- Email: [email protected]
Roll Stabilization Control of Sailboats
10th IFAC Conference on Control Applications in Marine Systems 10th IFAC Conference Control Applications in Marine Systems September 13-16, 2016.on Trondheim, Norway 10th IFAC on Control in 10th IFAC Conference Conference Control Applications Applications in Marine Marine Systems Systems September 13-16, 2016.on Trondheim, Norway Available online at www.sciencedirect.com September 13-16, 2016. Trondheim, Norway September 13-16, 2016. Trondheim, Norway
ScienceDirect Roll Roll Roll Roll
IFAC-PapersOnLine 49-23 (2016) 552–556
Stabilization Stabilization Stabilization Stabilization ∗
Control Control Control Control ∗,∗∗,1
of of of of
Sailboats Sailboats Sailboats Sailboats ∗∗
∗∗ Kristian L. Wille ∗∗ Vahid Hassani ∗,∗∗,1 ∗,∗∗,1 Florian Sprenger ∗∗ Kristian L. Wille ∗ Vahid Hassani ∗,∗∗,1 Florian Sprenger ∗∗ Kristian L. Wille ∗ Vahid Hassani ∗,∗∗,1 Florian Sprenger ∗∗ Kristian L. Wille Vahid Hassani Florian Sprenger ∗ ∗ ∗ Department of Marine Technology, Norwegian university of science Department of Marine Technology, Norwegian university of science ∗ of Marine Technology, Norwegian university of science ∗ Department and technology (NTNU), Trondheim, Norway Department of Marine Technology, Norwegian university of science and technology (NTNU), Trondheim, Norway ∗∗ ∗∗ and technology (NTNU), Trondheim, Norway Norwegian Marine Technology Research Institute ∗∗ The and technology (NTNU), Trondheim, Norway ∗∗ The Norwegian Marine Technology Research Institute Marine Technology ∗∗ The Norwegian (MARINTEK), Norway Institute The Norwegian Marine Trondheim, Technology Research Research (MARINTEK), Trondheim, Norway Institute (MARINTEK), Trondheim, (MARINTEK), Trondheim, Norway Norway Abstract: This paper reports a way of dynamically controlling the sail of a sailboat to reduce Abstract: This paper reports a way of dynamically controlling the sail of a sailboat to reduce Abstract: This paper reports a way dynamically controlling the sail of to the heel angle and roll motion the wind. This is mainly robustness and Abstract: This paper reports acaused way of ofby dynamically controlling the to sailincrease of aa sailboat sailboat to reduce reduce the heel angle and roll motion caused by the wind. This is mainly to increase robustness and the heel heel angle and roll rollsailboats motion caused caused by also the be wind. This is mainly mainly to increase increase robustness and safety for autonomous but could used to increase comfort for crew. The solution the angle and motion by the wind. This is to robustness and safety for autonomous sailboats but could also be used to increase comfort for crew. The solution safety for sailboats but could also be used to increase comfort for crew. consists ofautonomous a linear quadratic regulator (LQR) controlling the moment created byThe thesolution sail. A safety for autonomous sailboats but could also be used to increase comfort for crew. The solution consists of a linear quadratic regulator (LQR) controlling the moment created by the sail. A consists of quadratic (LQR) the moment created by sail. lookup table will then choose regulator the optimal anglecontrolling of the sail, maximum consists of a a linear linear quadratic regulator (LQR) controlling theoptimized moment for created by the theforward sail. A A lookup table will then choose the optimal angle of the sail, optimized for maximum forward lookup table then choose the optimal angle of sail, for forward acceleration, given the relative wind direction desired moment. Simulation results are lookup table will will then choose the optimal angle and of the the sail, optimized optimized for maximum maximum forward acceleration, given the relative wind direction and desired moment. Simulation results are acceleration, given the relative direction and presented to show of the approach. acceleration, giventhe theeffectiveness relative wind wind direction and desired desired moment. moment. Simulation Simulation results results are are presented to show the effectiveness of the approach. presented to show the effectiveness of the approach. presented to show the effectiveness of the approach. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Roll Roll Control, Control, Modeling, Modeling, Sailboat. Sailboat. Keywords: Keywords: Roll Roll Control, Control, Modeling, Modeling, Sailboat. Sailboat. 1. INTRODUCTION where S x is the force created by the sail in positive surge x 1. INTRODUCTION INTRODUCTION where S the force created by the sail in positive surge x is 1. where S is force created by the in surge direction (29)) and S the divided by wind x x rr is 1. INTRODUCTION is the the force created by force the sail sail in positive positive surge where Sxx (see direction (see (29)) and S the force divided by the the wind xr is direction (see (29)) and S is the force divided the wind x r issquared. is a by speed (relative to the boat) S function of x This paper is a continuation of Wille et al. (2016). The x direction (see (29)) and S the force divided by the wind r xr squared. Sx r is a function speed (relative to the boat) of This paper is a continuation of Wille et al. (2016). The r speed (relative to the boat) squared. S is a function of This paper is a continuation of Wille et al. (2016). The λ and β , and there will be an optimal λ given a β . model and the course controller presented in Wille et al. x r is a function ws ws ws ws speed (relative to the boat) squared. S This paper is a continuation of Wille et al. (2016). The xrλ given a βws . of λ and β , and there will be an optimal model and the course controller presented in Wille et al. ws λ and β , and there will be an optimal λ given a β . model and the course controller presented in Wille et al. ws ws (2016) is the foundation of the theory developed in this λ and βwsλ , and therebewill be anfreely; optimal λ given a βwsto. a model thefoundation course controller presented in Willeinetthis al. However, (2016) and is the the of the the theory theory developed cannot chosen it is restricted (2016) is foundation developed in this λ cannot be chosen freely; it is restricted to a paper. It is recommended the other paper (2016) is the foundation of of to theread theory developed in first this However, However, λ cannot be chosen freely; it is restricted paper. It is recommended