Optimization of Wave-Induced Motion of Ramp-Interconnected Craft for Cargo Transfer

Transcription

1 Optimization of Wave-Induced Motion of amp-interconnected Craft for Cargo Tranfer Mirolav Krtic, Jacob Toubi, Joe Doblack, lekander Vekler, Halil Baturk Keyword: Sea veel control, extremum eeking, cargo tranfer at ea, paive control, fin control. btract Cargo tranfer between two veel at ea require the ramp connection between the veel to be a table a poible. The complex nature of the ytem make employing control method difficult. Thi paper preent the reult of everal technique aimed to exploit dynamic of the ytem to reduce the relative movement of the adoining ramp and the two veel. The method include extremum eeking deigned to optimize wave heading angle and ramp length, fin control to alter veel heave to reduce pitch, and paive technique that invetigate the effectivene of pring-damper ytem at the oint of the ramp. We implement thee method on a SimMechanic model created to imulate the veel in ea tate four. We compare the imulation reult from the SimMechanic model to the olution of the equation of motion we derive uing Lagrangian mechanic that how continuity in the reult. The reult of the imulation with control algorithm implemented how the ucceful reduction of the pitch angle in the ramp. We provide the reult of thee imulation a well a a detailed account of how each apect of the ytem wa modeled.. INTODUCTION The tranfer of cargo over a ramp from a LMS (large, medium-peed, roll-on/roll-off) veel to a connector veel in high ea tate repreent ignificant challenge for hip and control ytem deigner. The goal of thi proect i to determine the actuation/ening requirement and to devie control and real-time optimization algorithm for reducing the ocillation of the pitch, roll, and yaw angle between the ramp and each veel. We invetigate a ytem that conit of a Sea Bae (the large veel) and a T-Craft (the connector veel) connected by a ramp that can vary in length. Due to the number of degree of freedom (DOF) involved in thi ytem, deriving the equation of motion i challenging, even when the hydrodynamic force law are known. an initial effort, we create a SimMechanic model to imulate the ytem and implement the control algorithm. The veel were modeled a half cylinder and the ramp a a rectangular olid a hown in Figure. Following thi, we derive the equation of motion for the ytem uing a Lagrangian mechanic approach. However, the SimMechanic model prove to be a impler way to implement control and Figure : Baic Diagram of T-Craft, ramp and Sea Bae ued in SimMechanic optimization technique without re- deriving the equation of motion. The SimMechanic model conit of a erie of rigid bodie acted on by a wave force model that i elaborated on below. The olution to equation of motion provide a cro reference with the reult of the rigid body imulation which validate each other. Three oint cae connecting each hip to the ramp are explored: Pitch only, Pitch-oll Joint, and Pitch-oll-Yaw Joint. Thee variou oint model influence the angle ocillation per DOF being conidered. Extremum eeking i ued to identify the optimal ramp length and ocean wave incidence from variou initial poition to minimize pitch angle amplitude. The goal i to employ thi method in ea tate where the optimal configuration i unknown. Paive control technique are modeled after an automotive hock aborber and are employed at the oint a an effort to reduce the angle amplitude for the variou DOF in the different oint cae. Finally, an active control approach i teted to control rotating fin in order to create variable lift and reduce the relative heave of the veel while connected in the pitch only oint cae. In all oint cae, imulation are conducted with and without control actuation to tet the effectivene of each algorithm. The reult are compared againt each other to determine the effectivene of each technique. full range of reult a well a explanation into each etup are explained in further detail.. MODELING WVES ND SHIP MOTIONS.. Modeling wave force Environmental diturbance for hip modeling are conidered a wave, wind and ocean current. In thi paper, direct effect of the wind and ocean current are neglected, only wave diturbance are conidered. Wave modeling for open ea condition i a very deep topic. Empirical model can be found in literature from Bretchneider, Pieron-Mokowitz, JONSWP and

