A control strategy for steering an autonomous surface sailing vehicle in a tacking maneuver

Syansk Universitet A control strategy for steering an autonomous surface sailing vehicle in a tacking maneuver Jouffroy, Jerome Publishe in: Proceeings of the IEEE International Conference on Systems, Man, Cybernetics Publication ate: 9 Document version Early version, also known as pre-print Citation for pulishe version (APA): Jouffroy, J. (9). A control strategy for steering an autonomous surface sailing vehicle in a tacking maneuver. In Proceeings of the IEEE International Conference on Systems, Man, Cybernetics IEEE. General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights. Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research. You may not further istribute the material or use it for any profit-making activity or commercial gain You may freely istribute the URL ientifying the publication in the public portal? Take own policy If you believe that this ocument breaches copyright please contact us proviing etails, an we will remove access to the work immeiately an investigate your claim. Downloa ate:. ec.. 18

A control strategy for steering an autonomous surface sailing vehicle in a tacking maneuver Jerome Jouffroy Mas Clausen Institute University of Southern Denmark (SDU) Alsion, DK-64 Sønerborg, Denmark e-mail: jerome@mci.su.k Abstract Sailing vessels such as sailboats but also lanyachts are vehicles representing a real challenge for automation. However, the control aspects of such vehicles were hitherto very little stuie. This paper presents a simplie ynamic moel of a so-calle lanyacht allowing to capture the main elements of the behavior of surface sailing vessels. We then propose a path generation scheme an a controller esign for a well-known an funamental maneuver in sailing referre to as tacking. Simulation results are presente to illustrate the approach. I. INTRODUCTION Whether they are aerial, marine, submarine or roa vehicles, autonomous vehicles are experimental platforms still wiely use in ifferent aspects of control research, such as motion planning, feeback, or formation control. Surprisingly, there is comparatively very little research on vehicles using a ow present in the environment as their main means of propulsion. Among these vehicles, one can obviously think of the well-known sailboats, but other vehicles, also evolving on a surface, such as kite-buggies or lanyachts, share the same basic ynamic properties of sailing vessels. Most stuies on these vehicles, such as [13], consier Articial Intelligencebase techniques for the control strategies an o not make use of the available ynamic moels that woul allow for further analysis to assess for example stability or performance. Another reference, [1], aopts a more traitional moel-base perspective, but its control esign is solely base on a linear moel structure, thus not allowing for the stuy of ynamical aspects an maneuvers that are specic to sailing vehicles, such as tacking, jibing or wearing. In sailing, one of the main concerns for the navigator is to plan maneuvers towar a given estination, taking into account these specicities. When consiering autonomous vehicles, another interest is to compute or plan the signals necessary to steer the vehicle on the esire paths. In many areas concerne with motion control, such as mobile robotics or marine control, this is usually one by guiance systems an trajectory generation, whose role is mostly to plan an compute esire trajectories that are feasible for the autonomous vehicle (see for example [6] for car-like robots, but also [, chapter 5] in marine control). In this paper, we propose to moel an stuy control aspects relate to motion planning issues of a lanyacht. A lanyacht is basically compose of a cart with three wheels Fig. 1. Picture of Ma Mas 1 from SDU. an a sail (see picture of our experimental platform Ma Mas 1 in Figure 1). This vehicle is controlle by changing the angle of the front wheel through peals for steering, an a simple rope attache to the sail for propulsion. After this introuction, section will be eicate to formalize a simple moel of a lanyacht. Like in mobile robotics, our moel is base on a basic kinematic structure, albeit complete with a simple ynamic equation to account for the specicities of the system responsible for propulsion. Incientally, other types of surface sailing vessels share similar characteristics. The following section is eicate to path generation an controller esign for an important maneuver in sailing: tacking. Of particular interest is a ynamic-relate conition for performing a tack maneuver, conition that is also relate to common practise in sailing. Simulations results are presente to illustrate effects linke with this conition an the behavior of the propose schemes. Finally, a few concluing remarks en the paper. II. MODELING OF A LANDYACHT In orer to stuy motion planning issues for our sailing vehicle, but also to gain insight on the problem at han, we woul like to have at our isposal a simple way to represent ynamically the relatively complex maneuvers that