to read the other paper first certain range depending on the wind direction and λto sata sat However, λ cannot be chosen freely; itdirection is restricted to a. paper. It is recommended to read the other paper first certain range depending on the wind and λ as theory and equations from the previous paper will be sat .. paper. It is recommended to read the other paper first certain range depending on the wind direction and λ as theory and equations from the previous paper will be sat This is because we only have control of a rope limiting |λ|, certain range depending on the windof direction and λsat . as theory equations from previous paper will referenced directly (Equations - 59) and notation is because we only have control a rope limiting |λ|, as theory and and equations from 11the the previous paper reused. will be be This This is we only control of limiting |λ|, referenced directly (Equations - 59) 59) and notation notation reused. and the rope only through tension. When the wind This is because because we works only have have control of aa rope rope limiting |λ|, referenced directly (Equations 1 and reused. and the rope only works through tension. When the wind referenced directlyarea (Equations 1 -provides 59) andthe notation reused. and the only works tension. When the is hitting from starboard, it follows that λ > 0 and that The large surface of the sail sailboat with the rope rope only works through through tension. When the wind wind is hitting from starboard, it follows that λ > 0 and that The large large surface surface area area of of the the sail sail provides provides the the sailboat sailboat with with and is hitting from starboard, it follows that λ > 0 and The the torque created by the sail around the mast has tothat be forward thrust. However, it also makes the boat vulnerable is hitting from starboard, it follows that λ > 0 and that The large surface area of the sail provides the sailboat with the torque created by the sail around the mast has to be forward thrust. However, it also makes the boat vulnerable the torque created by the sail around the mast has to be forward thrust. However, it also makes the boat vulnerable positive (meaning a positive angle of attack). When the against strong winds. This can cause large heeling angles, the torque created abypositive the sailangle around the mast When has tothe be forward thrust.winds. However, it can also cause makeslarge the boat vulnerable positive (meaning of attack). against strong This heeling angles, positive (meaning a positive angle of attack). When the against strong winds. This can cause large heeling angles, wind is hitting from port side, λ > 0 and the torque has aagainst lot of roll motion due to wind gusts, and in worst case positive (meaning a positive angle of attack). When has the strong winds.due This can cause large heeling angles, wind is hitting from port side, λ > 0 and the torque a lot of roll motion to wind gusts, and in worst case is hitting from side, λ > acause lot roll motion due wind and case to be negative the boat to capsize. Autonomous will never wind is hitting (negative from port portangle side,of λ attack). > 00 and and the the torque torque has has a lot of of motion due to to wind gusts, gusts,Sailboats and in in worst worst case wind to be negative (negative angle of attack). cause theroll boat to capsize. capsize. Autonomous Sailboats will never never to be negative (negative angle of attack). cause the boat to Autonomous Sailboats will be a viable option if they cannot handle a variety of to be negative (negative angle of attack). cause the boat to capsize. Autonomous Sailboats will never be aa viable viable option option if if they they cannot cannot handle handle aa variety variety of of The following equations describe the upper and lower be following equations describe the upper and lower different weather conditions. of aa asystem is an be a viable option if they Robustness cannot handle variety of The The following equations lower different weather conditions. Robustness of system is an limits of a stable angle ofdescribe the sail,the λ ∈upper [λll , λuuand ], given a The following equations describe the upper and lower different weather conditions. Robustness of a system is an limits of a stable angle of the sail, λ ∈ [λ , λ ], given a important aspect of autonomy. l u different weather conditions. Robustness of a system is an relative limits of a stable angle of the sail, λ ∈ [λ , λ ], given important aspect of autonomy. wind direction: l u limits ofwind a stable angle of the sail, λ ∈ [λl , λu ], given a a important aspect of autonomy. relative direction: important aspect of autonomy. relative The solution presented in this paper controls the roll relative wind wind direction: direction: The solution presented in this paper controls the roll The solution presented in this paper controls the roll motion by utilizing the sail. will reduce the heeling The solution presented in It theangle roll motion by utilizing utilizing the sail. sail. Itthis will paper reduce controls the heeling heeling angle motion It will reduce the angle and theby roll motionthe created by the wind. The controller motion by utilizing the sail. It will reduce the heeling angle if βws ws + π ws < 0 and the roll motion created by the wind. The controller β + π if β 0 βws and the roll motion created by the wind. The controller ws < design is simple but effective, and the states used in the λ = (61) u u = βws + π if β < and the roll motion created by the wind. The controller ws ws > λ (61) design is is simple simple but but effective, effective, and and the the states states used used in in the the u = 0 if < 000 βws + π β ws ws λ (61) design u 0 if β > feedback loop are easy to estimate. ws λu = 0 (61) design is loop simple but effective, and the states used in the if β > 0 feedback are easy to estimate. 0 if βws feedback ws > 0 feedback loop loop are are easy easy to to estimate. estimate. 2. SAIL CONTROLLER 2. SAIL SAIL CONTROLLER 2. 