting the ship in roll. It is thought that if the yawresponse is specified slow enough, the rudder lag problem may beavoided.25Part 3: Numerical Simulation3.1 Numerical ModelThe first attempt to obtain a numerical model involved the use ofdata available in the department from full scale tests performedin the summer of 1989. The data consisted of zig-zag trials ofan approximately 40' double ender cabin cruiser. A regressionanalysis was performed to obtain the yaw and roll parameters.The results for yaw were good and made physical sense. Theresults for roll were not as good and, more importantly, did notmake any sense physically. Therefore, a numerical model wasimplemented.The numerical model used in this section of tests is based on thelinear portion of Inoue's [22] manoeuvring model with additionalterms for the roll motion taken from Bhattacharya [21]. The26equations are (see Appendix IV for more information):( m + my) V = -22-- pLdV4v - ( mu + mxu - 1pL2d174) r + YR + Yw(I„ + j„ ) I = 1pLdV(L1■71„+Y,Nic]v + -1-pL2dV[ LIVir + Yx5 Jr1+ -2 p L2 dVN4) NR Nw(^+ J )4:1 - myz = -b(I) - gAGAI) - pLdV1,„zilv- [ mxu + pL2dV4^+ KR + KwThe hydrodynamic coefficients are estimated from basic shipcharacteristics alone. The coefficients were calculated from thedata of the Kynoc, a 60-foot west coast drum seiner and theEastward Ho, a 107-foot combination vessel that have been used inother research in the department. The coefficients werecalculated for different speeds and loading conditions. Fromthese coefficients a state space model was constructed for eachof the different conditions. The particulars and speeds for thedifferent conditions are given in the following tables.27Table 1. Kynoc ParticularsParticular Condition1Half LoadFull SpeedCondition2Half LoadHalf SpeedCondition3Full LoadFull SpeedCondition4Full LoadHalf SpeedLui (in) 17.25 17.25 17.4 17.4B1(m) 6.1 6.1 6.1 6.1T(m) 2.53 2.53 2.82 2.82T 1.24 1.24 1.41 1.41A(tonnes) 116.5 116.5 141 141Ca., 0.43 0.43 0.46 .46L/B 2.83 2.83 2.85 2.85LCG(m) -0.608 -0.608 -0.851 -0.851VCG(m) 2.22 2.22 2.13 2.13GM(m) 0.735 0.735 0.546 0.546Speed(kts)11 8 11 8Table 2. Eastward Ho Particulars.Particular Condition1Half LoadFull SpeedCondition2Half LoadHalf SpeedCondition3Full LoadFull SpeedCondition4Full LoadHalf Speed'Jut (m) 30.5 30.5 30.8 30.8But (m) 8.74 8.74 8.74 8.74T.(m) 3.45 3.45 4.04 4.04r 0.835 0.835 1.073 1.073A(tonnes) 445 445 572 572CI., 0.472 0.472 0.514 0.514L/B 3.49 3.49 3.52 3.52LCG(m) -1.405 -1.405 -1.893 -1.893VCG(m) 3.28 3.28 3.38 3.38GM(m) 1.161 1.161 1.268 1.268Speed(kts)13 10 13 928The matrices were discretized using a zero order hold with asampling period of 0.5s (See for example Ogata [18]).The rudder motion has two restrictions -amplitude and rate. Theamplitude is limited to ±40 degrees. The swing rate of therudder has a maximum of 5 degrees per second. If the demandedrudder angle is within the amplitude limitations, and thedifference between the previous rudder angle and the demandedrudder angle is less than or equal to the swing rate multipliedby the control cycle time, the demanded rudder angle isimplemented. Otherwise, the rudder angle implemented is theclosest value to the demanded angle that does not violate the tworestrictions.For the irregular wave disturbances used in some of the tests,digital filter approximations were constructed (see Appendix V).These approximations were based on excitation spectra calculatedusing the Bretschneider Spectrum. The approximations werematched to the spectra for shape qualitatively and for area underthe curve quantitatively. Examples of sway and yaw excitationare contained in Figures 3 and 4. Figure 5 shows an example ofwave slope. Roll Excitation was simply AGM x wave slope.29Figure 3. Sway Excitation SpectrumFigure 4. Yaw Excitation Spectrum30Calculated   Filter Approx.0.001 0.0009-0.0008-0.0007-0.0006-0.00040.00030.0002-0.0001-00 0.2^0.4^0.6^0.8^1Frequency ( rad. )45 degree headingSea State 3Speed 11 kts.1.2^14Figure 5. Slope SpectrumIn these tests, the state equations were:x(k+1) = Ax(k) + Bu(k) +^(k)where w(k) is a vector of the values from the three excitationfilters.3.2 Controller ParametersThe controller parameters that can be adjusted were set for alltests. The test used to specify these parameters consisted ofgiving the vessel an initial yaw offset of 100 so that thecontroller had to move the vessel to a zero yaw angle as quickly31and smoothly as possible.For the controllers under evaluation, simple autopilots wereconstructed to compare the amount of roll reduction produced.The parameters for the autopilots were as follows:- PID Controllers. The parameters of the PID controller forthe Kynoc were K=-1 ,TD=0.5 , and T1=30. The PID parametersfor the Eastward Ho were K=-1 ,TD=1.0 , and T1=45.- Minimum Variance Controllers. The ideal roll response wascharacterized by the equation 1 - 1.59z-1 + 0.764z-2,corresponding to a natural frequency of 7 seconds and adamping ratio of 0.3. The ideal yaw response for the Kynochad a characteristic equation of 1 - 1.6z^+ 0.6399z-2.The characteristic equation of the ideal yaw response forthe Eastward Ho was 1 - 1.8z^+ 0.8099z-2. Kr was set to 1for both vessels. The simple autopilots used the same idealyaw responses for the respective vessels. The Kynoc'sautopilot used p=0.0003, the Eastward Ho used p=0.00005.- GPC Controllers. For both vessels, the roll reductionautopilots used N=10. The simple autopilots for bothvessels used N=11.- VSS Controllers. The switching surfaces for the roll32reduction autopilot, the simple autopilot, and the 15°/srudder rate roll reduction autopilot were the same for bothboats. The nominal system used for rudder roll reductionwas:A=1.81-0.3-0.800.3001.762400-0.95610.0300.01 B=0.0300.010 0 1 0 0 00 0 0 0 1 1This system has a yaw rate time constant of 2.24 seconds, aroll natural period of 7 seconds, and a roll damping ratioof 0.05. The roll reduction autopilot's switching surfacewas G=[43.06 —38.32 16.69 —0.839 4.6215]. The boundary layerused for the Kynoc was 6=0.05; the boundary layer for theEastward Ho was 6=0.025. The nominal system used for thesimple autopilot was:0 1 0Ai 0 0 11 0Bi^ ^I0.8 -2.6 2.8 0.017The simple autopilot's switching surface was G=[107.8961—254.926 150.573]. The boundary layers used were 0.09 forthe Kynoc and 0.04 for the Eastward Ho. The 15°/s rollreduction autopilot had a switching surface of G=[14.24—12.67 5.56 —0.3188 1.536]. The boundary layers were 0.06for the Kynoc and 0.03 for the Eastward Ho.