Optimal ail angle computation for an autonomou ailboat robot Hadi Saoud 1, Minh-Duc Hua 1, Frederic Plumet 1 and Faiz Ben Amar 1 Intitute for Intelligent Sytem and Robotic (ISIR) UPMC Univ Pari 6, CNRS - UMR 7, 755 Pari - France Abtract A method to compute the optimal ail angle for an autonomou ailboat i propoed, which allow for maximizing the longitudinal velocity while maintaining afe ailing condition by limiting the roll angle. A implified 4- DOF dynamic model of the ailboat i ued to obtain the roll dynamic and longitudinal ail force. From thee expreion, we derive a cot function which depend on the thrut force. The cot need to be minimized under tability contraint in roll motion while the ytem i ubject to a bound on the roll angle. Numerical imulation, uing a full nonlinear dynamic model of the ailboat, how the improved performance of the propoed method, compared to traditional olution that can be found in the literature. I. INTRODUCTION Due to their low energy conumption, autonomou ailing robot provide a promiing olution for long-term obervation or monitoring miion in the ocean and a lot of ailing robot project were launched worldwide during the lat decade [1] [8]. The control of an autonomou ailboat (i.e. controlling it heading while enuring a good trimming of the ail) i challenging ince the thrut force depend on uncontrollable and partially unpredictable wind. Moreover, uch vehicle exhibit complex behaviour due to aero- and hydrodynamic propertie of their ail and hull. The problem of planning and controlling the heading angle of an autonomou ailboat ha been widely tudied (ee [1], [3], [8] [11] among other), but little attention ha been paid to the computation of the ail angle. On manned ailboat, ail trimming i mainly carried out on the bai of the experience and kill of the ailor. Sailor can alo ue VPP program to correct their ailing configuration (VPP are commercial product that give the theoretical maximum velocity of a ailboat for each wind condition [1]). Sailor kill and experience have been adapted to the cae of autonomou ailboat uing either a linear relationhip between the ail angle of attack and the ail angle (including or not aturation and hyterei [3], [13]) or by uing fuzzy logic controller [14] [15]. None of thee method i focued on the computation of a ail angle to maximize the longitudinal peed (urge) of a ailboat. Some other method have been propoed like the extremum-eeking [16] for the online peed optimization when ail model i unknown. Such method uually ha a long convergence time which could be annoying with fat varying wind condition [17]. Obviouly, maximizing the peed of a ailboat i important not only for racing purpoe but alo to be conitent with 1 All author, E-mail: latname@iir.upmc.fr miion objective for example. Maximizing the peed of a ailboat alo improve the manoeuvrability and the tability of the heading controller ince the torque produced by the rudder depend on the quare of the velocity of the fluid. On the other ide, when the peed of a ailboat increae, the force acting on the ail may increae the heeling (or roll) angle and thu the rik of capizing. In thi paper, we preent a method for calculating an optimal ailing angle that maximize the peed of a ailboat while enuring that the heeling angle remain bounded by a pre-defined value. Thi method i baed on the tudy of the teady tate behaviour in urge and roll and i formulated a an optimization problem (thrut force) ubject to equality contraint (roll tability). A major contribution of thi paper i to demontrate that the choice of uch an optimal ailing angle mathematically limit the teady heeling angle to a value maller than π/4 radian. Moreover, other value for the maximum teady heeling angle (le than π/4 radian) can be eaily introduced in the optimization proce in order to better reflect ailboat tability limit. In thi tudy, we aume that the ailboat ha only one mainail with well known lift and drag coefficient and that there i no aerodynamic interaction between the ail and the ret of the ailboat. The paper i organized a follow: we firt decribe a 4 DOF dynamic model of a ailboat and, under ome aumption, derive the expreion of the longitudinal ail force and equilibrium equation between the ail torque and retoring torque. Then, we formulate the computation of the optimal ail angle a a minimization problem ubject to equality contraint on the roll tability, firt without limiting the roll angle (ection VI-A) and then including a aturation of the roll angle (ection VI-B). A numerical tudy for a real ail i preented in the lat ection. Numerical imulation, baed on the full nonlinear dynamic model of a ailboat, how the performance of thi optimal ail computation, compared to other method decribed in the literature. II. NOTATION {e 1, e, e 3 } denote the canonical bai of R 3. For any x R 3, the notation x denote the kewymmetric matrix aociated with x, i.e. x y = x y, y R 3. For any affine vector x, x X denote the vector of coordinate of x in the bai of the frame X.