00 if βws 0 ws < 2. SAIL CONTROLLER CONTROLLER if ws < λ (62) l = 0 l 0 ·· 00 ws − π if β β ws < λ = (62) 2.1 Optimal angle of the sail for forward thrust β if > l < β ws > 0 ws − π λ ·· (62) 2.1 Optimal Optimal angle angle of of the the sail sail for for forward forward thrust thrust if β 0 ws ws λll = = β (62) 2.1 β − π if β > 0 ws ws 2.1 Optimal angle of the sail for forward thrust βws − π if βws > 0 The sail is used to create forward propulsion for the In addition to (61) and (62), λ The sail sail is is used used to to create create forward forward propulsion propulsion for for the the In addition to (61) and (62), λsat sat still applies. That is, The sat still sailboat. an optimal sail angle that gives The sail There is usedis to create forward propulsion for the In addition to (61) and (62), λ still applies. applies. That That is, is, λ ∈ [0, λ ] and λ ∈ [−λ , 0]. sailboat. There is an optimal sail angle that gives the sat to (61)ll and (62), sat λsat sat sat Inuuu addition sat still applies. That is, λ ∈ [0, λ ] and λ ∈ [−λ , 0]. sailboat. There is an optimal sail angle that gives the sat l sat highest forward acceleration for a given relative wind sailboat. There isacceleration an optimalfor saila angle that giveswind the λ ∈ [0, λ ] and λ ∈ [−λ , 0]. u sat l sat highest forward given relative λu ∈ [0, λsat ] and λl ∈of[−λ , 0]. can then be found by highest forward acceleration for aa given relative wind The optimal angle thesatsail direction. The objective is thus create aa map from highest acceleration for to given optimal angle of the sail can then be found by direction.forward The objective objective is thus thus to create relative map wind from The The optimal angle of the sail can be found by direction. The is to create a map from traversal of all viable λ, trying to maximize a given relative wind direction to the optimal angle of the sail. We x rr for x The optimal angle of the sail can then then S be found by direction. Thedirection objective is thus to angle createofathe map from for traversal of all viable λ, trying to maximize S a given relative wind to the optimal sail. We x r for a given traversal of all viable λ, trying to maximize S relative wind direction to the optimal angle of the sail. We x β . This can be computed off-line, and the results stored ws begin by defining a way of measuring the force created by r for a given ws . Thisof traversal all viable λ, trying to maximize S relative wind direction toofthe optimal angle of the sail. We x β can be computed off-line, and the results stored begin by defining a way measuring the force created by r ws . This can be computed off-line, and the results β stored begin by defining aa way of the created by wsa. lookup in table. In this paper, an optimal λ was found the sail the forward independent the wind This can be computed off-line, and the results stored begin byin waydirection of measuring measuring the force forceof by β wsa lookup in table. In this paper, an optimal λ was found the sail sail indefining the forward forward direction independent ofcreated the wind wind in a lookup table. In this paper, an optimal λ was found the in the direction independent of the for a step size of 1deg in β . In order to avoid unnecessary speed: ws ws in a lookup table. In this paper, an optimal λ was found the sail in the forward direction independent of the wind for a step size of 1deg in β . In order to avoid unnecessary speed: ws for a step size of 1deg in β . In order to avoid unnecessary speed: discontinuities in the control action, when there is only a ws for a step size of 1deg in β . In order to avoid unnecessary speed: ws discontinuities in the control action, when there is only a discontinuities in the control action, when there is only small difference in the optimal λs between a step change discontinuities ininthe control action, when there is change only a a S V x ws ws small difference the optimal λs between aa step x (β ws ,, λ, ws ) S (β λ, V ) small difference in the optimal λs between step change (60) S x rr (βws ws , λ) = x (βws ,2λ, Vws ) , in the lookup table, new optimal λ is found based on x small difference in theaa optimal λs between a step change x (βws (βws , λ) = S 2λ, Vws ) , (60) S in the lookup table, new optimal λ is found based on V x S , ws , r ws 2 , λ) = x ws (60) S the table, aa new is based linearized solution the twoλ points (thison is Vws (βws , (60) ain Sxx rr (β 2 in the lookup lookup table, between new optimal optimal λdata is found found based on ws ws , λ) = aa linearized solution between the two data points (this is V 2 1 ws 1 Corresponding author, (e-mail: [email protected]). V linearized solution between the two data points (this ws computed on-line). 1 Corresponding author, (e-mail: [email protected]). acomputed linearizedon-line). solution between the two data points (this is is 1 Corresponding author, (e-mail: [email protected]). computed 1 Corresponding author, (e-mail: [email protected]). computed on-line). on-line).
Copyright © 2016, 2016 IFAC 552Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 552 Copyright © 2016 IFAC 552 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 552Control. 10.1016/j.ifacol.2016.10.493
2016 IFAC CAMS Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
553
2.2 Relative moment of the sail In the same manner that we defined the relative force in surge direction, the relative moment in roll is defined as Sφr (βws , λ) =
Sφ (βws , λ, Vws ) , 2 Vws
(63)
where Sφr is calculated at each optimal λ and is added to the lookup table. This solution of λ will hereafter, be referred to as the unconstrained solution, λ100 . Before applying the control law to reduce roll motion and heel angle, we have to be able to control the amount of moment created in roll by the sail. This is done by finding new optimal λs, though with a restriction on the maximum relative moment created at each solution. To produce the desired result, the same traversal algorithm as discussed earlier is performed again, but with the restriction that the solution has to produce a relative moment of 67% or less compared to that of λ100 . This solution will hereafter be referred to as λ67 . Both λ67 and its relative moment is added to the lookup table. This is then repeated for a 33%- and a 0% (or as low as possible) constrained solution, referred to as λ33 and λ0 . However, a new constraint is added, which is that sign(λ100 − λ67 ) = sign(λ67 − λ33 ) = sign(λ33 − λ0 ). In more practical terms, this is to ensure that the different solutions all move the sail in the same direction compared to the previous solution. This is important when we try to find a continuous solution of the relative moment in the next step.