2 Figure 3: SimMechanic Viualization during imulation Figure : tranfer function and ine wave approach for decribing a wind-generated ocean wave Torethaugen Spectrum [3]. However, for cloed loop ytem and control analyi, linear wave repone model are developed. In thi approach, the empirically determined pectrum i approximated with a linear tranfer function with imilar power pectrum, driven by Gauian noie. Stated mathematically, the wave height y () i y( ) h( ) w( ) (0.) where w () i zero mean Gauian white noie proce with unity power acro the pectrum ( w( ) ) and h () i a tranfer function. The power pectral denity function for y () can be written a y h( ) (0.) The final nd-order linear wave repone tranfer function approximation with a damping term wa introduced by Saelid et.al (983) a K h () (0.3) 0 0 where the gain contant i defined a (0.4) K with being the wave intenity, the damping coefficient and 0 the dominating wave frequency. Hence, the power pectral denity function of the output y () i written a 4( 0 ) y h( ) (0.5) The parameter and can be varied to better fit the Pieron-Mokowitz empirical pectrum. In addition to thi, a impler approach can be ued. Figure how the output when white noie i filtered through the nd order tranfer function and a ine wave of the ame dominating frequency 0 with added tochatic white noie that act a 0 a rough approximation. For the ake of implicity, the wave model decribed by y( t) in( t ) w( t) (0.6) i ued intead of the nd order tranfer function approach. Thi facilitate the method we ue to define the time at which the wave hit each hip baed on modifying the phae in the ine wave. For example, if the wave hit T- Craft firt, there will be phae delay until it reache the Sea Bae. In addition, it i traightforward to ue thi approximation to orient the hip relative to the wave front by modifying the phae... Modeling hip motion due to wave Figure diagram the baic hape and connection of the veel. We apply the following aumption of the linear wave theory to thi ytem: the ea water i incompreible, invicid, there i no urface tenion, the fluid motion i irrotational, and the wave amplitude i ignificantly maller than the wavelength. Linear wave theory allow u to expre the wave model a a uperpoition of ine wave with differing frequencie. The reulting wave will produce an irregular pattern and i conidered a ueable approximation to model wave behavior. The ocean wave affect a emi-immered body by inducing force that act on the body that will influence it motion. The term ued to decribe thee force i retoring force []. Thi retoring force arie from the buoyancy and gravity force, which both depend on the wave. lthough generally a three dimenional body move in ix DOF, we will only model the three relevant DOF that are affected by the hydrotatic force and moment from the wave. The DOF conidered in the model are: roll, pitch and heave. Thee retoring force tend to return a floating body to it initial poition, while no oppoing hydrotatic retoring force exit in the urge, way or yaw direction. In other word, ince thee motion are analogou to a pringdamper ytem, their repective pring contant and damping coefficient are et to zero and therefore not conidered. n approximation to the equation of motion that govern the roll, pitch and heave take the form of an uncoupled econd order dynamical ytem []. The equation for roll motion with linear damping and linear wave i given a 0 3

3 i o b in Wt,, i J / Δ where b i a linear damping coefficient, natural frequency in roll, ggm (0.7) i the hip o i the wave amplitude, W i the wave angular frequency, GM i the metacenteric height in roll, i i the ma radiu of gyration in roll and equal w 3 where w = width of half cylinder, J i the ma moment of inertia, Δ i the ma diplacement from the floating body. The roll equation can be rearranged to o J Jb gδgm gδgm in W t (0.8) The undamped uncoupled pitch equation i in Et, E, TE (0.9) ggm P TE, c v co i where i the hip natural frequency in pitch, i the maximum pitch amplitude, E i the angular frequency of encounter (number of wave een by the hip per unit time), i i the ma radiu of gyration in pitch and equal P L 3 where L = length, v i the hip peed, c i the wave celerity, i the angle between hip peed and wave celerity. The pitch equation can be rearranged to yield: J gδgm P gδgm P in t (0.0) P The uncoupled heave equation i m z b z g z g co t (0.) z w w o E where z i the added ubmerged ma due to the heave, w i the waterplane area of the floating body, which both depend on the wave frequency of ocillation. The two metacentric height and waterplane area ued in the imulation are calculated for half-cylinder model of the two hip. Since no explicit expreion for the damping coefficient were available they are etimated in an ad hoc manner provided they comply with intuitive hip motion a a reult. Summarizing the reulting pring and damping contant we get k gδ GM, k gδ GM P, k g roll pitch heave w b J b, b b, b z 0.0 P E (0.) The above expreion are ued in the SimMechanic model to imulate the ocean wave diturbance by it analogou pring damper ytem. ˆ a co( t) Nonlinear Map f ( x, y) Figure 4: Baic extremum eeking loop k J h in( t) lthough thee equation were derived for a ingle hip, intead of an interconnected hip-ramp-hip ytem, neverthele the reulting pring and damper coefficient were ued a an etimate for the ytem motion. In reality there exit ome coupling between the variou motion, which thee equation do not capture. For example, the combination of roll and pitch motion will induce yaw and heave motion. lo, during the roll motion, the center of buoyancy will move and caue ome pitch motion. It i important to undertand the limitation of thee equation, which fall hort at decribing the full behavior of a floating body. 3. EXTEMUM SEEKING OPTIMIZTION 3.. Background and Implementation Extremum eeking i a powerful tool that i able to optimize parameter of a non-linear ytem around a local minimum or maximum. It ue a inuoidal perturbation to extract gradient information to achieve thi goal. mong it beneficial propertie i that it i not model baed. The motivation behind creating a SimMechanic model become evident when implementing thi optimization technique, which conit imply of tuning an appropriate loop intead of uing explicit model dynamic to find an optimal configuration. Figure 4 how the mot baic one parameter extremum eeking loop. The output i firt filtered through a high pa filter, demodulated, integrated with a gain, and then receive an added perturbation. Thi etimate i fed back into the ytem until the output reache a teady extremum. igorou proof for tability for variou ytem are given in []. Each ytem i unique, and for the purpoe of thi environment pecific apect are appropriately altered. In our model, we ue a modified two parameter extremum eeking loop to tune ramp length,, and hip orientation relative to the wave front, (), hown in Figure 4

4 ˆ ˆ Nonlinear Map f (, ) l l l l l l k k h P () l l in( t) J h co( t) Figure 5: Modified two parameter extremum eeking loop 5. The value of thee variable for thi particular etup are h.0, h 0, l, P.003,.04, k k.09,.0, 30 l Though ES i not model baed, it i important to undertand the effect of each tuned parameter. While ramp length ha an obviou effect on ramp pitch angle, the wave orientation relative to the hip ha a le intuitive impact. The goal i to be able to, in real-time, earch for an optimal configuration by varying each parameter until an aociated cot i minimized. Since the pitch angle decreae nearly monotonically with ramp length, a ramp penalty i impoed to create concavity in the cot map neceary for extremum eeking to function properly while alo avoiding extending the ramp to arbitrarily large value. It i alo important to note the pitch angle between the hip i an ocillatory ignal, therefore we extract the amplitude in order to get a ingle value for calculating cot. We accomplih thi by filtering the pitch angle for each configuration and multiplying the value by a function of the ramp length. Figure 6 how two different filtering reult, one where all of the ocillatory behavior i eliminated entirely for the cot map (top) and the other allowing ome ocillation to be ued in the extremum eeking loop (bottom). We repeat thi proce for varying ramp length of 5 to 0 meter and wave front angle of 0 to 90 degree to produce a three dimenional cot map that can be analyzed for local minima. Thi yield the cot map in Figure 7. The cot map create inight into the impact of each parameter a well a a way to verify the parameter value of the extremum eeking loop. With our configuration, the global minimum occur at a ramp length of ut under.5 meter and a wave front angle of ut under 8.5 degree a een in Figure 8 and 9 repectively. Some important deign criterion included the need to avoid rapid extenion and contraction of the ramp and orientation aligning. Extremum eeking require gradient Figure 6: The final value from the top i tored to form a cot map while the bottom ignal i fed into the ES loop Figure 7: Three dimenional cot map information that i typically provided through an added inuoidal perturbation. However, due to the periodic nature of the cot due to wave motion, the ignal naturally contain a inuoidal preence and i not added eparately in the ES loop a hown in. dditionally, in order to avoid rapid movement in the model, we filter out the high frequency content only leaving the lower component to actuate the ytem. Thee development are only made poible by the dynamic of the ytem and only by uccefully altering the algorithm are we able to ue them to our benefit. The reult i moother actuation that doen t ar the ytem back and forth while till operating in a functional manner. 3.. ES eult and Dicuion The hip were configured in a bow to tern orientation connected by a pitch ointed ramp. Figure 0 how the reult of a ytem beginning at a heading angle of 60 degree and an initial ramp length of 5 meter. The heading output decreae, low down lightly around 40 degree, and then continue until lightly overhooting then ettling back to 8.5 degree. The ramp length gently extend and 5