- Pole Placement Controllers. The roll poles, which werethe same for both boats, were placed at 0.798±0.3631. These33locations correspond to a natural frequency of 7 seconds anda damping ratio of 0.3. The Kynoc's yaw poles were placedat z=0.79 and z=0.81. The Eastward Ho's yaw poles wereplaces at z=0.89 and z= TestsAfter each controller was chosen, designed, and parameters set,the following tests were simulated using the numerical model:1. A 15 degree course offset. This offset is 50% larger than thedesign offset of 100. The half load full speed ship wasused for this test.2. A yaw velocity impulse of 5.7°/s. This test is the base casefor the following four tests.3. Change in speed. The speed was decreased to half speed, whichdecreases the effectiveness of the rudder and therebyincreases the settling time.4. Change in displacement. The full-load full-speed conditionwas used for this test. This condition increases the yawrate decay time and thereby increases the settling time.5. Change in speed and displacement. This test uses the full-load, half-speed condition. In this test the controller34response to both a change in speed and displacement wasmeasured.6. Rudder servo slows by 20%. The purpose of this test was tosee how well the controller works when the actuator'sperformance has been degraded slightly. This situationcould occur due to poor maintenance ( e.g., hydraulic fluidleakage or simple wear).7. A constant yaw moment. This test simulates effects such asprop walk, steady wind and current effects that wouldproduce a constant yaw moment. The half-load, full-speedcondition was used for this test.8. Irregular beam (90°) seas. This heading should produce theworst roll motions. The full-load, half-speed case was usedbecause this loading condition has the lowest rollstability. The modal frequency of the sea state was setequal to the roll natural frequency and the significant waveheight was one hundredth of the length of wave with a periodequal the roll period (1-11/2=0.01gT:/(27r) and Tm=T0) .9. 45 degree quartering seas. Quartering seas produce the mostexcitation for roll and yaw combined; thus three quarteringseas tests were run. The full-load, full-speed condition wasused for this test. Sea State 3 for the North Pacific wasused for the modal frequency and significant wave height(liti=0.88m and Tra=7 55) 3510. 60 degree quartering seas.^The half-load, full-speedcondition was used in this test. Sea State 3 for the NorthPacific was used for the modal frequency and significantwave height.11. 30 degree quartering seas. The half-load, half-speedcondition was used in this test. Sea State 4 for the NorthPacific was used for the modal frequency and significantwave height (H=1.88m and Tm=8 85) .Different load and speed conditions were used in the irregularsea tests to obtain a better idea of the range of behaviour. Theirregular sea cases (tests 8-11) were then rerun for both vesselsusing a rudder rate of 15°/s. Only the four better rollreduction autopilots, the VSS, the pole placement, the minimumvariance, and the rate constrained GPC, as well as the PID andthe pole placement autopilots were run.3.4 ResultsThe results are tabulated in Appendix VI. The impulse responsesfrom the Rynoc of the PID autopilot, the VSS roll reduction andsimple autopilots, and the pole placement roll reduction andsimple autopilots are shown in Figures 6 to 10. These figuresare examples of the deterministic tests and show the effect ofadding rudder roll reduction .36Figure 6. VSS Impulse Response for KynocFigure 7. VSS Roll Reduction Impulse Response forKynoc37Figure 8. Pole Placement Impulse Response for KynocFigure 9. Pole Placement Rudder Roll Reduction ImpulseResponse for Kynoc38Figure 10. PID Impulse Response for KynocThe more relevant tests are the irregular sea tests because inthese tests both roll and yaw are excited and because these teststypify the conditions where a ship's operator desires rollreduction. For convenience, the results from the irregular seatests for controllers with a 5°/s rudder rate are presented inFigures 11 to 18. The figures show the RMS roll and yaw angles,smaller values being better for each. The increase in rollreduction for the controllers tested with a 15°/s rudder rate arealso provided in Figures 19 to 22. All tests were run for a timeof 10 minutes. An angular value of 20 degrees for any of yaw,roll, or rudder angle was considered to exceed the valid range ofthe numerical model.In general, the simple autopilots had good results, often better39than their roll reduction counterparts.The adaptive controllers on the Eastward Ho tests did not dowell, often exceeding the valid yaw range of the numerical model.The constant yaw moment test on both vessels caused problems withmany of the controllers.One unusual result was that better course keeping resulted inbetter roll response. It was expected there would be a trade-offof worse yaw response for better roll response. The otherunusual result was that the increase in the rudder rate tofifteen degrees per second did not improve the roll results forany of the controllers. The controller helped most by theincreased rudder rate was the pole placement roll reductioncontroller; however, the increase in performance was notconsistent over all the tests.The results are summarized for each controller as follows:- The PID autopilot did surprisingly well, as it wasexpected to have problems with rudder phase lag in the moreextreme irregular seas. The actuated rudder angle alwaysmatched the command rudder angle from controller.- The minimum variance autopilot had excellent coursekeeping in almost every test.40- The VSS autopilot did extremely well, its course keepingwas phenomenal.- The pole placement autopilot had excellent results;however, the actuated rudder angle did not always match thecommand rudder angle from the controller.- Both GPC autopilots had fair results, but betterperformance was expected.- All the roll reduction autopilots did poorly. The bestroll reduction controller is the VSS roll reductioncontroller which outperforms the PID controller in six outof eight cases. The next best controller is the poleplacement roll reduction autopilot, which is better than thePID controller four out of eight times.41_IFS^fil_GPC_PAS^VSS_RRS^Poi RAS^SSAPCJIRS013_CPC^VSS Poi Pi^55_0PCControllerFigure 11. Kynoc 900 SeasFigure 12. Kynoc 45° Seas42I.Figure 13. Kynoc 600 SeasPD^IN_115^IR_CPC-RAS VSS-1115^Poi RAS^SS_CPC_DISIN^RLCPC^VSS^Pala Ph^SS_CPCControllerMg Yaw MI Roll2.001.601.600.40O.20O.00Figure 14. Kynoc 30° Seas43Figure 15. Eastward Ho 900 SeasFigure 16. Eastward Ho 450 Seas44Figure 17. Eastward Ho 600 SeasFigure 18. Eastward Ho 30° Seas4590 45 60 30 90 45 60 30Kynoc^Eastward HoHeadingFigure 19. VSS Roll Reduction Improvement 15°/s RudderRateFigure 20. Pole_ RR Roll Reduction Improvement 15°/sRudder Rate46Figure 21. MV_RR Roll Reduction Improvement 15°/sRudder RateFigure 22. SS_GPC_RR Roll Reduction Improvement 150/sRudder Rate473.4 Evaluation of Roll ReductionBased on the numerical simulations, rudder roll reduction doesnot work well enough for commercial implementation on fishingboats. None of the roll reduction controllers reduced the rollmotion below that of the PID controller in every irregular seacase.The result from the numerical simulations that rudder rollreduction does not work is suspect for several reasons. Thefirst reason is that some of the controllers exceeded the validyaw range of the numerical model in the Eastward Ho tests. Thereason for these results was found by comparing heeling momentproduced by the rudder to the wave induced roll moment. For awave induced roll moment produced by a wave with a period equalto the Eastward Ho's natural roll period and a wave height of onehundredth the wave length, the rudder angle