Fig. 2. Relative moment in roll caused by optimal sail position 3. MOMENT ANALYSIS WHEN BEAM REACHING Beam reaching describes the maneuver of sailing perpendicular to the wind. Figure 3 shows the result of the moment in roll when beam reaching at Um10 = 5 m s . It is easy to see that the sail and the restoring moment are dominating. Together they generate roughly 91% of the moment in roll. The keel produces another 8%. From (9) and (10) one can see that the two Coriolis terms provides no moment in roll, and it follows that the remaining 1% is caused by the rudder and the drag. 4. ROLL CONTROLLER The roll controller will be split into two mirrored controllers which will step in when the heeling angle gets too large. For now, we will look into the controller which will govern for roll (φ) bigger than zero. The heeling angle
Fig. 1. Optimal position of the sail We then have all that we need to reverse the process. By entering βwr one finds the solutions of λs that corresponds to that relative wind direction. Then, by applying the restriction of the desired Sφr , the ideal λ is found based on a linearization (computed on-line) between the different Sφr created by λ100 , λ67 , λ33 and λ0 . 553
Fig. 3. Moments while beam reaching
2016 IFAC CAMS 554 Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
should not be larger than 10 degrees for a comfortable ride, and the desired heel angle, φd , is thus set to 10 degrees. A linear quadratic regulator (LQR) will be used to control the roll motion. The first step is thus to make a linearized system of the roll motion about an equilibrium point, which is selected to be the desired heel angle. The linearized system can be written as φ x= (64) x˙ = Ax + Bu· (65) p Figure 3 and the moment analysis showed which of the moments that are important. The two biggest contributions are from the sail and the restoring moment. The keel also provides some moment, though estimating the moment created by the keel is fairly difficult as it requires measurements of the speed and side slip, see (32) and (33). However, it was argued in Wille et al. (2016) that the undesirable side force FU ((56) and (57)) is equal to the lifting force of the keel. Replacing KL with FU in (33) shows that when βck ≈ π then the moment created by the keel in roll is equal to −FU h2 . βck ≈ π is a good approximation when the drift angle is low and there is little current. The restoring moment (20) when linearized becomes Gφ = ρw g∇GMt ((1 − 2sin(φw )2 )φ + C1 ,
(66)
where φw is the equilibrium point and C1 is equal to (67) C1 = ρw g∇GMt cos(φw )sin(φw )· Making C1 and the moment generated by the keel a part of our control input u linearizes the system, and the linearized system model do then only includes the restoring moment. Matrices A and B in our linearized system then becomes
0 A = ρw g∇GMt (1 − 2sin(φw )2 ) − Ixx − Kp˙ 0 , 1 B= Ixx − Kp˙
1 0
(68)
quadratic cost caused by an error in x, and R weighs the quadratic cost of using the actuator. R was tuned to be R = 1e-10 and Q was tuned to 0.75 0 Q= · (71) 0 1 The LQR gain was then calculated in matlab by using the function lqr(A,B,Q,R,0). The result was K = [9.0e4 1.0e5]. Ki was manually tuned to be equal to 2500. The controller is very similar when the heeling angle is negative. The only difference is that φw and φd is changed to −10deg, and C1 is recalculated for the new equilibrium point (changes sign). In practice, only one controller is used, and the signs in front of φw , φd and C1 switches depending on the sign of the heel angle. The controller should only be turned on when the heel angle is above 10 degrees. Hysteresis has also been implemented to avoid the controller turning on and off too fast, causing discontinues in the control action. It is important to note that the controller would never actually be able to create more moment than the optimal unconstrained solution (λ100 ) gives, and it follows that the roll controller would not be able to increase the heeling angle further than what was previously possible. It follows that steps has to be taken to avoid integral build up if the wind speed is to low and the sail is not able to provide enough moment to keep the heeling angle at 10 degrees. To avoid integral build up the integral action is not allowed to further grow when λd = λ100 , i.e. when no more moment can be provided. It might seem counter-intuitive to keep the controller on if the heeling angle is lower than 10 degrees. Though this might be because the integral action has not yet reached a steady value, and the sail is then restricting the moment too much. Restricting the allowed moment in roll comes at a cost of reducing the force in forward direction, see Figure 4 and will thus in general decrease the speed of the sailboat.
(69)
where u is defined as
t φ − φd u = −K − Ki φ − φd dτ + C1 + FU h2 · (70) p 0 Integral control action is added to remove errors caused by the linearization of the system (and of linearization used when calculating optimal λ for a given βwr and Sφr ), and other system uncertainties. As a note, the moment generated by the keel is fairly steady while keeping a set course (Figure 3), and the error by not including it in our model could be handled by the use of the integral control action if a simpler controller design is desired, or if FU is difficult to compute. The LQR controller gain, K, were tuned by setting the matrices Q and R, where Q is the weighting matrix in the 554
Fig. 4. Relative force in surge
2016 IFAC CAMS Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
5. SIMULATION AND RESULTS Three sets of simulations have been studied in what follows. In the first simulation, the controller in roll has been turned off, meaning λ100 is always chosen as the desired angle of the sail. In the next simulation, the controller is turned on, but only the average wind speed is known. In the last simulation, the controller is on, and perfect wind estimation is used. The results will show the last 50 seconds of a 300 second simulation, ensuring that the top speed in surge has been reached and that the integral term in the controller has had time to reach a steady value. The boat is set to go on a course direction perpendicular to a mean wind of Um10 = 10 m s (beam reaching). Notice that βws = 90deg, see equation (26) and (27). Figure 5 demonstrates the effectiveness of the controller keeping the heel angle low. Furthermore, not only does
Fig. 5. Time evolution of φ, comparison of all simulations
Fig. 6. Time evolution of p, comparison of first and second simulation 555
555
the controller reduce the maximum heeling angle, it also heavily reduces the amplitude of the roll motion. In Figure 6 one can see that the rotational velocity in roll is reduced considerably, and Figure 7 shows that having perfect wind estimation improves the results even further. When perfect wind estimation is used we have an almost perfect result, and this justifies the simplifications made to the linearized version of the system as it do not effect the performance of the controller. The remaining error is likely caused by the delay in the sail actuator. Table 1. Results of roll controller test amplitude of roll motion [deg] ] maximum rotational velocity p [ deg s average speed in surge, u [ m ] s
sim 1 4.38 5.15 7.75
sim 2 1.18 1.42 6.59
sim 3 0.29 0.47 6.59
The results of the test is given in Table 1. The amplitude of the roll motion is reduced by 73% and the maximum rotational velocity by 72% when the average wind speed is known. However, reducing the heeling angle comes at a cost in speed. Restricting the moment created in roll comes at a cost in forward thrust from the sail, which directly effects the maximum speed. When the roll controller is enabled there is a 15% reduction in speed. While it in general is true that restricting the moment causes a lower speed it is not correct for extreme heeling angles. Figure 8 shows the top speed in surge when φd is set at a range of different values while beam reaching at very high wind speeds. It shows that the maximum speed is reached at roughly 34deg at Um10 = 13 m s and 35deg at Um10 = 16 m . This can be explained by equation s (26) and (27). When the heeling angle increases then βws approaches π (assuming positive speed in surge), i.e. as if the sailboat were sailing up-wind. The optimal heeling angle in terms of forward speed is however quite high, and keeping the heeling angle this large at such high wind speeds would be very dangerous.
Fig. 7. Time evolution of p, comparison of second and third simulation
2016 IFAC CAMS 556 Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
Fig. 8. Relation between roll angle and maximum speed in surge 6. CONCLUSION This paper presented a new roll reduction technique in sailboats by trimming the sail. The roll controller works as intended, keeping the heeling angle low and reducing roll motion. Having perfect wind estimation improves the result, but the controller is effective regardless. Reducing the heeling angle come at a small cost in speed, though the increase in robustness and safety will make it worth it in many applications. REFERENCES Wille, K.L., Hassani, V., and Sprenger, F. (2016). Modeling and course control of sailboats. 10th IFAC Conference on Control Applications in Marine Systems, Trondheim, Norway.