5 Figure 8: Wave front angle v. magnitude profile Figure 0: amp length and alpha tracking reult Figure 9: amp length v. magnitude profile low down until it ultimately ettle to a length of ut under.5 meter a expected. The pitch ocillation are effectively reduced from an initial amplitude of 5 degree to ut over 5 degree from beginning to end. Figure how the pitch angle evolution with no ES tuning (top) and it reduction while uing the ES algorithm (bottom). The reult how the favorable attribute we attempted to create while working on a low enough time cale to be realitically implementable. We choe the initial condition to allow optimum eeking while avoiding the local minimum in cot along the way. Since large perturbation aren t preent in the output of the ignal, it i more difficult to avoid local minima than when uing tandard ES algorithm. Care mut be given when electing gain and other tuning parameter. In order to avoid thee local minima, we ugget beginning eek from a relatively known poition. While ramp length begin at a fixed 5 meter, the veel hould begin ailing into the oncoming wave before beginning the earch. From there, mall variance in initial poition are acceptable a oppoed to beginning perpendicular to the wave front. Thi allow a gradual decent to the minimum while avoiding the encloed local minima. Figure : Pitch angle evolution for open and cloed loop imulation 4. PSSIVE CONTOL 4.. Implementation The goal of the paive control invetigation i to mimic automotive hock aborber with pring and damper. Thi reduce the angle amplitude of the ramp between the Sea Bae and the T-Craft. We invetigate three oint cae of the cargo tranfer ytem with thee paive control technique. The three type of oint are: pitch only oint (P-Joint), pitch/roll oint (P-Joint), and pitch/roll/yaw oint (PY-Joint). We ue the ame initial condition for each imulation. The ramp length i 5 meter and the wave front angle, (), i 45 degree. We ue a damping coefficient of 00 N/m in all cae. Thi magnitude i fixed to provide minimal impact on angle amplitude that intead will vary with pring rate. In the PY-Joint cae, we introduce pring in the roll and yaw DOF to tabilize the ytem. Otherwie, the two hip end up crahing during the imulation. In the PY- Joint cae, the T-Craft (TC) roll and yaw pring rate are x0 N/m and x0 6 N/m repectively, and the Sea-Bae 6

6 Figure : TC Joint ngle Summary Figure 4: Heave Summary for No borber, TC Joint borber, and SB Joint borber Figure 3: SB Joint ngle Summary (SB) roll and yaw rate are x0 5 N/m and x0 6 N/m repectively. We conider thee pring rate the minimum value to maintain tability and name them the uncontrolled cae. The controlled cae refer increaing pring rate beyond the uncontrolled cae. In each oint cae, we ground the Sea-Bae in the urge and way direction to improve ytem tability. We firt imulate the ytem with no pring/damper aborber on either oint in order to get an etimate of the baeline angle amplitude value during a table limit cycle. Thi i the open loop repone. In thi imulation we meaure the following variable: pitch, roll, yaw angle, and the difference in heave between the two hip. The difference in heave meaurement i calculated uing the abolute value of the ditance between the center of gravity of the T-Craft and Sea-Bae. Due to the hip different inertial and buoyancy propertie their z-direction poition i not necearily going to be equal. The relative ditance between the center of gravity of the two hip directly relate to the pitch angle ocillation. If the difference in heave decreae in magnitude, the pitch angle amplitude will decreae accordingly. The open loop imulation reult for the three cae for the TC and SB oint angle and heave Figure 5: Paive Control Effort Summary are ummarized on the firt row of Figure, Figure 3 and Figure 4 repectively. We then imulate the ytem applying a pring/damper to the TC oint that i analogou to poition and velocity feedback. For the pitch only cae, we only ue one pring/damper pair. We ue two for the pitch/roll cae and three for the pitch/roll/yaw cae, one for each DOF. During the imulation the following meaurement are taken: TC oint angle (pitch, roll, and yaw), difference in heave between the two hip and paive control effort (force from pring/damper). The reult are ummarized in the econd row of Figure, Figure 4 and firt row of Figure 5 repectively. In cae of aborber failure, the control effort data can be ued for the actuator requirement to provide the neceary torque for tability. Finally, we imulate the ytem applying a pring/damper to the SB oint. The reult of the angle, paive control effort and difference in heave and from the controlled SB oint cae are ummarized in the econd row of Figure 3, Figure 5 and third row of Figure 4 repectively. 7