Fig. 4. Tacking maneuvers to go upwin Fig.. The Dubins car moel The kinematic moel of the Dubins car is Fig. 3. Polar curve of the propulsive system such vehicles can perform. To this en, rst consier that our lanyacht is nothing but, roughly speaking, a car whose propulsive part is a sail. Hence, owing to the vast literature on the subject in mobile robotics (see for example [6][7][9][5]), introuce rst the wellknown kinematic car moel, also calle bicycle moel or Dubins car, represente in Figure, where the front wheel is steering the vehicle, while the rear center wheel is an approximation of the motion inuce by the two rear wheels of the cart. _x(t) = v(t) cos (t) (1) _y(t) = v(t) sin (t) () _(t) = v(t) tan (t) (3) L where x, y represent the position of the contact point between the rear wheel an the groun, an its heaing. The steering angle is normally irectly controlle by the pilot an is henceforth consiere as a control input. Parameter L is the istance between the two wheel axes (see Figure ). In a usual Dubins car moel, the longituinal velocity v(t) of the rear wheel axis is consiere as a control input, either irectly, or through an integrator (see [7]). However, because of the specicities of our propulsion system, i.e. a sail, the velocities v(t) that a vehicle can reach mostly epen on its orientation with respect to the win. Inee, an as is well-known by sailors, if the vehicle is too close to facing the win, it will loose propulsion. This is illustrate by the so-calle "no-sailing zone" or "no-go zone", as represente in Figure 3, assuming a win coming from the east. Despite this limitation, a sailing vehicle can reach a estination upwin by zigzagging in the irection of the win, i.e. by staying out of the no-go zone on long transients alternate by short crossings of the no-go zone, calle "tacks" (see Figure 4). To be able for the vehicle to perform a tack also inicates the presence of some intertia effect on the velocity v(t). Thus, to moel this funamental behavior in sailing, we propose the following equation. m _v(t) + v(t) = :((t); v s (t)) (4) where m an play the role of a mass an a amping coefcient to account for the global ynamics of the vehicle in a tack. Function ((t); v s (t)) plays the role of a performance

polar iagram in sailing by giving the maximum reachable velocity of the vehicle epening both on its orientation (t) an on the way the propulsion system is trimme, represente by the control input v s (t). Although ((t); v s (t)) can take many complex shapes epening on the characteristics of the sailing vehicle (see for example [8][1]), we will in the following assume that it is ene as if j(t)j L p((t); v s (t)) = (5) sat vmax (v s (t)) otherwise an constructe after the limits L an L of the nosailing zone, since we are mostly intereste in the role of this ea zone in our moel. v max is the maximum velocity that the vehicle can reach an is relate to win spee. In the following, we assume v max to be constant. Interestingly, the simple ynamical moel (1)-(4) raises important questions regaring controllability properties that can generally be relate to the behavior of sailing vehicles. Inee, performing a tack efciently is important if one wants not to lose too much spee while crossing the no-go zone, or worse to get stuck there. In the particular case of a lanyacht, it is especially relevant, because after it is in irons, there is sometimes no other way to restart than for the pilot to push his vehicle. This is clearly not suitable for an autonomous vehicle. To see the essential role playe by the no-go zone in our moel, rst consier the vehicle as it just enters the no-go zone, with a heaing of either (t) = L or (t) = L. In this situation, (4) reuces to m _v(t) + v(t) = (6) with an initial velocity v. Integration obviously gives