necessary for theheeling moment to counteract the wave moment is 51°. Theresulting heading angle is 60°. Both these values are welloutside the numerical model's range. In comparison, the Kynoc'srudder angle is 15° and the yaw angle is 19° for a wave with theKynoc's natural roll period. The reason for the large differencebetween the two boats is the Kynoc has a more effective rudder.The second reason comes from comparing the results in this studyto previous results in literature [3,5,8], where roll reductionwas reported successful. To compare the same qualities on48Table 3. Dimensionless QuantitiesDimensionlessquantityGM't/L zR/L AR/L2 Fo To/To 1/(6^x To)Kynoc 0.037 0.126 0.0046 0.32 - 0.21 0.0280.435Eastward 0.040 0.077 0.0027 0.26 - 0.23 0.029Ho 0.39USCGC 0.0096 0.037 0.0015 0.21 - 1.37 0.02Jarvis60.24Amerongen2 0.17 0.014Kallstrome 0.0026 0.031 0.0012 0.17 1.52 0.0054different ships, the technique of dimensional analysis was used.The dimensionless quantities are presented in Table 3. The threequantities vertical distance to rudder centre of effort tolength, total rudder area to length squared, and Froude Number(zR/L, AR/L2, and Fn) are measures of how effective the rudder isin roll and the two boats in this study would appear to have moreeffective rudders. The ratio of yaw-rate decay-time to rollnatural period (VT.) for the Kynoc and Eastward Ho are not anybetter or worse than the vessels used in previous studies. Therudder rate to roll period fraction indicates that the boats inthis study have slower rudder rates for their given roll periods;however, the rudder rates are not much slower than the USCGCJarvis's rudder rate and the Eastward Ho, which has a slightlyfaster rudder, had poorer results than the Kynoc.The third reason is the result that better course keeping givesbetter roll response. Intuitively, this result does not seemcorrect because there should be some trade-off between coursekeeping and roll reduction. Reasons for this result are unknown490.9^2y•^b2y400^500Time (s)boy6000.6-200at this time.The last reason to suspect the result from the numerical modeltests is that increasing the rudder rate did not improve the rollresponse. Previous results [2,3,5] indicate that increasing therudder rate always increases the roll response and intuitively,improving the actuator should improve the system response whensystem improvement is possible. Reasons for the lack ofimprovement are unknown at this time.The identification algorithm, used in the adaptive controllers,worked well (see Figures 23 and 24 ). The yaw parameter a2y wastypically slightly higher than expected, possibly due to thepresence of sway motion.Figure 23. Example of yaw parameter identification( random sea test - Eastward Ho)501i"bira2rIb2r-_- air _160 260 360 460 500. 60Time (s)-1- o0.!-0.5Figure 24 Sample Roll Identification (Random SeaCAse - Eastward Ho)The poor results with the adaptive controllers were first thoughtto be from the adaptive algorithm because at the same time intothe simulation when the yaw results started to exceed the model'srange, the yaw parameter a2y would rapidly change to a new value(see Figure 25). A sensitivity experiment was performed usingthe minimum variance roll reduction autopilot. The controllerparameter Kr was reduced, in effect reducing the importance ofroll reduction, and the results compared. When Kr was smallenough the numerical model did not exceed its range of validityand the a2y parameter did not have a rapid change (as in Figure23). Therefore, from this experiment and the comparison ofheeling moment to wave-induced roll moment, the problems with the51b 1 ,,0b 2 y160^260^360^460^500Time (s)-01 ^0 6001 0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 -adaptive controllers are either from the controller algorithms orthe numerical model and not from the adaptive algorithm or theprocess model used in the adaptive algorithm.Figure 25 Yaw Parameter EstimatesThe results for the VSS, the minimum variance, and the poleplacement autopilots are very good, not just because of theirgood yaw response, but because of the form of the algorithms.These controllers have just one "tuning knob", compared to thePID controller which has three, making it easier for the ship'soperator to adjust the autopilot. The minimum variance and poleplacement would appear to have more than one parameter to adjust;however, the pole locations could be chosen by the controllerbased on the identification results, most likely the Joh,parameter because it shows how effective the rudder is andtherefore how quickly the boat can respond.52Part 4: Physical ModelA physical model that had been used for other self-propelledmodel tests was prepared for autopilot testing. The model is a15:1 scale model of the Eastward Ho, a 107' steel combinationvessel. The principle particulars of the model are given in thefollowing table.Table 4. Model Particulars.ParticularLength (m) 1.805Beam (m) 0.54Depth (m) 0.35Draft (m) 0.22Ch 0.5A(kg) 102Preparations for the tests included re-surfacing the outer hullof the model with a fabric composite, splining the outer hull andinstalling a new power bus. Two 12V 24Ah batteries and one 12V7Ah battery were installed for power. One of the 24Ah batterieswas replaces with a 31Ah battery after the old battery becamedamaged. A choke and capacitor were installed as powerconditioners for the existing motor control unit and a capacitorwas installed as a power conditioner for the rudder servomotor,respectively.534.1 Yaw SensorA yaw sensor is necessary for measuring the model's direction.The first sensor installed for measuring yaw angle was aInternational Navigation INI-100 Digital Heading Sensor. Thisdevice is a fluxgate magnetometer, or a solid state compass.After many trials with poor results, it was discovered that therewas too much distortion of the earth's magnetic field within thetowing tank building.A directional gyroscope, an Aviation Instrument Manufacturing Co.205-2A, was then installed. This instrument is connected to amodified Wagner SEAutopilot Generation 2 autopilot to resolve thesynchronous output of the gyroscope to a dc signal for thetelemetry system.4.2 Roll SensorThe other sensor necessary in a roll reduction system is one tomeasure roll angle; therefore, a Humphrey VG24-0825-1 verticalgyroscope was installed. This gyroscope has a potentiometricoutput, i.e., dc signal output; however, the signal to noiseratio was too large. Therefore, an amplifier between thegyroscope and telemetry system was built.544.3 Telemetry SystemA Sigma Data Systems telemetry system was used to send the sensorsignals to the shore-based computer. The system consists of atransmitter installed on the model, a radio receiver and a readerunit on shore. The time delay from the sensor reading on themodel through to the output from the reader was approximately40ms.4.4 Data AcquisitionA Strawberry Tree ACJr-12 was used for data acquisition. Thecard has a maximum of eight channels of which two were used.With two channels used, the card has a sampling rate of 1 kHz ata resolution of 11 bits.4.5 ComputerA Toshiba T3200 portable computer was used as a host