R v,θ SO(3) denote the rotation matrix that rotate a vector by an angle θ around v. G: ailboat center of ma (CoM). G : center of preure of the ail (aumed to be fixed). I = {; ı, j, k }: inertial frame choen a the North-Wet-Up frame. B = {G; ı, j, k }: body frame fixed to the hull. S = {G ; ı, j, k }: ail-fixed frame. M R 3 3, J = diag([j 11 J J 33 ]) R 3 3 : ailboat ma and inertia matrice, including real-body and added-ma component. x, x : poition of G and G w.r.t. the inertial frame. ω : angular velocity of the body-fixed frame w.r.t. the inertial frame. v : linear velocity of G w.r.t. the inertial frame. v : linear velocity of G w.r.t. the inertial frame. v w : wind velocity w.r.t. the inertial frame. v a : apparent velocity of the ail. x, R, ω, v: hort notation of x I, RB I, ωb, v B. δ, α : ail angle and ail angle of attack. C L ( ), C D ( ): lift and drag coefficient of the ail. f(x) = f x φ, ψ: roll and yaw angle of the ailboat. c( ) co( ) and ( ) in( ) h := GG k : vertical ditance between G and G V long., V lat. : longitudinal and lateral velocity of the ailboat F, τ : um of force and torque applied on G l φ : metacentric height of the ailboat Θ V : angle of apparent wind peed w.r.t the body frame III. MODEL USED The 6-DOF model ued for the ailboat i a claical model in marine robotic [1], [18]. The dynamic can be ummarized a follow: M v = ω Mv + F B J ω = ω Jω + (1) τ B with v = [v 1, v, v 3 ] and ω = [ω 1, ω, ω 3 ]. By approximating the hull to a volume with three mutually perpendicular axe of ymmetry, the contribution of the offdiagonal element in the added ma matrix can be neglected, i.e. both ma and inertia matrice are diagonal. Aumption 1 The pitch motion and vertical motion are mall o that we can ue a 4 DOF dynamic model: x, y tranlation and roll, yaw (φ, ψ) rotation. Thu, kinematic equation of motion are given by: φ = ω 1, ψ = ω 3 (co φ) 1 The longitudinal and lateral velocitie V long., V lat. of the ailboat are defined a: V long. V lat. := R z,ψẋ = R x,φ v j v a G ψ k α v w v δ F D ı φ j k F L F F N F T ı Fig. 1: Frame orientation and wind effort From here, roll dynamic can be expreed a the following: φ = ω 1 = e 1 J 1 ω Jω + e 1 J 1 τ B = J + J 33 J 11 ω 3 tan φ e 1 J 11 τ B Auming that roll dynamic i much fater than yaw dynamic ( φ ψ), the lat one can be neglected leading to the following expreion: φ = e 1 J 11 τ B Thi dynamic equation will be ued later in the expreion of the equilibrium contraint of the ailboat. IV. PROBLEM FORMULATION By neglecting the angular velocity of the rotating ail, the apparent velocity of the ail w.r.t the urrounding air v a i defined by: v a = v v w Let define V a the reult of rotating va I around the k axi: V V 1 V = Rz,ψv a I = V long V lat Rz,ψv w I Then, the angle of apparent wind peed w.r.t the body frame i given by: Θ V atan(v, V 1 ) Note that, Θ V i related to the apparent wind angle (AWA) meaured by on-board anemometer by: tan(awa) = co φ tan Θ V