v(t) = v e m t (7) Because of the structure of moel (1)-(3), integrating in turn v(t) gives Z t v() = Z t p _x () + _y () := l(t) (8) which correspons to the arclength of the path followe by the vehicle. If the vehicle stays in the no-sailing zone, then the arclength will be (using (7)) m l(t) = v e m t + m (9) which, to the limit, gives l 1 := lim l(t) = v m t!1 (1) This in turn means that l 1 is the maximum istance the vehicle can travel without catching the win. In other wors, in a tacking situation, it shoul reach the other sie of the no-go zone in a shorter istance than l 1 not to get stuck (or to be "in irons" as is usually sai in the sailing wor). This basically means making sharper turns. However, because the steering angle (t) is typically constraine to lie within a sector [ L ; L ], where < L < =, there is also a limit in how sharp a turn can be, which in turn inuces a limit on how short a path in the no-go zone can be. Clearly then, if the initial velocity v is not sufcient, l 1 coul be lesser than the length of the path require to cross the no-go zone. In terms of controllability notions (see [11] for a very goo overview), this implies that there are situations where the point x = (x ; y ; < L ; v ) in the state-space cannot be controlle to x T = (x T ; y T ; T > L ; v T ) since no control input signals v s (t) an (t) exist for that purpose. Thus, the above simple observations imply that system (1)-(4) is not completely controllable. III. MOTION PLANNING FOR A TACKING MANEUVER Even though they are ynamically couple through moel (1)-(4), the effects of the control inputs v s (t) an (t) are expecte to be roughly separate as propulsion an steering, respectively. Assuming that the vehicle is to follow a preene path, the steering will mostly epen on the position of the vehicle along the path. Hence, it will in the following be assume that v s (t) = v max, i.e. the sail is trimme so that the maximum spee can be mae out of the win. Dening a path in the plane by x(), y(), where (t) is a scalar parametrizing the path, equation (1)-(3) can be simply transforme into x ((t)) _(t) = v(t) cos ((t)) (11) y ((t)) _(t) = v(t) sin ((t)) (1) v(t) ((t)) _(t) = tan (t) (13) L where () is ene from tan () = (y=)=(x=). Then, isolating (t) in (13) gives L (t) = arctan ((t)) _(t) (14) v(t) Squaring then (11) an (1), one obtains _(t) = v(t) r Finally, using (15) in (14) gives (t) = arctan @L ((t)) 1 (15) x ((t)) + y ((t)) " x y ((t)) + ((t)) (16) i.e. the steering angle is a function of the position of the vehicle on the path, represente by the path variable. As expecte, the evolution of the latter is essentially etermine by the velocity of the vehicle v(t) through equation (15) (we assume that () = ). Hence, equation (15) an (16) represent a feeforwar controller for the steering angle of the vehicle. In orer to implement a path for a tacking maneuver, we will in the following use of combination of two clothoi arcs an two straight lines, as picture in Figure 5. Clothoi arcs # 1 1 A

Fig. 5. Tacking maneuver path compose of two straight line segments an two clothoi arcs. present the avantage of having zero curvature at their starting point, this ensuring a continuous transition with a straight line segment. A clothoi is a curve whose tangent vector is a quaratic function of the path parameter, i.e. () = k + + () (17) where k is a shape parameter, the initial curvature, an () the initial heaing. From (17), the path followe by a clothoi arc can be escribe by an x() = A y() = A Z Z cos () + x() (18) sin () + y() (19) where is the path variable evolving between an 1, A is constant positive scaling factor, an x() an y() are the initial conitions of the arc. We now wish to etermine the ifferent parameters of a clothoi arc (starting an ening points, shape factors). To o so, we will loosely follow the metho from [1] (see also [4] for another reference on the use of clothoi arcs in robotics) an focus specically on the rst arc of Figure 5 (in re). As will be seen, the orientation of this particular arc will not result in a loss of generality, thanks to the structure of steering controller (15)-(16). First, enote by the angle between the two straight lines of the maneuver, intersecting at the origin (see Figure 5). Let the enpoint of the clothoi arc be situate at a istance b an an angle = from the origin. Then we have that x(1) = b cos () an y(1) = b sin (1) for the enpoint, while since the starting point