7 Figure 6: Summary of Max amp ngle uing borber 4.. Paive Control eult ummary of all the reult i given in Figure - Figure 6. Figure ummarize the TC oint maximum angle of the open-loop and controlled ytem for the three oint cae. Figure 3 ummarize the SB oint maximum angle of the open-loop and controlled ytem for the three oint cae. Figure 4 ummarize the heave difference for the three cae uing no control, TC oint control, and SB oint control. Figure 5 ummarize the paive control effort for the three cae uing TC oint control and SB oint control. Figure 6 ummarize the ramp angle for all the cae and aborber arrangement conidered. In the bar graph the abcia correpond to the three oint cae conidered, namely pitch only oint, pitch/roll oint, and pitch/roll/yaw oint repectively and the ordinate the correponding relevant value Dicuion of Joint ngle The above reult ummarize the motion at the oint for the variou cae mentioned with and without hock aborber. The open loop imulation how, for each of the cae in both the TC and SB oint, that the pitch angle ocillated between about ± 0 degree. In the P-Joint cae, the TC and SB oint experienced a roll angle ocillation of 0.34 and 0.89 degree repectively. ny difference in TC and SB angle ocillation may be due to their different dimenion, inertial propertie and the Sea-Bae being grounded in the urge and way direction while the T-Craft wa not. In the PY-Joint cae, the TC and SB roll angle were.93 and.76 degree repectively, and their repective yaw angle were.4 and.09 degree. The yaw angle ocillation in both oint are ignificantly more enitive. Thee DOF have a higher tendency to become untable, and therefore require larger pring rate in the oint for tabilization. Increaing the pring rate in the yaw DOF decreae it influence on the behavior of the ytem, hence cloely mirroring the P-Joint cae, which ha a deired table ytem repone. Furthermore, increaing the pring rate of the roll DOF make the ytem repone cloely mirror the P-Joint cae, which alo ha a table repone. The heave ummary plot in Figure 4 how the improved performance between the open and cloed loop imulation. With no aborber the average delta heave between the T-Craft and Sea-Bae were 3.89 m, 4.0 m, and 4.00 m for the P-Joint, P-Joint, and PY-Joint cae repectively; for an aborber on the TC oint the delta heave were 3.03 m, 3.07 m, 3.07 m for each cae repectively; for an aborber on the SB oint the delta heave were 3.0 m, 3.07 m, 3.07 m for each cae repectively. The important point to ee from thee reult i the decreae of about m in delta heave between the uncontrolled and controlled imulation, which primarily affect the pitch angle, reducing it a few degree. The reult of the controlled imulation how an improvement in the amplitude of angle ocillation for each of the DOF a hown in Figure and Figure 3. The amount of reduction depend on the pring contant choen. n arbitrary contant wa choen in the effective order of magnitude range for the imulation to how a reduction in angle amplitude. For all three controlled cae the pitch angle wa reduced to about ±0.0 degree for both the TC and SB oint with a gain value of K = -5x0 8. In the controlled P-Joint cae the roll angle ocillation reduced to about ±0. and ± 0.59 degree for the TC and SB oint repectively with a gain value of K = -x0 3. The maller ocillation wa ±0.06 and ± 0.0 degree for the TC and SB oint repectively with a gain value of K = -x0 6, and the yaw angle ocillation wa ±0.0 and ± 0. degree for the TC and SB oint repectively with a gain value of K = - x0 8. Comparing the P-Joint to the PY-Joint we ee that the addition of the yaw DOF ignificantly influence the gain tuning by a few order of magnitude to reach a imilar teady tate value for the roll angle in the P-Joint cae. aumed the complexity and unpredictability of the repone grow with the amount of DOF Dicuion of amp ngle nother relevant et of value to look at i the ramp angle ince the above oint data mie ome crucial information. For example, in each of the oint cae, the only meaured angle i the allowed DOF(). In the pitch oint cae, only the pitch angle i meaured while the roll and yaw angle amplitude of the ramp are unmeaured. If the ramp roll DOF i untable, it can be conidered unobervable in the above oint meaurement and would not be known if we only rely on the oint angle meaurement. Meauring the ramp angle directly eliminate thi problem. Figure 6 ummarize the ramp angle for each of the three cae and for four variou cenario, namely: no aborber ued in oint, an aborber on the T-Craft oint, an aborber on the Sea Bae oint, and an aborber on both oint. Matching our intuition, the ramp 8