for the dataacquisition card and to implement controllers in software. Thecomputer is based on the 16-bit Intel 80286 microprocessor. Theoperating system used was MS-DOS 4.0. The sampling period usedfor the controllers was 0.11s. This value, which is constrainedby the operating system, when added to the delay time of thetelemetry system, results in the closest value to the desiredsampling period of 0.13s (Froude scaling of the 0.5s full scale55sampling period).4.6 Radio ControlThe radio control system is based on Futaba R/C components. Thetransmitter is a Futaba FP-5FG/K Digital Proportional RadioControl. It is interfaced with the computer by means of a customcircuit board between the serial port of the computer and thetrainer port of the R/C transmitter.The rudder servo on the model has a maximum rudder rate of 91degrees per second. The constrained rudder rate of 19 degreesper s (Froude scaling of the 5°/s full scale rudder rate) wastherefore implemented in software.56A diagram of the complete system is provided in Figure 26.Figure 26. Model Testing Arrangement4.7 Towing TankThe location of model testing is the towing tank at the OceanEngineering Centre at B.C. Research. The tank is 220 feet long,12 feet wide, and 8 feet deep. Physical model testing wasdisrupted prior to completion by B.C.Research going intoreceivership.574.8 TestsOnly some preliminary tests were completed, due to the loss ofthe facility. These preliminary tests consisted of nine zig-zagtrials, in which only yaw motion was recorded, and one zero-speedroll-decay test.From the model tests, the yaw rate time constant was found to bearound two seconds. The model time constant corresponds to afull scale time constant of around seven seconds. The modeltests were made at speeds estimated at three to five knots (fullscale). No comparison was made to the numerical model becausethe numerical model did not produce good responses at these lowspeeds.From the roll decay test, the roll natural period was 1.49seconds and the damping ratio was 0.075. These values were foundtwo ways: first, using logarithmic decrement for the dampingratio and visual measurement of the period, then from regressionanalysis. The natural period of the model corresponds to a fullscale roll period of 5.77 seconds. A comparison to the numericalmodel roll period was not possible because the GM of the modelwas not measured. The damping ratio of the physical model isapproximately twice that of the numerical model. Thisdiscrepancy is most likely due to the bilge keels fitted on thephysical model, as it was assumed there were none with thenumerical model.58Part 5: Conclusions1. Based on the results from the numerical simulation, rudderroll reduction does not work well enough for commercialimplementation on fishing boats.2. The yaw motion resulting from the rudder action necessary toreduce roll motion on fishing boats can be too large forrudder roll reduction to be effective.3. The results from the numerical simulation are suspect because:- the results do not indicate better roll response with afaster rudder rate.- the results indicate a straighter course gives better rollresponse.- comparison to previous results indicates little differencein the relative capabilities of ships used in thesetests and ships used in previous, successfulimplementations.4. The VSS, minimum variance, and pole placement controllers hadexcellent results as autopilots with no roll reduction.59Part 6: Future WorkThe research that remains is:1. Confirm that rudder roll reduction does not work using modeltests and the two better rudder roll reduction controllers,i.e., the VSS rudder roll reduction and the pole placementrudder roll reduction controllers.2. Confirm the autopilot results using model tests.3. Explore the idea of combining the pole placement and minimumvariance controllers into one controller. The resultingcontroller should have smooth, exact control (like that ofthe pole placement controller but which the minimum variancetends to lack) without any rudder saturation (which the poleplacement has but the minimum variance does not).60Part 6: BibliographyOn Rudder Roll Reduction1. Roberts, G.N., and S.W. Braham."A Design Study on theControl Of Warship Rolling Motion Using Rudder andStabilizing Fins." 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New York:Wiley Interscience, 1978.22. Inoue, S., et al. "A Practical Calculation Method of ShipManeuvering Motion." International Shipbuilding Progress 28,No.325 (September 1981): 207-222.23. Inoue, S., et al. "Hydrodynamic Derivatives on ShipManoeuvring." International Shipbuilding Progress 28, No.321(May 1981): 112-125.24. Clarke, D., et al. "The Application of Manoeuvring Criteriain Hull Design Using Linear Theory." Transactions of theRoyal Institute of Naval Architects (1983): 45-68.25. Bass, D.W., and M.R. Haddara. "Roll and Sway-Roll Dampingfor Three Small Fishing Vessels." International ShipbuildingProgress 38, No.413 (1991): 51-71.26. Rydill, L.J. "A Linear Theory for the Steered Motion ofShips in Waves." Transactions of the Royal Institute ofNaval Architects (1959): 81-112.27. Reid, R.E., et al. "The Use of Wave Filter Design in KalmanFilter State Estimation of the Automatic Steering Problem ofa Tanker in a Seaway." IEEE Transactions on AutomaticControl AC-29, No.7 (July 1984): 577-584.On PID Autopilots28. Dickson, Bill, R&D Director, CompuNav Systems. TelephoneConversation with Author, 8 April 1992.29. Wagner, Paul, Wagner Marine Systems Inc. TelephoneConversation with Author, 8 April 1992.63Appendix I: Approximate Maximum Likelihood IdentificationThe following is a more detailed description of theidentification method used and of the simplification of theidentification of the yaw parameters.GiveniA (z-1) y =E Bi (z-1) uii=owhere j is the number of inputs and consider the cross-couplingterms as another form of input.A (z-1) =1 + aiz-1 + a2z-2 +... +anz-nB(z1) = boi + bliz-1+...+bmiz-mAML identification has the form :P(t+1) = P(t)- P(t)x.r(t+1)xr(t+1)P(t)1+xf(t+1)P(t)x1(t+1)P( t+1) xi ( t+1)1+4 (t+1) P(t)x.r(t+1)e(t+1) = y(t+1) - xf (t+1)0(t)0(t+1) = 0( t) +1C(t+1) e(t+1)to include the Exponential Forgetting and Resetting algorithm [8]the co-variance and gain equations are modified:= 1 aP(t)x1(t+1)x.r(t+1)P(t)A^1 +XI( t+1) P( t) Xi( t+1)-^ccP(t+1)x1(t+1)1+x. r(t+1)Px.r(t+1)x/ is the vector of inputs and outputs.e is the vector of the parameters being estimated.x . [-y(k-1) ,..., -y(k-n) , Ell (k-1) , ..., ul(k-m1) , ....1-, u2 (k-1) ,..., ui (K-mi) ,ri (k-1) ,...,11 (k-n)] T0 = [al, a2, --, an, bii, -, blm,,b12, •••, bimi, ci, •••, cn} TIC( t+1) -+111-8p2(t)64n are the a posteriori errors andn ( t) = y(t) - xr(t)0(t)For the yaw sub-system, it is assumed n=2, j=2, m1=1, and m2= .that is( 1 + al z-1 + a2 z-2) lir = bil z-1 u +b12 Z-1 04)A simplification for the identification of the yaw subsystem ismade by assuming the natural response of yaw is a decay term andan integrator :A(z1) = (1-z-1) (1-a2 z')or al= -(1+a2)therefore it is only necessary to identify a2.The following changes must be made to x1 and 8:xi = Dir (k- 1) - Ip. (k-2) ,u(k-1) ,4)(k-1)]T9 = [a21b111b12] Te(k) = lir (k) - Ili (k-1) -4965Appendix II: Generalized Predictive ControlThe following is a more detailed description of the state spacegeneralized predictive controller.Given a state space systemx(k+i) = Ax(k) +Bu(k)y(k) = Ca(k)where A is (nxn), B is (nxm), C is (pxn), x is a vector of lengthn representing the states, u is a vector of length m representingthe incremental inputs, and y is a vector of length prepresenting the outputs.Assuming a prediction horizon of length N, and a control horizonof length Nu, Nu N, the global predictive model is:IF= Hira yowhereCB^0^0CAB^CB %.