is on the x-axis, we have y() =. Now referring to equation (17), it is clear from Figure 5 that = an () = on the rst clothoi arc. Since () is tangent to the path, we also have that (1) = + () which in turn leas to k = (1). It remains to etermine x() an A. Using (19) at = 1, we have A = y(1)= Z 1 sin((1) ) (3) Similarly, use (18) to obtain the starting point of the arc x() = x(1) A Z 1 cos((1) ) (4) The other clothoi arc is simply obtaine by symmetry. The rst straight line segment can be simply compute with another parametrize curve x() = A Z + x() (5) meaning this time that scale factor A = x(1) x(), i.e. the istance between the two en points of the segment.the other straight line is also euce by symmetry. Note that because of the structure of controller (15)-(16) it is essentially parameters A an k that matter for each segment. Inee, consiering a straight line with an orientation c, this woul give x () = A cos c (6) an y () = A sin c (7) while, for a clothoi segment, we woul have x () = A cos () (8) an y () = A sin () (9) Similarly, on a straight line, we woul have while for a clothoi, () = (3) () = k (31) Thus, taking into account (6)-(31) an regaring this time as a continuous parametrization on [; 4] for all segments, with each segment corresponing to a unit interval for, the steering controller equations are now _(t) = v(t) A((t)) (3)

an L (t) = arctan A((t)) ((t)) (33) where the scale factor A() changes on each segment. The general aspect of the tacking maneuver path seems to be ecie upon the value of the parameter b, which we recall, quanties the istance between the outer point of the maneuver, an the intersection of the straight line segments. Thus, an as hinte at in the previous section, whether or not it is possible to cross the no-go zone while tacking coul be linke with the choice of parameter b. To see this, rst note that because of the straight line segments have their orientation outsie the no-go zone, entering it will happen while the vehicle is on the rst clothoi arc. Assuming the tacking maneuver starts from an angle (t) L, an noting by an 1 the start an en points of the arc, we have () = ( 1 ) + (34) an the place on the curve where the vehicle enters the no-go zone correspons to ning the value of, in (34), for which () = L. Let us note this value L. Similarly, leaving the no-sailing zone will happen on the secon clothoi arc (still referring to an 1 as the enpoints of the arc) () = 3( 1 ) + 3 ( 1 ) + (35) an the value L is obtaine by solving () = L. Values L an L are then use to compute the length of the path in the no-go zone. Inee, from (18) an (19) the length of each clothoi arc is A. From there, the length of the part of the rst clothoi lying in the no-go zone is A(1 L ), while for the secon one, it is A L, which nally gives us A(1 L + L ) for the length of the tacking path in the no-sailing zone. Since, from (1), l 1 is the maximum istance that the vehicle can travel without propulsion, this implies the conition Fig. 6. Trajectory followe by the sailing vehicle. A(1 L + L ) < l 1 (36) Note, from equation (3) that A epens on y(1) which epens on b from (1). Hence, conition (36) transforms into where b < l 1 k (1 L + L ) k = cos R 1 sin( 1 ) (37) (38) From the point-of-view of motion planning, b can thus be tune to satisfy (37). Note that l 1 has to be compute from (1) rst. Since the velocity of the vehicle before leaving the no-go zone, i.e. before v, is lesser or equal to v, we can choose to compute l 1 from a point situate before the rst clothoi arc. Conition (37) becomes a bit more conservative, but can be use irectly for on-line motion planning. Fig. 7. Parameter (t) an scale factor A((t)). To illustrate the behavior of ynamic controller (3)-(33), an of the role playe by parameter b an its associate conition (37), we present a few simulation results with the vehicle parameters m = 15, = 135 an L =. Maximum velocity v max is set to 15. The sailing vehicle starts at a position (; ) an performs a tacking maneuver, with a win coming from the East, to go to position (8; ). The trajectory followe by the vehicle, with b satisfying conition (37) (in this simulation, we have b = 1), is shown in Figure 6. Figure 7 shows the evolution of path variable (t) on the interval [; 4], switching from one segment of the maneuver to the next as (t) reaches the next unit interval. This can also be seen from the upper plot of Figure 7, where scalar factor A() is represente.