The angle of attack α can be defined a [1]: with in α = vs a, v a va, S = V 1 in δ + V co φ co δ v a = V = V1 + V To implify the problem, we did everal aumption: Aumption The wind velocity i aumed to be alway parallel to the water urface, which i alo aumed to be flat. Aumption 3 Amplitude of apparent peed i non-null ( V > ) for all time. Uually, to decribe effort on ail, we ue lift and drag coefficient [19] related to the force generated by the apparent wind in both apparent wind direction and normal to the apparent wind direction. By uing thee coefficient, effort on the ail F i given by [1]: F = λ ( C D (α ) C L (α ) tan α ) v a v a + λ C L (α ) co α v a j with λ = 1 ρ airs (ρ air : air denity, S : urface of the ail). The effort F can be written a the um of two term: one normal to the ail plan (parallel to j ) and one tangential to the ail (parallel to ı ). Thi lead u to compute a normal coefficient C N and a tangential coefficient C T : C T (α ) := C L (α ) in α C D (α ) co α C N (α ) := C L (α ) co α + C D (α ) in α The ail force can now be written a the following: F = + λ C N (α ) V j λ C T (α ) V ı From curve preented in [19], we noticed that C T C N on ail that repect the following condition: 1) high apect ratio (AR > 4), ) low camber (c < 1 15 ). Furthermore, one can note that the normal coefficient C N (α ) of thee kind of ail i a monotonically increaing function of in α. Aumption 4 The tangential component i aumed to be very mall with repect to the normal component o that we can neglect the tangential force: F λ C N (α ) V j Thi aumption i verified in almot all ail decribed in [19]. Aumption 5 The ail i fully actuated o that the ail angle can take any value, without contraint related to apparent wind direction, i.e. δ [ π, π] Θ V. The roll ail torque and the longitudinal ail force are given by: { τ B,1 = λ h C N (α ) V co δ F B,1 = λ C N (α ) V in δ Our idea i to write C N (α ) a a function of in α : C N (α ) = f(in α ) where f( ) : [ 1, 1] R i a differentiable odd function atifying the following propertie: ign(f(x)) = ign(x),f(x) = iif x = f( x) = f(x), f(x) = f( x) Thu, the roll ail torque and the longitudinal ail force can be rewritten a: { τ,1 = λ h f(in α ) V co δ F,1 = λ f(in α ) V in δ (3) We alo aume that, at roll equilibrium, torque effect from rudder, keel and hydrodynamic effort are neglected due to the ailboat tructure. The remaining ail torque τ B and retoring torque τ B re. give the equilibrium condition in roll motion: = φ = τ B rep,1 + τ B,1 = mgl φ in φ λ h f(in α ) V co δ (4) V. OPTIMAL SAIL ANGLE DETERMINATION Since we want to maximize the longitudinal velocity, we try to maximize the ail force F,1 = λ f(in α ) V in δ under the equilibrium condition in roll motion to avoid capizing. Thi can be formulated a a minimization problem ubject to an equality contraint: with min J (δ, φ) := f(in α ) in δ.t. : β in φ + f(in α ) co δ = in α = V 1 in δ V co φ co δ V 1, := V 1, & β := mgl φ V λ h V To take into account the equilibrium contraint in the minimization problem, we compute the Lagrangian: L = f(α )δ + λ(βφ + f(α ) co δ ) Finding the optimal value δ and φ can be done by finding the minima of L through computing it partial derivative and olving (6) imultaneouly. The global minimum i the olution of (6) that lead to a minimal value of the cot function. = L λ = βφ + f(α )cδ = L φ = f(α )( V φcδ )δ + λ[βcφ + f(α )( V φcδ )cδ ] = L δ = f(α )( V 1 cδ + V cφδ )δ + f(α )cδ + λ[ f(α )( V 1 cδ + V cφδ )cδ f(α )δ ] (6) () (5)