556
ScienceDirect Roll Roll Roll Roll
IFAC-PapersOnLine 49-23 (2016) 552–556
Stabilization Stabilization Stabilization Stabilization ∗
Control Control Control Control ∗,∗∗,1
of of of of
Sailboats Sailboats Sailboats Sailboats ∗∗
∗∗ Kristian L. Wille ∗∗ Vahid Hassani ∗,∗∗,1 ∗,∗∗,1 Florian Sprenger ∗∗ Kristian L. Wille ∗ Vahid Hassani ∗,∗∗,1 Florian Sprenger ∗∗ Kristian L. Wille ∗ Vahid Hassani ∗,∗∗,1 Florian Sprenger ∗∗ Kristian L. Wille Vahid Hassani Florian Sprenger ∗ ∗ ∗ Department of Marine Technology, Norwegian university of science Department of Marine Technology, Norwegian university of science ∗ of Marine Technology, Norwegian university of science ∗ Department and technology (NTNU), Trondheim, Norway Department of Marine Technology, Norwegian university of science and technology (NTNU), Trondheim, Norway ∗∗ ∗∗ and technology (NTNU), Trondheim, Norway Norwegian Marine Technology Research Institute ∗∗ The and technology (NTNU), Trondheim, Norway ∗∗ The Norwegian Marine Technology Research Institute Marine Technology ∗∗ The Norwegian (MARINTEK), Norway Institute The Norwegian Marine Trondheim, Technology Research Research (MARINTEK), Trondheim, Norway Institute (MARINTEK), Trondheim, (MARINTEK), Trondheim, Norway Norway Abstract: This paper reports a way of dynamically controlling the sail of a sailboat to reduce Abstract: This paper reports a way of dynamically controlling the sail of a sailboat to reduce Abstract: This paper reports a way dynamically controlling the sail of to the heel angle and roll motion the wind. This is mainly robustness and Abstract: This paper reports acaused way of ofby dynamically controlling the to sailincrease of aa sailboat sailboat to reduce reduce the heel angle and roll motion caused by the wind. This is mainly to increase robustness and the heel heel angle and roll rollsailboats motion caused caused by also the be wind. This is mainly mainly to increase increase robustness and safety for autonomous but could used to increase comfort for crew. The solution the angle and motion by the wind. This is to robustness and safety for autonomous sailboats but could also be used to increase comfort for crew. The solution safety for sailboats but could also be used to increase comfort for crew. consists ofautonomous a linear quadratic regulator (LQR) controlling the moment created byThe thesolution sail. A safety for autonomous sailboats but could also be used to increase comfort for crew. The solution consists of a linear quadratic regulator (LQR) controlling the moment created by the sail. A consists of quadratic (LQR) the moment created by sail. lookup table will then choose regulator the optimal anglecontrolling of the sail, maximum consists of a a linear linear quadratic regulator (LQR) controlling theoptimized moment for created by the theforward sail. A A lookup table will then choose the optimal angle of the sail, optimized for maximum forward lookup table then choose the optimal angle of sail, for forward acceleration, given the relative wind direction desired moment. Simulation results are lookup table will will then choose the optimal angle and of the the sail, optimized optimized for maximum maximum forward acceleration, given the relative wind direction and desired moment. Simulation results are acceleration, given the relative direction and presented to show of the approach. acceleration, giventhe theeffectiveness relative wind wind direction and desired desired moment. moment. Simulation Simulation results results are are presented to show the effectiveness of the approach. presented to show the effectiveness of the approach. presented to show the effectiveness of the approach. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Roll Roll Control, Control, Modeling, Modeling, Sailboat. Sailboat. Keywords: Keywords: Roll Roll Control, Control, Modeling, Modeling, Sailboat. Sailboat. 1. INTRODUCTION where S x is the force created by the sail in positive surge x 1. INTRODUCTION INTRODUCTION where S the force created by the sail in positive surge x is 1. where S is force created by the in surge direction (29)) and S the divided by wind x x rr is 1. INTRODUCTION is the the force created by force the sail sail in positive positive surge where Sxx (see direction (see (29)) and S the force divided by the the wind xr is direction (see (29)) and S is the force divided the wind x r issquared. is a by speed (relative to the boat) S function of x This paper is a continuation of Wille et al. (2016). The x direction (see (29)) and S the force divided by the wind r xr squared. Sx r is a function speed (relative to the boat) of This paper is a continuation of Wille et al. (2016). The r speed (relative to the boat) squared. S is a function of This paper is a continuation of Wille et al. (2016). The λ and β , and there will be an optimal λ given a β . model and the course controller presented in Wille et al. x r is a function ws ws ws ws speed (relative to the boat) squared. S This paper is a continuation of Wille et al. (2016). The xrλ given a βws . of λ and β , and there will be an optimal model and the course controller presented in Wille et al. ws λ and β , and there will be an optimal λ given a β . model and the course controller presented in Wille et al. ws ws (2016) is the foundation of the theory developed in this λ and βwsλ , and therebewill be anfreely; optimal λ given a βwsto. a model thefoundation course controller presented in Willeinetthis al. However, (2016) and is the the of the the theory theory developed cannot chosen it is restricted (2016) is foundation developed in this λ cannot be chosen freely; it is restricted to a paper. It is recommended the other paper (2016) is the foundation of of to theread theory developed in first this However, However, λ cannot be chosen freely; it is restricted paper. It is recommended to read the other paper first certain range depending on the wind direction and λto sata sat However, λ cannot be chosen freely; itdirection is restricted to a. paper. It is recommended to read the other paper first certain range depending on the wind and λ as theory and equations from the previous paper will be sat .. paper. It is recommended to read the other paper first certain range depending on the wind direction and λ as theory and equations from the previous paper will be sat This is because we only have control of a rope limiting |λ|, certain range depending on the windof direction and λsat . as theory equations from previous paper will referenced directly (Equations - 59) and notation is because we only have control a rope limiting |λ|, as theory and and equations from 11the the previous paper reused. will be be This This is we only control of limiting |λ|, referenced directly (Equations - 59) 59) and notation notation reused. and the rope only through tension. When the wind This is because because we works only have have control of aa rope rope limiting |λ|, referenced directly (Equations 1 and reused. and the rope