^0CBCAR-1B CAN-2B CA"uBCA={^ix(k)CANumr.(10 = [Au (10 Au (k+1) - Au(k+ Nu - 1) 12"The control law is to choose ü which minimizes:Jr = 1";IX + arnwhere P is a diagonal weighting matrix for the input.For the case of the rudder with the fixed swing rate, theincremental input has only three possible choices, swing left,swing right, or do not swing. In this case, and assuming acontrol horizon ( N4 ) of 1, it is easier to calculate J for eachof the three possibilities of input and choose the one whichHR. =and66minimizes J. When the rudder reaches its amplitude constraint,the number of choices decreases to two, decrease the rudder angleor let it remain.For the case of the rudder with a maximum swing rate, theincremental input which minimizes J is calculated from:a = {[H,TruHA H: Tu oThe calculated input is then checked against the maximum inputand if the calculated value exceeds the maximum, the maximum isimplemented. This setting the input to the maximum results fromassuming the constrained optimum lies on the boundary of the costfunction and the constraint.67Appendix III: Discrete-Time Variable Structure ControlA design equation will be derived for finding the discretesliding plane co-efficients for a state space system with boundedrelay control. The design equation has the form of a matrixRiccati equation.Given a state space system:x(k+1) = Ax(k)+Bu(k)^(AIII.1)where A is (nxn), B is (nxm) and each ui has the form:u1(k) = -Kisigm(Si(k))or u(k) = -Ksign(S(k))^(Anrr.2)where 17= diag(Ki)sign(Si(k))"sign (3(k)) =sign(Sm(k))_and Si (k) Gix^ (AIII.3)where Gi are row vector of size n.The Ricatti equation will be derived using the second method ofLyaponov.First, consider the stability of the state trajectories off thesliding subspace using a matrix Lyapanov function :V(x) =.KT(.1c)FX(k)then A.17(20 = v(x(k+1)) - v(x(k))= xr(k+i)px(k+1) -xT(k)Px(k)using (AIII.1):AV(x) = [Ax(k) + Bu(k)]TP[Ax(k) + Bu(k) - xr(k)Px(k)= xT(k) (ATPA-P)x(k) + xr(k)ATPBu(k)+ uT(k)BYPAx(k) + uT(k)BTPBu(k)using (AIII.2):AV(x) = XT(k) (ATPA-P)x(k) - xr(k)ATPBKsign(Gx(k))- [Ksign(Gx(k)) ]TBTPAx(k)+ f sign(G(x(k)) ]TBTPB[Ksign(Gx(k))68Assume G = BTPAV (x) = XT (k) (ATPA - P) x (k) - xT (k) ATPBKsign (BTPx (k) )- [ Ksign (BTPx(k) ) ] TB TPAx (k)+ [ Ks i gn (B TPx (k) ) ] TB TPB [ Ks i gn (B TPx (k) ) ]Letting sign(BTPx(k)) = 12-1BTPx(k) where R is an arbitraryweighting matrix of form diag( ri )A V (x) = XT (k) TPA- P) x(k) - XT(k) (ATPRKR-IBTP) x(k)- XT (k) (PB (R-1) TKBTPA) x(k)+ xT (k) (PB (1?-1)TKBTPBKR-1BTP) x (k)Let= _[ A TPA _ A TpBKR-lBTp _ pBK- ( R-1) TB TpApB T (R-1) TKB TpBKR-1B TplThe system is stable if AV(x) is negative definite, Q must bepositive definite. This leads to the following Ricatti equation:P = + A TPA _ A T pBR'KB Tp pB(R-1)7703 TPApB (R-1) TKB TpBR-1KBTp (AIII.4)The design procedure is specify a nominal system A,B with controlconstraints K, and specify the arbitrary weighting matrices Q andR. Then find P from (AIII.4), then G=BP and implementu=—Ksign(Gx(k)).69Appendix IV: Equations of the Transverse MotionsThis appendix presents the derivation of the state spaceequations for the transverse motions of a vessel. The equationswill be based on the linear equations of motion. A samplecalculation for one speed and loading condition will beperformed.From Inoue et al [22]. ( see also v.3, Chapter 9, section 16.7 of[20]) the transverse equations of motion are:Sway^m(V + ur) =^+ YR + Yw^(AIV.1)Yaw Izzt = NH ÷ NR NW (A/V. 2 )Roll^,x4) = KH KR + Kw^(A/V. 3)The force and moments due to wave action (.) have been added.The free response of the three motions will be developed first,starting with sway (AIV.1)1m(li+ur) = -m P--mxur + -2- pLdV2{117 + rL1 + YR + YwvV^r IriM M 1.7 —1 pLdITYfy + (mu + m^1u - —pL2 dVYIr) .r =YR+ Yw2^ 2Yaw (AIV.2):= -J„t+4pLdV2[Nlvvi+leirl+4pL2dV2[44)]+NR+Nw( Izz + Jzz) - 4 pLdVILNIv+ YlvKs I V —4 pL2dvtizer+Y/gcsli-- l_pL2dv244) = 1\TR + Ivw2Roll (AIV.3): J - N(4) - gAGZ(4)) - YHZH +KR +Kw(^+^)4) + N(44) + gAGZ (4)) + YyzH = KR + KwUsing linear damping and stiffness, as in Bhattacharya [21]( /xx + LT) + NI) + gAIGMA + YHzH = KR + KwExpand YH(^+J ) + bd) + gAMT24)+ [ -myx>. mxur + ipLdVYI,v+ pL2dVYIrr]zH = KR + Kw( Im,r+ J )(T) + b4) gAGMT4- myzHV + 4 p LdVY/vzilv+ [ -mxu + p L2 dVilzHr = KR + Kw70GeometriesxR - The rudder centre of pressure with respect to the ship'scentre of gravity. It is measured longitudinally, and isnegative aft (see Figure 27).zH - Location of the ship's centre of pressure with respect tothe ship's centre of gravity. This is the point at whichthe lateral hydrodynamic forces are assumed to act. Ismeasured vertically, negative up. For all ships in thisstudy, it was assumed to be on-half the mean draft.zR - Location of the rudder's centre of pressure with respect tothe ship's centre of gravity. It is measure vertically, andis negative up (see Figure 27).Figure 27. Geomerty of Rudder MomentsMasses and Moments of InertiaAdded mass in the y-direction.ny = k2 AAdded mass in the x-directionmx = ki AAdded yaw mass moment of inertiaJ = k'InzzThe co-efficients, k1, k2, and k' are found in Vol.3, Chapter 9,Section 9.4 of [20] . They are based on an ellipsoid of revolutionwith a major axis of one half the waterline length, and a minoraxis of the mean draft.Yaw mass moment of inertia.71The yaw radius of gyration is assumed to be approximately onequarter of the waterline length, i.e.,kzz = 0.25 4athen the yaw mass moment of inertia isIzz = Alc!zRoll mass moment of inertia.The roll radius of gyration is assumed to be:= CVB2 + .D2where C varies between 0.33 and 0.39.The added roll mass moment of inertia was assumed to be 15% of themass moment of inertia.Hydrodynamic DerivativesThe following are the linear hydrodynamic derivatives from Inoue etal. [23]:Y. = -1-17tk + 1.4C 411.0 + —\2T^‘2^BLA^31"/= 17k[1 + 0.81]^r ^4_ 0.2711 T= -[0 . 54k - k2][1.0 + 0.3]where^k- 2TL pp1 - ^13^[k + 1.4C 2-312B LT = T. - TFwDThe roll damping coefficient depended on the hull form. For hardchine hulls a damping ratio of 0.09 [25] was used. For round bilge72hulls the following formulae, from [20] were used:0.55(.0024LBZ0d554)1 Co - ^AB2)212 Fn 1+ Fn +2( FnCBCf = 0.00085H(-Lm CB^CBB G C = CO + CfwhereCo - zero speed damping ratioCf - forward speed correctionFn - Froude Number - VsIgT,d - distance the intersection of the waterlineand centreline to the turn of the bilgeThe critical roll damping was found from:= 2 VgAGMT(Ixx + Jxx)Rudder ForcesThe rudder forces and moments are :YR -(1 + aH)FNCOS8NR = -( 1 + aB) XRFNCOS8KR = (1 +^zHFNcosOwhere- 1 p  6.131  ADVsina R-^2 1+2.25 - -aH = 0.627CB- 0.153The method for finding VR in Inoue et al. is quite complicated, so73instead equation (169) from Vol.3, Chapter 9 of [20] was usedTIR=k(1 + Sa) Vwhere^0.8<k<1.0 depending how much of the rudder- is in the propeller slipstream.Pn VSa -For aR, assume aRz6. Also, it was assumed cos(5)=1.Sample CalculationsThe following are the necessary parameters needed to calculated thetransverse motion co-efficients.Vessel name & Condition:^Eastward HoShip DataFull load 13ktsLwl 30.7848 Cb 0.513686Bwl 8.738616 L/B 3.522846Depth 4.2672 B/T 2.163774Taft 4.54025 B/D 2.047857Tfwd 3.4671Tmean 4.0386tau 1.07315Displ (Vol.) 558.094Displ (Mass) 572.0463Engine^ PropellerType Dia.