Fig. 8. Velocity v(t) an steering angle (t). IV. CONCLUDING REMARKS This paper reporte on a stuy of nonlinear aspects for trajectory an reference input generation for sailing vessels. The propose moel both represents in a simple way ynamics that are common to a wie variety of surface sailing vessels (lanyachts, but also sailing boats, an systems with kites), an is also relate to car-like robots that are wiely stuie in the mobile robotics community. The essence of the ynamics of sailing vessels seem to lie in the connection between the non-holonomic constraints of the vehicle together with the nogo zone, well-known to sailors, an is at the origin of the nee for tacking maneuvers to go upwin. We propose a simple way to escribe paths associate with such maneuvers, an presente a simple controller to steer the vehicle on this path. A conition on the parametrization of the path for avoiing the vehicle to get stuck in the no-go zone was also introuce. Other maneuvers such as wearing or boxhauling have also been one for centuries in sailing vessels [3]. Further work will examine motion planning for such cases, as well as esign guiance systems combining such maneuvers. REFERENCES Fig. 9. Vehicle in irons The evolution of the vehicle longituinal velocity v(t) is shown in the lower part of Figure 8. Note the ecreasing values (after t = 4s) of v(t) as the vehicle enters the no-sailing zone, then the re-acceleration as it escapes from it (before t = 8s). The corresponing steering angle, as obtaine from controller (3)-(33) is shown in the upper part of the same gure. Figure 9 shows another simulation illustrating a case when conition (37) is not fullle, an the initial velocity of the vehicle is not sufcient to allow it to cross the no-go zone (in this case, we ha b = 14). The upper plot shows the angle of the vehicle trying to go to the limit L = =6 of the no-go zone but never really reaching it (=6 :536), while the lower plot shows the velocity ecreasing continuously until it reaches zero, inicating that the vehicle is stuck in the no-go zone. [1] S. Fleury, P. Soueres, J.-P. Laumon an R. Chatila, Primitives for smoothing mobile robot trajectories, IEEE Transactions on Robotics an Automation, vol. 11, no. 3, pp. 441 448, 1995. [] T. I. Fossen, Marine control systems: guiance, navigation an control of ships, rigs an unerwater vehicles. Marine Cybernetics,. [3] J. Harlan, Seamanship in the age of sail. Naval Institute Press, 1984. [4] Y. Kanayama an B. I. Hartman, Smooth local path planning for autonomous vehicles in Proc. IEEE Int. Conf. on Robotics an Automation, Scottsale, AZ, 1989. [5] U. Kiencke an L. Nielsen, Automotive Control Systems. Springer,. [6] J.-P. Laumon (E.), Robot motion planning an control. Springer, 1998. [7] S. M. LaValle, Planning algorithms. Cambrige University Press, 6. [8] C. A. Marchaj, Aero-hyroynamics of sailing. International Marine Publishing, 199. [9] R. M. Murray an S. S. Sastry, Nonholonomic motion planning: steering using sinusois, IEEE Transactions on Automatic Control, vol. 38, no. 5, pp. 7 716, 1993. [1] P. J. Richar, A. Johnson an A. Stanton, America's cup ownwin sails vertical wings or horizontal parachutes?, Journal of Win Engineering an Inustrial Aeroynamics, vol. 89, pp. 1565 1577, 1. [11] E. D. Sontag, Mathematical control theory (n e.). Springer, 1998. [1] C. M. Xiao an P. C. Austin, Yacht moelling an aaptive control in Proc. IFAC Conf. on Manoeuvering an Control of Marine Craft, Aalborg, Denmark,. [13] E. C. Yeh an J.-C. Bin, Fuzzy control for self-steering of a sailboat in Proc. of the Singapore Int. Conf. on Intelligent Control an Instrumentation, Singapore, 199.