only works through tension. When the wind referenced directlyarea (Equations 1 -provides 59) andthe notation reused. and the only works tension. When the is hitting from starboard, it follows that λ > 0 and that The large surface of the sail sailboat with the rope rope only works through through tension. When the wind wind is hitting from starboard, it follows that λ > 0 and that The large large surface surface area area of of the the sail sail provides provides the the sailboat sailboat with with and is hitting from starboard, it follows that λ > 0 and The the torque created by the sail around the mast has tothat be forward thrust. However, it also makes the boat vulnerable is hitting from starboard, it follows that λ > 0 and that The large surface area of the sail provides the sailboat with the torque created by the sail around the mast has to be forward thrust. However, it also makes the boat vulnerable the torque created by the sail around the mast has to be forward thrust. However, it also makes the boat vulnerable positive (meaning a positive angle of attack). When the against strong winds. This can cause large heeling angles, the torque created abypositive the sailangle around the mast When has tothe be forward thrust.winds. However, it can also cause makeslarge the boat vulnerable positive (meaning of attack). against strong This heeling angles, positive (meaning a positive angle of attack). When the against strong winds. This can cause large heeling angles, wind is hitting from port side, λ > 0 and the torque has aagainst lot of roll motion due to wind gusts, and in worst case positive (meaning a positive angle of attack). When has the strong winds.due This can cause large heeling angles, wind is hitting from port side, λ > 0 and the torque a lot of roll motion to wind gusts, and in worst case is hitting from side, λ > acause lot roll motion due wind and case to be negative the boat to capsize. Autonomous will never wind is hitting (negative from port portangle side,of λ attack). > 00 and and the the torque torque has has a lot of of motion due to to wind gusts, gusts,Sailboats and in in worst worst case wind to be negative (negative angle of attack). cause theroll boat to capsize. capsize. Autonomous Sailboats will never never to be negative (negative angle of attack). cause the boat to Autonomous Sailboats will be a viable option if they cannot handle a variety of to be negative (negative angle of attack). cause the boat to capsize. Autonomous Sailboats will never be aa viable viable option option if if they they cannot cannot handle handle aa variety variety of of The following equations describe the upper and lower be following equations describe the upper and lower different weather conditions. of aa asystem is an be a viable option if they Robustness cannot handle variety of The The following equations lower different weather conditions. Robustness of system is an limits of a stable angle ofdescribe the sail,the λ ∈upper [λll , λuuand ], given a The following equations describe the upper and lower different weather conditions. Robustness of a system is an limits of a stable angle of the sail, λ ∈ [λ , λ ], given a important aspect of autonomy. l u different weather conditions. Robustness of a system is an relative limits of a stable angle of the sail, λ ∈ [λ , λ ], given important aspect of autonomy. wind direction: l u limits ofwind a stable angle of the sail, λ ∈ [λl , λu ], given a a important aspect of autonomy. relative direction: important aspect of autonomy. relative The solution presented in this paper controls the roll relative wind wind direction: direction: The solution presented in this paper controls the roll The solution presented in this paper controls the roll motion by utilizing the sail. will reduce the heeling The solution presented in It theangle roll motion by utilizing utilizing the sail. sail. Itthis will paper reduce controls the heeling heeling angle motion It will reduce the angle and theby roll motionthe created by the wind. The controller motion by utilizing the sail. It will reduce the heeling angle if βws ws + π ws < 0 and the roll motion created by the wind. The controller β + π if β 0 βws and the roll motion created by the wind. The controller ws < design is simple but effective, and the states used in the λ = (61) u u = βws + π if β < and the roll motion created by the wind. The controller ws ws > λ (61) design is is simple simple but but effective, effective, and and the the states states used used in in the the u = 0 if < 000 βws + π β ws ws λ (61) design u 0 if β > feedback loop are easy to estimate. ws λu = 0 (61) design is loop simple but effective, and the states used in the if β > 0 feedback are easy to estimate. 0 if βws feedback ws > 0 feedback loop loop are are easy easy to to estimate. estimate. 2. SAIL CONTROLLER 2. SAIL SAIL CONTROLLER 2. 00 if βws 0 ws < 2. SAIL CONTROLLER CONTROLLER if ws < λ (62) l = 0 l 0 ·· 00 ws − π if β β ws < λ = (62) 2.1 Optimal angle of the sail for forward thrust β if > l < β ws > 0 ws − π λ ·· (62) 2.1 Optimal Optimal angle angle of of the the sail sail for for forward forward thrust thrust if β 0 ws ws λll = = β (62) 2.1 β − π if β > 0 ws ws 2.1 Optimal angle of the sail for forward thrust βws − π if βws > 0 The sail is used to create forward propulsion for the In addition to (61) and (62), λ The sail sail is is used used to to create create forward forward propulsion propulsion for for the the In addition to (61) and (62), λsat sat still applies. That is, The sat still sailboat. an optimal sail angle that gives The sail There is usedis to create forward propulsion for the In addition to (61) and (62), λ still applies. applies. That That is, is, λ ∈ [0, λ ] and λ ∈ [−λ , 0]. sailboat. There is an optimal sail angle that gives the sat to (61)ll and (62), sat λsat sat sat Inuuu addition sat still applies. That is, λ ∈ [0, λ ] and λ ∈ [−λ , 0]. sailboat. There is an optimal sail angle that gives the sat l sat highest forward acceleration for a given relative wind sailboat. There isacceleration an optimalfor saila angle that giveswind the λ ∈ [0, λ ] and λ ∈ [−λ , 0]. u sat l sat highest forward given relative λu ∈ [0, λsat ] and λl ∈of[−λ , 0]. can then be found by highest forward acceleration for aa given relative wind The optimal angle thesatsail direction. The objective is thus create aa map from highest acceleration for to given optimal angle of the sail can then be found by direction.forward The objective objective is thus thus to create relative map wind from The The optimal angle of the sail can be found by direction. The is to create a map from traversal of all viable λ, trying to maximize a given relative wind direction to the optimal angle of the sail. We x rr for x The optimal angle of the sail can then then S be found by direction. Thedirection objective is thus to angle createofathe map from for traversal of all viable λ, trying to maximize S a given relative wind to the optimal