^2.1336Bhp 850^Pitch 1.524RPM^1225 k 0.99Gear Red^4 RPM^300knots^m/sSpeed^13 6.68772Fn 0.384836Stability x - positive fwd;z - positive downLCG^-1.8928^VCG^3.377184GM 1.267968 zh 1.357884 below c.g.Round bilge or Hard Chine ( b or c )distance from C.L. @ W.L. to turn of bilge 4.60Pnwhere P - propeller pitchn - rev/ unittimeV - Ship velocity74Rudderheight^2.286^xr^-12.509Span 2.286 zr 2.36584Chord 1.2192Area^2.787091Aspect Ratio^1.875The added masses, mass moments of inertia and hydrodynamicderivatives were calculated using the above formulae:Added Mass x-direction^45.04215Added Mass y-direction^494.4331Yaw Radius of Gyration^7.6962Yaw Mass Moment of Inertia 33883.16Added yaw m.m. of i.^19587.96Roll Radius of Gyration^3.50094Roll Mass Moment of Inertia7011.333Added Roll M.M. of I. 1402.267Zero Speed Roll DampingCoefficient 0.011913Roll Damping 675.55470.262376L-beta 0.425741Yv prime -0.72545Yr prime 0.249876NV prime -0.21816Nr prime -0.07865Nphi prime -0.0076For the rudder forces:Sa 0.122346Rudder Velocity 7.430879Rudder Normal Force 219.7674ah 0.169081The co-efficients for the equations of motion are then found to be:Sway EquationSway acceleration 1066.479Sway Velocity 309.136Yaw rate 848.984275Yaw EquationYaw Acceleration^53471.11Yaw Rate^ 25557.17Sway Velocity 3447.02Roll Angle 666.7563Roll EquationRoll Acceleration^8413.6Roll Velocity 675.5547Roll Angle^ 7115.55Sway Acceleration^-671.383Sway Velocity -419.771Yaw Rate 4042.015Rudder forcesSway^ -256.926Yaw 3213.883Roll 607.8455The co-efficents are then arranged into proper matrices A and B:where the state vector is:Ac0 1 0 0 00 -0.47796 -0.06447 -0.01247 00 -0.79606 -0.28987 0 00 0 0 0 10 -0.54394 0.026761 -0.84572 -0.08029Bc00.060105-0.2409100.053022Kc o o o1 0 00 1 00 0 00 0.197267 1These matrices are discretized using a zero order hold to obtainthe discrete time matrices:A1 0.445675 -0.00711 -0.00142 00 0.792773 -0.02667 -0.00535 -0.00140 -0.32935 0.870468 0.001073 00 -0.06138 0.003745 0.897635 0.4730250 -0.23094 0.01553 -0.39928 0.859656760.0072480.028484-0.117630.0057230.0207890.11573 -0.00122^00.445675 -0.00713 0-0.08775 0.466396^0-0.01052 0.010064 0.121193-0.06138 0.040641 0.473025A comparison between the continuous time model (using a Runge-Kuttadifferential equation solver) and the discrete time model is givenin the following figures. As can be seen in the figures, not onlydo both models seem to be reasonable representations of thetransverse motions, for the purposes of this study the differencebetween them is unmeasureable.Figure 28. 10° Rudder Step Response.77Figure 29. 100/100 Zig-zag, rudder and yaw.Figure 30. 100/100 Zig-zag, roll.78Appendix V: Wave ExcitationThe wave action on a ship is highly non-linear, and is not easilysimplified. This appendix describes the steps used to obtain anumerical model of wave excitation for implementation in adiscrete simulation. The excitation is usually represented as aforce or moment equation that is dependent on frequency. Fromthese equations, digital filters must be obtained that have astheir inputs white noise.From ref. 26, the sway excitation force isYw = p ge-kZSirlX NCOS (ki dx trnsin ( wet)and the yaw excitation moment is:Arw = -pge-kzsinxfAxxsin (1c/x) dx fl,cos (co et)whereZ = distance of the M-3" below the waterlinekl= kcosxk = the wave numberx = the wave encounter angle ( 0° following seas, 1800 head seas)we = wave encounter frequencyA = 1,4 Ta _ 47ue )2-^L2In this simulation, the cross-sectional area Ax is that of anellipsoid of revolution, with the minor axis equal to the draughtand the major axis equal to one half the length.From [21] , the roll excitation moment is:= gAGMTsinx Esin(wet)The wave slope spectrum used in this simulation is based on theBretschneider Spectrum (wave height):_BA^4S(W) = w5 e "where A and B depend on the modal frequency and variance. Forwave slope spectrum, the Bretscneider Spectrum is modified:S(w) = W4 S(w) - Acog2eg279The spectrum must be further modified so it is based on thefrequency of encounter we:St (() e ) —S (c)- 2 6) ucosdwhere:()2 U(4)e = 16)^COS)]All of the forces and moments are an amplitude multiplied by thewave slope Esin(wet + e); however, both the sway and yawamplitudes are functions of the wave number and hence the wavefrequency. Therefore the spectra of these amplitudes must befound. They have the form:Se(we) = Ihre(wle) ps, (6)4)This complicates the calculation of the spectrum. Because each wehas three corresponding values of w, the amplitudes and waveslopes must be calculated for each of the three frequencies,multiplied, then summed ( see ref ).These spectra were then approximated by continuous time transferfunctions whose power spectra were matched for shape visually andarea under their curves quantitatively.The transfer functions were discretized using the bi-lineartransform with the appropriate sampling period.For beam seas ( x=90 ), the excitation was further simplified.The sway excitation force used in this case was:Yw-  mg  sin(wet)m + mThe yaw excitation was taken to be zero because the hull shapeapproximation of an ellipsoid of revolution means the hull hasfore-aft symmetry resulting in zero net excitation.The roll excitation remained the same.The only necessary spectrum calculation was then for wave slope,and in beam seas there is no complication with different wavefrequencies for encounter frequency, i.e. w=coe.80Appendix VI: Numerical Simulation ResultsK noc 15° OffsetController Final yawangleMaximum yaw angles Settlingtime (s)Rise Time(s)PID 8.50e-08 -2e-06 9e-06 45 5.5MV 0.125072 -5e-05 5e-05 20.5 20.5MV RRS -0.38812 -2e-04 2e-04 27.5 11.5BB GPC -0.90094 -2e-04 2e-04 22 6BB GPC RRS -0.38774 -5e-05 4e-05 42.5 7.5VSS 0.638019 -3e-05 3e-05 20.5 18.5VSS RRS 0.125071 -5e-05 5e-05 33.5 7Pole Plc -4.0e-09 -2e-06 2e-06 16 13.5Pole RRS 0 0 0 23.5 12SS GPC 1.00e-09 -2e-06 2e-06 21 7SS GPC RRS -2.4e-07 -3e-04 3e-04 43.5 7.5K noc 5.7° s ImpulseController Final yawangle_Maximum Yaw angles Settlingtime (s)Rise Time(s)PID 2.10e-08 -2e-06 5e-06 44 5.5MV -0.08354 -5e-06 4e-06 27.5 13.5MV RRS 0.429435 -6e-05 4e-05 39 7BB GPC 0.942373 -6e-05 5e-05 30 4BB GPC RRS -0.08349 -4e-05 4e-05 27 5VSS 0.429384 -5e-06 3e-06 27.5 18.5VSS RRS -0.0836 -4e-05 5e-05 47.5 10.5Pole Plc -1.2e-08 -6e-06 4e-06 27 7Pole RRS 0 0 0 33.5 6.5SS GPC -9.0e-09 -3e-06 2e-06 25 5SS GPC RRS 0 0 0 23 581.Kynoc Change in SpeedController Final yawangleMaximum yaw angles Settlingtime (s)Rise Time(s)PID 0.000968 -0.025 0.06 112.5 