One deduce from = L δ that: λ = f ( V 1 co δ + V co φ in δ ) in δ f co δ f ( V 1 co δ + V co φ in δ ) co δ f in δ Replacing λ into the expreion = L φ one obtain: f f ( V φcδ ) + βcφ[ f ( V 1 cδ + V cφδ )δ + fcδ ] = Replacing co δ = (β/f) in φ and V 1 in δ = in α + V co φ co δ into the above equation, we get: β f V φ+βcφ[ f ( V 1 cδ + V co φδ )δ +fcδ ] = that lead to a relation between the roll angle φ and the angle of attack α : tan(φ) = V ( ) f f (7) β α f + f In ummary, intead of olving (6), the following equation need to be olved in order to find the minima: in α = V 1 in δ V co φ co δ (8a) β in φ + f co δ = (8b) tan(φ) = V β ( f f in α f + f ) (8c) Remark 1 Relation (8c) implie that the optimal roll angle φ i bounded by π/4, i.e. φ π/4. Moreover, in the cae where f i an increaing function, one enure that ign(φ ) = ign( V ) Indeed, ince f i poitive a f i an increaing function, and ince f(in α ) ha the ame ign a in α, f f one verifie that in α i poitive. Thu, f+f ign(φ ) = ign(tan(φ )) = ign( V ). Lemma 1 If (δ, φ ) i the optimal olution to the minimization problem (5), then (δ ± π, φ ) i alo an optimal olution to (5). Proof: Let α uch that in α = V 1 in δ V co φ co δ. Since (δ, φ, α ) i the optimal olution to (5), they atify (8). Therefore, we only need to how that (δ ±π, φ, α ±π) alo atify (8) and the aociated cot function J (δ ±π, φ ) i equal to the cot function induced by the optimal olution (δ, φ ), i.e. J (δ ± π, φ ) = J (δ, φ ). Indeed, relation (8a) with (δ, φ, α ) = (δ ± π, φ, α ± π) write in(α ± π) = V 1 in(δ ± π) V co φ co(δ ± π) in α = V 1 in δ V co φ co δ which i exactly relation (8a) with (δ, φ, α ) = (δ, φ, α ). Similarly, uing the fact that f( ) i an odd function, one eaily verifie that relation (8b) and (8c) alo hold with (δ, φ, α ) = (δ ± π, φ, α ± π), provided that they are atified with (δ, φ, α ) = (δ, φ, α ). It i alo traightforward that J (δ ± π, φ ) = J (δ, φ ). Remark A a reult of lemma 1, if the ail angle i phyically limited to the interval [ π/, π/], there exit alway one olution inide thi interval. VI. SOLVING THE OPTIMAL SAIL PROBLEM A. Without aturation on optimal Roll angle Sytem of equation (8) need to be olved in real-time on the ailboat to continuouly get an optimal ail angle reference. Becaue thi computation can be time-conuming, we tranform the ytem to get a ingle nonlinear equation with a ingle variable. Denote: x := in α [ 1, 1] y := in(φ) ] 1, 1[ z := in δ [ 1, 1] The cot function i then: J (x, z) = f(x)z From (8a) and (8b) one deduce: in α = V 1 in δ + V β in(φ) f x = V 1 z + V β y f(x) = From here and by uing (8) and (9), one obtain the following equation: x = V 1 z + V β y (1a) f(x) β y 1 f(x) = 1 z y + 1 (1b) y = V f(x) f(x) (1c) 1 y β x f(x) + f(x) for x [ 1, 1], y ] 1, 1[, z [ 1, 1]. Letting: g(x) := f(x) f(x) x f(x) + f(x) > ḡ(x) := V β g(x) = λ h V V g(x) mgl φ one deduce from (1c): y = ḡ(x) 1 + ḡ (x), 1 1 y = 1 + ḡ (x) and, thu, from (1b): 1 z = One deduce from (1a): z = 1 V1 [ (9) βḡ(x) f(x)(1 + ḡ (x)) 1 4 (1 + 1 + ḡ (x)) 1 (11) x V g(x) f(x) 1 + ḡ (x) ] (1) Uing V 1 z + V 1 ( 1 z ) = V 1, one finally get the ingle nonlinear equation to be olved: [ xf(x) 1 + ḡ (x) V g(x)] + V 1 β ḡ (x) 1 + ḡ (x) 1 + 1 + ḡ (x) V 1 f (x)(1 + ḡ (x)) = (13)
Solving the nonlinear equation (13) get the ame reult than olving the nonlinear ytem (8). The reult are the local minima and maxima of the cot function. To find the global minimum, we can continue to evaluate the cot function for each value of x, in order to obtain the optimal value for x (and y and z accordingly). Lemma If x [ 1, 1] i a olution to Eq. (13), then x [ 1, 1] i alo a olution to Eq. (13). Thi lemma i a direct conequence of Lemma 1. It can alo be eaily proven uing the propertie () of f( ). Solving Eq. (13) take only few milliecond on an Arm Cortex-A9 @ 1 Ghz and thu can be eaily done on-line to continuouly compute optimal ail angle δ. B. Including aturation on Roll angle Limiting roll angle