sail. We x r for a given traversal of all viable λ, trying to maximize S relative wind direction to the optimal angle of the sail. We x β . This can be computed off-line, and the results stored ws begin by defining a way of measuring the force created by r for a given ws . Thisof traversal all viable λ, trying to maximize S relative wind direction toofthe optimal angle of the sail. We x β can be computed off-line, and the results stored begin by defining a way measuring the force created by r ws . This can be computed off-line, and the results β stored begin by defining aa way of the created by wsa. lookup in table. In this paper, an optimal λ was found the sail the forward independent the wind This can be computed off-line, and the results stored begin byin waydirection of measuring measuring the force forceof by β wsa lookup in table. In this paper, an optimal λ was found the sail sail indefining the forward forward direction independent ofcreated the wind wind in a lookup table. In this paper, an optimal λ was found the in the direction independent of the for a step size of 1deg in β . In order to avoid unnecessary speed: ws ws in a lookup table. In this paper, an optimal λ was found the sail in the forward direction independent of the wind for a step size of 1deg in β . In order to avoid unnecessary speed: ws for a step size of 1deg in β . In order to avoid unnecessary speed: discontinuities in the control action, when there is only a ws for a step size of 1deg in β . In order to avoid unnecessary speed: ws discontinuities in the control action, when there is only a discontinuities in the control action, when there is only small difference in the optimal λs between a step change discontinuities ininthe control action, when there is change only a a S V x ws ws small difference the optimal λs between aa step x (β ws ,, λ, ws ) S (β λ, V ) small difference in the optimal λs between step change (60) S x rr (βws ws , λ) = x (βws ,2λ, Vws ) , in the lookup table, new optimal λ is found based on x small difference in theaa optimal λs between a step change x (βws (βws , λ) = S 2λ, Vws ) , (60) S in the lookup table, new optimal λ is found based on V x S , ws , r ws 2 , λ) = x ws (60) S the table, aa new is based linearized solution the twoλ points (thison is Vws (βws , (60) ain Sxx rr (β 2 in the lookup lookup table, between new optimal optimal λdata is found found based on ws ws , λ) = aa linearized solution between the two data points (this is V 2 1 ws 1 Corresponding author, (e-mail: [email protected]). V linearized solution between the two data points (this ws computed on-line). 1 Corresponding author, (e-mail: [email protected]). acomputed linearizedon-line). solution between the two data points (this is is 1 Corresponding author, (e-mail: [email protected]). computed 1 Corresponding author, (e-mail: [email protected]). computed on-line). on-line).
Copyright © 2016, 2016 IFAC 552Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 552 Copyright © 2016 IFAC 552 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 552Control. 10.1016/j.ifacol.2016.10.493
2016 IFAC CAMS Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
553
2.2 Relative moment of the sail In the same manner that we defined the relative force in surge direction, the relative moment in roll is defined as Sφr (βws , λ) =
Sφ (βws , λ, Vws ) , 2 Vws
(63)
where Sφr is calculated at each optimal λ and is added to the lookup table. This solution of λ will hereafter, be referred to as the unconstrained solution, λ100 . Before applying the control law to reduce roll motion and heel angle, we have to be able to control the amount of moment created in roll by the sail. This is done by finding new optimal λs, though with a restriction on the maximum relative moment created at each solution. To produce the desired result, the same traversal algorithm as discussed earlier is performed again, but with the restriction that the solution has to produce a relative moment of 67% or less compared to that of λ100 . This solution will hereafter be referred to as λ67 . Both λ67 and its relative moment is added to the lookup table. This is then repeated for a 33%- and a 0% (or as low as possible) constrained solution, referred to as λ33 and λ0 . However, a new constraint is added, which is that sign(λ100 − λ67 ) = sign(λ67 − λ33 ) = sign(λ33 − λ0 ). In more practical terms, this is to ensure that the different solutions all move the sail in the same direction compared to the previous solution. This is important when we try to find a continuous solution of the relative moment in the next step.
Fig. 2. Relative moment in roll caused by optimal sail position 3. MOMENT ANALYSIS WHEN BEAM REACHING Beam reaching describes the maneuver of sailing perpendicular to the wind. Figure 3 shows the result of the moment in roll when beam reaching at Um10 = 5 m s . It is easy to see that the sail and the restoring moment are dominating. Together they generate roughly 91% of the moment in roll. The keel produces another 8%. From (9) and (10) one can see that the two Coriolis terms provides no moment in roll, and it follows that the remaining 1% is caused by the rudder and the drag. 4. ROLL CONTROLLER The roll controller will be split into two mirrored controllers which will step in when the heeling angle gets too large. For now, we will look into the controller which will govern for roll (φ) bigger than zero. The heeling angle
Fig. 1. Optimal position of the sail We then have all that we need to reverse the process. By entering βwr one finds the solutions of λs that corresponds to that relative wind direction. Then, by applying the restriction of the desired Sφr , the ideal λ is found based on a linearization (computed on-line) between the different Sφr created by λ100 , λ67 , λ33 and λ0 . 553
Fig. 3. Moments while beam reaching
2016 IFAC CAMS 554 Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
should not be larger than 10 degrees for a comfortable ride, and the desired heel angle, φd , is thus set to 10 degrees. A linear quadratic regulator (LQR) will be used to control the roll motion. The first step is thus to make a linearized system of the roll motion about an equilibrium point, which is selected to be the desired heel angle. The linearized system can be written as φ x= (64) x˙ = Ax + Bu· (65) p Figure 3 and the moment analysis showed which of the moments that are important. The two biggest contributions are from the sail and the restoring moment. The keel also provides some moment, though estimating the moment created by the keel is fairly difficult as it requires measurements of the speed and side slip, see (32) and (33). However, it was argued in Wille et al. (2016) that the undesirable side force FU ((56) and (57)) is equal to the lifting force of the keel. Replacing KL with FU in (33) shows that when βck ≈ π then the moment created by the keel in roll is equal to −FU h2 . βck ≈ π is a good approximation when the drift angle is low and there is little current. The restoring moment (20) when linearized becomes Gφ = ρw g∇GMt ((1 − 2sin(φw )2 )φ + C1 ,