118MV 0.137847 -3e-05 3e-05 18.5 8.5MV RRS -0.09171 -8e-05 9e-05 41.5 11.5BB GPC -0.01489 -0.393 0.389 599 6.5BB GPC RRS -0.09163 -2e-05 2e-05 32 8.5VSS -0.55074 -le-04 le-04 22.5 13.5VSS RRS -0.09163 -2e-05 2e-05 20 18.5Pole Plc -3.0e-09 -2e-06 2e-06 19 8.5Pole RRS 0 0 0 31 10SS GPC -6.0e-09 -3e-06 2e-06 30 7.5SS GPC RRS -2.6e-08 -le-05 8e-06 33 8.5Kynoc Change in DisplacementController final yawangleMaximum Yaw Angles Settlingtime (s)Rise Time(s)PID -8.2e-08 -2e-05 3e-05 49 5.5MV -0.3404 -4e-05 7e-05 35.5 13.5MV RRS 0.199443 -5e-05 9e-05 44 10BB GPC -0.88027 -le-04 2e-04 48 4.5BB GPC RRS 0.199578 -6e-05 5e-05 38 4.5VSS 0.199515 -2e-05 3e-05 35.5 22VSS RRS 0.199452 -3e-04 6e-04 78 12Pole Plc 7.80e-08 -3e-05 4e-05 35.5 6.5Pole RRS 0 0 0 31.5 6.5SS GPC 2.50e-08 -le-05 2e-05 33 5SS GPC RRS -1.0e-09 -le-06 le-09 34.5 4.582Kynoc Change in Speed and Dis lacemntController Final yawangle Maximum Yaw angles Settlingtime^(s) Rise Time(s)PID 0.000507 -0.048 0.108 118 9MV -0.72037 -3e-05 3e-05 22.5 8MV RRS -0.20882 -3e-05 8e-05 45.5 11.5BB GPC -0.28261 -10.03 11.2 186 7BB GPC RRS 0.047001 -6e-05 3e-05 55.5 8.5VSS 1.069895 -le-04 le-04 22 16.5VSS RRS -0.20894 -le-04 le-04 31.5 14.5Pole Plc -9.0e-09 -le-05 le-05 24.5 7.5Pole RRS 0 0 0 36 8SS GPC -2.9e-08 -2e-05 2e-05 37.5 7.5SS GPC RRS 3.05e-07 -0.002 0.002 53 8K noc Rudder Servo SlowsController Final yawangle Maximum Yaw Angles Settlingtime (s) Rise Time(s)PID 2.30e-08 -2e-06 5e-06 44.5 599.5MV -0.18619 -5e-05 5e-05 27.5 13.5MV RRS -0.18596 -le-04 le-04 28.5 7.5BB GPC -1.00688 -5e-05 5e-05 21 4.5BB GPC RRS -0.18634 -2e-04 2e-04 27 5.5VSS 0.634489 -5e-05 5e-05 27.5 15VSS RRS -0.18619 -5e-05 5e-05 35.5 11.5Pole Plc -1.0e-08 -5e-06 4e-06 27 11.5Pole RRS 0 0 0 31.5 6.5$S GPC -9.0e-09 -3e-06 2e-06 23 5.5SS GPC RRS 0 0 0 24.5 5.583Kynoc Constant Yaw MomentController Final yawangleMaximum Yaw Angle Settlingtime^(s)Rise Time(s)PID -1.9e-06 -le-06 3e-06 40.5 94MV 3.042305 -3.162 10.99 595.5 5.5MV RRS 148.3223 -126.2 179 520 76.5BB GPC 0.760904 -3.342 3.932 530.5 5.5BB GPC RRS 25.81668 -162.8 180.3 593 7VSS -0.1463 -0.132 0.129 597.5 16VSS RRS -3.4502 -0.143 0.133 599.5 6.5Pole Plc 0.613604 -1.095 1.169 583.5 32Pole RRS 0.824794 -11.78 5.514 584 10.5SS GPC 0.562783 -0.326 0.331 599 35SS GPC RRS 14.26879 -118.4 135.1 599.5 7.5The column %PID, in the tables of the irregular sea test results,is the amount of roll reduction using a controller as compared tousing the PID controller.K noc 900 SeasController RMSrudderangleRMS yawangleMaximum yawanglesRMSrollangleMaximum rollangle%PIDPID 1.68 1.53 -5.61 3.39 2.78 -8.28 8.45 0.0%MV 6.42 1.56 -5.12 5.71 3.37 -9.49 10.4 -21.3%MV RRS 5.68 1.64 -4.67 6.28 3.45 -9.07 9.27 -24.0%BB GPC 7.44 2.41 -9.08 10.9 3.3 -10.5 8.97 -18.7%BB GPC RRS 7.32 3.35 -13.4 11.9 2.82 -8.08 10.5 -1.3%VSS 4.98 1.07 -3.58 4.94 3.23 -9.74 8.51 -16.1%VSS RRS 4.68 1.72 -5.22 6.15 1.76 -5.27 5.11 36.8%Pole Plc 4.5 1.14 -5.54 4.37 3.28 -9.53 10.4 -17.9%Pole RRS 5.52 3.14 -10.3 12.6 3.03 -9.93 11.1 -8.9%SS GPC 2.47 1.17 -3.46 3.86 2.82 -7.86 8.37 -1.5%SS GPC RRS 5.09 1.91 -6.68 4.88 2.22 -5.74 8.05 20.1%84Rynoc 450 SeasController RMSrudderangleRMS yawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDPID 2.36 2.19 -5.71 5.36 2.59 -6.84 6.99 0.0%MV 2.08 0.27 -0.87 0.91 1.34 -3.57 3.95 48.3%MV RRS 2.3 1.37 -5.22 4.45 2.43 -7.11 8.05 6.2%BB GPC 2.19 0.42 -1.36 1.75 1.57 -3.48 4.48 39.3%BB GPC RRS 3.35 2.73 -9.85 8.93 2.37 -7.31 8.35 8.5%VSS 1.93 0.27 -0.85 0.67 1.3 -3.52 4.5 49.9%VSS RRS 5.27 5.91 -16.3 15.9 3.46 -10.5 9.84 -33.6Pole Plc 1.87 0.48 -2.11 2.08 1.61 -4.4 5.25 37.9%Pole RRS 2.42 1.13 -4.14 4.37 2.09 -9.85 8.48 19.3%SS GPC 2.41 0.88 -3.57 3.34 2.1 -7.65 7.38 18.9%SS GPC RRS 3.13 1.24 -3.47 7.35 3.12 -10.4 9.68 -20.6Kynoc 600 SeasController RMSrudderangleRMS yawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDPID 2.17 1.94 -6.67 5.43 2.56 -8.98 8.81 0.0%MV 2.78 0.4 -1.43 1.91 2.03 -6.53 6.46 20.5%MV RRS 3.15 1.46 -6.21 4.75 3.03 -9.02 8.66 -18.4BB GPC 2.64 0.51 -1.92 1.8 2.54 -8.33 7.8 0.6%BB GPC RRS 3.64 2 -9.41 7.12 3.14 -9.4 9.91 -22.9VSS 2.78 0.59 -2.27 1.91 2.29 -6.89 8.33 10.6%VSS RRS 5.18 5.64 -12.7 19.4 3.03 -8.4 8.38 -18.3Pole Plc 2.18 0.61 -3.06 1.5 2.16 -6.92 6.05 15.6%Pole RRS 2.46 1.09 -2.87 3.21 2.09 -6.99 6.28 18.2%1SS GPC 2.34 0.85 -2.87 2.43 2.37 -6.43 7.18 7.2%SS GPC RRS 4.08 1.42 -4.15 8.5 4.24 -14.4 11 -65.9,85Kynoc 30° SeasController RMSrudderangleRMS yawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDPID 1.63 1.44 -4.98 4.66 1.34 -3.9 4.81 0.0%MV 5.71 1.03 -5.08 4.37 1.9 -5.57 5.2 -41.6%MV RRS 4.81 0.85 -2.67 3.35 1.85 -5.89 4.88 -38.2%BB GPC 6.89 1.85 -6.01 9.23 1.85 -4.9 6 -38.0%BB GPC RRS 4.37 1.72 -5.23 6.74 1.45 -4.58 4.61 -7.8%VSS 4.55 0.53 -2.68 2.41 1.65 -4.46 4.1 -22.8%VSS RRS 3.43 1.84 -5.06 4.7 1.21 -2.95 3.51 9.8%Pole Plc 3.78 1.02 -8.71 3.72 1.61 -4.07 6.23 -19.7%Pole RRS 3.01 0.86 -3.42 2.53 1.35 -3.95 5.33 -0.6%SS GPC 2.1 0.92 -2.59 3.43 1.38 -4.23 3.93 -3.1%SS GPC RRS 3.93 1.05 -3.27 3.26 1.5 -4.62 4.6 -11.7%Eastward Ho 150 OffsetController Final yawangleMaximum yaw angles Settlingtime (s)Rise Time(s)PID -7e-06 -0.0006 0.0002 61.5 9.5MV -0.5592 -5e-05 0.0002 57.5 29.5MV RRS -0.7745 -141.09 142.86 599.5 9.5BB GPC -0.0004 -0.7497 0.7434 599.5 7BB GPC RRS 0.00436 -4.1305 3.1941 173 7.5VSS -0.5593 -0.0002 0.0005 45 20.5VSS RRS -0.0035 -7e-05 5e-05 54.5 10Pole Plc -0.0006 -0.0283 0.0088 112.5 13Pole RRS 3.0e-06 -le-05 0.0001 61 18SS GPC 1.6e-07 -0.0001 0.0002 46.5 7.5SS GPC RRS -7e-05 -0.3572 0.3867 125 7.586Eastward Ho 5.7°/s ImpulseController Final yawangleMaximum yaw angles Settlingtime (s)Rise Time(s)PID -6e-06 -0.0007 0.0002 53 11.5MV -0.6879 -0.0003 0.0004 52.5 26.5MV RRS -3.5587 -141.76 150.63 588.5 23BB GPC 0.7869 -62.223 61.5 599.5 13BB GPC RRS -0.176 -5.9902 4.4323 215.5 20VSS -0.1325 -0.0003 0.0007 37.5 27.5VSS RRS -0.1321 -0.0002 0.0066 86.5 13.5Pole Plc 0.00013 -0.0162 0.0488 105.5 19Pole RRS 8.5e-06 -4e-05 0.0005 64.5 27SS GPC -0.4564 -53.359 56.018 599.5 13SS GPC RRS -0.0215 -1.4335 1.4286 188 18Eastward Ho Chan e in S eedController Final yawangle_Maximum yaw angles Settlingtime (s)Rise Time(s)PID 0.00002 -0.0072 0.0213 82 18.5MV 0.02614 -0.0002 0.0019 59.5 94MV RRS 0.39059 -61.347 62.129 588.5 31BB GPC -0.0496 -41.354 37.701 599 17.5BB GPC RRS 0.15408 -23.311 24.146 410.5 25VSS 0.32531 -0.0008 0.0003 42 11VSS RRS 0.02617 -9e-05 0.0028 70.5 20Pole Plc 0.00237 -0.0476 0.1434 113 21.5Pole RRS 0.0001 -0.0005 0.0043 71 29.5$S GPC -0.1898 -35.549 34.262 599.5 17.5SS GPC RRS^_ 0.30052 -25.041 25.199 599.5 2587Eastward Ho Chancie in DisplacementController Final yawangleMaximum yaw angles Settlingtime^(s)Rise Time(s)P1D -7e-06 -0.001 0.0005 63.5 12MV -0.7322 -0.0005 0.0012 56 34.5MV RRS -0.1813 -16.977 14.463 599.5 25BB GPC 0.63329 -52.121 53.6 593.5 13.5BB GPC RRS 0.19987 -21.275 10.783 398 22VSS 0.23758 -0.0002 0.0005 40 14VSS RRS 0.23783 -0.0004 5e-05 64.5 12.5Pole Plc 0.00021 -0.0146 0.048 105.5 19.5Pole RRS 0.00001 -0.0001 0.0013 64.5 26.5SS GPC 0.61526 -52.615 52.128 599.5 13.5SS GPC RRS -0.4353 -43.571 42.295 595 19Eastward Ho Change in Speed and DisplacementController Final yawangleMaximum yaw angles-Settlingtime (s)Rise