may be a prime concern to avoid capizing. If we want to limit the optimal roll angle uch that φ φ max, with φ max < π/4, we can formulate the following minimization problem: min J (δ, φ) := f(in α ) in δ.t. : 1) β in φ + f(in α ) co δ = (14) ) φ φ max Thi i an optimization problem in a compact et. Therefore, a traightforward way to olve (13) i to compute a et Λ that include all potential olution of (5). To take into account roll limit φ max, the olution φ = ign( V )φ max mut be added to Λ. The correponding value of α and δ are obtained by olving numerically the roll motion equilibrium in 5 and in α = V 1 in δ V co φ co δ. By evaluating the cot of all potential olution in Λ and excluding olution where φ < φ max, we can extract the optimal ail angle δ o that equilibrium contraint i atified and φ < φ max. VII. NUMERICAL ANALYSIS AND SIMULATION A. Numerical cae tudy for a flat ail Let u conider a ail with no camber and AR = 5 [19, p. 86]. The experimental data for the coefficient C L and C D are depicted in Fig. a. Fig. b how the computed coefficient C N and C T veru the ine of the angle of attack (i.e. in α ). It can be oberved that the tangential component C T evolve near zero and i largely dominated by the normal component C N. Thu, Aumption 4 hold with good accuracy. An approximation by a polynomial of degree 5 for C N, (red curve in Fig. b), i given by: { C N p5 x (x) = 5 + p 4 x 4 + p 3 x 3 + p x + p 1 x if x p 5 x 5 p 4 x 4 + p 3 x 3 p x + p 1 x if x < where x = in(α ) and with p 5 = 11.5, p 4 = 33.79, p 3 = 39.45, p = 3.7 and p 1 = 7.15. The numerical tudy ued parameter defined in table I and have been done for two value of apparent velocity ( V = 5 and V = 8) and two value of metacentric height (l φ =.1 and l φ = 1). 1 1 1.8.6.4...4.6.8 1 in(α ) (a) Value of C L and C D v.. inu of the attack angle α (from [19, p. 86]). 1 1 1.8.6.4...4.6.8 1 in(α ) C L C D C N comp. C T comp. C N app. (b) Computed and approximated C N and computed C T v.. inu of the attack angle α. Fig. : Lift, drag, normal and tangential coefficient for a no camber ail with AR = 5 TABLE I: Numerical value of the parameter name m S g ρ air h value 15 1 9.81 1.5.8 For thi numerical tudy, equation (13) i olved numerically uing, for intance, the Secant method. It exit normally oppoite-ign pair of olution for x, but we know that the ign of x hould be oppoite to that of V. Therefore, for each Θ V, we have olved twice Eq. (13) with initial guee, repectively, cloe to zero and ±1 (depending on the ign of V ). The comparion of the two reulting value of the cot function allow u the find the optimal olution. We firt conider the cae where the roll angle i not limited and plot only the olution where δ [ π/, π/]. From 3b, we can note that, when wind peed i low, the relation between δ and Θ V can be approximated by a linear function (a in [13]). In cae of high wind peed, the curve begin to be aturated. Now, we introduce the limitation to the roll angle, a in ection VI-B. Fig. 4 how the effect of uch limitation on φ and δ. We oberve from 4a that a mall variation in δ lead to an important variation in φ. We alo notice that the main influence of limiting the roll angle i on the lope of the curve on it center part (when δ [ π/, π/]), i.e. when ailing upwind. Fig 3 and 4 have origin ymmetry. We oberve from Fig 3c that angle of attack i not contant. In other word,