8 Figure 9: Viualization taken during fin imulation Figure 7: Illutration howing and z Figure 8: Illutration howing x angle are motly maximal when no aborber are ued (with two exception a een in figure), and motly minimal when both oint have aborber. The addition of each DOF create unintuitive reaction in angle amplitude. For example, when no aborber i ued the roll angle i minimum in the P-Joint cae and greater in the P- and PY-Joint cae for no apparent reaon. future endeavor i to tune the damper uing ES in order to reduce the angle amplitude even further. 5. FIN CONTOL 5.. Control Plant Model In thi ection the ability of actuating the maller veel with fin in the pitch DOF to reduce the roll of the ramp i conidered. implified verion of the ytem i imulated, and an etimate for the neceary ize of the fin i provided. Preliminary analyi ha hown that forcing even the T-Craft in heave will require a prohibitive amount of force; therefore the control tak i limited to actuating the maller hip in pitch. To provide the fin ample pace, only the bow-to-tern configuration i conidered. In order to maximize the torque per area unit of the fin, four fin were propoed on each ide of the hip, with one pair cloe to the bow and another pair cloe to the tern. In hydrodynamic application, it i uual to eparate between the Proce Plant Model (PPM) and the Control Plant Model (CPM). The firt one, being more complex, intend to incorporate everything known about the plant dynamic and i ued to make high-fidelity imulation. Converely, a CPM need to be a imple model, only decribing the mot important dynamic of the ytem for ue in the controller. Thu, before we deign the controller, we implify the dynamic. Neither of the hip movement in roll, way and yaw affect the roll of the ramp. For thi reaon, only pitch, heave and urge are modeled in thi ection, reducing the model to two dimenion. Secondly, we linearize the dynamic. The reulting model i expreed uing Hamiltonian Mechanic. The tate variable x, z and for the Seabae are illutrated on Figure 7 and 8. Variable x, z and for the T-craft are defined imilarly. Defining tate vector x x z x z (0.3) Thi vector ha ix dimenion, while the ytem ha only five degree of freedom. We elect x to be expreed with other variable and define vector q x z z (0.4) Illutrated in Figure 7 and 8 along with d T q x z z dt (0.5) The kinetic energy can be expreed a T T 5x5 T( ) M, M M 0 (0.6) 5 where the conugate momenta p T / v i p M, yielding T T( p) p M p (0.7) uming linear box-haped veel (ame a [4] ection 3..3), retoring force potential can be expreed a 3 V ( q) LW wgz wgw L 6 (0.8) 3 LW wgz wgw L 6 with the Hamiltonian being H p, q T p V q and equation of motion H ( p, q) p Q q q H ( p, q) p T, for each DOF. The generalized force Q i defined x T (0.9) i Q Fi (0.0) i q 9

9 Figure 0: Ship movement in wave with fin inactive Figure : The torque on the fin a function of time coordinate ytem defined by(0.4). Numerical value for the imulation are ued from nondimenional model from the GNC toolbox, uing the length of the Sea Bae and T-Craft (00m and 40m repectively) to calculate the dimenionalized value in particular the Mariner cla veel. nap hot of the imulation i hown in Figure 9. Figure : Ship movement in wave with fin active where x i i a coordinate vector with i element and i ued to include the force reulting from damping a well a control input, which are not derivable from a potential. Hydrodynamic force i conidered to be dependent on velocity and acceleration of the veel [4]. Thee force are linearized a follow: FH FH FH(, ) M D (0.) M with M typically referred to a added ma that i generated by the inertia of diplaced water caued by the body moving through the liquid. gain referring to [4], M, unlike the ma tenor in (0.6), i generally not ymmetric but the aociated kinetic energy i T T TM ( M ) M (0.) 