Time(s)PID 0.00075 -0.0133 0.0498 89 20.5MV 0.89277 -0.001 0.0008 45 24.5MV RRS 0.35148 -7.8573 11.842 567.5 31.5BB GPC 0.5361 -37.423 37.03 588.5 19BB GPC RRS 0.02525 -21.816 21.485 597 26.5VSS 0.43468 -20.182 19.51 599.5 12VSS RRS 0.09858 -0.0057 0.0002 75.5 22.5Pole Plc -0.0015 -0.1442 0.2244 121.5 19.5Pole RRS 0.00051 -0.005 0.0293 85.5 29.5SS GPC -0.3188 -34.427 35.615 591 19SS GPC RRS 0.41512 -31.58 29.617 593.5 2788Eastward Ho Rudder Servo SlowsController Final yawangleMaximum yaw angles Settlingtime^(s)Rise Time(s)PID -6e-06 -0.0007 0.0002 53 11.5MV -0.3545 -0.0002 0.0004 58.5 34.5MV RRS 3.77801 -172.25 165.01 599.5 24.5BB GPC -2.0863 -92.61 93.464 599.5 12BB GPC RRS 0.06609 -2.5976 2.3221 325.5 22VSS -0.3546 -0.0001 0.0004 38.5 21VSS RRS 0.09006 -0.0006 3e-05 69 13.5Pole Plc 0.00013 -0.0162 0.0488 105.5 19Pole RRS 0.00001 -8e-05 0.001 65.5 26.5SS GPC 0.50356 -81.621 82.572 594.5 12SS GPC RRS -3.0707 -60.934 62.794 599.5 19Eastward Ho Constant Yaw MomentController Final yawangleMaximum yaw angles Settlingtime (s)Rise Time(s)PID 2.30345 -7e-05 8e-05 56.5 2MV -1.2041 -1.5041 1.3088 599.5 24.5MV RRS 125.524 -113.83 161.42 599.5 351.5BB GPC 0.19339 -0.1952 0.2128 599.5 6BB GPC RRS -2.005 -8.6811 11.364 577 5VSS 0.07789 -0.1144 0.0831 596.5 11.5VSS RRS -0.5758 -0.115 0.1003 596.5 13Pole Plc 0.54117 -0.458 0.6232 599.5 147Pole RRS 0.62503 -6.2533 6.4285 599.5 12SS GPC 0.11393 -0.1941 0.2111 475 7.5SS GPC RRS -7.709 -5.606 8.5922 599.5 56.589Eastward Ho 900 SeasController RMSrudderangleRMS yawangleMaximum yawanglesRMSrollangleMaximum rollangles%PIDPID 0.79 0.741 -1.71 2.34 4.264 -12.5 10.6 0.0%MV 4.42 0.564 -1.69 1.81 3.649 -12.3 12.6 14.4%MV RRS 23 40.66 -105 84.2 4.013 -12.6 12.4 5.9%BB GPC 22.8 19.32 -42.2 39.9 4.632 -13.1 12.7 -8.6%BB GPC RRS 26.8 23.7 -42.9 44.4 5.749 -16.1 18.7 -34.8%VSS 6.73 0.831 -4.09 2.96 3.873 -10.6 11.1 9.2%VSS RRS 5.33 2.043 -6.67 6.42 4.112 -12.1 12.1 3.6%Pole Plc 0.39 0.83 -1.81 2.9 3.748 -9.93 9.94 12.1%Pole RRS 12.8 20.51 -66.6 45.8 4.509 -11.9 11.9 -5.7%SS GPC 20.3 16.29 -37.4 38.3 4.557 -13.1 13.2 -6.9%SS GPC RRS 25.4 20.46 -39.4 37.1 4.272 -13 11 -0.2%Eastward Ho 450 SeasController RMSrudderangleRMS yawangleMaximum yawanglesRMSrollangleMaximum rollangle%PIDPID 1.68 1.639 -3.91 3.88 0.929 -2.66 3.14 0.0%MV 2.17 0.464 -1.26 1.45 0.97 -2.56 3.36 -4.5%MV RRS 2.63 2.821 -9.94 7.5 1.024 -3.37 3.2 -10.2%BB GPC 2.76 0.318 -1.52 0.81 1.177 -2.85 3.78 -26.7%BB GPC RRS 7.43 3.68 -18.6 15.5 2.375 -6.25 5.89 -156%VSS 2.21 0.105 -0.37 0.25 0.859 -2.59 2.69 7.5%VSS RRS 3.12 2.847 -6.39 7.61 0.828 -2.02 2.03 10.9%Pole Plc 2.16 2.341 -6.55 6.08 0.913 -3.46 2.96 1.7%Pole RRS 2.32 1.091 -3.46 3.79 0.923 -2.89 4.93 0.6%SS GPC 2.7 0.395 -1.49 1.4 1.06 -3.47 2.96 -14.1%SS GPC RRS 4.16 0.863 -3.61 3.33 1.812 -5.71 5.65 -95.1%90Eastward Ho 60° SeasController RMSrudderangleRMS yawangleMaximum yawanglesRMSrollangleMaximum rollangles%PIDPID 1.78 1.72 -4.96 4.53 1.959 -4.71 6.34 0.0%MV 2.75 0.487 -1.26 1.35 1.955 -6.03 7.03 0.2%MV RRS 2.47 2.394 -9.44 7.29 1.64 -4.25 4.74 16.3%BB GPC 3.14 0.38 -1.84 1.19 1.91 -6.32 7.02 2.5%BB GPC RRS 5.99 2.367 -8.47 8.27 2.855 -7.8 7.27 -45.8%VSS 2.71 0.118 -0.39 0.32 1.629 -5.05 5.03 16.8%VSS RRS 4.15 3.62 -9.16 9.29 1.267 -4.17 3.7 35.3%Pole Plc 2.15 2.726 -6.34 8.06 1.639 -4.55 5.27 16.3%Pole RRS 4.24 2.633 -8.05 7.95 1.634 -6.94 6.14 16.6%SS GPC 2.9 0.457 -2.06 1.92 1.663 -4.41 4.4 15.1%SS GPC RRS 4.82 1.177 -5.9 5.29 2.537 -6.66 7.68 -29.5%Eastward Ho 300 SeasController RMSrudderangleRMS yawangleMaximum yawanglesRMS rollangleMaximum rollangles%PIDPID 0.84 0.809 -2.08 1.68 1.004 -2.95 4.02 0.0%MV 3.03 1.393 -2.61 5.26 1.159 -3.48 3.13 -15.4%MV RRS 2.08 0.737 -2 2.81 1.004 -3.47 3.19 0.0%BB GPC 24.2 21.6 -42.1 40.5 2.474 -5.69 5.22 -146%BB GPC RRS 7.53 1.998 -6.61 6.04 1.968 -5.34 4.88 -96%VSS 4.66 0.212 -0.8 0.87 1.32 -3.54 4.06 -31.4%VSS RRS 2.85 1.199 -3.3 3.22 0.687 -1.77 1.88 31.6%Pole Plc 0.53 1.101 -3.6 2.02 0.918 -2.68 2.66 8.6%Pole RRS 5.09 1.813 -5.85 4.17 1.348 -3.51 3.72 -34.2%SS GPC 23.6 19.13 -36.6 37.1 2.269 -6.03 5.3 -125%91The following are results with a rudder rate of 15°/s.Kynoc 90° SeasController RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDVSS RR 6.26 1.79 -4.63 5.67 1.9 -5.51 6.29 27.0%Pole RR 9.41 2.42 -12.1 7 2.44 -8.81 9.71 6.3%MV RR 10.1 1.38 -3.64 7.03 4.28 -11.8 12.6 -64.4%SS GPC RR 8.35 2.2 -8.69 8.83 2.99 -11.5 12.6 -14.7%PID 2 1.79 -4.53 5.39 2.6 -9.32 9.18 0.0%Pole AP 5.91 0.56 -2.02 2 4.16 -13.9 15.1 -59.7%Kynoc 45° SeasController RMSrudderangleRMSyawangleMaximum yawanglesRMSrollangleMaximum rollangles%PIDVSS RR 6.46 6.12 -17.8 19.1 3.77 -11.2 10.3 -69.8%Pole RR 3.55 1.39 -4.6 13.6 2.23 -18 15.6 -0.6%MV RR 3.26 2.26 -8.57 6.53 2.56 -7.01 8.27 -15.2%SS GPC RR 2.68 0.85 -3.34 2.63 2.57 -7.81 8.3 -15.8%PID 2.11 1.98 -6.4 5.65 2.22 -7.07 6.53 0.0%Kynoc 60° SeasController RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDVSS RR 6.25 5.61 -19.8 17.7 3.33 -8.98 8.94 -40.6%Pole RR 3.37 0.96 -3.33 5.71 1.84 -12.4 12.2 22.3%MV RR 5.49 1.92 -6.45 6 4.34 -11.2 9.8 -83.6%SS GPC RR 3.08 0.96 -4.32 4.27 3.37 -10.8 9.89 -42.4%PID 2.17 1.96 -5.84 5.55 2.37 -5.88 8.7 0.0%Pole AP 2.27 0.59 -2.79 2.01 2.38 -7.28 7.9 -0.5%92Kynoc 30° SeasController RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDVSS RR 4.19 1.61 -4.08 4.61 1.21 -3.52 3.73 9.5%Pole RR 2.49 0.72 -2.69 2.7 1.3 -4.28 4.34 2.4%MV RR 6.56 0.92 -4.34 3.06 2.1 -7.21 6.76 -57.5%SS GPC RR 5.67 0.98 -4.9 2.99 1.71 -7.47 6.43 -28.5%PID 1.7 1.48 -3.69 4.51 1.33 -4.79 4.07 0.0%Pole AP 3.97 0.47 -2.44 1.98 1.54 -4.87 5.02 -15.5%Eastward Ho 900 SeasController RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDVSS RR 12.6 2.13 -9.07 5.41 4.37 -12 11.8_ 0.7%Pole RR 16.4 11.3 -33 31.2 4.81 -16.6 15.6 -9.5%MV RR 15.9 8.11 -23.6 30.9 3.72 -11.1 11.8 15.4%SS GPC RR 18.4 2.54 -7.94 10.4 4.07 -11.6 11.9 7.4%PID 0.73 0.68 -2.37 1.81 4.39 -12 12.3 0.0%Pole AP 0.46 0.98 -2.52 2.69 3.92 -12.1 11.4 10.7%Eastward Ho 450 Seas,Controller RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDySS RR 4.12 2.25 -6.89 7.74 0.72 -2.47 2.03 11.0%Pole RR 3.31 0.61 -2.41 1.65 0.71 -2.24 2.47 12.2%MV RR 12.8 19.4 -53.2 55.2 2.31 -6.69 6.94 -186.5%SS GPC RR 5.5 0.62 -2.65 3.3 2.12 -10.7 9.68 -162.6%PID 1.59 1.54 -3.58 4.02 0.81 -2.35 2.5 0.0%Pole AP 2.12 2.17 -6.7 4.55 0.83 -2.46 2.74 -3.6%93Eastward Ho 600 SeasController RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDVSS RR 5.46 3.53 -10.3 9.97 1.34 -4.1 3.74 20.1%Pole RR 5.15 0.68 -1.9 3.18 1.23 -3.42 3.78 26.8%MV RR 3.97 3.26 -12.6 8.62 1.83 -5.25 5.77 -9.0%SS GPC RR 5.9 0.84 -4.58 4.57 2.96 -11.6 11.5 -76.3%PID 2.04 1.97 -5.96 7.08 1.68 -5.43 5.67 0.0%Pole AP 2.45 2.75 -9.56 7.76 1.81 -4.22 5.03 -7.8%Eastward Ho 300 SeasController RMSrudderangleRMSyawangleMaximum yawangleRMSrollangleMaximum rollangle%PIDVSS RR 4.65 1.38 -3.85 2.68 0.78 -2.62 2.02 19.8%Pole RR 5.54 0.99 -2.87 3.89 0.98 -5.56 4.4 0.0%MV RR 2.16 1.32 -2.69 2.89 1 -3.08 3.66 -1.8%SS GPC RR 8.65 0.7 -2.96 2.89 2.29 -8.37 8.21 -134.2%PID 0.92 0.88 -2.34 1.97 0.98 -3.06 3.38 0.0%Pole AP 0.68 1.28 -2.97 2.93 0.92 -2.6 3.05 6.3%94


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