φ [deg] 4 c c 4 φ [deg] 3 1 c 4 6 8 1 1 14 16 18 (a) Optimal roll angle φ v.. apparent wind angle Θ V 4 6 8 1 1 14 16 18 (a) Optimal roll angle φ v.. apparent wind angle Θ V 9 9 δ [deg] 6 3 4 6 8 1 1 14 16 18 c c 4 δ [deg] 6 3 4 6 8 1 1 14 16 18 c α [deg] (b) Optimal ail angle δ v.. apparent wind angle Θ V 3 6 9 4 6 8 1 1 14 16 18 (c) Optimal angle of attack α v.. apparent wind angle Θ V Fig. 3: Optimal angle v.. relative wind angle Θ V without limitation on φ ( : V = 5 & l φ = 1, c : V = 8 & l φ = 1, : V = 5 & l φ =.1, c 4 : V = 8 & l φ =.1. c c 4 (b) Optimal ail angle δ v.. apparent wind angle Θ V Fig. 4: Optimal angle v.. relative wind angle Θ V with limitation on φ (with V = 5 & l φ =.1, : φ max = 45, c : φ max =, : φ max = 1 ) δ [deg] 9 6 (15, 15)(45, 15) 3 (15, 9) 4 6 8 1 1 14 16 18 Fig. 5: Sail angle δ a a function of the apparent wind angle Θ V, adapted from [3] maintaining a contant angle of attack i not an optimal method for ail trimming. B. Simulation In order to evaluate the performance of thi optimal ail election, we perform numerical imulation and compare the reult with the ail trimming method ued in [3]. The ail trimming method ued in [3] i ummarized on Fig. 5: the ail angle i limited to ±π/ when Θ V > 5π/6 to ave energy by limiting ail trimming. For the ame reaon, the ail angle remain contant when Θ V [π/1, π/4]. For the imulation, we ued an implementation of the full nonlinear 6-DOF model given in [1]. For thee imulation, the real wind v w velocity i equal to 4m/ and the real wind angle vw B i a function of time a follow: 1 if t [, [ vw B = 45 if t [, 4[ 8 if t [4, 6] In both cae, a heading controller have been ued to maintain a zero heading angle. Simulation reult (fig. 6) how that both method perform equally when ailing downwind (t [, ]) but the gain on velocity appear when ailing beam reach (t [, 4]) and even more when ailing upwind (t [4, 6]). Figure 7 how the maximum velocity that can be reached by the ailboat a function of vw B the relative angle between the heading and the true wind when v w i 4 m/. Curve S i for maximum velocity when uing the preented trimming method and φ max = 3 while curve B m i for maximum velocity when trimming the ail a in 5. They
v [m/].5 1.5 S Bm 1 3 4 5 6 t[ec] (a) Boat peed v over time tudying the longitudinal force and torque generated by the apparent wind on the ail, it value depend on the apparent wind velocity, which i more conitent with the ailor practice. Numerical imulation how the performance of thi method, compared to more empirical one. Future work will focu on deigning a roll controller taking the optimal roll angle φ a the reference and the ail angle δ a the input to dynamically control the roll equilibrium during tacking manoeuvre for example or in cae of wind gut. φ[deg] 1 3 1 3 4 5 6 t[ec] (b) Roll angle φ over time Fig. 6: Comparion between ail trimming method (S: optimal ail angle, B m ail trimming a in [3]) are compared with the theoretical maximum velocity (vel max ) that i obtained by imulating the ytem with every poible ail configuration for each wind condition. We oberve that trimming the ail with δ give better reult than with [3] and that it i cloe to the maximum theoretical value. 