4 T which i non-negative. The term M M can be added to the regular rigid body ma tenor in(0.6), thu utifying the terminology. Damping D i calculated a a generalized force in D 5.. Control Input and Controller The control in pitch i performed by configuring the fin in front and in the back of the T-craft to generate force in oppoite direction. far a the controller i concerned, the required torque wa the only input needed. We alo calculate the ize of the fin needed to generate the required torque and dicu it further on. The controller itelf i an infinite horizon continuou time LQ. The mot important cot function to be minimized i ( z z), ince z z i a diffeomorphim on the roll angle of the ramp. Thi require a tranformation of the tate pace equation o a to include z z in the tate pace. Both the tranformation and implementation of LQ are done uing Matlab and are not dicued in further detail Fin Control eult Behavior of the two-hip ytem without any actuation i illutrated on Figure 0. The line indicate the difference z z ). The maximal between height of attachment point ( difference i approximately. meter. Since the wave are modeled a noie-driven linear filter, it take ome time before the wave model i fully excited. Figure how the fin actuated ytem and the ubequent reduction in ramp roll. Figure illutrate the actuation torque required to achieve the reduction. The maximal torque requirement i 60 MN m. Since the T-craft i 40 meter long, the fin can maximally be placed 0 meter away from the center of 0

10 gravity. Thu, the total force the fin provide along the length of the T-craft i 3MN. The required fin area at a given peed i calculated uing tandard formulae from either aerodynamic or hydrodynamic literature. The relationhip between kinematic vicoity of water and air mean that water flow around a fin at 0 m/ i imilarr to air flow around a wing at about 00 m/, the exact number being temperaturedependent. Thi mean that aerodynamic literature uch a [6] can be applied directly. The non-dimenional lift coefficient i defined a L C L 0.5v (0.3) where L i the lift force, i the denity of the working fluid, v i the peed and i the area of the fin. From Table 7.0 in [6] it may be een that the lift coefficient i about. in for a foil with apect ratio of ix. Subtituting kg / m and v 0 m /, yield 58m. Thi area can be provided with four fin of.6m 9.3m each, a feaible ize. lthough the etimation work had to be rough due the ytem being in conceptual tage, the reult achieved ugget that a fin-actuated T-craft can be built yielding ignificant boot in overall ytem performance. utilize port to tarboard tranfer of water out of ync with the roll of the hip. modification of thi technique for pitch tabilization during the cargo tranfer between the two hip will alo be conidered in future work. Finally, thee method are being teted on a time domain ea keeping program that more accurately model the dynamic of the ytem. Thi will help to verify the reult in a more rigorou imulation environment. 7. EFEENCES [] K. B. riyur, M Krtic. eal-time Optimization by Extremum-Seeking Control. John Wiley & Son, New York, 003 []. Biran. Ship Hydrotatic and Stability. Oxford, Butterworth-Heinemann, M, 003 [3] T. I. Foen. Guidance and Control of Ocean Vehicle. John Wiley & Son, New York, 994 [4] T. I. Foen. Marine Control Sytem, Tapir Trykkeri, Trondheim, Norway, 00 [5] John J. Bertin, erodynamic for Engineer, Prentice Hall, 00 [6] Odd M. Faltinen, Hydrodynamic of High-Speed Marine Vehicle. Cambridge Univerity Pre, CONCLUSIONS ND FUTUE WOK In thi paper, pitch, roll and heave motion of two hip with a ramp connection were modeled a a erie of rigid bodie acted on by a wave force model analogou to a pring-damper ytem. Contant for the pring and damper were derived and the model wa implemented in SimMechanic for imulation. Firt, we optimized the hip orientation relative to the wave heading angle and ramp length by extremum eeking. The optimal value for heading angle and ramp length were correctly identified a 8.5 degree and.5 meter, repectively. Secondly, we invetigated paive control technique to tabilize the ramp making cargo tranfer afer and eaier. Three different oint cae were imulated and the reult how that putting an aborber on both oint provide the mot reduction in angle amplitude. Finally, an active control approach wa invetigated uing fin control to reduce the relative movement of the connection point between the ramp and each of the hip. Thi method howed a great reduction in pitch amplitude of the T-craft, which greatly reduced the pitch amplitude of the ramp. We did not preent a ytematic way to chooe the aborber contant for paive control technique. In future reearch we will optimize the pring contant and damper coefficient to achieve the deired angle amplitude reduction. In addition, roll tabilization technique uch a an anti-roll tank, which i baed on a Frahm vibration aborber, will be incorporated into thi reearch. Thi idea