(66)
where φw is the equilibrium point and C1 is equal to (67) C1 = ρw g∇GMt cos(φw )sin(φw )· Making C1 and the moment generated by the keel a part of our control input u linearizes the system, and the linearized system model do then only includes the restoring moment. Matrices A and B in our linearized system then becomes
0 A = ρw g∇GMt (1 − 2sin(φw )2 ) − Ixx − Kp˙ 0 , 1 B= Ixx − Kp˙
1 0
(68)
quadratic cost caused by an error in x, and R weighs the quadratic cost of using the actuator. R was tuned to be R = 1e-10 and Q was tuned to 0.75 0 Q= · (71) 0 1 The LQR gain was then calculated in matlab by using the function lqr(A,B,Q,R,0). The result was K = [9.0e4 1.0e5]. Ki was manually tuned to be equal to 2500. The controller is very similar when the heeling angle is negative. The only difference is that φw and φd is changed to −10deg, and C1 is recalculated for the new equilibrium point (changes sign). In practice, only one controller is used, and the signs in front of φw , φd and C1 switches depending on the sign of the heel angle. The controller should only be turned on when the heel angle is above 10 degrees. Hysteresis has also been implemented to avoid the controller turning on and off too fast, causing discontinues in the control action. It is important to note that the controller would never actually be able to create more moment than the optimal unconstrained solution (λ100 ) gives, and it follows that the roll controller would not be able to increase the heeling angle further than what was previously possible. It follows that steps has to be taken to avoid integral build up if the wind speed is to low and the sail is not able to provide enough moment to keep the heeling angle at 10 degrees. To avoid integral build up the integral action is not allowed to further grow when λd = λ100 , i.e. when no more moment can be provided. It might seem counter-intuitive to keep the controller on if the heeling angle is lower than 10 degrees. Though this might be because the integral action has not yet reached a steady value, and the sail is then restricting the moment too much. Restricting the allowed moment in roll comes at a cost of reducing the force in forward direction, see Figure 4 and will thus in general decrease the speed of the sailboat.
(69)
where u is defined as
t φ − φd u = −K − Ki φ − φd dτ + C1 + FU h2 · (70) p 0 Integral control action is added to remove errors caused by the linearization of the system (and of linearization used when calculating optimal λ for a given βwr and Sφr ), and other system uncertainties. As a note, the moment generated by the keel is fairly steady while keeping a set course (Figure 3), and the error by not including it in our model could be handled by the use of the integral control action if a simpler controller design is desired, or if FU is difficult to compute. The LQR controller gain, K, were tuned by setting the matrices Q and R, where Q is the weighting matrix in the 554
Fig. 4. Relative force in surge
2016 IFAC CAMS Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
5. SIMULATION AND RESULTS Three sets of simulations have been studied in what follows. In the first simulation, the controller in roll has been turned off, meaning λ100 is always chosen as the desired angle of the sail. In the next simulation, the controller is turned on, but only the average wind speed is known. In the last simulation, the controller is on, and perfect wind estimation is used. The results will show the last 50 seconds of a 300 second simulation, ensuring that the top speed in surge has been reached and that the integral term in the controller has had time to reach a steady value. The boat is set to go on a course direction perpendicular to a mean wind of Um10 = 10 m s (beam reaching). Notice that βws = 90deg, see equation (26) and (27). Figure 5 demonstrates the effectiveness of the controller keeping the heel angle low. Furthermore, not only does
Fig. 5. Time evolution of φ, comparison of all simulations
Fig. 6. Time evolution of p, comparison of first and second simulation 555
555
the controller reduce the maximum heeling angle, it also heavily reduces the amplitude of the roll motion. In Figure 6 one can see that the rotational velocity in roll is reduced considerably, and Figure 7 shows that having perfect wind estimation improves the results even further. When perfect wind estimation is used we have an almost perfect result, and this justifies the simplifications made to the linearized version of the system as it do not effect the performance of the controller. The remaining error is likely caused by the delay in the sail actuator. Table 1. Results of roll controller test amplitude of roll motion [deg] ] maximum rotational velocity p [ deg s average speed in surge, u [ m ] s
sim 1 4.38 5.15 7.75
sim 2 1.18 1.42 6.59
sim 3 0.29 0.47 6.59
The results of the test is given in Table 1. The amplitude of the roll motion is reduced by 73% and the maximum rotational velocity by 72% when the average wind speed is known. However, reducing the heeling angle comes at a cost in speed. Restricting the moment created in roll comes at a cost in forward thrust from the sail, which directly effects the maximum speed. When the roll controller is enabled there is a 15% reduction in speed. While it in general is true that restricting the moment causes a lower speed it is not correct for extreme heeling angles. Figure 8 shows the top speed in surge when φd is set at a range of different values while beam reaching at very high wind speeds. It shows that the maximum speed is reached at roughly 34deg at Um10 = 13 m s and 35deg at Um10 = 16 m . This can be explained by equation s (26) and (27). When the heeling angle increases then βws approaches π (assuming positive speed in surge), i.e. as if the sailboat were sailing up-wind. The optimal heeling angle in terms of forward speed is however quite high, and keeping the heeling angle this large at such high wind speeds would be very dangerous.
Fig. 7. Time evolution of p, comparison of second and third simulation
2016 IFAC CAMS 556 Sept 13-16, 2016. Trondheim, Norway
Kristian L. Wille et al. / IFAC-PapersOnLine 49-23 (2016) 552–556
Fig. 8. Relation between roll angle and maximum speed in surge 6. CONCLUSION This paper presented a new roll reduction technique in sailboats by trimming the sail. The roll controller works as intended, keeping the heeling angle low and reducing roll motion. Having perfect wind estimation improves the result, but the controller is effective regardless. Reducing the heeling angle come at a small cost in speed, though the increase in robustness and safety will make it worth it in many applications. REFERENCES Wille, K.L., Hassani, V., and Sprenger, F. (2016). Modeling and course control of sailboats. 10th IFAC Conference on Control Applications in Marine Systems, Trondheim, Norway.
556