18 135 5 9 1 7 45 315 v w velmax Fig. 7: Maximum velocity a function of relative angle between the followed direction and the true wind (S: optimal ail, B m ail trimming a in [3], (vel max : maximum theoretical velocity) VIII. CONCLUSION In thi paper, a method wa deigned to optimally tune the ail angle of a ailboat in order to maximize it velocity along it current heading. Thi optimization take into account a afety criterion to avoid capizing. We how that, uing thi method, the heeling angle i intrinically limited to ±π/4. Moreover, other value for the maximum heeling angle (le than ±π/4) can eaily be introduced in the optimization proce to better reflect ailboat tability limit. A a conequence, ince thi optimal ail angle i computed by S Bm S Bm REFERENCES [1] G. Elkaim, The atlanti project: A gp-guided wing-ailed autonomou catamaran, Journal of the Intitute of Navigation, vol. 53, pp. 37 47, 6. [] M. Neal, A hardware proof of concept of a ailing robot for ocean obervation, IEEE Journal of Oceanic Engineering, vol. 31, no., pp. 46 469, 6. [3] Y. Briere, IBOAT: An autonomou robot for long-term offhore operation, in Electrotechnical Conference, 8. MELECON 8. The 14th IEEE Mediterranean, May 8, pp. 33 39. [4] N. Cruz and J. Alve, Autonomou ailboat: An emerging technology for ocean ampling and urveillance, in MTS/IEEE OCEANS, 8. [5] R. Stelzer, K. Jafarmadar, H. Haler, R. Charwot, et al., A reactive approach to obtacle avoidance in autonomou ailing. in International Robotic Sailing Conference, 1, pp. 33 39. [6] H. Ercken, G.-A. Buer, C. Pradalier, and R. Siegwart, Avalon: Navigation trategy and trajectory following controller for an autonomou ailing veel, IEEE Robotic Automation Magazine, vol. 17, no. 1, pp. 45 54, 1. [7] C. Petre, M.-A. Romero-Ramirez, F. Plumet, and B. Aleandrini, Modeling and reactive navigation of an autonomou ailboat, in 11 IEEE/RSJ International Conference on Intelligent Robot and Sytem (IROS), Sept. 11, pp. 3571 3576. [8] L. Jaulin and F. Le Bar, An interval approach for tability analyi: Application to ailboat robotic, IEEE Tranaction on Robotic, vol. 9, no. 1, pp. 8 87, 13. [9] R. Stelzer and T. Proll, Autonomou ailboat navigation for hort coure racing, Robotic and Autonomou Sytem, vol. 56, no. 7, pp. 64 614, 8. [1] H. Saoud, M.-D. Hua, F. Plumet, and F. Ben Amar, Modeling and control deign of a robotic ailboat, in International Robotic Sailing Conference. Springer, 13, pp. 95 11. [11] L. Xiao and J. Jouffroy, Modeling and nonlinear heading control of ailing yacht, IEEE Journal of Oceanic Engineering, vol. 39, pp. 56 68, 14. [1] Boehm, C., A Velocity Prediction Procedure for Sailing Yacht with a hydrodynamic Model baed on integrated fully coupled RANSE- Free-Surface Simulation, Ph.D. diertation, Techniche Univeriteit Delft, 14. [13] K. Legurky, A modified model, imulation, and tet of a full-cale ailing yacht, in Ocean, 1, Oct. 1, pp. 1 7. [14] E. Yeh and J.-C. Bin, Fuzzy control for elf-teering of a ailboat, in Singapore International Conference on Intelligent Control and Intrumentation, 199. SICICI 9. Proceeding, vol., Feb. 199, pp. 1339 1344. [15] R. Stelzer, T. Proll, and R. John, Fuzzy logic control ytem for autonomou ailboat, in IEEE Fuzzy Sytem Conference, July 7, pp. 1 6. [16] L. Xiao, J. Alve, N. Cruz, and J. Jouffroy, Online peed optimization for ailing yacht uing extremum eeking, in Ocean, 1, Oct. 1, pp. 1 6. [17] K. Treichel and J. Jouffroy, Real-time ail and heading optimization for a urface ailing veel by extremum eeking control, in 55th International Scientific Colloquium (IWK), Ilmenau, Germany, 1. [18] T. I. Foen, Handbook of Marine Craft Hydrodynamic and Motion Control. Wiley-Blackwell, Apr. 11. [19] C. A. Marchaj, Sail Performance : Technique to Maximize Sail Power, nd ed. International Marine/Ragged Mountain Pre,.